Texte de l’article qui a été présenté aujourd’hui par ma collègue Yu Li de l’Université de Picardie, au congrès Unilog 2022 qui se tient à Chania en Crète. À terme, ce texte servira de préambule à l’exposé que je suis en train de bâtir sur la conjecture P vs NP. Une esquisse en français du texte ici a déjà été publiée sur le blog : Que démontrent les mathématiciens, et le font-ils d’une manière digne de ce nom ?, le 5 mars 2022.
What makes a demonstration worthy of the name?
Paul Jorion *
* ETHICS – EA 7446, Université catholique de Lille, 60 Bd Vauban, 59800 Lille, France
Subject: Foundations of mathematics
Cite as: https://www.pauljorion.com/blog_en/2022/04/09/unilog-2022-what-makes-a-demonstration-worthy-of-the-name-by-paul-jorion/
Three ways of ascertaining the truth
One reads in an introductory book to the work of Kurt Gödel:
« … Aristotelian logic rests on two pillars: a set of premises, or assumptions, which are taken to be true without proof, and a collection of rules of inference, by which we transform one true statement into another […] As long as we can convince ourselves that the premises are true, then the conclusions [are] as solid and inescapable a fact as there ever will be. It is a certainty that follows from the semantic content of the premises and from the process of deduction laid down in our minds and formalised by Aristotle.
What Gödel discovered is that even if there exist true relations among pure numbers, the methods of deductive logic are just too weak for us to be able to prove all such facts. In other words, truth is simply bigger than proof » (Casti & DePauli 2000: 4-5).
Although widely accepted, what is asserted here in the second paragraph is based on a major misunderstanding as that truth is bigger than proof is no discovery of Gödel’s, as it was established by Aristotle twenty-five centuries ago and was the starting point of his groundbreaking reflection on truth and falsehood.
This fundamental misunderstanding, however, is far from trivial: it has undermined the pursuit of the mathematical venture in several ways. Indeed, mathematicians have attempted to prove statements which were in no need to be proved in the first place as their truth was established at little cost by other means, and in so doing they have at times introduced erroneous demonstrations aimed at proving… the undemonstrable, or hardly demonstrable (e.g. that there exists in the Amazon a seven black-dotted yellow ladybird).
« Truth is simply bigger than proof », indeed, according to Aristotle, proof was but one of the three available sources of truth; since his days, none has been added to his brief list. Aristotle determined that there were three known ways for accessing the truth:
1° Truth known from perceptual evidence arising from our five senses, providing it can be corroborated so that facts are distinguished with certainty from illusions.
2° Truth derived from commonly accepted definitions, by which we assign to a single word or expression the meaning of a phrase: « the fawn is the young of the deer and the doe »; henceforth, whenever I am tempted to say « the young of the deer and the doe », I can say « fawn » instead. In mathematics, definitions are called « axioms ».
3° Truth reached as the conclusion of a well-formed syllogism, i.e. syntactically correct, by valid deductions from true premises. In mathematics, we call the equivalent of sequences of conclusions of valid syllogisms « theorems ».
Transposed into the language of mathematics, these three modes of accessing the truth become
1° Truth resulting from observations of the physical world expressed as models in mathematical language (mathematics as « virtual physics »)
2° Truth established by axioms, which are definitions expressed in mathematical language
3° Truth obtained by the demonstration of a theorem, which is the conclusion of the proof provided.
Because in recent centuries mathematicians have ignored corroborated sensory evidence as the main and primary source of truth, they have attempted to prove by means of axioms and theorems solely, a considerable number of things that appear to be true, neglecting the « obvious » truths that come to us as trusted perceptions by our senses, i.e. as data captured from the world surrounding us as it stands. E.g. Goodstein’s theorem which « proves » … an observation about natural numbers. Thus, instead of saying: « Here is a curiosity that arises when we define a sequence of natural numbers in a certain way », Goodstein thought it appropriate to demonstrate a theorem stating what we observe, taking axioms as the starting point for his demonstration. The fact that Goodstein’s theorem can be demonstrated within one mathematical framework (second-order arithmetic), but not in a closely related one (Peano’s arithmetic), betrays the contrived nature of trying to mathematically prove a fact of observation.
In their desire to prove everything, mathematicians have produced erroneous proofs: clumsy and error-ridden attempts to prove what was in no need to be demonstrated and was often essentially unprovable due to the complexity of a demonstration candidate. Indeed, most of our surrounding empirical world requires in order to be proven an overwhelming number of elements requiring to be proven upstream. The mere fact of an observation as such summons the existence of an observer who would need to be proven in the first place.
Since sensory evidence has done the job satisfactorily in most instances, this wealth of unnecessary proofs were achieved through resorting to various tricks such as starting from false premises (paradoxes, conclusions of fallacious reasonings), or presenting as the conclusion of a demonstration worthy of the name some truth being in disguise either corroborated sensory evidence translated into mathematical language (virtual physics) or an axiom. Such a « theorem-in-no-need-to-be-demonstrated » happens then to be true
1° either because it is in fact an axiom in disguise (i.e. a definition) where the demonstration is a petitio principii: a circular reasoning, where what one claims to have discovered in fine was known from the start, the demonstration having been but an artful reformulation of the axiom
2° or because an observation derived from sensory evidence, an empirical fact, is hidden in one of the axioms; in such a case, mathematicians have used mathematics to build a « virtual physics » and it is their intuition of the world around us that has guided them unbeknownst.
In one particular instance what is usually the « stowaway » presence of empirical facts within mathematical reasoning has been deliberate and proclaimed, I’m referring here to Hilbert’s 6th proposition:
« Mathematical Treatment of the Axioms of Physics. The investigations on the foundations of geometry suggest the problem: To treat in the same manner, by means of axioms, those physical sciences in which already today mathematics plays an important part; in the first rank are the theory of probabilities and mechanics. »
I reported in Comment la vérité et la réalité furent inventées (2009) that in the historical case of the invention of the infinitesimal calculus, the central issue was not of the protagonists at work, Newton’s and Leibniz’ intuition, whose workings would have remained opaque to their conscious self but of their quite explicit experimental validation all along of the mathematics whose methodology they were developing. Indeed, they defined the rules of this new method of calculation in such a way that their model of a planet’s orbit accurately reflected the observed motion, revealing the pre-eminence they implicitly gave to physics over mathematics in the development of their new tool (Jorion 2009: 332-346). The alternative approach would have been to demand mathematical rigour above all, which would have betrayed slight anomalies in the orbit of the planets, the pre-eminence being given in this case to mathematical precepts rather than to an accurate account of physical phenomena.
What requirements for a demonstration to be « worthy of the name »
In my analysis of Gödel’s demonstration of his incompleteness theorem of arithmetic in Comment la vérité et la réalité furent inventées (2009: 285-326), I show in particular how at a critical juncture Gödel cheats by surreptitiously slipping into his demonstration that a proposition is true when its truth comes neither from being an axiom nor a demonstrated theorem, but from resulting from empirical evidence. In other words, at the moment when he finds himself mired in a paradox of his own construction, Gödel sneaks physics into mathematics in order to extricate himself from the pit he’s dug himself in.
Gödel’s proof of his incompleteness theorem of arithmetic fails to be « worthy of the name » for two types of reasons: for using arguments (transformation rules) of too low a level of validity (demonstrability) at different stages of his proof and for relying on an intuitive notion of truth whose precise concept varies according to his whim in different contexts.
Instead of sticking to arguments deemed by Aristotle to be of an « analytical » standard of logic, i.e. worthy of being used in a true demonstration with « scientific » status, Gödel went astray by resorting to arguments deemed more modestly to be of a « dialectical » standard of logic, i.e. acceptable in the courts or within the political debate, such as the reductio ad absurdum, or even going so low as to use purely « rhetorical » arguments, i.e. acceptable only in everyday conversation. Of only a rhetorical standard in Gödel’s demonstration is the recourse to a proposition asserting about itself that it is unprovable, having no other evidence to support this assertion than « its own word », the near equivalent in everyday conversation of « It’s true because I’m telling you so ». Attention is often drawn to the fact that the weakness here in the demonstration is due to the assertion being self-referential, but the bottom line is that there is nothing to support this pseudo-affirmation of a proposition about itself: a proposition cannot be « taken at its word » because there is no way a metamathematical argument can be held accountable and shamed for having lied.
When Gödel says of that proposition, as we’ve seen, that it is true even though it is impossible to prove this means out of necessity that it can only be true for one of the other two possible reasons why a proposition may be true: either that it is an axiom (i.e. a definition) or that it derives from corroborated sensory evidence. Since the context of the demonstration excludes the proposition in question from being a vast tautology: a hidden or unrecognised axiom, the only remaining option is that its truth derives from sensory evidence. But this would imply that the proposition has as its reference the empirical world around us, offering of it a true account, which would mean that Gödel is unwittingly bringing physics to bear on what he claims is mathematics only.
