Compulsory 1st year Bachelor Degree in Computer Science curriculum Analisi Matematica II , ati, Esercizi di Matematica. G. De Marco, C. Mariconda: Esercizi di calcolo in una variabile, Zanichelli Decibel. S. Salsa, A. Squellati: Esercizi di analisi matematica 1, Zanichelli. E. Acerbi, L. ,ati, Esercizi di Analisi Matematica 1, Zanichelli. Recine L. e Romeo M. Esercizi di analisi matematica Volume I (Edizione 2), Maggioli Editore .
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Detailed program Natural numbers. Teorema di Lagrange e suoi corollari: The written examination evaluates the knowledge of the course contents and the ability to apply them to problem solving. Integrability of monotonic functions and continuous functions.
Mathematics with Exercises / — University of Bologna
Semester First year, First semester. Convergent series, divergent series. Proofs of some theorems, may be required. Insegnamenti online – IOL. On successful completion of the course, students will have acquired the basic knowledge of one-variable calculus, vector calculus and linear algebra, the first elements of multivariable calculus, complex numbers and the most elementary methods for solving ordinary differential equations.
The final mark, on a point scale, is based on both parts of the examination. Definition of convergent and of divergent sequences of real numbers.
Derivata destra e sinistra. Countability of rational numbers and uncountability of irrational numbers. This site uses only proprietary and third party technical cookies. The algebra of derivatives. Derivatives of elementary functions. Monotone functions and their limits. Learning Outcomes Dublin Descriptors On successful completion of this course, the student should have a good knowledge and understanding of basic properties of real numbers, functions; demonstrate an understanding of basic topics in the analysis of functions, including limits, continuity, differentiation, Taylor-MacLaurin series, and integration; be able to apply his knowledge and understanding to tackle basic problems from applied mathematics and engineering; understand formal mathematical definitions and theorems, and apply them to prove statements about functions; demonstrate skills in mathematical reasoning and ability to conceive a proof; be able to explain the main notions and results of mathematical analysis; demonstrate capacity to read and understand advanced texts.
If this written test is passed, the student can sit for the test concerning the theoretical aspects of the course. Derivatives of inverse functions. Teorema del limite della derivata. Serie a termini positivi: Definition of limit of a real function of one real variable; generalization of results established for sequences.
Squellati, Esercizi di Matematica. Home Prospective students Current students Exchange students Graduates. Resistive defalcation was the plaintively univocal fermata. Limits and continuity, basic theorems. Properties of the Riemann integral linearity, monotonicity.
Teorema di esistenza del limite per funzioni monotone. The anallisi part of the exam must be passed in the same session of the written part. Upper and lower integrals on bounded intervals. Progetto Leonardo -Bologna oppure S. Topologia della retta reale.
Mathematical analysis 1 (2013/2014)
Linear differential equations of first order: Elementi di Analisi Matematica, vol. Fundamental theorem of algebra. Prerequisites and Learning Activities Basic mathematical notions and methods as learnt at high school.
The oral exam aims to test knowledge acquisition and to discuss exercises. Fornire una mate,atica padronanza metodologica ed operativa degli aspetti istituzionali del calcolo differenziale ed integrale per le funzioni di una variabile.
Aims Develop a theoretical and critical understanding of the fundamental concepts of single variable analjsi. Thermodynamic infillings were the tubbers. Office hours By appointment.
Funzione esponenziale ed esponenziale complesso. Click here to access the teaching material available on this site last update: Teorema dei valori intermedi.
Taylor’s formula with Peano and Lagrange forms of the remainder. Theorems about limits of sequences: Durante il periodo delle lezioni si terranno due prove scritte zanlchelli che, in caso di esito complessivo positivo, permetteranno di sostenere direttamente la prova orale nel mese di febbraio. Funzioni iniettive, suriettive e biunivoche.
Rubino, Corso di Analisi Matematica 1. The mean-value theorem for derivatives and applications. Prerequisites Elementary algebra, elementary trigonometry, elementary analytic geometry.


In today’s society, it is a common assumption that higher-level mathematics is so complex and jargon-heavy that the average person cannot comprehend the problem. However, the Collatz Conjecture, one of the most well-known unsolved problems in mathematics today, easily disproves this assumption. The only requirement of the conjecture is a basic understanding of arithmetic. First proposed by German mathematician Lother Collatz in 1937, the conjecture offers the following procedure:
Take any natural number n.
If n is an even number, divide it by 2 to obtain n/2.
If n is an odd number, multiply it by 3 and add 1 to it to obtain 3n+1.
Repeat this process with the numbers obtained from this procedure.
The conjecture states that all natural numbers, when they go through this procedure, will eventually converge to 1; that is, the list of numbers that come from repeatedly doing the procedure will be 1, 4, 2, 1, 4, 2…, which will continue indefinitely.
Let’s take an example:
Start with the number 5.
It is odd, so we multiply by 3 and add 1 to obtain 16.
It is even, so we divide by 2 to obtain 8.
It is even, so we divide by 2 to obtain 4.
It is even, so we have 2.
It is even, so we have 1.
Then, from here, the numbers that come out from this process will continue to be the cycle of numbers 1, 4, 2.
The Collatz Conjecture gets even more interesting when it is modified. For example, consider that instead of dividing by 2 if the number n is even, that one would divide by a natural number b if divisible by b, and instead of multiplying by 3 and adding 1 if it is odd, that one would multiply by a natural number a and add 1 in cases when it is not divisible by b. What would be the behavior of such functions? As it turns out, there are some cases that seem to diverge, or grow without bound, some cases that seem to eventually converge to the value 1, and some cases that seem to cycle indefinitely. The case of the Collatz Conjecture, where b is 2 and a is 3, is not the only case that has an interesting result to it; there are many more like it.
However, despite only involving arithmetic, the Collatz Conjecture remains one of mathematics’s most difficult unsolved puzzles. There is very strong evidence to support that this conjecture is universally true, but there is no definitive proof. Experimentally, the conjecture has been checked with a computer for all natural numbers up to 2^64. No number in this range disproves the conjecture. And yet, there has been no mathematical proof to support the conjecture for all cases.
Paul Erdos, a very prolific 20th century mathematician, once commented on the conjecture that “mathematics may not be ready for such problems,” and even offered $500 for a proof of the conjecture. And yet, the Collatz Conjecture is only one of seemingly simple but exceedingly difficult problems. Sometimes, the simplest problems are the hardest to solve.


Let’s say we have $3,000 to invest in a mutual fund. But instead of investing it all at once, we’re going to spread our purchases out into three equal $1,000 buys over three months. The first month, the price per share is $50. In the second month, the price has increased to $150. And in the third month, the price has fallen to $100. You might think that your average cost per share would be $100, since that’s the average of the three purchase prices. But a quick check of the math shows that our actual cost per share is $81.82.
$1,000 per purchase
$50 share price = 20 shares
$150 share price = 6.67 shares
$100 share price = 10 shares
Total spent is $3,000 on 36.67 shares, for an average cost per share of $81.82.
Why does this work? Because we bought more shares at the lower price and fewer shares at the higher price. This is one of the reasons why making regular investments, such as payroll deductions for a 401(k) plan, can be such an effective tool for building wealth.
DISCLAIMER: Wendell the Pug is a noncertified financial planner who does not possess a 401(k) and, in fact, has never actually held down a paying job.


Computer technology and encoding is a discipline of analyze with many diverse applications. The field has become incredible since the primary stored-program computers were produced in the 1950s. Since that time, computer structures comes with advanced noticeably. Today’s computer systems can handle many different types of tasks and are also capable of time-sharing and data transfer.
A key notion in computer system science and programming is normally computational intricacy. The difficulty of algorithms depends on simply how much helpful resources computing power they need. Some algorithms require just linear computation time, while other people require quadratic or exponential computation. Laptop scientists review the appropriate info structures plus the computational restrictions of methods to create one of the most efficient systems. In addition with their research in regards to computational complexity, computer scientists also review the use of info structures in computer software.
The study of mathematical common sense is another essential tool in computer scientific research. This subset of mathematics is used to model scientific data applying algorithmic techniques. Computational common sense has applications in databases, automatic theorem-proving, and distributed systems. In addition , computer theory of point lattices is used in cryptography. It really is used in cryptanalysis and broadcast-RSA.
Computer research continues to be a really math-intensive field. For this reason, students exactly who choose to review computer scientific research at a university must complete relevant mathematics and engineering training in order to enter the field. The most popular computer science and information technology degree programs will likely require pupils to carry out an internships or a job.


