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The binomial distribution is quite common as it models chain of independent events each with binary distribution, it can also be used when each binary event has different probability this gives the ability to support iterative nature of events. However, the problem arises when we look on the whole picture of n events, in this case the variance is fixed without ability to control it, unlike normal distribution.
The question is how can we remain the iterative nature of the binomial distribution while controlling the variance of the total n events?
The general idea is to change the probability of the event based on the history. For example, if p=0.6 in the first even we use p0=0.6, and let's say we got 0 so in the next event we'll use a raise the probability, say p1=0.66, and if we got another 0 we'll continue to raise the probability, for example, p2=0.69. And the other way around, if we get 1 we will reduce the probability to get 1, so one can summarize that we take chances from the unlikely high to the unlikely low, and this is where the probability got its name from (it is important to add that increasing the variance is also possible when going the other way around)
We need to perform these changes in the probabilities in a manner that we would define a distribution with expected values and variance, SB(n,p,v). The expected value should equal to binomial distribution, E(B(n,p)) = E(SB(n,p,v)) = np. The variance will be extension of the binomial variance (V(B(n,p)) = npq) and will be defined as V(SB(n,p,v)) = pq*f(n,v).
Instead of using a constant p for all events we will define p(n,k)=p+(k-np)*v.
Using this probabilities for each event we will get that expectation is the same and the variance will be defined are Var(SB(n,p,v)) = sum((1-2v)^n) = ((1-2v)^n+1)/2v.
It is interesting to see that (1+2v)^n is a binomial series sum.
Here are the graphs of the SB(100,6,v) where v goes from -0.02 : 0.015
It seems like very nice Gaussians as expected from binomial distributions and we can see that the variance is controlled.