How is it possible that Gödel did not notice such a confusion in his argument between mathematics and physics? Because it is common knowledge that for him the two are coalesced: Gödel was a Platonist by ideological creed, convinced that the world is made up of numbers in its intimate build-up and that, as one of the consequences of this belief, everything we know in the world by sense evidence can also be proved mathematically. Bertrand Russell said of him: « Gödel turned out to be an unadulterated Platonist, and apparently believed that an eternal ‘not’ was laid up in heaven, where virtuous logicians might hope to meet it hereafter » (in Russell’s Autobiography, quoted by Dawson 1988: 8).
I am arguing here, to the contrary, that to be « worthy of the name », a mathematical demonstration is not allowed to import surreptitiously elements of proof borrowed from physics: it is required to confine itself entirely to the formal domain of mathematics. It is easy to understand that were this distinction not upheld, the notion of an « adequate mathematical model of physical reality » would lose all meaning as stowaway empirical evidence might be lurking in the model at undisclosed locations, blurring the distinction between which elements of the physical world are properly modelled and which have straightforwardly been imported, i.e. simply depicted instead of accounted for.
In their book quoted earlier, Casti & DePauli, unintentionally betray their unease in this respect when they write: « What Gödel discovered is that even if there exist true relations among pure numbers, the methods of deductive logic are just too weak for us to be able to prove all such facts. In other words, truth is simply bigger than proof. This fact does not seem too astonishing when put into the context of everyday life. We are all aware of things that we ‘know’, but that we feel can never be logically deduced in a formal, Aristotelian fashion. In fact, the Oxford don and philosopher J. L. Austin, when first told of Gödel’s result, remarked: ‘Who would have ever thought otherwise?’ It seems likely that the average person on the street would say the same thing if someone announced that not everything that’s true can be known by following a process of logical deduction. But not so for mathematicians! » (2000 : 4-5). Casti & DePauli should have written more specifically: « But not so for mathematicians who are devotees of the Platonic creed » whose belief is that mathematics and physics are one and the same, or at least don’t care distinguishing one from the other. Hence their indifference to the quality of the proof mobilised at various steps in their demonstrations, and Gödel’s specific legerdemain in this respect with his « I can’t care less » attitude.
That the crux of the issue, the source of all his confusion, lies in Gödel’s Platonic ideological commitment, Casti and DePauli bring it out even more – albeit accidentally – in a passage where they attempt to clarify his approach with the help of an illustration featuring… chocolate cakes:
« Truths = all conceivable cakes satisfying the chocolate cake test.
Proof = all recipes for actually making chocolate cakes with the Chocolate Cake Machine.
Now comes the Big Question: Is there a recipe for every conceivable chocolate cake? Or, equivalently, is every true statement provable? What we’re asking here is whether there are ‘honest-to-god’ chocolate cakes in the Platonic universe of cakes for which no recipe can ever be given » (ibid. 19-20).
« In the Platonic universe… », one could not have stated it any better: Gödel’s demonstration only makes sense and represents a meaningful progress in understanding on the stage of a theatre where the Real would be entirely made out of numbers and where, as an entailment, two representations are fused: that of the empirical world surrounding us, which is accounted for by physics, and that of the world of mathematical entities, tools that we devised in order to offer a useful, if only stylised, vision of the world around us.
Mathematicians’ intuition and « virtual physics »
Mathematics allows us to design models of physical reality. Some of our models are causal: if one thing happens, another or others will follow. We can also do simulations: we know what elements of different types are involved in a phenomenon of a collective nature and how they interact: let’s act it out, let numerical instances of objects interact for a while and see what happens.
It is often in no time that new mathematical objects and methods are mobilised for new advances in physics. This is no accident: they get immediately tested by various researchers for possible use. Einstein, for example, turned to tensors as soon as that mathematical tool became available.
Barrow reports: « The development of non-Euclidean geometry as a branch of pure mathematics by Riemann in the nineteenth century, and the study of mathematical objects called tensors was a godsend to the development of twentieth-century physics. Tensors are defined by the fact that their constituant pieces change in a very particular fashion when their coordinate labels are altered in completely arbitrary ways. This esoteric mathematical machinery proved to be precisely what was required by Einstein in his formulation of the general theory of relativity. Non-Euclidean geometry described the distortion of space and time in the presence of mass-energy, while the behaviour of tensors ensured that any law of nature written in tensor language would automatically retain the same form no matter what the state of motion of the observer. Indeed, Einstein was rather fortunate in that his long-time friend, the pure mathematician Marcel Grossmann, was able to introduce him to these mathematical tools. Had they not already existed, Einstein could not have formulated the general theory of relativity » (Barrow  1992: 189)
Not all of mathematics lends itself however with equal ease to the construction of physical models. Is there something then about a new mathematical tool that predisposes it to be summoned for modelling physical phenomena?
How are axioms defined? In Comment la vérité et la réalité furent inventées I argued that they are quite often not entirely abstract and seem to display an inclination towards the easy generation of physical models. I pointed out that this was then no deliberate strategy, the mathematician author remaining in all likelihood unaware of the fact. When Stephen Wolfram writes « Maybe there’s something special about the particular axioms used in mathematics. And certainly if one thinks they’re the ones that uniquely describe science and the world, there might be a reason for that » (Wolfram 2020: 559), this is precisely what I’m having in mind when I talk of mathematics as « virtual physics ». By taking as a starting point axioms that are already de facto « virtual physics », theorems are generated that remain all along « virtual physics » so to say by construction.
When I mentioned the attempts of mathematicians to demonstrate what is in no need to be demonstrated because it is in truth a straightforward fact of observation, something that our senses have allowed to establish, I already pointed out that in certain cases, the axiom makes it possible to surreptitiously import into the theorem an element of physics: « because hidden in one of the axioms is an observation derived from sensory evidence, an empirical fact, is hidden in one of the axioms; in such a case, mathematicians have used mathematics to build a « virtual physics » and it is their intuition of the world around us that has guided them unbeknownst ».
Curiously, if one thinks of the fact that they are working on formal objects whose syntax and semantics alone should be relevant, mathematicians readily admit that they rely on their intuition, which they suggest is a very abstract thing that is very difficult to pin down more precisely. For example, Alan Turing, in his 1938 thesis, Systems of Logic Based on Ordinals, remarked: ‘The activity of intuition consists in the production of spontaneous judgments which do not result from conscious trains of reasoning […] I shall not attempt to explain this idea of intuition more explicitly » (in Copeland 2004: 192). In fact, the intuition of mathematicians owes its very shape to the world as it stands around us. And what mathematicians produce, therefore, since the origins of their discipline, and especially when they refer to their intuition as the source of their inspiration, is mathematics that constitutes, until better informed, a « virtual physics ».
This I set out systematically in Comment la vérité et la réalité furent inventées, taking as an example the birth of the infinitesimal calculus mentioned above, when it could be observed that the mathematics newly conceived by Newton and Leibniz were abused (the rigour asserted being sacrificed in the process) until they « stuck » to the celestial mechanics for which they were developed. The famous bishop cum philosopher George Berkeley (1685-1753) denounced then the deception: a « compensation of errors » involving « Ghosts of Departed quantities » (Berkeley  1992: 199). He pointed out that « in every other Science, Men prove their Conclusions [the ‘theorems’] by their Principles [the ‘axioms’], and not their Principles by their Conclusions. But if in yours you should allow your selves this unusual way of proceeding, the Consequence would be that you must take up with Induction and bid adieu to Demonstration. And if you submit to this, your Authority will no longer lead the way in Points of Reason and Science. I have no controversy about your Conclusions but only about your Logic and Method » (ibid. 180). And Berkeley cruelly wondered « whether such Mathematicians as cry out against [divine] Mysteries, have ever examined their own Principles? » (ibid. 220).
You mathematicians, Berkeley argued, should take your starting point in mathematically meaningful axioms (« Principles ») and see what theorem (« Conclusions ») you can derive from them. But what you do instead is that you predefine the theorem (« Conclusions »), and then you make up the axioms (« Principles ») that will allow you to prove it. In so doing, you do not proceed as scholars do, according to Aristotle, by deduction, within the framework of the « analytical » standard of logic, but by induction, which Aristotle has shown to be a mode of proof proper to the « dialectictical » standard of logic which is of choice for lawyers and politicians, who care only to not contradict themselves, but have no regard for the truth, unlike genuine scholars.
Barrow, John D., Theories of Everything. The Quest for Ultimate Explanation, London: Vintage 1990
Berkeley, George,  1992 De Motu and The Analyst. A Modern Edition, with Introductions and Commentary, ed. and transl. from latin by Douglas M. Jesseph, Dordrecht (Holland): Kluwer Academic Publishers
Casti, John L. & Werner DePauli, Gödel. A Life of Logic, Cambridge (Mass.) Perseus: 2000
Copeland, B. Jack, The Essential Turing, Oxford: Oxford University Press 2004
Dawson, John W., « Kurt Gödel in Sharper Focus », in S. G. Shanker, Gödel’s Theorem in Focus, London: Croom Helm 1988
Jorion, Paul, Comment la vérité et la réalité furent inventées, Bibliothèque des sciences humaines, Paris: Gallimard 2009
Wolfram, Stephen, A Project to Find the Fundamental Theory of Physics, Wolfram Media 2020