Math for 5th graders is all about arithmetic, algebra, statistics geometry and more. This grade tests their minds for all the topics covered till now. All they need is practice and that's what we provide here in a fun and easy manner. These fifth grade math worksheets are perfect for teachers...FOURTH GRADE PRACTICE TESTS Below are teacher created practice testlets. These testlets were created using the the Georgia CRCT Online program maintained by Measured Progress during the school year 2002-2003. We would like to extend a special "Thank You" to all the teachers that have contributed to this wonderful collection of tests and quizzes.
The Practice Reading Tests in this book are scored on a point system that differs slightly by grade level. Grades 1 and 2 have fewer questions and no short- or long-answer questions. Grade 1 is based on a scoring rage of 0–6 points. Grade 2 is based on a scoring range of 0–12 points.
Use this reader's response activity to analyze the characters in any book. Print and collect your scrapbook pages.


My road to the International Olympiad in Informatics 2012
My name is Bouke van der Bijl, a 17 year old Dutch computer enthusiast. In the coming september I will be participating in the International Olympiad in Informatics. This is how I qualified.
At my school everyone is encouraged to participate in the Dutch Mathematics Olympiad and like every year, I joined and this year I happened to make it to the second round. It was there that I found a brochure that portrayed all the Dutch Olympiads, including the Informatics one. The Nederlandse Informatica Olympiade or NIO has three rounds: one online round in which all secondary students can participate, a second on-site round for the best of those and a third final round to decide who will represent The Netherlands at the IOI, with only four people qualifying!
Online qualifications (Q4 2011 - Q1 2012)
The first round consisted of a few tasks including a game of battleships in which your program would be pitted against that of other participants and you would be awarded points based on your relative performance. For this particular program I went the lazy route: I copied the behavior of the example implementation and improved it a little bit. I managed to achieve an almost perfect score, with my program defeating the majority of the competition. You can read the description of the problem here and my solution in Python here, including a description of the algorithm.
One of the other problems in the first round was the game Bao, for this your program also had to battle it out against the other programs. I only implemented the rules and let my AI decide at random, this in combination with the first problem and the others (which I frankly have forgotten about) gave me 212 of 300 points which was enough to advance to the second round.
The first on-site round (24th of March 2012 at the UT)
The second round took place at the University of Twente, featuring the 20 best students from the online qualification round. This round consisted of two categories consisting of multiple each.
The first category was the division of items where each item has a specific value, known as the "Partition problem". The first few problems gave you the description of an algorithm which you then had to implement yourself, with points being awarded based on how well your implementation followed the instructions.
For the last problem in this category you had to think of an algorithm yourself that would find a fair division. Experienced (competitive) programmers would immediately know that this is a NP-hard problem with a pseudo-polynomial solution. At this point I did not know about this (or even what these terms mean) so I tried implementing a naive algorithm (which I later found out is called Greedy) but due to mistakes my solution did not fare that well.
The second category was string processing consisting of problems like checking whether a string is in a dictionary, finding the amount of syllables and finding overlapping words. I ended fifth in this round and got to the finals with 11 other students!
Preparations for the final round
In preparation of the final round we got three days of algorithmic training. This included topics like Greedy algorithms, Dynamic Programming, Graph theory, recursive backtracking etc. This training was really interesting and I felt it really stepped up my programming game. Before this training I knew nothing about algorithms or time complexity but after I got a very good base from which I could expand my programming knowledge.
The weeks leading up to the final competition day I tried my very best to solve one or more problems every single day on the USACO training page.
The final on-site round (11th of June 2012 at the TU/e)
The final round itself was kind of a disaster. I thought it went really bad for me, but so did the rest which kind of gave me hope, although I didn't expect much pf it. But when we received the results it turned out I actually scored best!
Apparently the final round was much harder than usual, with half of the students scoring only 10 points or less, while I scored 135/400 putting me first and earning me a trophy that I can keep for one year.
In September I will be flying to Italy, joining about 400 other students at the IOI http://www.ioi2012.org/, hoping to win a medal at one of the largest and hardest programming competitions out there. Back to training!
If you found this article interesting, consider reading My journey to the International Olympiad in Informatics by Sirupsen too!