Almost everyone has an intuitive notion of what a random number is. For example, consider these two series of binary digits:
01010101010101010101The first is obviously constructed according to a simple rule; it consists of the number 01 repeated ten times. If one were asked to speculate on how the series might continue, one could predict with considerable confidence that the next two digits would be 0 and 1. Inspection of the second series of digits yields no such comprehensive pattern. There is no obvious rule governing the formation of the number, and there is no rational way to guess the succeeding digits. The arrangement seems haphazard; in other words, the sequence appears to be a random assortment of 0's and 1's.
The second series of binary digits was generated by flipping a coin 20 times and writing a 1 if the outcome was heads and a 0 if it was tails. Tossing a coin is a classical procedure for producing a random number, and one might think at first that the provenance of the series alone would certify that it is random. This is not so. Tossing a coin 20 times can produce any one of 220 (or a little more than a million) binary series, and each of them has exactly the same probability. Thus it should be no more surprising to obtain the series with an obvious pattern than to obtain the one that seems to be random; each represents an event with a probability of 2−20. If origin in a probabilistic event were made the sole criterion of randomness, then both series would have to be considered random, and indeed so would all others, since the same mechanism can generate all the possible series. The conclusion is singularly unhelpful in distinguishing the random from the orderly.
Clearly a more sensible definition of randomness is required, one that does not contradict the intuitive concept of a ``patternless'' number. Such a definition has been devised only in the past 10 years. It does not consider the origin of a number but depends entirely on the characteristics of the sequence of digits. The new definition enables us to describe the properties of a random number more precisely than was formerly possible, and it establishes a hierarchy of degrees of randomness. Of perhaps even greater interest than the capabilities of the definition, however, are its limitations. In particular the definition cannot help to determine, except in very special cases, whether or not a given series of digits, such as the second one above, is in fact random or only seems to be random. This limitation is not a flaw in the definition; it is a consequence of a subtle but fundamental anomaly in the foundation of mathematics. It is closely related to a famous theorem devised and proved in 1931 by Kurt Gödel, which has come to be known as Gödel's incompleteness theorem. Both the theorem and the recent discoveries concerning the nature of randomness help to define the boundaries that constrain certain mathematical methods.
Suppose, on the other hand, your friend is interested not in trigonometry but in baseball. He would like to know the scores of all the major-league games played since he left the earth some thousands of years before. In this case it is most unlikely that a formula could be found for compressing the information into a short message; in such a series of numbers each digit is essentially an independent item of information, and it cannot be predicted from its neighbors or from some underlying rule. There is no alternative to transmitting the entire list of scores.
In this pair of whimsical messages is the germ of a new definition of randomness. It is based on the observation that the information embodied in a random series of numbers cannot be ``compressed,'' or reduced to a more compact form. In formulating the actual definition it is preferable to consider communication not with a distant friend but with a digital computer. The friend might have the wit to make inferences about numbers or to construct a series from partial information or from vague instructions. The computer does not have that capacity, and for our purposes that deficiency is an advantage. Instructions given the computer must be complete and explicit, and they must enable it to proceed step by step without requiring that it comprehend the result of any part of the operations it performs. Such a program of instructions is an algorithm. It can demand any finite number of mechanical manipulations of numbers, but it cannot ask for judgments about their meaning.
The definition also requires that we be able to measure the information content of a message in some more precise way than by the cost of sending it as a telegram. The fundamental unit of information is the ``bit,'' defined as the smallest item of information capable of indicating a choice between two equally likely things. In binary notation one bit is equivalent to one digit, either a 0 or a 1.
We are now able to describe more precisely the differences between the two series of digits presented at the beginning of this article:
01010101010101010101The first could be specified to a computer by a very simple algorithm, such as ``Print 01 ten times.'' If the series were extended according to the same rule, the algorithm would have to be only slightly larger; it might be made to read, for example, ``Print 01 a million times.'' The number of bits in such an algorithm is a small fraction of the number of bits in the series it specifies, and as the series grows larger the size of the program increases at a much slower rate.
For the second series of digits there is no corresponding shortcut. The most economical way to express the series is to write it out in full, and the shortest algorithm for introducing the series into a computer would be ``Print 01101100110111100010.'' If the series were much larger (but still apparently patternless), the algorithm would have to be expanded to the corresponding size. This ``incompressibility'' is a property of all random numbers; indeed, we can proceed directly to define randomness in terms of incompressibility: A series of numbers is random if the smallest algorithm capable of specifying it to a computer has about the same number of bits of information as the series itself.
This definition was independently proposed about 1965 by A. N. Kolmogorov of the Academy of Science of the U.S.S.R. and by me, when I was an undergraduate at the City College of the City University of New York. Both Kolmogorov and I were then unaware of related proposals made in 1960 by Ray J. Solomonoff of the Zator Company in an endeavor to measure the simplicity of scientific theories. During the past decade we and others have continued to explore the meaning of randomness. The original formulations have been improved and the feasibility of the approach has been amply confirmed.
The ensemble theory of probability, which originated in the 17th century, remains today of great practical importance. It is the foundation of statistics, and it is applied to a wide range of problems in science and engineering. The algorithmic theory also has important implications, but they are primarily theoretical. The area of broadest interest is its amplification of Gödel's incompleteness theorem. Another application (which actually preceded the formulation of the theory itself) is in Solomonoff's model of scientific induction.
Solomonoff represented a scientist's observations as a series of binary digits. The scientist seeks to explain these observations through a theory, which can be regarded as an algorithm capable of generating the series and extending it, that is, predicting future observations. For any given series of observations there are always several competing theories, and the scientist must choose among them. The model demands that the smallest algorithm, the one consisting of the fewest bits, be selected. Stated another way, this rule is the familiar formulation of Occam's razor: Given differing theories of apparently equal merit, the simplest is to be preferred.
Thus in the Solomonoff model a theory that enables one to understand a series of observations is seen as a small computer program that reproduces the observations and makes predictions about possible future observations. The smaller the program, the more comprehensive the theory and the greater the degree of understanding. Observations that are random cannot be reproduced by a small program and therefore cannot be explained by a theory. In addition the future behavior of a random system cannot be predicted. For random data the most compact way for the scientist to communicate his observations is for him to publish them in their entirety.
Defining randomness or the simplicity of theories through the capabilities of the digital computer would seem to introduce a spurious element into these essentially abstract notions: the peculiarities of the particular computing machine employed. Different machines communicate through different computer languages, and a set of instructions expressed in one of those languages might require more or fewer bits when the instructions are translated into another language. Actually, however, the choice of computer matters very little. The problem can be avoided entirely simply by insisting that the randomness of all numbers be tested on the same machine. Even when different machines are employed, the idiosyncrasies of various languages can readily be compensated for. Suppose, for example, someone has a program written in English and wishes to utilize it with a computer that reads only French. Instead of translating the algorithm itself he could preface the program with a complete English course written in French. Another mathematician with a French program and an English machine would follow the opposite procedure. In this way only a fixed number of bits need be added to the program, and that number grows less significant as the size of the series specified by the program increases. In practice a device called a compiler often makes it possible to ignore the differences between languages when one is addressing a computer.
Since the choice of a particular machine is largely irrelevant, we can choose for our calculations an ideal computer. It is assumed to have unlimited storage capacity and unlimited time to complete its calculations. Input to and output from the machine are both in the form of binary digits. The machine begins to operate as soon as the program is given it, and it continues until it has finished printing the binary series that is the result. The machine then halts. Unless an error is made in the program, the computer will produce exactly one output for any given program.
Any minimal program is necessarily random, whether or not the series it generates is random. This conclusion is a direct result of the way we have defined randomness. Consider the program P, which is a minimal program for the series of digits S. If we assume that P is not random, then by definition there must be another program, P', substantially smaller than P that will generate it. We can then produce S by the following algorithm: ``From P' calculate P, then from P calculate S.'' This program is only a few bits longer than P', and thus it must be substantially shorter than P. P is therefore not a minimal program.
The minimal program is closely related to another fundamental concept in the algorithmic theory of randomness: the concept of complexity. The complexity of a series of digits is the number of bits that must be put into a computing machine in order to obtain the original series as output. The complexity is therefore equal to the size in bits of the minimal programs of the series. Having introduced this concept, we can now restate our definition of randomness in more rigorous terms: A random series of digits is one whose complexity is approximately equal to its size in bits.
The notion of complexity serves not only to define randomness but also to measure it. Given several series of numbers each having n digits, it is theoretically possible to identify all those of complexity n−1, n−10, n−100 and so forth and thereby to rank the series in decreasing order of randomness. The exact value of complexity below which a series is no longer considered random remains somewhat arbitrary. The value ought to be set low enough for numbers with obviously random properties not to be excluded and high enough for numbers with a conspicuous pattern to be disqualified, but to set a particular numerical value is to judge what degree of randomness constitutes actual randomness. It is this uncertainty that is reflected in the qualified statement that the complexity of a random series is approximately equal to the size of the series.
Another binary series that can be profitably analyzed in this way is one where 0's and 1's are present with relative frequencies of three-fourths and one-fourth. If the series is of size n, it can be demonstrated that its complexity is no greater than four-fifths n, that is, a program that will produce the series can be written in 4n/5 bits. This maximum applies regardless of the sequence of the digits, so that no series with such a frequency distribution can be considered very random. In fact, it can be proved that in any long binary series that is random the relative frequencies of 0's and 1's must be very close to one-half. (In a random decimal series the relative frequency of each digit is, of course, one-tenth.)
Numbers having a nonrandom frequency distribution are exceptional. Of all the possible n-digit binary numbers there is only one, for example, that consists entirely of 0's and only one that is all 1's. All the rest are less orderly, and the great majority must, by any reasonable standard, be called random. To choose an arbitrary limit, we can calculate the fraction of all n-digit binary numbers that have a complexity of less than n−10. There are 21 programs one digit long that might generate an n-digit series; there are 22 programs two digits long that could yield such a series, 23 programs three digits long and so forth, up to the longest programs permitted within the allowed complexity; of these there are 2n−11. The sum of this series (21 + 22 + ... + 2n−11) is equal to 2n−10−2. Hence there are fewer than 2n−10 programs of size less than n−10, and since each of these programs can specify no more than one series of digits, fewer than 2n−10 of the 2n numbers have a complexity less than n−10. Since 2n−10 / 2n = 1/1,024, it follows that of all the n-digit binary numbers only about one in 1,000 have a complexity less than n−10. In other words, only about one series in 1,000 can be compressed into a computer program more than 10 digits smaller than itself.
A necessary corollary of this calculation is that more than 999 of every 1,000 n-digit binary numbers have a complexity equal to or greater than n−10. If that degree of complexity can be taken as an appropriate test of randomness, then almost all n-digit numbers are in fact random. If a fair coin is tossed n times, the probability is greater than .999 that the result will be random to this extent. It would therefore seem easy to exhibit a specimen of a long series of random digits; actually it is impossible to do so.
It is in the realm of mathematical proof that Gödel's incompleteness theorem is such a conspicuous landmark; my version of the theorem predicts that the required proof of randomness cannot be found. The consequences of this fact are just as interesting for what they reveal about Gödel's theorem as they are for what they indicate about the nature of random numbers.
Gödel's theorem represents the resolution of a controversy that preoccupied mathematicians during the early years of the 20th century. The question at issue was: ``What constitutes a valid proof in mathematics and how is such a proof to be recognized?'' David Hilbert had attempted to resolve the controversy by devising an artificial language in which valid proofs could be found mechanically, without any need for human insight or judgement. Gödel showed that there is no such perfect language.
Hilbert established a finite alphabet of symbols, an unambiguous grammar specifying how a meaningful statement could be formed, a finite list of axioms, or initial assumptions, and a finite list of rules of inference for deducing theorems from the axioms or from other theorems. Such a language, with its rules, is called a formal system.
A formal system is defined so precisely that a proof can be evaluated by a recursive procedure involving only simple logical and arithmetical manipulations. In other words, in the formal system there is an algorithm for testing the validity of proofs. Today, although not in Hilbert's time, the algorithm could be executed on a digital computer and the machine could be asked to ``judge'' the merits of the proof.
Because of Hilbert's requirement that a formal system have a proof-checking algorithm, it is possible in theory to list one by one all the theorems that can be proved in a particular system. One first lists in alphabetical order all sequences of symbols one character long and applies the proof-testing algorithm to each of them, thereby finding all theorems (if any) whose proofs consist of a single character. One then tests all the two-character sequences of symbols, and so on. In this way all potential proofs can be checked, and eventually all theorems can be discovered in order of the size of their proofs. (The method is, of course, only a theoretical one; the procedure is too lengthy to be practical.)
Gödel's proof of the incompleteness theorem is based on the paradox of Epimenides the Cretan, who is said to have averred, ``All Cretans are liars'' [see ``Paradox,'' by W. V. Quine; Scientific American, April, 1962]. The paradox can be rephrased in more general terms as ``This statement is false,'' an assertion that is true if and only if it is false and that is therefore neither true nor false. Gödel replaced the concept of truth with that of provability and thereby constructed the sentence ``This statement is unprovable,'' an assertion that, in a specific formal system, is provable if and only if it is false. Thus either a falsehood is provable, which is forbidden, or a true statement is unprovable, and hence the formal system is incomplete. Gödel then applied a technique that uniquely numbers all statements and proofs in the formal system and thereby converted the sentence ``This statement is unprovable'' into an assertion about the properties of the positive integers. Because this transformation is possible, the incompleteness theorem applies with equal cogency to all formal systems in which it is possible to deal with the positive integers [see ``Gödel's Proof,'' by Ernest Nagel and James R. Newman; Scientific American, June, 1956].
The intimate association between Gödel's proof and the theory of random numbers can be made plain through another paradox, similar in form to the paradox of Epimenides. It is a variant of the Berry paradox, first published in 1908 by Bertrand Russell. It reads: ``Find the smallest positive integer which to be specified requires more characters than there are in this sentence.'' The sentence has 114 characters (counting spaces between words and the period but not the quotation marks), yet it supposedly specifies an integer that, by definition, requires more than 114 characters to be specified.
As before, in order to apply the paradox to the incompleteness theorem it is necessary to remove it from the realm of truth to the realm of provability. The phrase ``which requires'' must be replaced by ``which can be proved to require,'' it being understood that all statements will be expressed in a particular formal system. In addition the vague notion of ``the number of characters required to specify'' an integer can be replaced by the precisely defined concept of complexity, which is measured in bits rather than characters.
The result of these transformations is the following computer program: ``Find a series of binary digits that can be proved to be of a complexity greater than the number of bits in this program.'' The program tests all possible proofs in the formal system in order of their size until it encounters the first one proving that a specific binary sequence is of a complexity greater than the number of bits in the program. Then it prints the series it has found and halts. Of course, the paradox in the statement from which the program was derived has not been eliminated. The program supposedly calculates a number that no program its size should be able to calculate. In fact, the program finds the first number that it can be proved incapable of finding.
The absurdity of this conclusion merely demonstrates that the program will never find the number it is designed to look for. In a formal system one cannot prove that a particular series of digits is of a complexity greater than the number of bits in the program employed to specify the series.
A further generalization can be made about this paradox. It is not the number of bits in the program itself that is the limiting factor but the number of bits in the formal system as a whole. Hidden in the program are the axioms and rules of inference that determine the behavior of the system and provide the algorithm for testing proofs. The information content of these axioms and rules can be measured and can be designated the complexity of the formal system. The size of the entire program therefore exceeds the complexity of the formal system by a fixed number of bits c. (The actual value of c depends on the machine language employed.) The theorem proved by the paradox can therefore be stated as follows: In a formal system of complexity n it is impossible to prove that a particular series of binary digits is of complexity greater than n+c, where c is a constant that is independent of the particular system employed.
The complexity of the formal system has such an important bearing on the proof of randomness because it is a measure of the amount of information the system contains, and hence of the amount of information that can be derived from it. The formal system rests on axioms: fundamental statements that are irreducible in the same sense that a minimal program is. (If an axiom could be expressed more compactly, then the briefer statement would become a new axiom and the old one would become a derived theorem.) The information embodied in the axioms is thus itself random, and it can be employed to test the randomness of other data. The randomness of some numbers can therefore be proved, but only if they are smaller than the formal system. Moreover, any formal system is of necessity finite, whereas any series of digits can be made arbitrarily large. Hence there will always be numbers whose randomness cannot be proved.
The endeavor to define and measure randomness has greatly clarified the significance and the implications of Gödel's incompleteness theorem. That theorem can now be seen not as an isolated paradox but as a natural consequence of the constraints imposed by information theory. In 1946 Hermann Weyl said that the doubt induced by such discoveries as Gödel's theorem had been ``a constant drain on the enthusiasm and determination with which I pursued my research work.'' From the point of view of information theory, however, Gödel's theorem does not appear to give cause for depression. Instead it seems simply to suggest that in order to progress, mathematicians, like investigators in other sciences, must search for new axioms.
Algorithmic definition of randomness relies on the capabilities and limitations of the digital computer. In order to produce a particular output, such as a series of binary digits, the computer must be given a set of explicit instructions that can be followed without making intellectual judgments. Such a program of instructions is an algorithm. If the desired output is highly ordered (a), a relatively small algorithm will suffice; a series of twenty 1's, for example, might be generated by some hypothetical computer from the program 10100, which is the binary notation for the decimal number 20. For a random series of digits (b) the most concise program possible consists of the series itself. The smallest programs capable of generating a particular series are called the minimal programs of the series; the size of these programs, measured in bits, or binary digits, is the complexity of the series. A series of digits is defined as random if series' complexity approaches its size in bits.
Formal systems devised by David Hilbert contain an algorithm that mechanically checks the validity of all proofs that can be formulated in the system. The formal system consists of an alphabet of symbols in which all statements can be written; a grammar that specifies how the symbols are to be combined; a set of axioms, or principles accepted without proof; and rules of inference for deriving theorems from the axioms. Theorems are found by writing all the possible grammatical statements in the system and testing them to determine which ones are in accord with the rules of inference and are therefore valid proofs. Since this operation can be performed by an algorithm it could be done by a digital computer. In 1931 Kurt Gödel demonstrated that virtually all formal systems are incomplete: in each of them there is at least one statement that is true but that cannot be proved.
Inductive reasoning as it is employed in science was analyzed mathematically by Ray J. Solomonoff. He represented a scientist's observations as a series of binary digits; the observations are to be explained and new ones are to be predicted by theories, which are regarded as algorithms instructing a computer to reproduce the observations. (The programs would not be English sentences but binary series, and their size would be measured not in characters but in bits.) Here two competing theories explain the existing data; Occam's razor demands that the simpler, or smaller, theory be preferred. The task of the scientist is to search for minimal programs. If the data are random, the minimal programs are no more concise than the observations and no theory can be formulated.
Illustration is a graph of number of n-digit sequences as a function of their complexity. The curve grows exponentially from approximately 0 to approximately 2n as the complexity goes from 0 to n.
Random sequences of binary digits make up the majority of all such sequences. Of the 2n series of n digits, most are of a complexity that is within a few bits of n. As complexity decreases, the number of series diminishes in a roughly exponential manner. Orderly series are rare; there is only one, for example, that consists of n 1's.
Three paradoxes delimit what can be proved. The first, devised by Bertrand Russell, indicated that informal reasoning in mathematics can yield contradictions, and it led to the creation of formal systems. The second, attributed to Epimenides, was adapted by Gödel to show that even within a formal system there are true statements that are unprovable. The third leads to the demonstration that a specific number cannot be proved random.
Unprovable statements can be shown to be false, if they are false, but they cannot be shown to be true. A proof that ``This statement is unprovable'' (a) reveals a self-contradiction in a formal system. The assignment of a numerical value to the complexity of a particular number (b) requires a proof that no smaller algorithm for generating the number exists; the proof could be supplied only if the formal system itself were more complex than the number. Statements labeled (c) and (d) are subject to the same limitation, since the identification of a random number or a minimal program requires the determination of complexity.