Course Number: C250
Recommended Grades: 11,12
Description: This course is a college-level Calculus course. Course topics include functions and limits, differentiation, applications of differentiation, integration, applications of the definite integral, and techniques of integration. Graphing calculators will be used extensively throughout the class. Students are expected to take the National AP Calculus AB examination in May. Students are required to complete a summer assignment preceding the course. Prerequisites: A high level of success (a grade of B or better) in Precalculus Honors and a teacher recommendation.


Class 8 Assignment Math Answer & Solution 2021 for highschool college students has now obtainable. The authorities has offered the first week task activity for the category 8 college students beginning on 20 March in 2021. The task for the category eight can be of all topic based on a brand new syllabus. So, arithmetic stands for essentially the most troublesome topics within the task syllabus revealed by the directorate of secondary and better training. This submit is now an try to offer direct reply and resolution for the class 8 math assignment 2021.
Class 8 Math Assignment
Class 8 Assignment Math is within the syllabus of 3rd week thus far in accordance to the discover of the authority. This task for sophistication eight math course would be the second task in 2021. Every pupil at the highschool in Bangladesh are afraid of arithmetic task. It isn’t completely true that class 8 math topic may be very laborious of not simple to know. So, examine the beneath math task to know.
Actually our class 8 college students are not looking for, because it appears to me, to know their class math. Rather, they are saying they’re afraid of math and try to memorize the reply. That is the fundamental drawback for a lot of the class eight college students who’re set for sitting JSC Exam 2021 a couple of days later in 2021. That is why now we have introduced some solutions of the category 8 task 3rd week math to make their rigidity ease.
Math Assignment Class 8
Math Assignment Class 8 has been influencing the scholars on the rise of rigidity. However, our greatest suggestion is that don’t panic however take a look at the mathematics query in your task syllabus very rigorously. First attempt to understand what’s requested within the task activity from you. Then discover out is that they any hints of any mathematical time period within the given task.
Class 8 Math Textbook has quite a lot of chapters and sections that one must undergo first. If they fight, our class 8 college students can discover some instance math’s of their textbook that may be very useful for the answer. What is within the least task given on this 3rd week is about to note from the a part of the learners. We don’t inform anybody to repeat the precise math task reply from our place.
Class 8 Math Assignment 2021
Class 8 math task 2021 is now our subject for this submit as you see right here with the doable assertion. The authority of the training ministry has made the conclusion relating to math task for sophistication 8. The primary message that has been issued following this week is the task class 8 math 2021. So, the beloved learners of the actual class must demand the next arithmetic task query.
The loving info for the aim of the large subject on this conclusion is the mathematics topic task. In 2021, there are various duties for the class 8 books by college students. However, the duty of writing reply is the should work as it’s from the college authority by the category 8 task math reply. To anybody, you’ll be able to deny following their instructions. But, none can dare to keep away from to hear the task of math from the college academics.
Class 8 Assignment Math 3rd Week
Class 8 Assignment Math 3rd Week is now obtainable for everybody who’re underneath the highschool training. 3rd week task is now very particular for the learners at current scenario because it consists of the mathematics topic within the listing. Mathematics is a quite common and broadly referred to as one of the best topic on this planet. Therefore, the authority can’t however embrace this math topic within the class 8 3rd week task 2021 by hook or prepare dinner.
3rd assignment class 8 math reply has gone into impact from in the present day with quite a lot of questions and options from the facet of the learners. Novice of the highschool is now looking for the class 8 task math 3rd week 2021 from any standpoint. The most and outnumber incident is that math task class 8 3rd week reply. So, getting the category 8 task 3rd week reply is your prime wanting on the good second.
Class 8 Assignment Math Answer
Class 8 assignment 2021 math reply may be very simple for many who are very environment friendly on this subject. However, the scholars who usually are not skilled within the math subject will discover this 3rd week task very laborious to contemplate. So, for each degree learners our staff has provide you with the category eight math assignment answer in 2021. The college students of our nation must be within the line with the instructions of the authority for getting the reply.
Assignment class 8 2021 reply consists of one of the best variations and version in our web site. Going for the mathematics reply of the prevailing week specifically referred to as 3rd one must be essentially the most exhausting factor. The novice of this planet need to observe with the principles of the (*8*). So, discover essentially the most unique part of this submit that claims a couple of PDF obtain hyperlink for everybody in 2021.
Class 8 Assignment 3rd Week Math Solution 2021
Class 8 Assignment 3rd Week Math Solution 2021 has about to make its finish inside a couple of days. So, class 8 math assignment solution must be very environment friendly in each week. Our staff has additionally reached to the distraction of the answer from the relative sector. The precious contributions that the scholars is about to make is the growth and improvement of the information within the math topic.
Students ought to try to kindle the curiosity of their academics and classmates indefatigably all by means of the interval. They also needs to really feel enormously dismayed by the social, tutorial, and political scenes which flood within the Class 8 task math 3rd week 2021 reply. Novice my really feel discouraging all of the honest, sincere and hard-working patriots for class 8 assignment 3rd week math resolution.


Introduction Preparing for the ASVAB (Armed Services Vocational Aptitude Battery) can be daunting, especially when it comes to the math sections. Understanding the structure and content of the Arithmetic Reasoning (AR) and Mathematics Knowledge (MK) sections is crucial for success. This comprehensive guide is designed to help you navigate these areas with confidence, providing insights, strategies, and practice to boost your scores.
Understanding the ASVAB Math Sections The ASVAB test includes two math-focused sections:
Arithmetic Reasoning (AR): This section tests your ability to solve basic arithmetic problems encountered in everyday life. It includes 16 questions to be answered in 39 minutes.
Mathematics Knowledge (MK): This section measures your knowledge of mathematical concepts and applications. It comprises 16 questions with a 20-minute time limit.
Key Topics to Cover To excel in these sections, you should be comfortable with a range of mathematical concepts. Here’s a brief overview:
Basic Arithmetic: Addition, subtraction, multiplication, and division
Fractions, Decimals, and Percentages
Ratios and Proportions
Algebra: Basic equations, inequalities, and factorization
Geometry: Understanding shapes, area, volume, and perimeter
Basic Statistics: Mean, median, mode, and range
Strategies for Tackling ASVAB Math Questions
Understand the Question: Read each question carefully to grasp what is being asked.
Identify Keywords: Look for words that indicate mathematical operations or relationships.
Practice Mental Math: Enhance your ability to calculate quickly and accurately without a calculator.
Break Down Word Problems: Convert words into mathematical expressions or equations.
Time Management: Allocate your time wisely, spending more time on harder questions.
Effective Study Tips
Start with Basics: Ensure a strong foundation in basic arithmetic and algebra.
Use ASVAB Study Guides: These guides are tailored to the exam format and offer practice questions.
Review Mistakes: Understand why an answer was incorrect to avoid repeating mistakes.
Join Study Groups: Collaborate with others to gain different perspectives and explanations.
Practice Questions Here are some sample questions to get you started. Answers will be provided at the end of the post.
Arithmetic Reasoning: If a shirt costs $20 and is on sale for 25% off, what is the sale price?
Mathematics Knowledge: Solve for x: 2x + 5 = 11.
Conclusion Success in the AR and MK sections of the ASVAB requires a solid grasp of basic math concepts, strategic preparation, and practice. By understanding the structure of these sections, focusing on key topics, and employing effective study techniques, you can enhance your math skills and boost your overall ASVAB performance. Remember, consistent practice and a positive attitude are your best tools for success.
Answers to Practice Questions:
$15 (25% of $20 is $5, so $20 – $5 = $15)
x = 3 (Subtract 5 from both sides to get 2x = 6, then divide by 2)
Encouragement Every question you practice, every concept you master, brings you one step closer to achieving your goals in the ASVAB. Stay motivated, stay focused, and your hard work will pay off!
Pass your ASVAB Exam
Preparing for your ASVAB exam? Sure you can study the dusty textbooks, but they don’t give you a pass guarantee. We do. Pass using our fast and efficient method, or your Premium membership is free.


By Jean-Pierre Petit
Read Online or Download Les Aventures d'Anselme Lanturlu - Le Geometricon PDF
Best geometry and topology books
Quantum drinks in limited geometries show a wide number of new and fascinating phenomena. for instance, the interior constitution of the liquid turns into extra suggested than in bulk beverages while the movement of the debris is particular through an exterior matrix. additionally, loose quantum liquid droplets let the research of the interplay of atoms and molecules with an exterior box with out problems coming up from interactions with box partitions.
This booklet offers a strong solution to learn Einstein's specified idea of relativity and its underlying hyperbolic geometry during which analogies with classical effects shape the precise instrument. It introduces the inspiration of vectors into analytic hyperbolic geometry, the place they're referred to as gyrovectors. Newtonian pace addition is the typical vector addition, that is either commutative and associative.
- Encyclopedie des sciences mathematiques. Geometrie algebrique dans l'espace
- Sub-Riemannian Geometry: General Theory and Examples
- Lectures on the Geometry of Position Part I
Additional info for Les Aventures d'Anselme Lanturlu - Le Geometricon
29) convex polygon ABCDE, and polygonal line A B C D E F G A which surrounds it completely. We extend side AB in both directions until it intersects the surrounding polygon at M , N . By the preceding result, the length of the path AEDCB is less than AM B A G N B, and therefore the perimeter of AEDCBA is less than the perimeter of the polygon N M B A G N . This perimeter, in turn, is less than the surrounding line because the part M B A G N is common to both, and M N < M C D E F N . 28. Theorem.
The angle B can therefore be arbitrary (at least within certain limits). A quadrilateral which can be deformed under these conditions is called an articulated quadrilateral. This notion is important in practical applications of geometry. 1 47. Theorem. In a parallelogram, the two diagonals divide each other into equal parts. In parallelogram ABCD (Fig. 46), we draw diagonals AC, BD, which intersect at O. Triangles ABO, CDO are congruent because they have equal angles, and an equal side AB = CD (by the preceding theorem).
In rectangle ABCD (Fig. 47), the diagonals are equal because triangles ACD, BCD are congruent: they have side DC in common, ADC = DCB since they are both right angles, and AD = BC since they are opposite sides of a parallelogram. Corollary. In a right triangle, the median from the vertex of the right angle equals half the hypotenuse. This is true because if we draw parallels to the sides of the right angle through the endpoints of the hypotenuse, we form a rectangle, in which the median in question is half the diagonal.
Les Aventures d'Anselme Lanturlu - Le Geometricon by Jean-Pierre Petit


My research is mostly in "applied theoretical computer science".
DBLP - GScholar - MSFT Acad - RG - cvLehman Computer Science is hiring! (tenure track / open area / open rank) Come join us!
More specifically, my group studies algorithmic problems motivated by real-world applications, but algorithmic solutions with provable optimality or approximation guarantees. This includes projects on networking and sensor networks, power plants, batteries and environmental optimization, and security. Also, algorithmic game theory, computational geometry, and massive datasets.
Our research is supported by grants from the NSF, PSC-CUNY, and ASRC.
- PhD Program in Computer Science
CUNY Graduate Center
Office: Room 4327
365 Fifth Avenue, New York 10015
- Department of Mathematics and Computer Science
Lehman College, CUNY
Office: Gillet 220B
250 Bedford Park Boulevard West, Bronx, NY 10468


Explore Uses for a Scientific Calculator
Without a doubt, a calculator has been used for several years in classrooms, especially top scientifics calculators, have completely changed the way students learn. This device has become an indispensable part for students to learn complex math as well as science topics. It comes with a lot of functions that can help students to understand concepts quite well.
The first thing is first…what is a scientific calculator?
Well, scientific calculator performs the same functions as its standard counterparts. However, it also has myriad other advanced features that you may don’t know. The scientific calculator is the only device that can deal with certain functions or features in fields like chemistry, engineering, trigonometry, physics, and chemistry.
The scientific calculators have extra functionalities that let you work with logs and exponents that need a certain amount of memory to execute functions and provide you the accurate results.
Uses for a Scientific Calculator
Following are the users for a scientific calculator you should know before you purchase it-
- Basic Functions & Exponents
Besides basic functions such as subtraction, addition, division, and multiplication, there are other functions that scientific calculator can perform. No doubt, exponents are used in any math course, and an only scientific calculator is capable of performing any algebraic function. Basic calculators are quite good at solving simple equations, but when it comes to solving complex equations, a scientific calculator is definitely required.
Once you’ve enhanced your knowledge and you’re ready to learn new concepts related to trigonometry and calculus, it’s probable that you’ll need to learn about logarithms. In the past, these formulas were performed completely by hand. But, it’s quite tricky and time-consuming to perform by hand. Happily, the scientific calculator has made the process so easier for you.
In fact, some teachers have already started explaining logarithms with the help of scientific calculator for better understanding. Moreover, the scientific calculator comes with an in-built memory that let you save certain equations.
- Scientific Notation
You surely need a scientific calculator if you’re working in a field related to engineering, science, or mathematics in order to deal with your homework. The scientific calculator is capable of doing scientific notation as it can handle larger values, unlike basic calculators. Moreover, top scientifics calculators can also help you to calculate negative scientific notation.
- Sine, Cosine & Tangent Functions
If you’re planning to take a trigonometry/calculus course that involved sine, cosine and tangent functions, you surely need a scientific calculator to deal with your regular work. With the help of a scientific calculator, you can ease your work and you can get the answer quite fast and saving your time.
As you can now see that scientific calculator can come quite in handy if you’re in the field of engineering, physics, or mathematics. So, take advantage of a scientific calculator to handle your regular work with ease.