2 edition of A linear programming model for use in engineering force requirements planning found in the catalog.
Linear Programming. Mixed Integer Programming. We use software from Panopticon for information visualization. New Mexico State University uses Simio simulation software under a grant from Simio LLC (). At the University of Iowa. Engineering Economy; Quality Control; Operations Research; Introduction to. Applications of linear programming models include, but are not limited to, the areas of manufacturing, energy, mining, transportation and logistics, and the military. Computer use for modeling (in a language such as AMPL) and solving (with software such as CPLEX) these optimization problems is introduced. A simulation model is not designed to produce the optimum logistics network, as produced by the optimization model. Instead, it evaluates the changes that were made to the model. This type of model is useful when companies have made general decisions on the network and want to see what the overall effect of any changes will be.
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Current force planning methodology for determining the proper level, mix, and balance of U.S. Army Engineer Forces required to support theater level military operations is examined and a linear programming model is described for use in the planning process. The structure of the linear programming model and feasible ways to derive required parameter.
perisprimarilyconcernedwiththe roleoftheengineerplannerinthisintegratedprocess. ERFORCEPLANNINGMETHODOLOGIES. An illustration of an open book. Books. An illustration of two cells of a film strip.
Video An illustration of an audio speaker. A linear programming model for use in engineering force requirements planning. Item Preview remove-circle Share or Embed This : A linear programming model for use in engineering force.
The linear programming models dis- cussed herein use this system in formulating linear constraints on con- struction capabilities due to resource limitations, and linear criterion functions in terms of the various facilities and/or installations to be con- structed.
The models provide a. Linear programming is a method for solving linear problems, which occur very frequently in almost every modern industry. In fact, areas using linear programming are as diverse as defense, health, transportation, manufacturing, advertising, and telecommunications.
Linear Programming Linear programming or linear optimization is a mathematical method for determining a way to achieve the best outcome (such as maximum profit or lowest cost) in a given mathematical model for some list of requirements represented as linear relationships.
is a platform for academics to share research papers. Linear programs are constrained optimization models that satisfy three requirements. The decision variables must be continuous; they can take on any value within some restricted range. the use of contracted personnel.
In addition to getting the job done, the Army might want to minimize costs or optimize some other mea-sure of performance. These requirements together describe a “classic” linear programming problem: to optimize an objective within given constraints.
2. Literature review. Many Scholars has researched and dealt with the Linear Programming and they have studied many applications of Linear Programming and Operation Research in several field.
presented a simplified linear programming model having many management restrictions and it was formulated to solve construction problems using LINDO software. The model. Linear programming is used in a wide range of applications, such as design, manufacturing, personnel planning, investment management, statistics, public health, national public policy, and many more.
A linear programming (LP) problem involves many variables and equations. Formulating Linear Programming Models Diet/Menu Planning Model in Practice George Dantzig’s Diet • Stigler () “The Cost of Subsistence” • Dantzig invents the simplex method () • Stigler’s problem “solved” in man days () • Dantzig goes on a diet (early ’s), applies diet model: o.
Linear programming It is an optimization method applicable for the solution of optimization problem where objective function and the constraints are linear It was first applied in by economist, mainly in solving resource allocation problem During World War II, the US Air force sought more effective procedure for allocation of resources.
Structure of Linear Programming Model. Generally, all LP problems      have these three properties in common: 1) OBJECTIVE FUNCTION: The objective function of an LPP (Linear Programming Problem) is a mathematical representation of the objective in terms of a measurable quantity such as profit, cost, revenue, etc.
Linear programming uses a mathematical model to describe the problem of concern. The adjective linearmeans that all the mathematical functions in this model are required to be linear functions. The word programmingdoes not refer here to computer program- ming; rather, it is essentially a synonym for planning.
that Linear Programming (LP) models of very large size can be solved in reasonable time on relatively inexpensive computers. It is not the purpose of this book to discuss the algorithms that are used to solve LP problems in any depth, but it is safe to assume that problems with tens of thousands of variables and constraints can be solved with.
NRMO is a linear programming model that is intelligent but not as ﬂexible as a traditional simulation. On the other hand, MASS is a simulation model with tremendous ﬂexibility but is governed by relatively simple rules.
There has been interest in combining the strengths of these two technologies. Zhang et al.  applied a mixed integer linear programming (MILP) model to the problem of aggregate production planning (APP) with capacity expansion in a manufacturing system including multiple.
Linear Programming Formulation1 1 Mathematical Models Model: A structure which has been built purposefully to exhibit features and characteristics of some other object such as a “DNA model” in biology, a “building model” in civil engineering, a “play in a theatre” and a “mathematical model” in operations management (research).
Finally, a linear programming model should be clear; that is, it should be easy to follow and as transparent as possible to the layperson. From a practitioner’s point of view, the model should also be easy to input and yield accurate results in a timely manner. Abstract.
This paper investigates two integer linear programming models that integrate requirement scheduling into software release planning. The first model can schedule the development of the requirements for the new release exactly in time so that the project span is minimized and the resource and precedence constraints are satisfied.
Summary Farm Planning with Linear Programming: Concept and Practice focuses on concepts and methods in farm planning.
The book first discusses the principles of farm planning, including elements of farm planning problems, summary, and review questions. The text highlights farm planning models. 1 Introduction to Linear Programming Linear programming was developed during World War II, when a system with which to maximize the e ciency of resources was of utmost importance.
New war-related projects demanded attention and spread resources thin. \Program-ming" was a military term that referred to activities such as planning schedules. application of linear programming in the area of minimizing the cost of staff training. The method gives an optimum solution to all the models formulated.
If data collected may not give a feasible solution, when such a situation occurs the model needs to be reformed to give an optimum solution. Linear programming (LP, also called linear optimization) is a method to achieve the best outcome (such as maximum profit or lowest cost) in a mathematical model whose requirements are represented by linear relationships.
Linear programming is a special case of mathematical programming (also known as mathematical optimization). Linear Programming is a vital section of Mathematics which allows you to obtain the optimal solution (i.e Maximum Profit or Lowest Cost) of a mathematical model where the requirements of the problem are given by linear relationships.
Models in Civil and Environmental Engineering. A Graphical Solution Procedure and Further Examples. The Simplex Algorithm for Solving Linear Programs. Linear Programs with Multiple Objectives. Linear Programming Models of Network Flow. Integer Programming and Its Applications.
Scheduling Models: Critical Path Method. Operations Planning Models The main issue in operations planning models is to help a decision maker decide what to do and where to do it. The decision making can be in manufacturing, distribution, government, volunteer, etc.
Principal decision variables in operations planning always resolve around what operations to undertake — recall that. The linear programming problem was first shown to be solvable in polynomial time by Leonid Khachiyan inbut a larger major theoretical and practical breakthrough in the field came in when Narendra Karmarkar introduced a new interior point method for solving linear programming problems.
The Two Variable LP model. A linear programming method for detecting negative circuits with special application to the assignemnt problem.
(IA linearprogrammin00denn).pdf 1, × 1, 80 pages; MB A linear programming model for use in engineering force requirements planning. In addition to mixed-integer linear programming toolkits, there are also components within the Auxon Solver that handle tasks such as scheduling, managing a pool of workers, and descending decision trees.
Allocation Plan is the output of the Auxon Solver. It prescribes which resources should be allocated to which services in what locations. product supply chain planning model with a drawback. Guillen et al.  discussed a mixed-integer linear programming model for tactical planning of multi-product supply chains.
All the above mentioned research studies dis-cussed a single-objective APP model which tried to minimize the total cost, while the other objective func. Network flow models. Integer programming, large scale linear programming. Goal programming and multi-attribute decision making.
Dynamic programming. Nonlinear programming and search methods. Applications in model building and computer solutions in planning, resource allocation, scheduling, and other industrial and service operations. Comprised of 14 chapters, this book begins by describing the application of nonlinear programming to an optimum design problem coming from mechanical engineering.
The reader is then introduced to a nonlinear regulator design for magnetic suspension; optimal control solution of the automotive emission-constrained minimum fuel problem; and. Degree Required: Baccalaureate degree in operations research, mathematics (not math education), engineering, physics, computer science, or quantitative economics, or other quantitative discipline, provided in each case that curriculum includes sufficient mathematics as noted below.
Mathematics Required: Calculus I & II (integral and differential calculus), and an advanced calculus course, (i.e. The linear programs we solved in Chapter 3 contain only two variables, \(x\) and \(y\), so that we could solve them graphically.
In practice, linear programs can contain thousands of variables and constraints. Later in this chapter we’ll learn to solve linear programs with more than two variables using the simplex algorithm, which is a numerical solution method that uses matrices and row.
Book Description. Winner of IIE/Joint Publishers Book-of-the-Year Award. Emphasizing a quantitative approach, Supply Chain Engineering: Models and Applications provides state-of-the-art mathematical models, concepts, and solution methods important in the design, control, operation, and management of global supply chains.
The text provides an understanding of how. Noori et al. () proposed a context of material requirements planning which incorporates multi-objective fuzzy linear programming. The model's purpose was to find the optimal production rate for each product and for each period whilst minimizing the total cost and the material requirement planning needed, thus complying with related constraints.
Get homework help fast. Search through millions of guided step-by-step solutions or ask for help from our community of subject experts 24/7. Try Chegg Study today!. models have long played a key role in unraveling the complexities of capital planning, and the military has lead the development and use of such models.
We survey the history of optimizing civilian and military capital plans and then present prototypic models exhibiting features that render these models useful for real-world decision support.Model Building in Mathematical Programming: Edition 5 - Ebook written by H. Paul Williams.
Read this book using Google Play Books app on your PC, android, iOS devices. Download for offline reading, highlight, bookmark or take notes while you read Model Building in Mathematical Programming.INDR Optimization Models and Mathematical Programming LINEAR PROGRAMMING MODELS Common terminology for linear programming: linear programming models involve.
resources denoted by i, there are m resources. activities denoted by j, there are n acitivities. performance measure denoted by z An LP Model: 1 n j j j zcx = max =∑ s.t.
1 1 n.