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3-In-Through LIVE sessions, students are able to talk about Trigonometry issues with solutions. 4. 4The professors also offer Trigonometry notes for exams for defence.1 Are Trigonometry included in the syllabus for defense exams in India?
Through a rigorous set of trigonometry exercises and exams The BYJU’s Exam Prep will prepare students for the most difficult questions on examinations for defence. Absolutely, Trigonometry is a part of the syllabus in all defence entrance tests in India.1 Trigonometry Commonly Asked Questions. 1. Trigonometry for Dummies cheat sheet. What are the most fundamental trigonometric ratios?
A lot of the formulas utilized in trigonometry can also be found in analytic and algebraic geometry. The six Trigonometric Ratios-sine, cosine and cosine as well as cosecant, tangent secant and cotangent constitute the foundation of Trigonometry.1 However, trigonometry has its own specific formulas that are usually only found in these discussions. 2. Formulas provide you with an equation or rule that can be relied upon to be successful, each and every time. What are the uses of Trigonometry in the real world? Formulas provide a connection between specific numbers and units.1
The application of Trigonometry as well as its formulas in the real world is linked to all fields where distance and height calculations are needed. The most important thing to remember when formulas is knowing what each letter represents. 3. In the formulas provided in this article, you will find the following formulas: the r (radius) and the term d (diameter or length) and B (base or measurement of the side) and h (height) and a B , C (measures of the sides) and x , y (coordinates in graphs) and m (slope) and M (midpoint) and h K (horizontal as well as vertical distances to the central point) and the (angle theta); as well as s (arc long).1
Who’s responsible for the origins of Trigonometry? The trigonometric formulas include sin (sine), cos (cosine), and the term tan (tangent) however only sin is used here. Hipparchus who was from Nicaea is believed to be the founder of Trigonometry.
Particular right triangles. 4. Each right triangle is characterized by the fact that the total of squares on the two legs equals the sum of squares on the hypotenuse (the longest side).1 Are Trigonometry included in the syllabus for defense exams in India? The Pythagorean theorem can be written as: A 2 + b 2 = c 2 . Absolutely, Trigonometry is a part of the syllabus in all defence entrance tests in India. What’s unique with the right-hand triangles illustrated in this example is that there’s an additional relationship between the measurements of the sides, one that is more than (but is still compatible in conjunction with) that of the Pythagorean theorem.1 If you are dealing with a 30-60-90 right-angled triangle, the measurement of the hypotenuse will always be twice that of the side with the shortest length, and the opposite side is never. Trigonometry To Dummies Cheat Sheet. approximately 1.7 times larger than the side with the smallest length.
Many of the formulas that are used in trigonometry also appear in analytic geometry and algebra.1 When you look at the isosceles right triangle that is the case, both legs are identically, and the hypotenuse will always be. But there are also particular formulas which are typically only in these discussions. which is about 1.4 times longer than the legs.
A formula gives you an equation or rule you can trust to perform, every time.1 Definitions of right triangle for trigonometry calculations. Formulas establish a relation between certain amounts and units. The fundamental trig functions can be described using ratios that are created by dividing the dimensions of each side of the right-angled triangle, in a particular order. The key to formulas is to understand the meaning behind each letter.1 The label for hypotenuse is always the same – it’s the side with the longest length.
In the formulas provided below, you’ll find the following: (radius); r (radius) (radius); D (diameter of distance) (diameter or distance); the term b (base or measurement of an aspect) (base or measure of a side) (height) (height); a , the b, and the c (measures of side) (measures of sides) (coordinates in graphs) (coordinates on a graph) (slope) (slope); M (midpoint) (midpoint); h , the k (horizontal or vertical distances away from center) as well as the (angle theta); (height); and (arc the length).1 However, the names of adjacent and opposite can differ depending on the angle you’re talking about at the moment. The formulas that are specific to trigonometry comprise sin (sine), cos (cosine), and Tan (tangent) but the sin formula is the only one that is utilized here. It is always the side that doesn’t create the angle, while the side that is adjacent is one of the angles’ sides.1 Right triangles with special right angles. Definitions of coordinates for trigonometry functions. Each right triangle is characterized by the fact in that the product of the squares of its 2 legs are equal to that of the hypotenuse (the longest side).
The trig functions are defined by measuring the sides of the right triangle.1 The Pythagorean theorem is writtenas follows: A 2 + B 2 = c 2 . However, they also provide useful definitions based on the coordinates of the points on graphs. What’s special in the 2 right triangles displayed here is that they have an even more unique relation between the dimensions of the sides.1
Then, let the vertex of the angle be located at its origin (the zero point (0,0) -and let the first part of the angle be on the positive x-axis and the other side be an anticlockwise motion. This relationship exceeds (but does not completely break together with) what is known as the Pythagorean theorem.1 When the point ( x, y ) is located on a circle which is intersected by the terminal side, the trig functions are identified by the following ratios which the radius is the diameter of the circular. If you have a 30–60-90 right triangle length of the hypotenuse is double the measurement of the shortest side and the opposite legs are always.1 Trigonometry-related functions are evident within quadrants. which is about 1.7 times the size of the side with the smallest length. An angle is in its standard position when its vertex is located at the point of origin, and its primary side is located on the positive x-axis and its terminal side is rotated clockwise from the original side.1
The isosceles right triangular and the two legs measuring exactly the same. The location on the side that is terminal determines the meaning of all trig functions associated with the angle. The hypotenuse will always be.
This table reveals the functions that are positiveand then you can conclude that the rest of the functions are negative in the quadrant.1