The Ultimate Poker Club cardroom uses a RNG (random number generator) to ensure that all players are dealt a fair deal hand each and every game. The RNG system produces innumerable permutations and combinations in the 52-card deck that is virtually impossible to predict. The numbers must be unpredictable and not related to each other in any way to be considered truly random.
The Ultimate Poker Club random number generator, has been tested and certified by top regulatory agency as a part of our regulatory process. The random number generator has been found to deal cards from the deck such that each card has an equal chance of being selected unless it had already been dealt in the poker game.


Hey there, gamers! Have you heard about Cool Math Games? It’s one of the most popular game websites on the internet and offers a wide variety of puzzles, educational games, and more. Whether you’re looking for a challenging brain teaser or just want to pass the time with a few silly games, Cool Math Games has something for everyone.
You might think you know all there is to know about Cool Math Games—but do you really? Believe it or not, there’s a lot about this website that even its most dedicated fans don’t know. Ready to get your math geek on? Here are seven facts about Cool Math Games that may surprise you.
The Origin Story of Cool Math Games
You may not realize it, but Cool Math Games has been around for quite a long time. This interactive gaming website first launched in 1997, and quickly became the world’s most popular math-based online entertainment platform. At the time, there were no other websites offering educational gaming experiences for kids—but Cool Math Games changed that.
The site was founded by two brothers from Louisiana, who were inspired to create it after seeing how their own children enjoyed playing video games together at home. The brothers had some unique ideas about how to make math learning more engaging, like using themes and stories to make the material fun and relatable.
Today, Cool Math Games remains true to its roots, with a wide array of math-based games and activities designed to get kids thinking beyond traditional school lessons. The website is updated regularly with new content and features that keep users coming back for more!
How Cool Math Games Are Made
We’ve all heard of Cool Math Games, but do you know how the games are actually made?
It turns out that it’s much more than randomly slapping together a few math-related activities. Every game is actually designed with a specific purpose in mind—whether it’s to teach problem solving skills or to introduce a new and interesting math concept.
The developers start every game by brainstorming a concept and figuring out what type of game play would be most suitable for it. Next, they write the code to create the game, followed by very detailed play testing. The games are tested many times to make sure that they meet the company’s strict standards for quality control. Finally, they are released on the Cool Math Games website for you to enjoy!
The Different Types of Math Games Available
You might already be familiar with Cool Math Games, but did you know that there are different categories and types of math games to choose from? From games designed for younger players, to ones specifically geared towards adults, Cool Math Games has something for everyone.
Here are some of the types of math games available:
- Logic and problem solving: These are puzzles that require players to use their mathematical skills to think logically and solve problems. Examples include Tower Blaster and 2048.
- Mental arithmetic: Test your mental calculation skills by playing a game like Falling Fractions or Pumpkin Man 2.
- Educational: If you’re looking for something more educational, try one of the educational math games such as Fruit Splat Subtraction or Base 10 Blocks Addition.
- Strategy: Want to sharpen your strategy skills while having fun? Give Bloons Tower Defense 5 or Defend Your Nuts a try!
- Interactive apps and videos: There are plenty of interactive apps and videos available, like 7th Grade Math Fun and Math Attack!
So don’t be afraid to explore—with so many different kinds of math games available at Cool Math Games, you’re sure to find something that fits your interests!
Why Kids Love Cool Math Games
Have you ever wondered why kids love Cool Math Games? Well, you are not alone!
There are a few reasons why these games have become so popular:
Fun and Exciting
Cool Math Games are designed to be fun and exciting. The games feature bright colors, catchy music, and entertaining levels that challenge players to think creatively in order to complete their mission. Plus, there’s something for everyone — from classic puzzle and strategy games to funny brain-teasers — so no matter what kind of game your kids like to play, there’s sure to be something for them.
Another great thing about Cool Math Games is that they offer educational benefits for your kids. While the games are obviously meant to be enjoyed as entertainment, they can actually help your kids learn problem solving, strategic thinking, and critical analysis skills. Kids can even use the games as an introduction to coding — several of the titles feature basic programming elements that teach them how programming works!
Finally, Cool Math Games have custom rewards that make playing even more fun! When players complete certain objectives or missions in the game, they receive badges or virtual currency they can use to purchase new items or levels. Plus, many of the games also allow players to track their progress so they can see how far they’ve come and what they’ve achieved!
The Educational Benefits of Playing Cool Math Games
When you think of gaming, you probably don’t immediately think of the educational benefits it could have. However, Cool Math Games offer a variety of ways in which gamers can learn while playing. Not only are the puzzles and games fun, but they also help sharpen problem solving skills and develop critical thinking skills.
Some of the benefits that come with playing Cool Math Games include:
- Developing memory retention – as gamers must remember sets of instructions and apply them to each game level.
- Strengthening quick thinking – as puzzles quickly changes or levels become more challenging, gamers must be able to think quickly and adapt to changing scenarios.
- Improving logical reasoning – puzzles often require players to assess a situation, analyze it, and come up with a solution to progress further in the game.
- Enhancing concentration skills – as players work through levels, concentration is key if they are going to succeed in solving challenging puzzles.
Cool Math Games provide an exciting and engaging way for children (and adults!) to learn without being bored or intimidated by too much information all at once. So why not try one out today and reap the rewards?
What Other Games Does Cool Math Offer?
Did you know that Cool Math Games offer more than just the iconic one you know and love? They actually offer a wide variety of math and logic games that are perfect for sharpening your skills, no matter your age or proficiency. Whether it’s counting, problem-solving or even strategy-based, they’ve got something for everyone.
Here are some of the cool math games they offer:
- Bloxorz: This puzzle game requires you to get a block to the goal without falling off the edge. It’s fun and surprisingly challenging!
- Math Millionaire: Just like the TV show ‘Who Wants to Be A Millionaire?’ but with math questions instead. A great way to review and learn concepts from your math classes.
- Fireboy and Watergirl: These two characters need to work together to finish puzzles, relying on each other’s strengths and weaknesses. It encourages cooperation as well as logical thinking.
- 2048: Based on the concept of doubling numbers, this game is simple yet addictive! It tests your ability to add up numbers quickly as well as strategize for future moves.
No matter your age, playing these cool math games offered by Cool Math Games can help make learning more fun while building up your skillset at the same time!
How to Install Cool Math Games
Did you know there are hundreds of Cool Math Games available online? And did you know they’re absolutely free?
It’s true! Here’s how to install and get started playing Cool Math Games:
- Visit the official website. You’ll want to make use of the search bar on the homepage to find a game that interests you.
- Once you choose a game, just click the “Play Now!” button to start playing.
- You may be asked if you want to add the game to your favorites list—so don’t forget to check that box if it appears!
- To download a game, simply click on the green “Download Now” button underneath the image of the game. The download will start automatically and installation will begin once it is complete.
- You can also access your favorite games from your computer’s desktop, as most games come with an icon that appears on your screen after installation is complete.
- For further help with installation, take a look at our help page for more detailed instructions and troubleshooting tips!
- Last but not least: HAVE FUN!
From its history to its library of games, there’s plenty to learn about Cool Math Games. Whether you’re a parent, educator, or game enthusiast, Cool Math Games is a fantastic resource to find fun and educational games. At the same time, its secure browser offers a safe online environment for kids.
So, the next time someone asks if you’ve heard of Cool Math Games, you can give them an informed answer. It’s a great online resource for both kids and adults. With a collection of over 2,000 math and logic-based games, it’s an excellent source for online entertainment with educational value.