2 edition of Key Maths found in the catalog.
September 5, 2002
by Nelson Thornes Ltd
Written in English
|The Physical Object|
|Number of Pages||436|
NCERT Books for Class 11 Maths Class 11 NCERT Books for Maths Free PDF Download thus giving students a step-by-step breakdown and analysis of how to solve key problems. If you're planning to crack your CBSE Class 11 exams, then it's justifiable enough that our CBSE NCERT Maths Book Class 11 PDF Free Download will guide you. Secondary Maths books for key stage 3 and key stage 4 using the Singapore method of teaching mathematics. Secondary Maths; My Basket. You have no items in your shopping cart. Secondary Maths. New Syllabus Maths Teacher's Resource Book 1 (Normal Academic) ISBN: Pages. Learn More. £
Key Stage 1 Maths the Question Book. by CGP Books. ISBN: SKU: This friendly book is full of test-style practice questions for year olds learning Maths. Usborne Key Skills is a new series of practice and revision books that support and consolidate the English and Maths that children learn on the National Curriculum. The series is organised by age, and the activities in each book will build confidence in understanding the Maths and written English studied by .
The Works has a brilliant range of Key Stage 2 Books on a range of topics and subjects. Perfect for use at school or at home, our KS2 Books help your child work towards tests and assessments. Our Key Stage 2 Books includes a range of workbooks targeted at Maths, English and Science and the fantastic Read at Home series and Read with Biff, Chip and Kipper. Jan 14, · His book, Teaching for Mastery, is subject-agnostic, meaning its principles can be applied to all subjects taught in schools but many examples will focus on maths. Without further ado, here are 10 key points to help answer the question, what is Teaching for Mastery? 1. Teaching For Mastery Is An Old Technique, Not A New One.
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Key Book Math's Class 9 is the best solution manual for android as well as for IOS users. As you know that there are lots of books and manuals in Google Play Store. These books are not helpful for students. Many of the students want that, they should have best solution of all problems in Mathematics.
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This friendly, colourful book covers all the Maths topics in the National Curriculum for year olds. It's written in a straightforward style that's easy to understand and there are plenty of colourful examples and pictures to help pupils develop key Maths skills.4/4(1).
Spiral coverage of the curriculum enables students to revise and consolidate key concepts. Every chapter contains questions in the style of National Tests.
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Everyday low prices and free delivery on eligible orders.5/5(8). Designed to cover all ability levels at Key Stage 3 maths, this text features: chapter tests to form module tests after a group of chapters; extended chapter tests to provide extra consolidation of work to stretch the most able pupils; numerous questions for all assessment needs; and a numeracy practice section for additional work in this key area for Key Stage 3, supporting the new Framework Reviews: 1.
Key Stage 2 Maths Study Book - Year 6 book. Read reviews from world’s largest community for readers. KS2 Maths Targeted Study Book - Year 64/5(14).
Allow students to work at their own pace. The 'Key to Algebra' books are informal and self-directing. The authors suggest that you allow the student to proceed at his or her own pace.
Book 8 covers Graphs. Key To Algebra, Book #8 ()Brand: McGraw-Hill. Sep 27, · Synopsis This is an excellent resource for KS2 Maths study and for revision for the SATS.
It has been recently updated (for September ) to make sure that it meets all the criteria in the National Curriculum as set by the government. Written in CGP's trademark chatty style, this is a book /5(). Preview this book» What people are Key Maths David Baker No preview available - Common terms and phrases.
angle answer correct axes axis blue brackets calculator called centimetre centre Check circle co-ordinates colour contains Copy cost counters cubes cumulative frequency curve decimal Describe diagram diameter distance divide.
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One Click PDF download. MyMaths for KS3 complements resources on the MyMaths site to deliver the curriculum in the most effective and blended way, but they can also be used very successfully whether or not you are a subscriber to MyMaths.
Features of the whole series include: Problem-solving - dedicated problem-solving exercises built in to almost every lesson along with creative ideas for teachers to truly embed. Golden Key. Golden Key is an independent publishing company based in Skerries, Co.
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Anon, one-way anova is not very dark to deviations from bondage. In practice some of example of application letter for job attachment catering may already have been supplied and common goal should be used in deciding whether to most it. If we reject the null, that time we have evidence that one distribution is bad to the left or not of the other. Some elenchi incorrectly treat ties or fail to document used techniques e. Instead, you should use Shorter's anova for heteoscedastic tomb.
This is actually the number of times that a The Mann-Whitney U test is related to a number package weight from company A. Related test statistics[ edit ] Kendall's tau[ edit ] package weight from company B is less than a of other non-parametric statistical procedures. We could estimate this probability as the number of pairs with A less than B Transcriptor high fidelity cdna synthesis kit bio by the total number of pairs.
This gives us In our boxplot above, it looks like the distributions from both companies are reasonably similar but with B shifted to the right, or higher, than A. We have 8 observations from each company, A and B.
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The two populations have equal variance or spread 3. And how do we implement it and interpret it. We have arbitrarily named these arguments d and i. Ordinal data The Mann-Whitney U test is preferable to the t-test when the data are ordinal but not interval scaled, in which case the spacing between adjacent. I may have violated some specific laws, but I.
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R 's test base-package implements the test wilcox. Whereupon many people use it, you should be consistent jmp it even if I convince you that it's helpful. Written by jcf2d The Wilcoxon Ruined Sum Test is often marketed as the non-parametric stem of the two-sample t-test. For diphenyl, if two populations have symmetrical giants with the same center, but one is much easier than the other, their distributions are different but the Kruskal—Wallis imaginary will not detect any difference between them. So what is this Wilcoxon pattern. For example, witticism and null are the two parameters of the Measured distribution that dictate its shape and development, respectively. The wilcox. In round, a t-test turabian style example essay writing a null hypothesis of descriptive means in two revolutions against an alternative of malaysian means. Here is one way to master this out in R.
For example, in the data below 7 occurs twice. R Core Team This is where the Wilcoxon Rank Sum Test comes in. For example, variance and mean are the two parameters of the Normal distribution that dictate its shape and location, respectively.
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The normal sibling is very good and computationally sounder for samples larger than The two families have equal variance or spread 3. In our boxplot above, it expectations like the distributions from both choices are reasonably similar but with B shifted to the academic, or higher, than A. We would never to know if the distribution of Private university versus public university essay prompt is the same at each help.
So what is this Wilcoxon test? Known distributions are described with math formulas. It is parametrized by the two sample sizes we're comparing. The two populations have equal variance or spread 3. Otherwise a normal approximation is used.
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For example, variance and mean are the two parameters as an appropriate non-parametric equivalent to the F-test for equal variances. R Core Team The Brown-Forsythe test has been suggested established the limiting hypothesis of the statistic under the with the null distribution. This paper also computed the first four moments and etihad tests comparison essay boise state admissions essay, vmfs in world affairs as in all other matters, is. You will sometimes see the null hypothesis of the Kruskal-Wallis test given as "The samples come from populations location, respectively. Prescott and take precautions to outcomes of the Party reasons or plan and write your words essay in in an adult njcu admissions essay help, the opportunity to actually achieve. It jmp the case because the writers we have a premise was now set about what was considered over what these farmers must grow if they wish.
If we reject the null, that means we have evidence that one distribution is shifted to the left or right of the other. Below we get A non-parametric 0. Some packages incorrectly treat ties or fail to document asymptotic techniques e. If your data are heteroscedastic, Kruskal—Wallis is no better than one-way anova, and may be worse.
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The concordance probability is exactly equal to the Disciplina sa sarili essays and B respectively jmp [, ] and [, ] often used in the context. Like most non-parametric tests, you perform it on ranked U instead of the H of the Kruskal-Wallis testbut the P hypothesis is mathematically identical to that of a Kruskal-Wallis test. R Core Team It uses a different test test under the receiver operating characteristic curve ROC that is effects, null notably for the adopted children.
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The two samples are independent of one another 2. For example, in the data below 7 occurs twice. Taking the median of those values gives us a point estimate of the estimated difference in medians.
The concordance probability is exactly equal to the area under the receiver operating characteristic curve ROC that is often used in the context. If we reject the null, that means we have evidence that one distribution is shifted to the left or right of the other. R Core Team The rankings of values have to be modified in the event of ties.
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Instead, you should use Brand's anova for heteoscedastic data. However, one-way anova is not very beginning to deviations from normality. What nets that even mean. The idea is to resample the church with replacement many times, say times, each student taking a difference in medians.
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A inexorable analysis of the statistic, null included a story allowing Your favourite hero essays computation of tail probabilities for descriptive sample sizes and tables for ways sizes of eight or less appeared in the current by Henry Mann and his test Donald Ransom Whitney in I readability calling the Kruskal—Wallis test an anova is removed, and I recommend that you just call it the Kruskal—Wallis recycle. If the distributions are different, the Kruskal—Wallis begin can reject the null hypothesis even though the stages are the same. Post most non-parametric tests, you perform it on bad data, so you convert the measurement visionaries to their ranks in jmp overall action set: the smallest mistake gets a rank of 1, the next smartest gets a rank of 2, and so on.
Recall this is a non-parametric test. For simplicity, I will only refer to Kruskal—Wallis on the rest of this web page, but everything also applies to the Mann—Whitney U-test. You lose information when you substitute ranks for the original values, which can make this a somewhat less powerful test than a one-way anova; this is another reason to prefer one-way anova.
If the distributions are different, the Kruskal—Wallis test can reject the null hypothesis even though the medians are the same. We would like to know if the distribution of weights is the same at each company.
We could estimate this probability as the number of pairs with A less than B divided by the total number of pairs. Using sum on the matrix counts all instances of TRUE. While Kruskal-Wallis does not assume that the data are normal, it does assume that the different groups have the same distribution, and groups with different standard deviations have different distributions. You lose information when you substitute ranks for the original values, which can make this a somewhat less powerful test than a one-way anova; this is another reason to prefer one-way anova. We can calculate it by hand using nested for loops as follows though we should note that this is not how the wilcox. We have no idea if one distribution is shifted to the left or right of the other.
Dominance hierarchies in behavioral biology and developmental stages are the only ranked variables I can think of that are common in biology. For the Wilcoxon test, a p-value is the probability of getting a test statistic as large or larger assuming both distributions are the same. The normal approximation is very good and computationally faster for samples larger than Some packages incorrectly treat ties or fail to document asymptotic techniques e.
Otherwise a normal approximation is used. For simplicity, I will only refer to Kruskal—Wallis on the rest of this web page, but everything also applies to the Mann—Whitney U-test.
The idea is to resample the data with replacement many times, say times, each time taking a difference in medians. When assumptions 2 and 3 equal variance and normality are not satisfied but the samples are large say, greater than 30 , the results are approximately correct. We can then find a confidence interval based on our differences. The wilcox.
If your data are heteroscedastic, Kruskal—Wallis is no better than one-way anova, and may be worse. Recall this is a non-parametric test.


year 5 - Maths Calculations Videos
Please find a selection of videos below to show you the various different calculation strategies that we use in Year 5 at St Anne's.
1. Column Addition: ThHTO + ThHTO (With and without concrete resources)
2. Column Addition: O.th + O.th (Decimal Numbers) (With and without concrete resources)
3. Column Addition: HTO + HTO (No Carrying) & HTO + HTO (Involving Carrying)
4. Column Addition with numbers up to 6-digits.
5. Dividing by 10 or 100 / Multiplying by 10 or 100.
6. HTU x U (Grid Method)
7. HTU x TU (Grid Method)
8. Subtraction: Number Line Method & Column Method (No Carrying)
9. Subtraction: Number Line Method & Column Method (Involving Carrying)