This year, our school has chosen to work on addition and subtraction word problem types in our grades 3-5. You may think that it seems strange to work on add/sub in the upper grades, but I assure you it is quite a complicated endeavor. Read my previous post on the various problem types here: Word problems with addition and subtraction are easy, right?
Our initiative exposes all of our students to the variety of addition/subtraction word problem types (then we’ll go on to mult/div) as based on the research in the book Cognitively Guided Instruction (and described in the CCSS) and have used the tape diagram as the way to model these situations. We have followed the rectangular tape diagram as found on Greg Tang’s math website within his word problem generator program. (If you haven’t checked out this part of his site, I would encourage you to do so. You can select the problem type, the unknown, and the number range. Awesome resource to create differentiated Guided Math workstations). There is a large rectangle on top and beneath it are two smaller boxes whose size equals the large rectangle and hence the values of the two smaller boxes must equal the value in the top box. I find it an incredibly powerful model that allows us to teach many mathematical concepts. Here is an example from the site at http://www.gregtangmath.com.
First, the tape diagram provides us a visual model to teach the relationship between addition and subtraction since the values of both boxes (and eventually more when there are more addends introduced) must equal the value of the large box. I love using Cuisenaire rods to model this using numbers within 10 – even for our upper elementary students since our goal initially is not about the computational skills to get the numerical answer, but the underlying structure of the situation. If the top box is unknown, students will need to add the two smaller boxes. If one of the smaller boxes is unknown students have a choice of adding OR subtracting. There is no dictate in terms of how they can find the numerical answer. The problem may be an “Add to” problem type, but they could use subtraction to solve it if one of the addends is unknown.
Secondly, the tape diagram makes complicated-sounding word problems much more manageable…
For example, there were 5 more sheep than cows on a farm. There were 7 sheep. How many cows were there?
Before my work with CGI and the tape diagramming it would take me a bit to figure out exactly what is going on. Now, though, I know that there is a bigger amount and a smaller amount being compared as well as a difference between those two. I actually see the tape diagram in my head. After I read the first sentence, I know one of the smaller boxes is the difference of 5 and I can even label the small box cows the larger box sheep since I know there are more sheep than cows. Once I read that there are 7 sheep, I would put that value in the big box and then be left to my own decision as to how I want to compute the answer of how many cows there are. For example, I may want to use 7-5 or I may want to think 5 + ? = 7. That’s where the student choice comes in. When numbers become larger, students would then choose if they want to use an open numberline, act out with blocks, use a 100’s chart, draw a picture, etc. Total freedom to figure out the numerical answer, yet the model is the same.
Thirdly, these problem types will follow our students throughout many years of their math journey. It’s just that the numbers will get larger or become fractions and decimals. Yet the underlying structure is always there. If we teach the tape diagram from the earliest years, I feel like we are providing them with a tool in understanding the structure for years to come. They will then bring to the table their own inventive strategies for calculating the answers.
Finally, having learned this particular model for the add/sub problem types, students will have developed the mathematical thinking to find answers where they may not need to find a numerical answer at all. I’ve seen many questions on our district standardized testing measure NWEA as well as SBAC where it gives the students the situation and then asks them to choose an equation that could be used to solve it. There’s no number answer at all. Inherent in the given equation choices are understandings of the commutative property, the equal sign as meaning “the same as”, the inverse relationships between addition and subtraction, etc. All powerful foundational mathematics concepts we are able to discuss while working on this…and the exact same model is used for all the add/sub problem types!
So, while I’m not in favor usually of mandating things, we are mandating the model including labeling the diagrams with words from the word problem, but in no way are we mandating the computational methods. In fact, in many instances we don’t find the answers at all. We have students determine which equations, of many given, could be used to solve the problem. We put the answers on the left side of the equal sign, we change the order of the addends, we have some with the inverse operation, and we even have some with the smaller number minus the larger number to discuss that we COULD do that math but we’d end up in negative numbers and not find the answer we are looking for (we live in a cold area so weather below zero is very familiar to them). Students are doing a lot of discussing about why given equations will solve the situation as well as why some will not. I get the chills listening to them support their answers.
I have struggled with the idea of “mandating” as well, but I feel like the tape diagram specifically provides such rich mathematical concepts that it is worth it. I’m not seeing nearly as much as I used to of students mindlessly plucking numbers from word problems and finding a key word that would suggest an operation with no thought to the context in the problem. I think that right there is a win.
I’m hoping that in this process, not only are we setting the foundation of the structure of addition/subtraction word problems but we are encouraging the thinking process needed to diagram word problem situations that we can then apply to more difficult problem types they will encounter down the road like multiplication and division of whole numbers, decimals and fractions.