LES ULTRAS est désormais compatible avec l'extension FastNews.kiwi disponible pour votre navigateur. Avec cette extension, vérifiez s'il y a des nouveaux sujets sur ce forum en un clic depuis n'importe quelle page !Cliquez ici pour en savoir plus.
Saifurs Math 100+ 0.16 0.01 . Saifurs Book Free Download 10+ 9.27 0.25 Saifurs Book Download 10+ 0 0.08;. Where do I download free SAT ebooks from Barron and the Princeton Review . Im going to warn you that the only way you can download it online for free is by .. New York Times Book Review . Free Resources on the Internet . Over 1000 Math and Verbal practice questions, .. FreeMathBooks.com Download Legally Free Math Books. Categories. eBooks (83) . Understanding Calculus is a complete online introductory book that focuses on concepts.. By analyzing the massive data associated with the keyword Bsc Maths Books Free Download, . 9th Class Math Key Book Free Download 100+ 0.41 0.07. Bank Recruitment Guide is About Bank Exam Questions and Answers, Sample Questions and Solve, Math, English Updates and Exam, Test Question Patterns.. offer in this article about saifurs math solution you know really that this book . Saifurs math solution, [download] . pdf free pdf printer you can use . download .. english grammar book, . saifurs math pdf free download. Posted by Bd Boy Share to: Next . BookRix.com - Read books online free and download eBooks.. Discover your favourite saifurs math solution book right here by downloading and getting the . saifurs math solution [free download] saifurs math solution ebooks .. Saifurs Math Book Free Download 10+ 0 0.21 Www Saifurs Spoken English Com 10+ 5.06 0.22 Saifurs Spoken English Course In Bangladesh 10+ 0.1 0.29 Sign In for More .. Permutations and Com binations (Assisted with TI-83+ silver) . Book -Discrete . Under the MATH key there is a PRB tab.. The following keyword list is recommended by the search term Sagir Ahmad Maths Book Pdf Free Download, . Singapore Math Books Free Download Pdf 100+ 0.68 0.3. GRE Mathematics Test Practice Book . MATHEMATICS TEST 5 PRACTICE BOOK 007624-72506 GRE Math Practice Book Hel, .. BCS books free download. MP3 BCS guide book, . Bangla pdf download, mp3 math review pdf free download, .. GEE Bangladesh, A Global Education Explorer; explore your education research and career. BCS books free download. MP3 BCS guide book, . Bangla pdf download, mp3 math review pdf free download, .. Mathematics books. Need help in math? . Download all Mathematics books in . Algebra is one of the main branches in mathematics. The book series of .. Looking for books on Geometry? Check our section of free e-books and guides on Geometry now! This page contains list of freely available E-books, Online Textbooks and .. Now, the presented saifurs math solution is a book that you . saifurs math solution [free download] saifurs math solution ebooks Page : 1.. Find this Pin and more on Bangla Book Free Bangla Boi Download by banglabook. . Present Perfect English Translation English Language Bengal Detective Math Comic .. By analyzing the massive data associated with the keyword Vedic Maths Books Free Download, . Saifurs Math Pdf Free Download .. PRACTICE BOOK 007624-72506 GRE Math Practice Book Hel, . The questions in the practice test in this book illustrate the types of multiple-choice questions .. Download >> Download Saifurs math book pdf . saifurs math pdf download saifurs bank written math . saifurs math solution pdf free download .. Free Download Saifur's Zero to Hero . Book Name: Saifur's Zero to Hero . Saifurs Publication publishes various types of jobs seeking and university admission test.. reading all content of the book. saifurs math solution is also one of the windows to reach and . saifurs math solution [free download] saifurs math solution ebooks .. Looking for books on Geometry? Check our section of free e-books and guides on Geometry now! This page contains list of freely available E-books, Online Textbooks and .. Free-eBooks.net is the internet's #1 source . Read & download eBooks for Free: anytime! Menu . This book or "spiritual journal" as I like to call it is about .. Top keyword related from Search Engine of saifurs book free download. Saifurs Math 100+ 0.16 0.01 Saifurs English 100+ 0.51 0.13 Saifurs . Saifurs Book Free Download 10+ 9.27 0.25 Saifurs Book Download 10+ 0 0.08. Saifurs Math Book Free Download 10+ 0 0.21 Www Saifurs Spoken English Com 10+ 5.06 0.22 Saifurs . Saifurs Book Download 10+ 0 0.08. Online books on:Free Mathematics Books for Download and Ebooks Online Textbooks tutorials downloadable e-books downloads zip chm rar. 1. saifurs vocabulary pdf free download amader-kotha.com. Link: . 8b9facfde6 jo nesbo the cockroaches pdf downloadshogun james clavell download pdfstephen coonts epub collection downloadphysical education class 12 book free downloaden iso 10628 pdf downloaddan brown inferno ebook free download nederlandssimon sinek leaders eat last epubsignal and system book by tarun kumar rawat pdf downloadcracking the pm interview ebook downloadmenschen a2.2 lehrerhandbuch pdf free


Use this math lesson to introduce compound events! Students will build on their understanding of probability as they learn how to find the sample space and probability of compound events given real-world examples.
Informational Essay: Getting Organized Before Writing
Students will continue their informational writing project by organizing the information they gathered through research. They will use a graphic organizer to organize their ideas and sort their research notes into meaningful sections.
Students begin their informational writing process by gathering information through research. They will use a graphic organizer to record important information and sources and reflect on how to present the information.
Students will have the opportunity to strengthen their informational essay drafts by correcting errors in conventions and mechanics. They’ll use a checklist to guide this critical step in the writing process.
Students will have the opportunity to fine-tune their writing by adding, deleting, or reworking content in their informational essay drafts. They’ll use a checklist to guide this critical step in the writing process.
This lesson will help students understand how to solve one-step addition, subtraction, multiplication, and division inequalities using inverse operations, as well as how to graph inequalities on a number line.


An experienced educator with career spanning over 16 + years.
She has an experience of teaching mathematics to students from middle to senior secondary level across various educational boards.
Apart from teaching she has varied experience in research, designing curriculum and authoring books.
She is one of the author for Class X NCERT Mathematics Text book and a co-author for Teaching and Learning of Mathematics course book (MESE-001) for IGNOU.
She has many research publications in International journals and has also written Teachers Resource Material for renowned publishers.
She is highly passionate about teaching and has a potential to keep the students motivated to achieve their goals.
She creates liking towards Mathematics and ensures that the learners are able to link the concepts to the real world.


Thanks to home loans, purchasing a house is easy and convenient today! You no longer need to exhaust all your life savings or wait for years to collect enough money to buy a house. However, a home loan continues for a long time and can keep you indebted for almost two decades until you repay the loan amount. Naturally, it can be stressful if you are unable to manage your loan well.
It is easier to plan your finances if you know beforehand how much you would need to pay. Lenders give you an option to repay the loan in EMIs
What is an EMI?
Equated monthly instalments are the monthly payments you make towards the loan repayment every month. An EMI is calculated as per your loan amount, loan tenure and interest rate. Each EMI consists of a part of the principal amount repayment and the interest on the outstanding amount. You can change your EMI amount by altering your loan tenure. If you want a higher EMI, you can opt for a shorter tenure and vice versa.
How to calculate home loan EMI?
Though the lender would inform you about the loan EMI, you can also calculate the EMI on your own with the help of an EMI calculator and through a mathematical formula.
Through the EMI calculator
You can find a home loan calculator easily online. It is an online tool to help you determine the EMI with the help of loan tenure, interest and loan amount. You can find this calculator on your lender’s website. Often, an EMI calculator also gives you a repayment pattern in the form of interest payout, principle payout and total loan payout.
Through mathematical formula
You can calculate EMI using the following formula.
E = P * r * (1+r)^n / ((1+r)^n-1)
E is the equated monthly instalment amount or EMI
P is the principal loan amount or the loan you want to take
r is the expected rate of interest calculated monthly (dividing the value by 12)
Benefits of calculating home loan EMI
Fast and easy – Using a home loan EMI calculator is convenient and quick. Just input the value in the calculator or in the formula and wait for the results to come. If you are using an EMI calculator, you do not even need to do complex calculations all by yourself. It makes all your EMI calculations fast. You can search for an EMI calculator online and calculate the EMI amount instantly.
Accurate – You no longer need to rely on the lender to tell you the EMI value as you can easily do that yourself. With a calculator, there are no chances of any error or mistakes and you can be rest assured that your results are correct.
Aids in planning – By calculating the EMI, you can plan your finances in a better way. You can know beforehand if you will be able to repay the loan with the current tenure. If there is any difficulty, you can simply alter the tenure of the loan to get an EMI amount that fits your budget.
Simply put, using an EMI calculator to calculate a home loan EMI even before applying for a home loan can be beneficial for you at various levels. It helps you avoid going through several processes before applying for any type of loan. Not just for home loan, an EMI calculator is a necessary tool to use before availing any type of loan.


If you are thinking about becoming an accountant, you may be wondering what types of mathematics skills you will need. There are a variety of jobs in the accounting profession, and with an accounting degree you can choose which one suits you best. You may be surprised to learn that although maths is important, it is not the most important skill that is needed in an accounting degree.
Mathematics and Other Skills
The main use of maths in accounting is using algebra, statistics and basic formulas to make sense of the data. More importantly than maths, computer and calculator skills are important as well as the ability to manage and analyse data. Computers and technologies are more or less able to handle most of the mathematical work, so the most complicated parts of accounting are usually setting up the programs, inputting data, running the spreadsheets and analysing the date.
Knowing how to interpret data is a talent taught in degrees and used in accounting. Logic and problem solving abilities are important when taking your accounting degree. Accountants Kent and nationwide should have basic maths skills to go alongside their other skills of analysing, data management and being able to problem solve to help companies and individuals reach their financial goals. To go further in your accounting career once qualified, there may be extra skills and procedures that you need to study depending on the type of accountant you wish to become.


Donna Cox, George Francis and Ray Idaszak, NCSA, University of Illinois at Urbana-Champaign, Tom DeFanti and Dan Sandin, Electronic Visualization Lab, School of Art and Design, University of Illinois at Chicago.
Special thanks to Larry Smarr, NCSA, University of Illinois at Urbana-Champaign
Kodak Fast Film, Kodalith, Plexiglas
PHSColograms by (Art)n, Fermilab Gallery, Fermi National Accelerator Laboratory IL, November 11, 1987–January 6, 1988
The Interactive Image, Museum of Science and Industry, Chicago, IL, October 1987–January 1988
Nature, Feature, Chicago, IL, April 1987
"Magritte distorted and inverted the relationship between art and mathematics in order to liberate his art."
The Last Judgement (Beersel)
1935/2-1/4"x3-1/4" Gelatin silver print
from The Sandor Family Collection
A virtual photograph of a Romboy Homotopy, a four-dimensional mathematical object.


Each matrix class has a conversion operator so that the programmer can explicitly cast one type of matrix to another (the casting is done by copying). It would have been possible to make conversions occur automatically when needed, but because these conversions can be quite extensive for large matrices and because unexpected results might occur if the programmer did not intend for a conversion to occur, the conversions have to be used explicitly.
operator XXXMatrix () const
Example: FloatMatrix C = A * (FloatMatrix)B;
Convert a matrix of one type into another. These conversions allow the various arithmetic operators, such as * and +, to be used on matrices of different type. For example, if A in the example above is a (3, 3) FloatMatrix and B is a (3, 2) IntMatrix, then C is a FloatMatrix with dimensions (3, 2).