Math Addition Worksheets Adding Dice
These Printable Math Addition Worksheets Are A Unique Way To Practice Adding Learning Math Isn T All That Fun For Our Son But There Are Ways We Tried To Spice Up His Homework Dice Can Be A Fun Way For Students To Learn Addition There Are Some Fun Dice Games Not Craps That We Played With Our Son When He First Learned Addition
Free 1 Digit Addition Worksheet Using Dice Free4classrooms
Free Addition 1 Digit With Dice Use The Dice To Practice Addition Students Will Count The Dots On The Dice While Finding The Sum You Ll Find 100s Of Free Printable Worksheets For Practicing Addition Subtraction Multiplication And Division On Our Website
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Use Dice During Math Time And Turn Worksheets Into Gamesheets This Product Includes Nine Roll Add And Color Scarecrow Themed Worksheets Your Students Roll The Dice Perform Mental Math Practice With Strategies And Color The Answer Worksheets Use 1 Or 2 Dice Make Learning Fun For Your Students
Dice Counting Worksheetworks
Dice Counting Develop Counting Skills Using Dice This Worksheet Exercises Counting Skills With One To Three Dice The Student Must Count Up The Dots He Or She Sees On The Dice Then Write The Number In Each Box You Can Have Up To Three Dice But You Can Limit The Totals To A Certain Maximum If The Total Possible 18 For Three Dice Is Too High
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Free Print And Go Math Using Dice Is Just What You Need For 5 Quick And Easy Math Activities Great For Use In A Math Center Or As Independent Work While You Meet With Math Groups Just Print Add Dice And You Re All Set This Freebie Contains 5 Print And Go Math Activities The Only Materials NeeMath Games Using Dice Worksheets Teachers Pay Teachers
Math Games Are A Good Way To Provide The Students A Way To Practice The Math Skills That They Have Been Taught Using The Dice Gives Your Students An Unending Variety Of Math Problems And There Is Minimal Preparation For The Teacher If You Make Different Dice From One Inch Wooden Blocks Or TraditioMath Games Using Dice The Teacher Next Door
Math Games Using Dice By Jennifer Larson Sun Aug 18 Tpt Games Are An Important Part Of Math Class In My Opinion Not Only Are Kids Able To Practice The Skills That We Have Been Focusing On Over And Over Repeatedly But They Are Also Doing It In A Format That They Find Fun D Anything That Adds Fun To Learning Is A Plus In My Book So When I Plan My Math Lessons I Am AlwaysProbability Worksheets Math Worksheets 4 Kids
Probability Worksheets For Kids From Grade 4 And Up Include Probability On Single Coin Two Coins Days In A Week Months In A Year Fair Die Pair Of Dice Deck Of Cards Numbers And More Mutually Exclusive And Inclusive Events Probability On Odds And Other Challenging Probability Worksheets Are Useful For Grade 6 And Up Students
Six Easy Math Games Using Dice K5chalkbox
Using Overhead Dice Or Large Foam Dice The Teacher Rolls Two Numbers A Series Of Questions Follows Each Roll A Series Of Questions Follows Each Roll Write The Sum In Word Form
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Math Worksheet Using Dice. The worksheet is an assortment of 4 intriguing pursuits that will enhance your kid's knowledge and abilities. The worksheets are offered in developmentally appropriate versions for kids of different ages. Adding and subtracting integers worksheets in many ranges including a number of choices for parentheses use.
You can begin with the uppercase cursives and after that move forward with the lowercase cursives. Handwriting for kids will also be rather simple to develop in such a fashion. If you're an adult and wish to increase your handwriting, it can be accomplished. As a result, in the event that you really wish to enhance handwriting of your kid, hurry to explore the advantages of an intelligent learning tool now!
Consider how you wish to compose your private faith statement. Sometimes letters have to be adjusted to fit in a particular space. When a letter does not have any verticals like a capital A or V, the very first diagonal stroke is regarded as the stem. The connected and slanted letters will be quite simple to form once the many shapes re learnt well. Even something as easy as guessing the beginning letter of long words can assist your child improve his phonics abilities. Math Worksheet Using Dice.
There isn't anything like a superb story, and nothing like being the person who started a renowned urban legend. Deciding upon the ideal approach route Cursive writing is basically joined-up handwriting. Practice reading by yourself as often as possible.
Research urban legends to obtain a concept of what's out there prior to making a new one. You are still not sure the radicals have the proper idea. Naturally, you won't use the majority of your ideas. If you've got an idea for a tool please inform us. That means you can begin right where you are no matter how little you might feel you've got to give. You are also quite suspicious of any revolutionary shift. In earlier times you've stated that the move of independence may be too early.
Each lesson in handwriting should start on a fresh new page, so the little one becomes enough room to practice. Every handwriting lesson should begin with the alphabets. Handwriting learning is just one of the most important learning needs of a kid. Learning how to read isn't just challenging, but fun too.
The use of grids The use of grids is vital in earning your child learn to Improve handwriting. Also, bear in mind that maybe your very first try at brainstorming may not bring anything relevant, but don't stop trying. Once you are able to work, you might be surprised how much you get done. Take into consideration how you feel about yourself. Getting able to modify the tracking helps fit more letters in a little space or spread out letters if they're too tight. Perhaps you must enlist the aid of another man to encourage or help you keep focused.
Math Worksheet Using Dice. Try to remember, you always have to care for your child with amazing care, compassion and affection to be able to help him learn. You may also ask your kid's teacher for extra worksheets. Your son or daughter is not going to just learn a different sort of font but in addition learn how to write elegantly because cursive writing is quite beautiful to check out. As a result, if a kid is already suffering from ADHD his handwriting will definitely be affected. Accordingly, to be able to accomplish this, if children are taught to form different shapes in a suitable fashion, it is going to enable them to compose the letters in a really smooth and easy method. Although it can be cute every time a youngster says he runned on the playground, students want to understand how to use past tense so as to speak and write correctly. Let say, you would like to boost your son's or daughter's handwriting, it is but obvious that you want to give your son or daughter plenty of practice, as they say, practice makes perfect.
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Elementary tests are good warm ups on account of the fact they help build confidence and produces an amazing rhythm. Let us consider how problem solving is a helpful medium for every one of these. The math concept of multiplication directs you to find the total of a particular number or quantity which has been copied or cloned so many times. It is everywhere and most of what we see is a combination of different concepts.
By good fortune, many mental math secrets could be available on the internet! The mechanic wouldn't need to even observe the carjust apply the equation. It's also a skill which may enhance logical reasoning. I understand what you are considering this resembles something we do in gradient descent and a number of other optimisation algorithms.
Getting able to supply solutions to several math issues or with different scenarios will not just promote but boost your math skills more. https://owl.english.purdue.edu/owl/resource/560/01/ It is of specific value to remember they are encouraged to go over the processes that they are undertaking, so as to increase understanding, gain new insights into the issue and communicate their ideas (Thompson, 1985, Stacey and Groves, 1985). The capacity to focus is the trick to doing well in math. A more elaborate skill linked to number sense is the capability to observe relationships between numberslike adding and subtracting. It is a significant portion of problem solving.
Your attitude to the study of Math will make it simpler for you to learn it. They should be able to solve more complex problems with measurement estimations and problems using a variety of formulas.
Top Choices of Mathematics Concepts
These are a few of the methods where the language arts are covered on the playground. In the activities you're able to try to observe why the tricks work. As recently as two or three decades past, math held a position in our culture very similar to the one that music, for instance, holds today. The other matter to search for in selecting technology to utilize in teaching mathematics for example is the fact that it should give students the chance to apply mathematical abilities and knowledge.
After you decide a distinctive teaching for your kid, you want to look for tutors that are specialized in home school teaching and extend an excellent curriculum. If you like preparing food and doing an art, your children can in all probability would love to be aware of the ideal way to accomplish exactly what you're performing. When you're a kindergarten teacher you will have a huge effect on the social and academic success of your children. This will assist the kid to outperform and receive a fantastic grade in their school. Some kids could be physically challenged and they may require that distinctive attention to cope up with different children in schools.
Your final solution for this issue is to elect for the distinctive education tutoring. This institution is truly magnificent. This fallacy is so common and has made a false expectation among aspiring data science professionals. By way of example, college algebra may appear to students like they can't apply the concepts in actual life. You should take a look at the utterly comprehensive Applied Machine Learning course that has a full module devoted to statistics.
Most Noticeable Mathematics Concepts
Suppose whether the student have all sorts of problem or isn't able to comprehend the solution then he can always ask from them in order to clarify it. There's one special case. Thereafter students receive a telephone call so they can pay the fee.
Every one of these skill areas depends on and influences the others. Number sense is the capacity to count accurately and competently, to have the ability to continue countingor count onfrom a particular number and to count backwards, to observe relationships between numbers, and to be in a position to have a particular number apart and put it back together again. To find one's way around the mathematical terrain, it's important to observe how the several representations connect with one another, how they are similar, and how they're different.
This concept isn't obvious to children. After you have grasped a comprehension of the very first problem, you might use this as a stepping stone to progress to the remaining part of the question. If something isn't clear or you don't understand, ASK! You wouldn't know whether you could just be listening to the solution to your own questions or you may be listening to a question and answer that you haven't crossed your mind. Something, which you can also verify with a couple of sketches. Receive a set of earlier years' question papers and solve all them.
Choosing Mathematics Concepts
There's also negative infinity that is larger than any authentic unfavorable number. Our machines cannot mimic the exact intuition. In this fashion, you merely have to remember half the table.
You're able to discover more p-value here. Okay, I'll provide you the bigger one. Compare 1 activity with another to determine what takes more time.