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What do you see when you look at the stars? Benjamin Banneker saw astronomical patterns from which he could make calculations and predictions. A mathematician and astronomer, Benjamin Banneker was born on November 9, 1731, in Ellicott’s Mills, Maryland.
Largely self-taught, Banneker was one of the first African Americans to gain distinction in science. His significant accomplishments include the successful prediction of a solar eclipse, publishing his own almanac, and the surveying of Washington, D.C. Banneker spent most of his life on his family’s 100-acre farm outside Baltimore. There, he taught himself astronomy by watching the stars and learned advanced mathematics from borrowed textbooks…
(Click on link below to read the entire article)
(Watch video below on the life/history of Mr. Banneker)


The proceedings of the 24th International Conference on Difference Equations and Applications (ICDEA 2018) will be published in a special volume of the series Springer Proceedings in Mathematics & Statistics entitled Difference Equations and Discrete Dynamical Systems with Applications, Editors: Martin Bohner, Stefan Siegmund, Roman Šimon Hilscher and Petr Stehlík.
The volume will include new and significant contributions in the field of difference equations, discrete dynamical systems and their applications to various sciences. A central aim is to disseminate recent studies and related results and in the same time to promote the recent progress in the field of difference equations and discrete dynamical systems.
All participants are invited to submit their contributions via e-mail to firstname.lastname@example.org
The manuscripts will be evaluated through a peer review process. The referees will evaluate the scientific quality of the papers, the originality of the work, the novelty and the impact to the further development in the area.
|Opening of submission:||
May 31, 2018
|Deadline for submission:||
December 1, 2018 (
The papers must be written in English and prepared using LaTeX2e style files which can be downloaded by following the link "LaTeX Package" on the website
Authors need to prepare their contributions using the file author.tex. Please note that Springer Proceedings in Mathematics & Statistics uses the Reference Style MathPhysSci. The authors are kindly asked not to change the commands and settings of the style files, as this may cause errors during compilation of the final manuscript.
After the acceptance, the authors need to submit all final source files, including figures, and the pdf file of the final version of their paper.
When the authors submit their accepted, final revised paper and its source files, each corresponding author must complete and send the copyright transfer agreement: Consent to Publish.
The authors should obtain permissions from the previous publisher for any material they want to reuse – please see Permission Request Form. The completed form must be submitted by the authors simultaneously with the final version of the paper.


Whether you’re on top of your house with rectangles and triangles, or estimating from the ground with pitch, here are two ways to calculate the area of your roof.
You need to get a new roof. Yes, the estimates you get from professional roofers will include a calculation of your roof area. But a conscientious homeowner will do his own roof area measurement to ensure that no costly mistakes are made.
To measure the roof, you’ll need to know how to find the areas of basic rectangle and triangle shapes, or perhaps how to calculate roof pitch.
Always add a little more
When calculating roof size, you should always take into account wastage during construction. So for a very simple roof, you should add five per cent for waste. This should go up to 10 per cent for a slightly more complicated roof, and 15 per cent for ones even more so (with lots of valleys and dormers, or complicated shapes, for example).
On the roof — get the picture
If you are comfortable getting right up on the roof, and can do so safely, then this is the most accurate way to measure its area. But before you climb up, make a rough diagram of your roof, dividing it into rectangle and triangle areas.
Calculating the areas
Once on the roof, use a tape measure to calculate the length and width of each rectangular area. Then multiply these two numbers together to give you the rectangle’s area. Note these figures on your diagram.
For a triangular area, multiply the width of the base by the triangle’s height and divide this number by two to give you the area. Again, make a note on your handy diagram.
Add it all up
Add up all the area values to give you your roof’s square footage. Roofing materials are determined by the number of “squares” a roof is. To calculate the square of a roof, take the number of square feet you came up with and divide it by 100. A 3,000-square-foot roof would be 30 squares.
From the ground
While measuring from the ground isn’t as accurate as going up on the roof itself, it is safer and does give you a rough idea to compare with roofer estimates. Assuming you have a rectangular shaped house, use a tape measure to measure its length and width from the edges of the eaves. Then multiply these two numbers together.
If the house shape is complex, you will have to calculate each wing or room separately.
What’s the pitch?
But hold on, you’re not finished yet. Unless you have a flat roof, you’ll need to calculate your roof pitch.
Roof pitch is determined by the number of inches it rises vertically for every 12 inches it extends horizontally. We’re afraid you’ll brieflyhave to go onto the roof itself for this. Mark a level at 12 inches, hold it perfectly level and measure from the roof surface to the 12 inch mark. A roof with a 3 inch rise (known as the roof’s slope or its slant) over 12 inches would be marked as having a pitch of 3/12. You would then use a calculator to determine the decimal figure — 1.03.
To get your final calculation
Take the original number for your ground level square footage and multiply it by your pitch decimal figure to give your “roof square.”
With final number proudly in hand, see how your estimate squares with the roofer’s, and then let the building begin!


Sooooo fun and funny! Bethany proves once again that she can take a “scary” subject and make it fascinating!
Chock full of facts about math and its applications, from music to cookie baking!
Perfect for teachers and librarians looking for an entertaining way to add to math curriculum.
Meets the demand for STEM & STEAM books for elementary school students.
Sure to entertain both math lovers and math scaredy-cats!
Bethany Barton has forged a name for herself in the young non-fiction genre, and everyone’s taking notice!
Award-winning previous books in this evolving series. Awards include Children’s Choice Award, as well as many state award nominations.
“Full of interesting and amazing facts and the many ways math is crucial in our
everyday lives, this entertaining, vibrant text is sure to get young readers excited about the subject.”—Booklist
Praise for Give Bees a Chance:
“Even the most bee-phobic readers will have a hard time resisting this swarm of humor and fact.”—Kirkus Reviews
“Refreshing [and] fun…A must for the lighter side of nonfiction.”—School Library Journal
“Colorful artwork, plenty of facts, and an entertainingly irreverent approach…[make this a] playful take on nonfiction.”—Booklist
Raves for I’m Trying to Love Spiders:
A 2016 CHILDREN’S CHOICE BOOK AWARD WINNER!
* “A winner.”—Publishers Weekly, starred review
* “Simultaneously cringe-worthy and cackle-inducing.”—Kirkus Reviews, starred review
“Zippy text and hilarious visuals.”—School Library Journal
“Fact-filled and funny arachnology.”—Booklist
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ming | playgrd presents
While modelling the weather, Edward Lorenz formulated a set of equations and visualized it in the 3-dimensional space. The computer drew a fascinating curve with overlapping spirals that resembled the wings of a butterfly. These lines never intersected but looped infinitely in interesting patterns. This was the Lorenz attractor. It is by no means the only set of equations that could give rise to such wondrous patterns. There are others, and they are known collectively as strange attractors. These equations describe a dynamic system, which is at once chaotic and ordered.
In this collection, we present:
50 ethereal strange attractors generated in code, sold in 1/11 editions
Each of these pieces is a programmatically generated video loop of the strange attractors with different energy levels, based on different artistically chosen equations, frequencies and coefficients.
The 50 gifs below are lower resolution previews of the high resolution videos. A preview of the video that will be sold as the NFT can be viewed at the bottom of the page.


数学系10月13日学术报告《The stable derived category of an isolated singularity》 [2009-05-07]
告人： Dr. Daniel Murfet（波恩大学数学系）
报告时间： 10月13日 16：00-17：00
报告地点： 理化大楼 16008
题目：The stable derived category of an isolated singularity
摘要：There has been recent interest in categorical invariants of singularities in algebraic geometry. One associates to any local noetherian ring a triangulated category called the "stable derived category", which reflects certain homological
information about the singularities of the local ring. We will discuss two natural properties of this invariant: the effect of completing the local ring, and the role of singular points as generators.


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The works of american artist Josh Sommers carve out their context, somewhere along the boundaries of landscape photography, art, mathematics, and technology. There’s a crispness to these photos, in that they appear to have been planned from the beginning instead of having an effect slapped on afterward.
The stereographic projection, in geometry, is a particular mapping (function) that projects a sphere onto a plane. The projection is defined on the entire sphere, except at one point — the projection point. It is conformal, meaning that it preserves angles. It is neither isometric nor area-preserving: that is, it preserves neither distances nor the areas of figures.
After one look at this planet any visitor from outer space would say ‘I want to see the manager. William S. Burroughs