The international team of researchers could speed up the modelling of large geospatial datasets with benchmark accuracy when implemented on a computing system based on highly parallelized graphics processor units.
Computers can perform large calculations in a short time. But due to the constraint on the way the digital system stores data, calculation resulting in a longer sequence of digits will be curtailed. Accumulation of precision errors in geospatial datasets can lead to faulty modelling outcomes. The latest development in the field seems to be an elegant solution for this problem.
“For decades, modelling of environmental data relied on double-precision arithmetic to predict missing data,” says the team. “Today, there is high-performance computing hardware that can run single- and half-precision arithmetic with a speedup of 16 and 32 times compared with double-precision arithmetic. To take advantage of this, we propose a three-precision framework that can exploit the acceleration of lower precision while maintaining accuracy by using double-precision arithmetic for vital information.”
“The main goal of this project is to leverage the recent parallel linear algebra algorithms developed by KAUST’s Extreme Computing Research Centre to scale up geospatial statistics applications on leading-edge parallel architectures,” they added
The next step for the team will be integrating approximations with mixed precision to lessen memory footprint and decrease calculation time.


Math Achievement Program
The Math Achievement Program (MAP) is closed for the summer. Please check back soon for the summer math tutoring support schedule, offered from 5/31-8/3.
At MAP, you can work with a peer tutor who specializes in MAT 121, 136, 137, 238 or STA 270. No appointment is necessary. This is group, drop-in tutoring. Come solo or with your study group, we look forward to assisting you with your course!
Looking for in-person tutoring? Please call us at 928-523-5524 to sign up for an appointment.
¿Busca apoyo académico en español? ¡Los empleados con “habla español” junto a su nombre pueden ayudar!
Spring 2021 Schedules
STA 270 Accordion Closed
Ruth – (Hablo Espanol)
PMA Ruth’s hours (all online): Wednesdays from 4pm-7pm, Thursdays from 5pm-8pm.
Other 1:1 and drop-in tutoring is available. Please give us a call at 928-523-5524 or visit nau.edu/asc for more information.
Get help when you need it
Stop in to work on homework, ask for help from a Peer Math Assistant, or attend a test review!
MAT 121, 136, 137, 238
Virtual MAP room
Monday to Thursday 10 a.m. – 6 p.m.
Friday 10 a.m. – 2 p.m.
Maximize your time- come prepared!
- attempt your homework first
- think of questions to ask your tutor based on your homework
- bring your books and notes
- bring previous tests, if applicable
Looking for help with MAT 100, 108, 114, 125?
Visit the Lumberjack Math Center website for hours and room locations.


HOME PAGE ONLINE MATH TUTORING ABOUT ME STUDY TIPS CONTACT ME LINKS
CREATIVE MATH VIDEOS & TUTORING BY COLLEGE MATH PROFESSOR
Electronic Flashcards - This device is amazing. A must have!! You will actually enjoy doing time table drills. Simply amazing.
Math Videos and Online Tutoring - Math videos and
tutoring for all levels through graduate school. All tutoring by college math
professor. After tutoring session you get a tape copy of the entire


Office: Room 5846 East Hall.
Office Phone: (734) 764-0372
Some information about me:
My CV is here.
to see a list of my publications.
Here is a brief description of my research and research interests:
Corrections and improvements to the book "Bruhat-Tits theory: a new approach" PDF
- Prasad: Number Theory in Geometry (Myhill lectures at SUNY, Buffalo)
- Prasad: A new approach to unramified descent in Bruhat-Tits theory
- Prasad: Tamely ramified descent in Bruhat-Tits theory
- Prasad, Rapinchuk: Developments on the congruence subgroup problem after the work of Bass, Milnor and Serre
- Prasad, Rapinchuk: On the conguence kernel for simple algebraic groups
- Prasad, Rapinchuk: Generic elements of a Zariski-dense subgroup form an open set
- Prasad, Yeung: Fake Projective Planes
- Prasad, Yeung: Arithmetic fake compact Hermitian symmetric spaces of type A3
- Conrad, Prasad: Structure and classification of pseudo-reductive groups
- Prasad, Rapinchuk: Corrections to: Weakly commensurable arithmetic groups and isospectral locally symmetric spaces, Publ. Math. IHES 109(2009).


As a high school student in Mount Pearl Newfoundland, a small community of St. John’s Newfoundland, Mark Penney’s nascent interest in mathematics and science was set afire by his chemistry teacher, who spent his lunch breaks helping a few interested students do Advance Placement university level courses, and by his mathematics teachers who encouraged Mark to compete in local and provincial mathematics problem-solving contests.
Mark studied applied mathematics and physics at the Memorial University of Newfoundland, theoretical physics at the Perimeter Institute for Theoretical Physics in Waterloo, Ontario and mathematics at the University of Toronto. These studies paved the way for what Mark calls the most fulfilling and productive years of his life, studying the mathematical aspects of topological phases at Oxford University.
During his time at Oxford and during sojourns at the Mathematical Sciences Research Institute in Berkeley California and the Hausdorff Research Institute for Mathematics in Bonn Germany Mark has had the opportunity to meet and work with world-class researchers; Oxford’s unique college system has afforded Mark the opportunity to meet and become friends with many brilliant and inspiring people who are studying topics ranging from classics to history to the arts.
Quantum computing is the love child of physics, mathematics and computer science. Working on the minute scale of atoms, quantum computers could work millions of times faster than conventional computers and look at problems that are so complex, that they require more computing power than any modern day computer could supply.
Despite the immense efforts being undertaken by research groups around the world, there are significant hurdles in the path to realizing this technology. One of the great challenges of quantum computing is to protect the computers from external environmental factors, which can negate any advantage they have over conventional computers. During his time at Oxford, Mark has has focused on removing this obstacle by doing research on materials that are highly resistance to influences from their environment. These materials, known as topological phases, are typically so thin that they are effectively two-dimensional. As you can imagine, this extreme thinness makes engineering materials to exhibit topological phases very difficult. Mark’s project has been to develop the mathematical models and tools needed to study real world, three-dimensional topological phases. This research directly contributes to the broader theoretical background that is attempting to ultimately solve the dilemma of protecting quantum computers from external environmental factors.
In the long term Mark would like to move from the theoretical to the more practical side of this subject. We do not know today what the quantum computer of tomorrow will look like or what it will be capable of achieving, but Mark’s work is bringing the future world of quantum computing that much closer to reality.
Mark, who is so accomplished in his field, is a tender 26 years old as this profile is being written. He plans to do post-doctoral work in North America after he is awarded his DPhil in Mathematics from Oxford University in 2017. Ultimately, he would like to be a professor in a mathematical area related to quantum computing, and return to Canada with the new skills and knowledge that he has acquired abroad.
The CCSF is proud to have Mark Penney among our distinguished scholars.


Download Free the latest version of Number Match – Logic Puzzle Game
Try the mobile version of the pen and paper game since you were a kid. Now you can take your number game with you wherever you go. Solving free number match puzzles on mobile is much easier than using pencil and paper.
Immerse yourself in the world of number games! Whenever you get tired or bored, take a break and play the number match puzzles. Solve addictive logic puzzles, match numbers and refresh! If you like classic board games, try Number Match. Enjoy the magic of numbers and give your brain a great time.
Number Match is an easy-to-learn brain puzzle that makes gray matter work. Merge numbers to clear the board. Adjust your eyes, hands, and mind. Enjoy hours of this free number game. Time goes by when you have fun! Install it now and give it a try. It cannot be stopped.
how to play:
• The goal is to clear the board.
• Find pairs with the same number (1 and 1, 7 and 7), or pairs that add up to 10 (6 and 4, 8 and 2) in the number grid.
• Tap the numbers one by one to strikethrough and earn points.
• You can connect pairs at adjacent horizontal, vertical, diagonal cells, and at the end of one line and the beginning of the next.
• When it gets stuck, you can add more rows by adding the remaining numbers below.
• If you get stuck, use hints to speed up your progress.
• You win after all the numbers have been removed from the number puzzle grid.
Break your score
The empty the board, the higher the score. Strikethrough all numbers on the field (+150 points) and remove rows (+10 points) to get the most points. Connect numbers that are far apart to get +4 points.
There are many ways to solve the puzzle. But it’s not as easy as it looks. Make fun of your brain and have a fascinating experience!
What you get:
• Easy to learn logic puzzles
• Gameplay time for you to enjoy
• Tips for achieving your goals faster
• There is no time limit so you don’t have to rush. Just relax.
• New number game by top developers!
Challenge your brain with number match puzzles and have fun! Play number games anytime, anywhere!
Download now the Free APP Number Match – Logic Puzzle Game to install on your mobile Android. The installation process of the program depends on its developer. Sometimes it will be just Download APK, sometimes it will be Download apk + obb.
Tags: #Android #eeuu #PlayStore #APP #english #usa #APK #OBB #Number #Match #Logic #Puzzle #Game
Security, availability of the latest version and download speed depend on the software administrator.


Answer to Puzzle #10
Well, it took us a long time to come back to this one. We were hoping for some answers, but evidently the problem seemed too complex. Here’s the problem for those of you who didn’t read it:
Happy Mother’s Day!
Little Johnny wanted to get his mother a gift for Mother’s Day. Unfortunately he only had 96 cents. Johnny decided that he had 4 possible solutions to his problem:
Spend his money on his lottery ticket. In this scenario there is a 9/10 chance he will lose all his money, a 1/20 chance he will double his money, an 0.049999999 chance he multiply his money by 10, and a 1 in 1 billion chance he will multiply his money by 1 million.
Invest his money in a fast yielding government bond, ensuring him 2% of interest. This is, of course, excepting the unlikely event that the U.S. government would be destroyed before Johnny could claim his yield, a 1 in 10 billion chance resulting in Johnny losing all of his money as well as some other problems . Note: You cannot get a deal like this in the real world. It was a special limited offer for children with epithet, “Little.”
Borrow some money from the neighbor kid~$8 dollars to be exact. The neighbor kid only deals in whole dollar multiples of 8 and only has 10 dollars. There is, however, a 1/10 chance that Johnny will fail to pay off his debt a weak later and get beat up.
Go scrounging around the local park for loose change. There is a 40% chance he will find a dollar’s worth, a 30% chance he will find between 20 and 80 cents, a 10% chance he will find $5 worth, a 19.8% chance he will find nothing, and a 0.2% chance he will find $50 dollars’ worth. Don’t ask how I know all this.
Johnny will pick which plan to carry out randomly, each plan having an equal chance. The gifts Johnny has in mind for his mother are a cheap yo-yo (97 cents), an issue of People magazine ($1.84), a bouquet of tulips ($7.84), a pair of nice sunglasses ($28.98), a tempurpedic pillow ($40.99), or a rustic hunting lodge in Wyoming ($400,000). All prices include tax. Johnny will buy the highest price gift he can afford. What is the probability that Little Johnny’s mom will receive each of these gifts?
Little Johnny needs $400,080.62 in order to buy all the gifts. The only way to get that much money would be to buy a lotter ticket (option 1). The chance he wins the lottery is 1 in 1 billion, and the chance option 1 is picked is 1/4. Therefore the chance he would win is 1 in 4 billion or 0.000000025%. Note this problem is really easy, just looks tricky because of options 2,3,4.