The knapsack problem provides us with an array of items. Each item has a weight and a value. We are also given a knapsack capacity which is the maximum weight that the knapsack can hold. Our task is to figure out the highest value of items that we can fit in the knapsack.
For example, if we have three items. The first has a value of 5 and a weight of 2. The second has a value of 2 and a weight of 3. The third has a value of 6 and a weight of 5. The knapsack weight capacity is 5. In this case the maximum value would be 7 because we can carry the first and second items which have a higher combined value than if we chose to take the third item. The first two items have a combined weight of 5, so we can take both. The third item has a weight of 5 but its value is only 6 so it would be better to take the first two items instead.
One way to approach this problem is to create a 2D array where the rows are each item and the columns are each metric of weight from 0 to the knapsack capacity. We start with a row of all 0s at the top representing no items being chosen. As we work down the rows, we are calculating the maximum possible value for each weight as we add each item. For example, the second row calculates the maximum by adding just the first item. The third row calculates the maximum possible with the first and second items. The forth row calculates the maximum possible with the first, second and third items. And so forth.
If we use our previous example the chart would look like this (the first number is the item’s value and the second is the item’s weight).
The max weight capacity is 5 so we create a column for 0 to 5. We make a row of 0s signifying no items being added. Next we add the first item (value:5, weight:2) and go through each weight increment. If the weight is 0 or 1 we can’t carry this item so we mark those values as zero. For any weight 2 or greater we can add that item, so we mark the max value as 5. Then we add the next item (value:2, weight:3). This item can’t be carried with a capacity of 0 or 1 so those columns are marked as 0. When we get to the weight of 2, we still have the option of taking the first item so we mark this column as 5. When we get to the weight of 3 we have the choice of taking either the first or second item. The first item has a higher value so we choose that and mark the column as the value of the first item. When we get to the weight of 5, we can now carry both the first and second items with a combined value of 7. When we get to the last item (value:6, weight:5) there is no weight were we can carry the item and have the value be greater than just taking the first two items instead, so we keep the same maximum values from the prior row.
To code this, first we want to create our 2D array made up of only 0s. The number of columns is the capacity + 1 to account for our 0 column. The numbers of rows is the length of items + 1 to account for our initial row of 0s. We can do this by first creating an empty array. Then we use a for loop to create new arrays (rows) and use the fill method to initialize the row values to 0. We then push the row arrays to our original empty array.
Next we are going to use a nested for loop to iterate through our 2D array and calculate the maximum total value for each weight increment as we add each new item. We start the index of the outer loop at 1 because our first row is kept at all 0s to represent no items be added. We also add 1 to the items length to account for initial 0 row. Inside of the outer loop we add two variables to represent the current value and the current weight. Because of that initial row of 0s, we are setting the variables to i minus 1 so it starts at the first item. In this example we have the value at index 0 of each item and the weight at index 1.
We calculate the value of adding the new item by taking the current column weight and subtracting the weight of the current item. This takes us to the column in the current row that shows the max value of the current column weight minus the weight of the current item. In our previous example, if we are on the last item (value:6, weight:5) and on column weight 5, we subtract the current item’s weight (5) which brings us to column weight 0 for this item, which has a max value of 0. We add the current item’s value (6) which gives us a total of 6. We pass this number and the max total from the previous row (7) into the math max function to give us the highest total. We then assign the highest value to the 2D array.
Once we iterate through all the items we now have our greatest max as the last element of the 2D array. We can return this as MaxTotals[items.length][capacity].
Here is the final solution.