Math 317 - Theory of Linear Algebra - Sections 2 and 3
Instructor: Jonas Hartwig firstname.lastname@example.org Office: Carver 470, office hours TF 3-4.
TA/Grader: Issac Odegard. Office 477, office hour W 2-3.
Course Log and Homework Assignments
Click here for a list of recommended practice problems.
Aug 26-30: §1.1, §4.1, §1.2. HW: §1.1 #11,20,22a
Sep 2-6: §1.4, §1.5, §2.1. HW: §1.2: #8,10; §1.4: #6,13b
Sep 9-13: §2.2, §2.3, §2.4. HW: §1.5: #28ab; §2.1: #5
Sep 16-20: §2.4, §3.1, §3.2. HW: §2.2: #9,11a; §2.3: #8b; §2.4: #7b,14bc
Sep 23-27: §3.4 Thursday Exam 1 on Ch.1 & Ch.2 HW: §3.1: #5b,9b,11b; §3.2: #2d,5,7
Sep 30-Oct 4: §3.4, §4.1. HW: §3.4: #1bd, 2b, 6, 13
Oct 7-11: §4.1, §4.2. HW: §3.4: #4g,5a; §4.1: #8,11,13a
Oct 14-18: §4.3, §4.4, §4.5. HW: §4.2: #2fg,3cdh,19
Oct 21-25: §4.5, §4.6, §4.7. HW: §4.3: #8,9; §4.4: #2a,4f,5,14
Oct 28-Nov 1: Review. Thursday Exam 2 on Ch.3 & Ch. 4 (excluding transition matrices). HW: §4.5: #1c,7; §4.6: #1b,5b,10b,11b; §4.7: #1fi
Nov 4-8: §5.1, §5.2. HW: §5.1: #16,17,23
Nov 11-15: §5.3, §5.4. HW: §5.2: #7b, 12ab,16abc;
Nov 18-22: §5.5. Thursday Exam 3 on Ch.5(§5.1-§5.5) HW: §5.3: #1bd,9; §5.4: #2b,4ab,5ab
Nov 25-29: Thanksgiving break
Dec 2-6: §5.6, §6.1 HW (due Monday Dec 9): §5.6: #1b, 2a, 10
Dec 9-13: Euclidean spaces; Review. HW §6.1: #2d,4bd,9ab;
Dec 16-20: Finals week
MATH 317 Section 2: Thursday December 19 at 12:00 PM
MATH 317 Section 3: Tuesday December 17 at 2:15 PM
MATH 317. Theory of Linear Algebra.
(4-0) Cr. 4. F.S.SS. Credit or enrollment in MATH 201
Systems of linear equations, determinants, vector spaces, inner product spaces, linear transformations, eigenvalues and eigenvectors.
Emphasis on writing proofs and results. Only one of MATH 207 and MATH 317 may be counted toward graduation.
Andrilli and Hecker
Elementary Linear Algebra
Chapter 1: Vectors and Matrices (6 lectures)
Chapter 2: Systems of Linear Equations (7 lectures)
Chapter 3: Determinants and Eigenvalues (8 lectures)
Chapter 4: Finite-Dimensional Vector Spaces (16 lectures)
Chapter 5: Linear Transformations (15 lectures)
Chapter 6: Orthogonality (7 lectures)
In-class activities: 15%
Exams 1-3: 15% each
Objectives for Math 317
Be able to:
- use vector algebra, matrix algebra and dot products to manipulate vector and matrix equations.
- find the solution set to a given linear system of equations in parametric form.
- compute the echelon and reduced echelon forms of a matrix .
- compute row space, column space, null space, left null space, rank of a matrix.
- compute inverse matrices.
- compute orthogonal projections on to vectors and hyperplanes.
- compute determinants, and understand the basic properties of determinants.
- compute orthogonal complements of a subspace
- determine the dimension of a vector subspace
- compute the standard matrix for a given linear transformation.
- compute the matrix for a linear transformation with respect to a given basis.
- compute an orthogonal basis from one that is not orthogonal.
- find an orthogonal matrix that diagonalizes a given symmetric matrix.
Be able to prove simple theorems on fundamental properties of linear algebra. These could include the following.
- Prove a given set is a subspace (or prove it is not).
- Prove a given set of vectors is linearly independent (or prove it is not).
- Prove a given transformation is linear (or is not).
- Use key theorems such as the Dimension Theorem to deduce properties of a given linear transformation.
- Prove whether a set of vectors forms a basis.
Sample Final With Solutions
Official Math Department Policies
The Math Department Class Policies page describes the official policies that all instructors have to follow. It covers rules on make-up exams, cheating, student behavior, etc.
Iowa State University is committed to assuring that all educational activities are free from discrimination and harassment based on disability status. Students requesting accommodations for a documented disability are required to work directly with staff in Student Accessibility Services (SAS) to establish eligibility and learn about related processes before accommodations will be identified. After eligibility is established, SAS staff will create and issue a Notification Letter for each course listing approved reasonable accommodations. This document will be made available to the student and instructor either electronically or in hard-copy every semester. Students and instructors are encouraged to review contents of the Notification Letters as early in the semester as possible to identify a specific, timely plan to deliver/receive the indicated accommodations. Reasonable accommodations are not retroactive in nature and are not intended to be an unfair advantage.
Additional information or assistance is available online at www.sas.dso.iastate.edu, by contacting SAS staff by email at email@example.com, or by calling 515-294-7220. Student Accessibility Services is a unit in the Dean of Students Office located at 1076 Student Services Building.


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College level textbooks might be useful tool for preparing home school students to go into the college arena. As a result, they develop confidence and start liking the subject. For instance, most students find it simple to remember 30 and 60.
Today it’s possible to define mental math in various distinctive ways. Attached you’ll discover an example Github project that you may fool around with. Others are going to delight in programming games.
The Prime Numbers can at times be tough and complex to learn. Excel offers lots of built-in functions that handle trigonometry. The Investment Calculator is invented to assist with the ROI results before you begin your business enterprise.
Calculus is about the rate. Practice all the fundamentals of geometry, Geometry is closely associated with trigonometry. Quadratic Calculator makes it possible to solve quadratic equations The quadratic equations may also be really time-consuming.
You may see the comprehensive history along with the way the equation was solved. Learning how to write out long-form computations and equations is a must, which will not only help you as soon as you don’t have a calculator handy, but gives a better illustration of a particular scenario or problem you are attempting to work through. The general way of solving an equation is to convert it into the shape of a single ratio only.
Otherwise, division will do the job just in combination with a number of the other operators or functions. It is equal to 625. So, you’ve got to be cautious when calculating the results.
Your work is done and now you merely have to watch for the results. Sometimes two solutions are possible. So, it is possible to just finish your exam paper easily.
To address a triangle method to determine angles and sides of a specific triangle when a few of the sides and or angles are given. 45 In this instance, the triangle is isosceles. For instance, the next triangles are alike.
Since similar triangles always have the very same angles, then the ratio is exactly the same for any specific angle. The idea of Pi has also existed for some time. They are independent of the unit used to measure the sides as long as the same unit is used for all the sides.
The Advantages of Trigonometry Calculator
It’s this fusion of algebra and geometry that produces trigonometry so intriguing. Trigonometry help isn’t hard to discover and should you believe that’s what you require, then don’t wait till the year ends. Which can help you to increase your trigonometry abilities.
It’s significant since it shows the attractiveness of pure mathematics. Search for trigonometry resources on the internet to discover extensive material you’re able to access anytime. Be certain to use precise expressions for your answers.
Finding the Best Trigonometry Calculator
You’ll like you may make use of these calculators right here on this page or you could get the program so you don’t even have to use the world wide web. Other people would reach the conclusion of the exact same on-line course and stare at a blank text editor, not making the relation between the syntax they memorized and the functional application of the code they are attempting to write. A help manual delivers detailed information concerning the available commands in the event that you don’t figure out how to decode them by yourself.