Published January 30, 1982
by Chapman & Hall/CRC .
Written in English
|The Physical Object|
|Number of Pages||345|
Abstract Harmonic Analysis: Volume I Structure of Topological Groups Integration Theory Group Representations Edwin Hewitt, Kenneth A. Ross Springer Science & Business Media, Dec 6, - Mathematics - pages. When we accepted the kind invitation of Prof. Dr. F.K. SCHMIDT to write a monograph on abstract harmonie analysis for the Grundlehren der Mathematischen Wissenschaften series, we intended to write aH that we could find out about the subject in a text of about printed pages. We intended that our book should be accessible to beginners, and we hoped to make it useful to specialists as weH. This volume contains the Proceedings of the 9th International Conference on Harmonic Analysis and Partial Differential Equations, held June , , in El Escorial, Madrid, Spain. Included in this volume is the written version of the mini-course given by Jonathan Bennett on Aspects of Multilinear Harmonic Analysis Related to Transversality. Articles in the present volume are based on talks delivered by plenary speakers at a conference on Harmonic Analysis and Ergodic Theory (DePaul University, Chicago, December 2–4, ). Of ten articles, four are devoted to ergodic theory and six to harmonic analysis.
The principal features of the book are as follows: a thorough treatment of the representations of the Heisenberg group, their associated integral transforms, and the metaplectic representation; an exposition of the Weyl calculus of pseudodifferential operators, with emphasis on ideas coming from harmonic analysis and physics; a discussion of. This book contains an exposition of some of the main developments of the last twenty years in the following areas of harmonic analysis: singular integral and pseudo-differential operators, the theory of Hardy spaces, L\sup\ estimates involving oscillatory integrals and Fourier integral operators, relations of curvature to maximal inequalities, and connections with analysis on the Heisenberg group. Zygmund's two volume books on trigonometric series are good, but I would tackle a few other books on harmonic analysis before going for it. It is quite complex in comparison to the other references and will not help much if you do not already have a foundation in harmonic/Fourier analysis. The Norbert Wiener Center for Harmonic Analysis and Applications provides a state-of-the art research venue for the broad emerging area of mathematical engineering in the context of harmonic analysis. This two-volume set consists of contributions from speakers .
The Norbert Wiener Center for Harmonic Analysis and Applications provides a state-of-the-art research venue for the broad emerging area of mathematical engineering in the context of harmonic analysis. This two-volume set consists of contributions from speakers . Read more about this volume This book contains lectures presented by Hugh L. Montgomery at the NSF-CBMS Regional Conference held at Kansas State University in May The book focuses on important topics in analytic number theory that involve ideas from harmonic analysis. Harmonic Analysis: Smooth and Non-smooth About this Title. Palle E.T. Jorgensen, University of Iowa, Iowa City, IA. Publication: CBMS Regional Conference Series in Mathematics Publication Year: ; Volume ISBNs: (print); (online). This book is a continuation of Volume I of the same title [Grund lehren der mathematischen Wissenschaften, Band ]. We constantly 1 1. The textbook Real and cite definitions and results from Volume abstract analysis by E. HEWITT and K. R. STROMBERG [Berlin Gottin gen Heidelberg: Springer-Verlag ], which appeared between the publication of the two volumes of this work, .


BMI or Body Mass Index is a mathematical formula derived by Adolphe Quetlette in the mid 1800s. Yes, that's right folks the BMI is over 200 years old.
BMI is s a value derived from the mass (weight) and height of an individual. The BMI is defined as the body mass (your weight) divided by the square of the body height in meters.
For example I am 168 cm tall (1.68 meters), and I weigh (at last weigh in) 79 kg my BMI is 79/2.8224 = 27.99 according to the body mass index table, I am over-weight.
As a healthy, active individual - it's clear that the BMI isn't taking much of my lifestyle into account, so let's look at what is wrong with BMI:
There is no physiological reason to square someone's height.
This part of the formula alone is bunkum. It's basically there to massage the numbers!
The "Formula" was designed to assess the collective mass of an entire population
Quetlette was purportedly attempting to study the mass of the population of Belgium. Not assess the fatness of individuals.
It doesn't take into account more accurate measures of health
The BMI doesn't actually measure fat - a Dexa scan is currently the only accurate way of measuring body fat.
It doesn't take into account waist to hip ratios - which are commonly regarded as a good indicator of health
Nor does it take into account the weight of muscle vs fat....
The categories of underweight, overweight, and obese are bunkum
Sweepingly applying a label to all bodies based on a mathematical equation that to begin with makes no sense is, well, nonsense.
If that isn't enough to convince you that that particular number on paper is a very poor summation of your overall health, I'm not sure what else will....
Photo Credit: Charlotte Astrid


On Wednesday May 15th at approximately 9am, grade 7 and 8 students all across North and South America wrote the 2013 Gauss Mathematics Contest (Eastern Hemisphere wrote the contest the next day). A growing number of keen students in Grade 6 also wrote the Grade 7 contest. The Gauss Mathematics Contest is a great opportunity to inspire students who already have a strong interest in mathematics and also help to increase confidence and ability in mathematics for other students. Although the emphasis is on participation for the sake of enrichment, rather than competition, 31 students from Our Lady of Mount Carmel took on the challenge and did quite well. The University of Waterloo, Mathematics Department awarded Certificates of Recognition to all contestants, with several students being awarded Certificates of Distinction for ranking in the top 25%!!! Way to go Mathletes!!!


Applied Mathematics Option
The Applied Mathematics Option is available to honours students in any discipline. The primary focus is on methods commonly used in modern mathematical models in science, especially in relation to Kinesiology, Physics and Computing, Chemistry, Biology and Psychology. However, the option can also be tailored to meet the needs of other areas of study, where quantitative methodology is an important part of the program.
A specific set of courses that give the student a broad introduction to mathematical methods, and to the theory that underlies these methods, is required. The option also allows students the opportunity to explore areas involving more advanced models than normally encountered in an undergraduate program, through the MA360 and MA480* (or MA485) courses. Also, where appropriate, software commonly in use will be incorporated into the program of study.
- Differential Equations I
- Introduction to Analysis
- Multivariable Calculus


Highlands Elementary-4th Grade
All instruction is based on District and State Standards
Language Arts: (Reading, Writing, Listening, and Speaking)
- Reading: Benchmark Advance Program (District adopted)
- Writing; Writers Workshop, Letter Writing, Writing Process, Grammar, Spelling, and Academic Vocabulary
- Listening and Speaking: Oral Reading, Listening Activities, Sharing, Oral Presentations
Math: Zearn Math (District adopted)
- Mastering Multiplication
- Daily reteach and practice
Grade 4 mathematics is about:
Developing understanding and fluency with multi-digit multiplication, and developing understanding of dividing to find quotients involving multi-digit dividends.
Developing an understanding of fraction equivalence, addition and subtraction of fractions with like denominators, and multiplication of fractions by whole numbers.
Understanding that geometric figures can be analyzed and classified based on their properties, such as having parallel sides, perpendicular sides, particular angle measures, and symmetry.
Science: TWIG (District adopted) Units:
- Magnetism and Electricity
- Earth Materials
Social Studies: California History
- Native California Tribes
- Gold Rush
Visual and Performing Arts
- Art in Action
P.E.: Activities for the development of fine and gross motor skills with emphasis on and refinement of: team building, skills, organized games, sportsmanship, and student participation.
Study Trips/Special Activities:
- San Mateo County History Museum
- Mission San Juan Bautista
- Sanchez Adobe
- Coloma Outdoor Discovery School
- 4th Grade Fun Day


Students worked on the Maths Fair project in Ms Venkataya's classroom for two weeks and then they presented in the Palm Pavilion where parents and students were able to join them to try to solve the problems they had tackled for themselves.
Students developed the following skills:
* critical thinking by solving the mathematical problems logically
* communication by working in groups to solve the problem and sharing their learning experience
with their peers, teachers, parents, and administration
* teamwork by working in pairs to create helpful hints, and a step by step process for others
* creative thinking and designing of the poster board
* perseverance through the notion of a growth mindset
A note from Ms Venkataya:
Thank you so much parents, students and teachers for making the first AISL Math Fair a HUGE success. I hope parents that you enjoyed solving a couple of challenging mathematical problems as well!
The Leopard's Tale is our main medium to keep our families informed on such things as the day to day happenings on campus, after school activities, summaries of any arts and sports events, helpful resources, and important dates and reminders.
Parent Information Sessions
Here are the upcoming sessions – all parents are welcome and encouraged to attend:


Kalles Fraktaler is a free Windows-based fractal zoom computer program used for zooming into fractals such as the Mandelbrot set and the Burning Ship fractal at very high speed, utilizing Perturbation
Electric Sheep is a volunteer computing project for animating and evolving fractal flames, which are in turn distributed to the networked computers, which display them as a screensaver.
Fractal-generating software is any type of graphics software that generates images of fractals. There are many fractal generating programs available, both free and commercial. Mobile apps are availabl
Fractint is a freeware computer program to render and display many kinds of fractals. The program originated on MS-DOS, then ported to the Atari ST, Linux, and Macintosh. During the early 1990s, Fract
Apophysis is an open source fractal flame editor and renderer for Microsoft Windows and Macintosh. Apophysis has many features for creating and editing fractal flames, including an editor which allows
Chaotica is a commercial fractal art editor and renderer extending flam3 and Apophysis's functionality. There is also a free version with limited render resolution and animation length.
openPlaG is a PHP based function graph plotter for use on websites. It was first released in April 2006. In June 2007 its source code was published under the GNU GPL license. PlaG is an abbreviation f
XaoS is an interactive fractal zoomer program. It allows the user to continuously zoom in or out of a fractal in real-time. XaoS is licensed under GPL. The program is cross-platform, and is available
Bryce, also often referred to colloquially as Bryce3D, is a 3D modeling, rendering and animation program specializing in fractal landscapes. The name is taken from Bryce Canyon—a rugged region with ma
Vista and Vista Pro are 3D landscape generation programs originally for the Amiga, Apple Macintosh and Microsoft Windows (PC) range of home computers. It was written by John Hinkley and its own About
Terragen is a scenery generator program for Microsoft Windows and Mac OS X developed and published by Planetside Software. It can be used to create renderings and animations of landscapes.
Sterling is a fractal-generating computer program written in the C programming language in 1999 for Microsoft Windows by Stephen C. Ferguson. Sterling was initially released as freeware but is not fre
Ultra Fractal is a fractal generation and rendering software application. The program was the first publicly available fractal software which featured layering methods previously only found in image e
MojoWorld was a commercial, fractal-based modelling program for the creation of digital landscapes, and attracted a following among artists who create space art and science fiction scenes. Originally


1.) Consider 2D steady state conduction heat transfer in a long rectangular bar. The bar has a height, h, of 10 cm, and a width, w, of 5 cm. The top of the bar is held at a temperature, T1, of 600 K while the remaining 3 sides are held at a temperature, T2, of 300 K. a.) State the governing partial differential equation. b.)
Apr 30, 2020 · Solution of 2D Heat Conduction Equation. Domain : Mechanical Engineering, Aerospace Engineering, Thermal Engineering. Benefits : In this project you will solve the steady and unsteady 2D heat conduction equations. You will implement explicit and implicit approaches for the unsteady case and learn the differences between them.
In this report the mathematical and numerical models for heat transfer are described. A code prototype is developed in MATLAB. For comparison di erent approximations with VIRSuite data structure, a utility translating the VIRSuite triangular mesh equipped with thickness to a 3D mesh composed of triangular prisms is also developed. A copy of the ...
Stack Exchange network consists of 176 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers.
Jan 04, 2013 · If you think you have played the best MATLAB game before. Think again after you try this one. _____ Experiments with MATLAB. Cleve Moler. Apps from Experiments with MATLAB. _____ Numerical Computing with MATLAB. Cleve Moler. Apps from Numerical Computing with MATLAB. _____ GUI 2D Heat Transfer. Dominik Gibala. This GUI presents 2D heat tranfer ...
2d Fem Code Matlab
Diffusion only, two dimensional heat conduction has been described on partial differential equation. I am currently writing a matlab code for implicit 2d heat conduction using crank-nicolson method with certain Boundary condiitons. From a practical point of view, this is a bit more is the alternating direction implicit (ADI) method.


Sir Charles ... The formula constant is T=150(d/r) for velocities from about 1500 to 2800 fps. Substitute 180 for the 150 value for velocities exceeding 2800 fps.
And for velocities below 1500 fps, substitute 125 for the T constant
According to Hatcher, the bullet used to derive the constant was a Krag 220 grain, L=1.35in., v=2000ft/sec, d=10.9 gm/ml, p=1.221gm/l For i=.61, the constant comes out to 150.


Cracking the Code of Good results My Math Research Unveiled
Arithmetic has usually been a matter that equally intrigues and difficulties students of all ages. From basic arithmetic to advanced calculus, math is an integral element of the academic journey. Among the quite a few assignments and duties that students experience, “my math homework” frequently stands out as a formidable adversary. In this write-up, we will investigate the significance of math research, its benefits, and approaches to tackle it effectively.
The Significance of My Math Homework
Arithmetic is not just a subject to be examined in the classroom it needs constant apply and reinforcement outside of faculty hrs. My math research plays a crucial role in solidifying the concepts taught in class. It allows college students to apply what they have uncovered independently, assisting them to internalize mathematical rules. Moreover, math research serves as an evaluation tool for teachers, enabling them to gauge a student’s comprehension and progress.
Advantages of Finishing My Math Homework
Finishing math homework delivers many advantages. To start with, it fosters willpower and time administration expertise. someone do my math homework Persistently doing work on assignments teaches pupils the importance of placing apart committed examine time. Secondly, it promotes dilemma-solving skills. Math research provides numerous issues that stimulate vital pondering and creativity. By tackling these problems, pupils increase their problem-solving skills, which are a must have in every day existence. Lastly, homework prepares students for upcoming assessments and examinations, allowing them to carry out better academically.
Techniques for Good results
Beating the hurdles of my math homework demands a strategic method. Here are some suggestions to assist you be successful:
Recognize the Principles: Make sure you understand the underlying concepts just before attempting any math homework. Seek clarification from your trainer or use on the internet sources to fortify your comprehending.
Organize Your Workspace: Create a thoroughly clean, properly-lit, and arranged review place free of charge from interruptions. A conducive setting can substantially boost your productiveness.
Prepare Your Time: Produce a review timetable that includes focused time for math homework. Consistency is important.
Practice Frequently: Mathematics is a topic that needs exercise. Will not shy absent from repetitive difficulty-solving it’s the essential to mastery.
Seek out Help When Necessary: If you come across troubles, will not hesitate to question for help. Your instructor, classmates, or online community forums can give valuable guidance.
In summary, “my math research” could seem complicated at instances, but it performs a pivotal position in mastering mathematics. Embrace it as an opportunity for progress, and with the proper approaches and frame of mind, you can excel in this difficult subject. Don’t forget, accomplishment in math is not just about resolving equations but also about constructing crucial expertise that will reward you all through your academic and professional journey.


If you’re thinking about installing artificial grass in your outside space, you may feel overwhelmed by the thought of knowing how much you’ll need to order. In this post, we’ll help calm any worries that you have by providing a simple artificial grass calculator. Read on to find out the best way to measure fake grass.
Measuring for artificial grass
When thinking about your artificial grass calculator, you will need to think of measuring your outdoor space in metre squares, much like a carpet!
The range of artificial lawns at SayGrass is all supplied in three ways. They can either be supplied as a full roll measuring 4 metres by 25 metres. Or the fake grass can be cut to size using either 2-metre or 4-metre widths.
You will need to measure your garden most effectively based on these sizes. You are aiming for the least number of joins, which will make the installation process easier.
Gardens come in a range of shapes and sizes. The easiest method for calculating how much fake grass you need is the basic square or rectangle-shaped garden.
Artificial grass calculator for square or rectangle shaped garden
For your fake grass calculator in this shaped garden, simply measure the length of your garden by the width and multiply these two measurements together to give you the area. For example, a garden measuring 5 metres long by 3 metres wide will need 15 square metres of artificial grass to cover this area.
Don’t forget that we only supply in widths of 2 metres or 4 metres which may mean you need to round up one of your measurements. In our above example, you would need to order a piece of false grass measuring 4 metres by 5 metres giving you a 20-metre square area of synthetic grass.
Artificial grass calculator for angled shape garden
If you have a garden that tapers off at the top or bottom, you can still use the fake grass calculator as mentioned above. Measure your space, in the same way, making sure to measure the widest and longest part in metres. There will be some leftover grass once you have finished installing.
Artificial grass calculator for a circular shaped garden
For circular-shaped gardens, you can use the same artificial grass calculator as above. Simply measure the width and the length of your circular-shaped garden and multiply these measurements together to get the area of your lawn.
Your synthetic grass will be supplied as a rectangular shape which you will then need to cut into shape.
Artificial grass calculator for an L-shaped garden
L-shaped gardens are not as complicated as you might think. First, split your outdoor shape into two rectangle shapes. Use the same fake grass calculator as above to calculate the longest rectangular section first. If for example, this measures 3 metres in width and 6 metres in length, this gives you 18 metres square for the first rectangle.
Use the same artificial grass calculator for the shorter rectangle of the L. For example, let’s say this measures 2 metres in width and 3 metres in length, it will give you an area of 6 metres square for the second rectangle.
Now add the two area rectangles together to give you the total amount of square metres that you need – in this example – 24 metres square.
Use our artificial grass calculator when working out how much fake grass you need from Say Grass ltd
Measuring fake grass doesn’t need to be difficult when you use the artificial grass calculator as shown.
At SayGrass our top-quality artificial grass range all comes by the roll, so it is supplied at unbeatable prices. It can also be cut to your specified measurements; take a look at the drop-down menu on our false grass product page.
For help or questions about measuring your synthetic grass, get in touch with our helpful team.


The Development of Analytical Skills in Mathematics of Grade 6 Students
This research is aimed at developing analytical skills in mathematics of Grade 6 students and evaluating the results of the development of these analytical skills in the learning of mathematics by integrating collaborative learning and case-based learning in the learning process. This research was onducted in the academic year 2012. The target population group consisted of 61 Grade 6 students at Satit Bilingual School of Rangsit University. The research instruments used in this study were: 1) the mathematics teaching process which integrated collaborative learning and case-based learning, 2) a case-study handbook, 3) mathematics lesson plans, 4) an evaluation form for analytical skills, 5) a mathematics test, and 6) an evaluation form concerning students’ attitudes towards the mathematics learning process. The data obtained were analyzed by using mean scores, standard deviation, and percentages. The results of the study were as follows. First, the development of the analytical skills was brought about by integrating collaborative learning and case-based learning in the learning process. This process consisted of the following four stages: a) elicitation of prior knowledge, b) presentations of case studies, c) case analysis through collaborative learning, and d) conclusion and application of the knowledge gained. Second, the average score for analytical skills of the targeted group of students was at a high level, and the development of their analytical skills increased steadily throughout the study. Third, the average score for knowledge of mathematics in the target group of students after the learning process (post-test) was higher than their average score prior to the learning process (pre-test). Last, the average score for attitude toward the mathematics learning process of the target student group was at the highest level.