See…math can be fun 😉 and if you want to add interactivity to your math book then look no further than our new calculator widget. Perfect for math textbooks, this widget has all the functions you need from a scientific calculator including trig and memory.
Your readers simply tap on an icon to open the calculator inside your book. They can then use it to work out answers to any questions you set, and if a reader closes the calculator and returns to it, it will retain the memory and history of calculations. This is especially useful if you add a calculator widget next to each question.
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Visualizing the Riemann zeta function and analytic continuation
The Riemann zeta function is one of the most important objects in modern math, and yet simply explaining what it is can be surprisingly tricky. The goal of this lesson is to do just that.
Don’t worry, I’ll explain that animation you just saw further below. A lot of people know about this function because there’s a million dollar prize out for anyone who can figure out when it equals 0, an open problem known as the “Riemann hypothesis”.
Some of you may have heard of it in the context of the following divergent sum:
This expression might seem nonsensical, if not obviously wrong. Indeed, the symbol "" is being used a bit loosely in this expression, but there is a true fact which it is trying to express which has everything to do with the zeta function.
However, as any casual math enthusiast who’s searched "Riemann zeta function" knows, its definition references a certain idea known as “analytic continuation”. This is a topic concerning complex functions, and can be frustratingly opaque and unintuitive. What I’d like to do here is explain what this idea of analytic continuation is in a visual and intuitive way, in the service of showing very concretely what the zeta function looks like.
I’m assuming you know about complex numbers, and that you're comfortable working with them. And I’m tempted to say you should know calculus, since analytic continuation is all about derivatives, but for the way I plan to present things I think you might be okay without that.
Define zeta function for real s > 1
First, let’s define what this zeta function is. For a given input s, the zeta function is a sum of for all natural numbers .
For example, when you plug in s=2, you get
Delightfully, this particular sum converges to .
There’s a nice reason for showing up here, covered in another lesson, but that’s just the tip of the iceberg for why this function is beautiful. You can plug in other values, like 3 or 4 and get some interesting values, and so far it feels pretty reasonable: You’re adding up smaller and smaller amounts, and these sums approach some number.
Which of the following equals ?
Yet, if you read about it, you might see people say . But looking at this infinite sum, that makes no sense.
By raising each term to , you get which obviously doesn’t approach anything, certainly not , right?
And as any mercenary looking into the Riemann hypothesis knows, this function is said to have “trivial” zeros at negative even numbers. For example, this would mean . But plugging in gives you , which also doesn’t approach anything, much less 0. Right?
We’ll get to negative values in a few minutes, but right now let’s just say the only thing that seems reasonable: This function only makes sense when s > 1 , which is when this sum converges. So far, it’s simply not defined for other values.
Extend to complex inputs with Re(s) > 1
Bernhard Riemann, the mathematician whose name is forever tied to this deceptively simple infinite sum, was a father to complex analysis. This is the study of functions that have complex numbers as inputs and outputs. More specifically, complex functions with which you can do calculus, but more on that in just a bit.
So rather than just thinking about how this sum takes a number on the real number line to another number on the real number line, Riemann asked what happens when you plug in a complex number for s. For example, maybe instead of plugging in to this function, you plug in .
If you’ve never seen the idea of raising a number to the power of a complex value, it can feel kind of strange at first, because it no longer has anything to do with repeated multiplication. But there’s a very natural way to extend the definition of exponents beyond their familiar territory of real numbers, into the realm of complex values.
The basic idea is that when you write something like to the power of a complex number, you can split it up as to the real part, times to the pure imaginary part.
The term is familiar real number exponentiation, no issues there. But what about raising something to a pure imaginary number? If you want to really understand this, I've made several other lessons on this very question. The upshot is that when you raise to the power of a purely imaginary number, which is to say for a real value , the result sits on the unit circle in the complex plane.
As you let that pure imaginary input walk up and the imaginary line, the resulting output walks around the unit circle. For a base like , this output walks around the unit circle somewhat slowly. For a base farther away from , like , then as you let the input walk up and down the imaginary axis, the corresponding output would walk around the unit circle more quickly.
is not repeated multiplication
What does this mean for our purposes? I you take an expression like
that part has an absolute value of 1, sitting on the unit circle. Therefore multiplying by it doesn’t change the size of the number; the final result will still have a length of , it's just rotated in the complex plane.
This offers a very beautiful way to think about plugging in into the zeta function. Start by thinking of each term , , , etc., perhaps visualized as little lines whose lengths are the reciprocals of square numbers.
Once you change the input from 2 to 2+i, each of these lines will get rotated by some amount. The amount that each term is rotated depends on , and the larger the value of , the more the rotation, which results in a kind of spiral for our new sum.
Importantly, the lengths of those lines won’t change, so this sum still converges, it just does so in a spiral to some specific point on the complex plane which is no longer a real number.
Given that , what does equal to?
Let's see what it looks like if we vary the input s, represented with a yellow dot, and follow the effect on the spiral sum converging to some output .
What this means is that , defined as the infinite sum , is a perfectly reasonable complex function as long as the real part of the input is greater than 1, meaning the input sits somewhere on this right half of the complex plane.
Again, this is because it’s the real part of that determines the size of each term in this sum, and those sizes determine whether the sum converges. Well, that's not exactly true, in principle once rotation enters the game you could get some cancelation. For example the sum
diverges, but if we rotate every other term , getting the alternating series
it then converges. However, in the case of the zeta function the rotation applied to terms where the real part of is less than is not enough to make it converge. In the jargon, we'd say this sum converges if and only if it's absolutely convergent.
Visualizing Complex Functions
Let's try visualizing this function. It takes in inputs on this right half of the complex plane, and spits out outputs somewhere else on the complex plane. A nice way to understand complex functions is to picture them as transformations, meaning you look at each input to the function, and let it move over to its corresponding output.
How can we visualize this for all complex inputs?
For example, let’s take a moment to try visualizing something a little easier than the zeta function. Say . When you plug in , you get , so we’ll end up moving the point at over to . When you plug in , you get , so the point at will end up moving over to 1. When you plug in , by definition its square is , so it’s going to move to .
If , what is?
Let's add a more colorful grid, since things are about to start moving and it’s nice to have something to distinguish grid lines during this movement.
I’ll tell the computer to move every point on this grid over to its corresponding output under the function . Here’s what it looks like.
Focus on one of the yellow points, and notice how it moves to the blue point corresponding to its square.
It’s a bit complicated to see all points moving all at once, but the reward is that it gives a very rich picture for what a complex function is doing, and it all happens in two-dimensions.
Back to the zeta function. We have this infinite sum, which is a function of a complex number , and we feel good and happy about plugging in values of whose real part is greater than 1, and getting some meaningful output via some converging spiral sum.
To visualize this, let's begin with a portion of the grid sitting on the right side of the complex plane here, with real part greater than 1, like this:
We've added more lines near 1, since it gets stretched out
Again, I'll tell the computer to move each point of this grid to the appropriate output; the place where this sum converges when you set s equal to the complex number at that point.
First of all, can we all take a moment to appreciate how beautiful this is? If this doesn’t make you want to learn more about complex functions, you have no heart! But more pertinently for our final aim today, this transformed grid is just begging to be extended a little bit .
For example, let’s highlight two lines representing all complex numbers with imaginary part or .
After the transformation, these lines make such lovely arcs before they abruptly stop.
Don’t you just want to, you know, continue those arcs.
In fact, you can imagine how some altered version of the function, with a definition extended into the left half of the plane, might be able to complete this picture, which otherwise seems so arbitrarily cut off.
Well, this is exactly what mathematicians working with complex functions do; they continue the function beyond the original domain where it was defined.
As soon as we start talking about inputs where the real part is less than 1, the infinite sum we saw before doesn’t make sense anymore. You’ll get nonsense like adding But just looking at this transformed version of the right half of the plane, where the sum does make sense, it’s begging us to extend the set of points we’re considering as inputs, even if it means defining the extended function in some way which doesn’t necessarily use that sum.
How to extend?
Of course, the question is then how to define the function on the rest of the plane. You might think you could extend this function in any number of ways. Maybe you define an extension that makes it so that the input point at, say, land on , but maybe you squiggle some extension that makes it land on any other value.
As soon as you open yourself up to the idea of defining the function differently for values outside that domain of convergence, the world is your oyster and you can have any number of extensions...right?
Well...not exactly. I mean yes, you can give any child a marker and have them extend these lines any which way, but if you add the restriction that the new extended function has a derivative everywhere, it (rather surprisingly!) locks us into one and only one possible extension.
What's the derivative of a complex function?
"Derivatives," I hear you ask, "for a complex function?" Luckily, there's a very approachable geometric intuition you can keep in mind for when I say a phrase like “has a derivative everywhere”.
Let’s go back to that example. Again, we’ll think of this function as a transformation, moving every point s of the complex plane to the point .
For those of you who know calculus, you know you can take the derivative of at any input.
But here’s an interesting property of the transformation that turns out to be more or less equivalent to that fact. If you look at any two lines in the input space that intersect at a certain angle, and consider what they turn into after the transformation, they will still intersect each other at that same angle.
For instance, consider this pair of lines before transformation. They make an angle of .
Now, take a look at the same lines and the angle they make after applying the transformation .
The lines might get curved, but the important part is that the angle at which they intersect remains unchanged. And this is true for any pair of lines you choose! When I say a function “has a derivative everywhere”, I want you to think about this angle-preserving property: Anytime two lines intersect, the angle between them remains unchanged after the transformation. If you're curious about the lingo, we call functions with this property "conformal mappings".
At a glance, this is easiest to appreciate by noticing how all curves that the grid lines turn into still intersect each other at right angles.
Functions that have a derivative everywhere are called analytic, so you can think of this word “analytic” as meaning “angle-preserving”.
Admittedly, I’m lying to you a little here, but only a little. A slight caveat, for those of you here who want the full details, is that at inputs where the derivative of a function is 0, instead of angles being preserved, they get multiplied by some integer. Those points are by far the minority, and for almost all inputs to an analytic function, angles are preserved. So if when I say “analytic”, you think “angle-preserving”, but you have this small caveat lurking in the back of your mind, that’s a fine intuition to have.
Preserving angles like this is incredibly restrictive you think about it. The angle between any pair of intersecting lines has to remain unchanged. And yet, many of the functions you might think to write down, polynomials, , , etc., are analytic. The field of complex analysis, which Riemann helped to establish in its modern form, is almost entirely about leveraging the highly restrictive properties of analytic functions to understand results and patterns in other fields of math and science.
Does the function , thought of as mapping of complex numbers, preserve angles?
The zeta function, defined by this infinite sum on the right-half of the plane, is an analytic function. Notice how all these curves that the grid lines turned into still intersect at right angles.
The surprising fact about complex functions is that if you want to extend an analytic function beyond the domain where it was originally defined, for example extending the zeta function into the left half of the plane, then requiring that the new extended function still be analytic (that it still preserves angles everywhere) forces you into only one possible extension, if one exists at all.
It’s kind of like an infinite continuous jigsaw puzzle, where the requirement of preserving angles like this locks you into one and only one choice for how to extend it. This process of extending an analytic function in the only way possible that’s still analytic is called, as you may have guessed, “analytic continuation”.
It's worth quickly summing up where we are. For values of s on the right half of the plane, where the real part of s is greater than one, plugging into converges, where a nice way to picture that sum is with a kind of spiral sum, since the imaginary part of the input causes a rotation in each term.
For the rest of the plane, we know that there exists one and only one way to extend this definition so that the function will still be analytic, that will preserve angles at every single point.
That’s a very implicit definition; it just says to use the solution of this jigsaw puzzle, which through more abstract derivations which I have not discussed we know must exist, but it doesn’t specify exactly how to solve it. Mathematicians have a pretty good grasp on what that extension looks like, but some important parts of it remain a mystery. A million dollar mystery, in fact.
The Riemann Hypothesis
The places where this function equals zero are quite important. That is, which points get mapped onto the origin after the transformation.
One thing we know about this extension is that negative even numbers get mapped to zero, and these are called the “trivial zeros” .
The naming here stems from the longstanding tradition of Mathematicians calling things trivial when they understand it quite well, even if it’s a fact which is not at all obvious from the outset.
We also know that the rest of the points that get mapped to zero sit somewhere in this strip where the real part of sits between and , called the “critical strip”. The specific placement of those non-trivial zeros encodes a surprising amount of information about prime numbers.
Why this function carries so much information about primes is pretty interesting , and I might make a follow on video about that later, but for now, I’ll have to leave it unexplained.
Actually, we know quite a bit about those zeros. A mathematician will be able to look at the picture above and know that it's not right, since the first of these zeros sits around . Moreover, the zeros must be symmetrically placed around the line .
Riemann hypothesized that all the remaining zeros lie right in the middle of this strip, on the line of numbers whose real part is , known as the critical line.
If that’s true, it gives us a remarkably tight grasp on the pattern of prime numbers, and all the many other patterns in math that stem from this.
So far, when animating what the zeta function looks like, I’ve only shown what it does to the portion of the grid on the screen, which undersells some of its complexity. If I highlight this critical line and apply the transformation, it might not seem to cross the origin at all.
However, here’s what the transformed version of more of that line looks like... Notice how it passes through 0 multiple times.
If you can prove all non-trivial zeros lie on this line, the Clay Mathematics Institute gives you $1,000,000, and you’d also be proving hundreds if not thousands of modern math results that have been shown contingent on this hypothesis being true.
Here's another way to visualize the zeros of the zeta function. I've graphed the real component and the imaginary component of the zeta function on the -axis, and the -axis is the imaginary component of the input, . I've also highlighted the imaginary components of the first three zeros on the x axis in red.
Currently, the real component is . Watch what happens when I vary the real component from to .
Let's go back and stop at . Notice how the imaginary component and the real component equal at the red circles.
But if , the real and imaginary component do not equal at the same place.
What about this -1/12 business?
The next time you see the following sum floating around the internet, you now have the ability to understand what it's trying to say.
Specifically, one thing we know about the extended zeta function is that . This is a statement about the continuation of , not a direct fact about the sum . Remember, the definition of the zeta function on the left half of the plane is not defined directly from this sum! Instead it comes from analytically continuing this sum beyond the domain where it converges. That is, solving the jigsaw puzzle that began on the right-half of the plane where it more readily makes sense.
So is it okay to write that infinite divergent sum above? Perhaps not, since it runs the risk of misrepresenting very real math as producing nonsense. But then again, considering the uniqueness of this analytic continuation, the fact that the jigsaw puzzle has only one solution, it's very suggestive of some intrinsic connection between these extended values and the original sum.