With Teacher of Sacred Geometry
Sunday, November 14
12:00 pm PT / 1:00 pm MT / 2:00 pm CT / 3:00 pm ET / 8:00pm GMT
Sacred Geometry & Universal Order: Shape Matters
A profound order pervades the universe. It stands independent of chaos, space, time and even creation itself. However, it is invisible and exists in a state of potential. Yet its complexities can be captured in a circle and sphere, and revealed through geometry as elegant shapes, patterns and forms. The patterns describe continuous, timeless, universal actions that are otherwise invisible and can only be known to us through geometry. Many universal principles are portrayed, such as equilibrium, fractalization, resonance, optimization, synthesis, symbiosis, etc.
Each shape is a mathematical program, containing numerous properties and dynamics such as inter-angular relationships, vectors, vertices, radials, meridians, perimeters, ratios, proportions, root powers and truncations. Each shape is also a fractal that can potentially create an endless number of smaller shapes and patterns, all perfectly connected and aligned to a higher order. Alternately, they can project outward, in a perfectly ordered and unending progression.
Sacred Geometry & Universal Order is a visual presentation that demonstrates how geo-harmonic shapes, patterns and forms can help attune our consciousness to the harmony of universal order. We are naturally attracted to these geometries because we contain the same mathematical properties in us.
Through the power of visualization and conscious awareness, we can integrate the essence of these realms into our mental and emotional patterns and introduce healthy working systems into our reality. The more we understand universal law the more we can apply the harmony that is apparent in nature as a more powerful force in our own lives.
“Because our world is constructed from geometric relations like the Golden Ratio or the Fibonacci series, by thinking about geometry often enough you can organize and harmonize your life with the structure of the world.” – Buckminster Fuller
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About Jonathan Quintin
Jonathan Quintin was born and raised in provincial New Zealand but has spent much of his life overseas. His interest in sacred geometry began as a child enthralled with the patterns in nature.
His international travels were focused on visiting ancient sites around the world. He lived in the Middle East for several years working for the United Nations. During that period, he was stationed throughout Israel, Lebanon, Syria, Jordan and Egypt… providing many opportunities to visit ancient monuments relics.
He was greatly influenced by how much the ancient civilizations were aware of universal law and natural order, which they expressed through their respective architectures, artwork and in the teachings of their mystery schools.
Sacred geometry was an important element in many of these schools. Often the goal was for the student to transition from the state of maya (illusion) to the state of samadhi (enlightenment), a process sometimes referred to as “the journey home”
For the past 35 years, Jonathan has been teaching sacred geometry as a tool to higher consciousness.


Physics takes pride in being the foundational science—an anchor of mathematical laws that describe the functioning of the physical world and that relate one phenomenon to another.
Albert Einstein’s 1917 equation E=mc2, for example, is elegant in its simple statement of the relationship between energy to mass via the constant “c,” the speed of light. Because mass is related to gravity, the theory of relativity, which describes how gravity operates at cosmic distances, is deeply connected to this equation. If new observations and measurements over the past 90 years had required regular tinkering with the equation by adding new factors and adjustments, we would easily discern that the theory attached to it was at best incomplete.
The term “theory” in science is lavished only on the big ideas that appear to have predictive power. Atomic, nuclear and even evolutionary theory are given high regard, because they each present a worldview that is consistent with a wide range of observations. Unfortunately, as astrophysicist J.V. Narlikar states, “you can always fit theory to data by adding a large number of parameters.” But doing science this way is like performing a secret-number trick: a magician, by getting an audience member to apply a certain sequence of mathematical operations to a secret number, amazingly comes up with the hidden number.
With regard to the big bang theory, research astronomer Tom Van Flandern finds it wanting on several fronts, but what concerns him most is its preservation through the invention of new mathematical factors or parameters. These, he believes, simply shave the square pegs of new observations to fit the round holes of the big bang model. Van Flandern argues that, like the magician who has established an arithmetic sequence that will invariably yield the answer he’s looking for, the big bang proponent makes the observation fit the theory through mathematical sleight of hand.
“Big bang models now use an ever increasing variety of free parameters to maintain consistency with various observational constraints,” he writes on his MetaResearch website. “Related to origin and expansion conditions alone, we now have the Hubble constant h (= expansion rate); the cosmological constant Λ (= pressure resisting gravity); the cosmic deceleration parameter q0 (= expansion deceleration); the density parameter Ω (= ratio of actual matter density to density needed for flat universe), subdivided into the density for ordinary matter and that for invisible dark matter; and the bias parameter b (= measures lumpiness of matter distribution). The hypothetical dark matter is itself a fudge factor required to obtain agreement with observations that were not in accord with big bang expectations. . . .
“If the field of astronomy were not presently over-invested in the expanding universe paradigm, it is clear that modern observations would now compel us to adopt a static universe model as the basis of any sound cosmological theory.”


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Applied Statistics, Certificate
Certificate Requirements (12 Hours)
The PBACC requires at least 12 semester hours of graduate credits in statistics, at least half of which must be courses at the 700-level or above with the STAT designator, completed within a period of six years before the award of the certificate.
The 12 Hours must include the following:
Basic Data Analysis (6 Hours)
|STAT 700||Applied Statistics I (Or the equivalent)||3|
|STAT 701||Applied Statistics II (Or the equivalent)||3|
|Total Credit Hours||6|
Additional Courses (6 Hours)
- At least 6 Hours of additional courses must be selected with the approval of the graduate director or the director for part-time students.
Note: Up to 6 semester hours of approved statistics courses may be taken from other departments and/or by transfer credit. With the approval of the department, and subject to the regulations of The Graduate School, courses taken in a certificate program can be applied toward other graduate degrees.


Learn C Basic Programs - Computer Programming
Tutorial of Simple C Programs with examples
1  C program to find the area of a triangle with three sides ?
2  c program to find a circles area with radius given by user ?
3  c program to find simple interest with user input principle amount ?
4  c program to check the entered number is odd or even ?
5  c program to check whether the number is negative or positive ?
6  Find the largest of three numbers given using c program ?
7  write a c program to find roots of a quadratic equation ?
8  C program for making a simple calculator for doing mathematical calculations ?
9  calculate the sum of n natural numbers using c program ?
10  generate a Fibonacci series of n numbers using c program ?
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C programming tutorial.com
. All rights reserved. Design by


A simple change of perspective is apparently enough to get a grip on the percentage calculation, which is hated by many. One of these appeared in 2019 Math trick as a post on Twitter, the reactions to it were sometimes violent. Even better: To this day, the rule for Percentage calculation Validity, so you can try it out right away.
Do you know this math trick for calculating percentages?
Since the 2019 post, Ben Stephens’ math trick has been shared more than 8,000 times and liked almost 23,000 times. At first glance, the formula doesn’t look particularly spectacular, but it does generate strong reactions. “You changed my life”, “It just broke my brain” or “Good Lord” are just a few of them. One user even confessed to having a degree in applied mathematics but not knowing this method.
Stephens had the math trick back then with the simple words “Fascinating life hack for percentage calculation” including the corresponding Formula “x% of y = y% of x” and an example posted:
“So, for example, if you have to compute 4% of 75 in your head, just turn it and do 75% of 4, which is easier.”
If you have problems with percentage calculations, the method is definitely a great help. It gets a little more complicated as soon as the result is no longer a natural number, but the math trick also works perfectly in these cases. As a simple example, Stephens gave the calculation “4% of 75”, which is more difficult to solve in your head than the reverse formula “75% of 4”. In both cases the result is “3”, but the time for the respective calculation should differ.
How new this process actually was for many is shown by further comments on Stephen’s “Life Hack”:
- @ BenCooper1975: “You won Twitter today, Mr. Stephens !!”
- @WillKanyusik: “That just destroyed my brain.”
- @RuthMCallaghan: “I taught math reporters (…) and wish I had that explanation in my pocket. I had other tricks for math phobics, but this one is far more elegant. “
- @theroxyd: “What the HELL”
- @ markab29: “Damn it! It’s so simple and useful. Thank you!!!”
Fascinating things from mathematics
In addition to such helpful math tricks, there are always exciting findings in the field of research. So a whole new prime was discovered a while ago. Other things continue to fascinate to this day, such as the mysterious number 137.
Sources: Twitter / @ stephens_ben


Florida's Florida Lotto lottery draw is a pick 6 game with a pool of 53 numbers to choose from.
The odds of winning the top prize are 22,957,480 to 1
The odds of winning any single prize is 67.36 to 1
The official website can be found www.flalottery.com/lottoMain.do
LSA can not improve the odds of winning the top prize, no software could do that!.
What LSA can do and does very well is improve the odds of winning smaller prizes, and on a more regular basis.
LSA is able to build advanced predictions for the next Florida Lotto draw, then enter the findings into
MNA's or wheels
In Florida's Florida Lotto there are 22,957,480 possible 6 ball combinations, LSA has the capability of analysing every single one while comparing past draw history, all done in a few milli-seconds.
This unique ability makes LSA's statistics the best available, by analysing everything there's no stone left unturned so to speak..
This chart below is an example of the powerful information and statistics LSA has to offer. The chart displays the top 5 appearing numbers from each of the 6 ball placements taken from the ENTIRE Florida Lotto draw as ball 1, ball 2 etc, so the left hand side of the = sign is the ball number the right hand side is the total number of times that number appears in the entire draw.
Top 5 appearing numbers for all 6 ball placements in the Florida Lotto draw
See LSA for the full list off all 6 ball positions and 53 numbers.
Table explained. Lets pick the top left of the above chart, 01=2598960 is ball number one, ball placement 1 (lowest number from every possible draw combination with the number 1 in it) appears more times as ball one than any other number, obvious for ball 1 as no number could be lower but for other numbers this type of information is a massive help in predicting numbers and their final placement, this information alone improves the odds substantially, but knowing the most likely numbers to appear in each ball position is just a tiny part of how LSA predicts the next draw.
LOWESS, an exciting and a very powerful regression algorithm that is used in modern statistics for predicting future events where the data analysed is seminally random.
The chart below is a screen capture of LSA performing a LOWESS prediction, this is for ball placement 3, the blue line is the LOWESS curve, the yellow band on the right represents the prediction or next draw if you like. The numbers on this graph represent the past 50 draws.
When using the LOWESS feature in LSA you can adjust the smoothness of the blue plot line, the number of draws analysed and the amount of times the LOWESS algorithm is run on the data before producing the plot line. By having this ability of tweaking the prediction you can fine tune the prediction to match the trend of your draw, this is very powerful and a lot of users are reporting excellent results from the predictions.
That's not all, knowing the best numbers to choose for each ball placement, the running trend of recent draws are just part of how LSA predicts the next draw. We need a system to provide maximum coverage of the predicted numbers and this is where wheels come into play. However good you think a wheel is, it is not. LSA uses another system called "Maximized Number Arrangments" or MNA's that provide upward of 40% more coverage than the best standard wheels. and injects the predicted numbers into the MNA system to provide a wheel that covers as many possible winning combinations for the money you bet.
It's really simple, the more winning combinations you can cover for your money the better the return will be and MNA's do this for you.
Download a copy of LSA, you get 15 free try's with absolutely no obligation to buy at the end of the trial.


McCalc® Fusion Pressure Calculator
Use McElroy's fusion pressure calculator to quickly find the right fusion pressure for your job. To properly fuse pipe the fusion pressure must be adjusted so the pipe manufacturers recommended interfacial pressure is achieved.
We have several convenient ways to calculate fusion pressure. You can use our online tool below, download the Fusion Pressure Calculator App for your mobile device, or order a sliding calculator.
Wondering how it all works? Learn how to calculate fusion pressure.
McCalc Mobile App
Always have a fusion pressure calculator with you no matter where you are.
Learn To Calculate Fusion Pressure
Using a 28 Machine from Chart A (at right) we will determine our gauge pressure.
= 8" IPS
OD of Pipe
DR of Pipe
IFP = 75 PSI
= 30 PSI
= Outside diameter
= Wall thickness
= Dimensional ratio
= Manufacturers recommended interfacial pressure
= Total effective piston area
= Force required to move pipe
Total effective piston area
||High Force Standard
||Medium Force High Velocity
||Low Force Extra High Velocity
Step 1. - Calculating Wall Thickness:
To determine wall thickness we will be using the Example (above).
| = Wall thickness
Now the Example and the formula above we will calculate the wall thickness.
Step 2. - Finding Total Effective Piston Area
Use Chart A (above) to find the TEPA of your machine.
TEPA = 4.710 (we are using the 28 as an example)
Step 3. - Calculating Gauge Pressure
Using the information we have gathered from above.
|(OD - T) x T x PI x IFP
|+ DRAG = Gauge Pressure
Step 4. - Completed Formula
Now using all the information and the formula above we will find the gauge pressure.
|(8.625 - .784) x .784 x 3.1416 x 75
| + 30 PSI =


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Sacred Geometry Symbols and Meanings
Have you ever wondered what those geometric patterns that decorate your favorite festival outfits or jewelry, or the symbols Dr. Strange uses in the Marvel comics and movies were. It’s known as Sacred Geometry, an idea that nature and all its mathematical principles are based on geometric shapes. What are they though and what’s their meaning? Were gonna break down a couple of the most popular symbols and their meanings so you can understand the significance and pick out your favorites.
The Flower of Life:
Known as the beginning, or the creation of all matter. The shape of a circle is associated with divinity, or the higher power. As the circle is repeated you can draw several other geometric shapes from the flower of life, such as a square, a triangle, hexagon, and more. Plato spoke of 5 Perfect Solids, the hedrons (tetrahedron, Icosahedron, dodecahedron, and octahedron) and the cube. You can find all these within the flower of life by combining the center of each sphere in different ways.
The body, the mind, the spirit or also known as the will, the manifestation, and the divinity. Named after the archangel Metatron (from Jewish traditions) who was seen as in charge of all creation. Thirteen circles appear of the same size, extending lines from the center of every circle to the center of the other twelve circles. It is sometimes referred to as the fruit of life. This symbol is for the balancing of energies, representing the harmony and balance within nature.
The equilibrium and creation of the universe. Four upright triangles are a representation of Shiva, the five downward are a representation of Shakti, the male and female embodiments. Fourty-three smaller triangles intersected by nine major ones are the “Cosmic Uterus”, the universe. This symbol represents coexistence of male and female, evil and good, duality, black and white.
This symbol holds the geometry of the vacuum of all matter. Also known as the “vehicle of light” this symbol is made of two tetrahedrons, one facing down the other up, interconnected through their planes. A complete equilibrium that all life is based and all creation is based on. 64 is the smallest number with exactly seven divisors, 64 codons in a humans DNA, 64 places on a chess or checker board. If you were to place a circle around each of the tetrahedrons in the star tetrahedrons grid you obtain a 2D Flower of life.
I hope this gathering of information helps you understand the strange geometry in the life all around and helps you pick out your favorites for you to wear from your favorite small business companies to your next festival.
-By Liz Colston


Pricing Modelling Analyst - London, United Kingdom,
Cashflow Modelling Analyst
A model development role for a statistically strong analyst with experience building financial models to price or evaluate investment opportunities in a deadline driven environment.
If you have a passion for coding, mathematical modelling and diving fast paced solutions this role might be of interest.
An industry leader operating in a niche space within financial services, now adding top class analysts to their team due to the growth of the business.
- Use mathematical modelling and machine learning techniques to predict the value of investment portfolios in a deadline driven environment
- Enhance existing pricing model with advanced analytical and statistical techniques
- Evaluate portfolio performance from mass unstructured data, using SQL, R, Python, SAS
- Present and explain valuations and predictions to Investment Committee and Stakeholders
- Increase accuracy of models by researching and incorporating machine learning and advanced analytical techniques
YOUR SKILLS AND EXPERIENCE
- A numeracy-based degree is a minimal requirement, with post-graduate degrees preferred
- Experience evaluating investment opportunities, acquisitions or pricing portfolios using advanced statistical methods and mathematical techniques is required
- Strong programming experience using SQL or R (or SAS, Python or similar)
Strong communication skills with ability to present to stakeholders
Portfolio Evaluation, Financial Modelling, Investment Models, Mathematics, Financial Services, Quantitative Analyst


By Chris Alu
Nokia, has unveils a game application for Nigerian children in primary schools, to learn mathematics. The App known as SmartKid is available for free download on the Windows Store.
SmartKid Maths is an educational game created by Finnish digital company, SkillPixels which contains complete preschool, 1st and 2ndgrade mathematics with three years’ worth of curriculum-based math
The managing director Nokia West and Central Africa, Nick Imudia, explained that the SmartKid Maths is the first learning game app featuring Learning Analytics which shows users their progress as they proceed to the next stage.
According to him,the game shows users how quickly or slowly they learned concepts; it identifies their strengths while also helping them get better in their weak areas amongst others exciting features.
“This app will help build a strong foundation for children in the area of Mathematics and get them acquainted with the subject before starting with the basics in school simply by teaching them about
shapes, coordinates, numbers, time, addition, subtraction, equations, money, multiplication, division, fraction, and much more. It is a simple, easy and fun app that we are really proud of and believe will
impact immensely in children, making school work easy for the parents and teachers to teach” Imudia stated.
He said that SmartKid Maths will enables children learn the concept of numbers, counting, order, distance, numbering system, place value and fractions, adding, that the app will help parents and teachers to know and identify their children personal strengths and pinpoint individual bottlenecks in learning and help to correct them.