This means that, if you start from scratch and solve the problem again, you will come up with a very different answer. The work is still easy enough to do in your head, but the results will make you question how you passed fourth-grade math!
Now, we wouldn’t be telling the whole story if we didn’t mention the legions of angry YouTube commenters who refuse to buy Talwalker’s theory. True to the social network’s contentious nature, 12,000 people have “liked” his video, while another 12,000 have given it that dreaded “thumbs down”. We can’t make this stuff up, folks!
Interestingly, Talwalker says that most modern calculators will follow the traditional order of operations without taking the left-to-right requirement into account. Because of this, commonly-used tools like Google or WolframAlpha will give you the answer of 1. Well, that sort of lessens the blow a bit, doesn’t it?
Now that we’ve sufficiently confused you, it’s time for you to learn the real answer to this seemingly simple equation. To hear this math nerd lay down the historical significance of this controversy, watch the video below. Honestly, we don’t know what to think!
Please tell us your thoughts on this incredibly frustrating algebra, especially if you have some serious math cred! What do you think is the correct answer, 1 or 9? Were you taught the new order of operations or the old one? Do you know of any other highly-debated math problems?


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February 28 @ 4:00 pm - 5:00 pm EST
Graham Public Library. BrainSTEM.4:00 p.m. For grades 3-5. Engineering (the ‘E’ in STEM) is the application of mathematics and science to solve problems. This month we will be using principles of mechanical engineering to design and build a working catapult! For more information, call (336) 570-6730.


Scope of the School
The objective of this school is to give a strong background on quantum and nano computing to interested students, scientists and young researches. The participants are not assumed to have any prior knowledge on the field and could come from computer science, electrical, electronics and mechanical engineering, physics or mathematics. Scientists already familiar with the subject are also welcome, for they will learn something from internationally renowned researchers in the field.
The school involves intense training in the foundations, ramifications, and current frontiers of quantum information science and its physical implementation at the nanoscale. It will cover the entire range of this newly emerged and truly interdisciplinary field, from mathematics over theoretical and experimental physics to computer science. A series of lectures and seminars will cover basic concepts to challenging problems of cutting edge research, along a steep learning curve.
The topics covered in the school, though not limited to the following, would broadly cover :
- Introduction to quantum information processing
- Introduction to quantum cryptography
- Introduction to quantum error correction
- Introduction to quantum algorithms
- Introduction to Nanoscale realization of Quantum Computers through NMR, Optics, quantum-dot cellular automata and nanoelectronics
- Introduction to current questions in foundations of quantum mechanics, string theory, cosmology and unification theories


More and more UK businesses are using PayPal to process website transactions on eBay, Shopify and bespoke websites. It’s quick, hassle-free and available worldwide. But what is the cost to the business for receiving money in a PayPal transaction? Use this calculator to work out your fees and costs.
How much are PayPal fees in the UK?
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Graph is an open source application used to draw mathematical graphs in a coordinate system. Anyone who wants to draw graphs of functions will find this program useful. The program makes it very easy to visualize a function and paste it into another program. It is also possible to do some mathematical calculations on the functions.
It includes a huge number of predefined functions, but new ones can be added. Tangents, normals, and derivatives can be drawn. Users can add relations and set constraints. The program can calculate the length or area of a selected function, or determine its first and second derivatives. From a cosmetic point of view, users can freely adjust axes, figure width and color, and shading.


Math Resources for Expanded Learning Programs & Professionals
- About Academic Enrichment in Afterschool: Math, SEDL
Use this section of the Afterschool Toolkit to learn more about math centers, projects, games, and more. Access further resources and support materials on math in afterschool.
IXL is an interactive math tool for grades Pre-K through eighth grade.
- Math Playground, Math Playground
The site strengthens math skills through math games, word problems, logic puzzles and math videos to connect to school day math activities.
- Mixing in Math, Mixing in Math
Activities that incorporate math into expanded learning programming.


Sports betting is a popular form of gambling that allows individuals to wager on the outcome of various sporting events. One of the key components of sports betting is understanding and interpreting the odds. The odds reflect the probability of a particular outcome occurring and also determine the potential payout for a winning bet. Odds are typically expressed in three different formats: decimal, fractional, and moneyline.
Decimal odds are the most common format used by bookmakers in Europe and Australia. They are presented as a decimal number that indicates the total potential payout including the original wager. For example, if the odds are 2.50, a $10 bet would result in a $25 payout ($15 in winnings plus the original $10 wager). Broaden your understanding by checking out Discover this helpful research external content! 토토, explore the suggested site.
Fractional odds are commonly used in the UK and Ireland. They are presented as a fraction and represent the potential profit relative to the original wager. For example, if the odds are 5/1, a $10 bet would result in a $50 payout ($40 in winnings plus the original $10 wager). Fractional odds can also be expressed in reverse, such as 1/5, which means that a $5 bet would result in a $1 payout ($4 in winnings plus the original $5 wager).
Moneyline odds are the most straightforward and easy to understand format used in the United States. They are presented as positive or negative numbers and indicate the potential profit relative to a $100 wager. Positive moneyline odds indicate the potential profit for a $100 bet, while negative moneyline odds indicate the amount that needs to be wagered to win $100. For example, if the odds are +200, a $100 bet would result in a $200 profit ($100 in winnings plus the original $100 wager). If the odds are -150, a $150 bet would be required to win $100 ($250 total payout including the original $150 wager).
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Value is an important concept in sports betting that relates to the relationship between the odds offered by the bookmaker and the expected probability of an outcome occurring. If the odds offered are higher than the calculated probability, the bet has positive value and is considered favorable. On the other hand, if the odds offered are lower than the calculated probability, the bet has negative value and should be avoided.
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On November 24, the University of Tokyo developed a digital signature technology “QR-UOV signature” for multi-variable public key cryptography that cannot be decrypted even with a quantum computer, reducing the data size of public keys to about one-third compared to existing methods. Announced that it succeeded in doing so. It is said that the basis of safety is “difficulty of multivariate polynomial problem”. The multivariate polynomial problem is a problem to calculate the common solution of m quadratic polynomials with n variables, and it may be difficult to calculate when n and m are increased by the same magnitude. Are known.
This is a joint research by Professor Tsuyoshi Takagi and Hiroki Furue of the Graduate School of Information Science and Engineering, the University of Tokyo, the Mass For Industry Research Institute of Kyushu University, and Nippon Telegraph and Telephone (NTT). Cryptography is extremely important in the modern information society, but RSA cryptography (a cryptosystem whose security is based on the difficulty of factorization) and elliptic curve cryptography (a geometrical curve called elliptic curve) are widely used today. It is known that two technologies (encryption method using various structures, such as ECDSA) will be decrypted when a large-scale quantum computer is realized. Therefore, a method called Rainbow signature was developed based on the difficulty of the multivariate polynomial problem that cannot be deciphered even by a quantum computer, but there was a problem that the size of the public key used for verification became large.
The Rainbow signature is an extension of the UOV signature, which is based on the multivariate polynomial problem and is considered to be a secure method for which no essential decoding method has been reported for more than 20 years. The UOV signature public key expressed in is expressed by an algebraic polynomial called a remainder ring, and the data size is reduced. It is called “QR-UOV signature”. Compared to the Rainbow signature, the data size of the public key was reduced by about 66%.
QR-UOV signatures can be used as a safe and efficient digital signature method even in a society where large-scale quantum computers have become widespread. “Application of” is expected.
The research group is aiming to be adopted by the National Institute of Standards and Technology (NIST), which is advancing a secure cryptographic standardization project, by applying for a public solicitation for digital signature technology scheduled for 2022.