MATHS PROJECT : MATHS PROJECT BINOMIAL THEOREM BINOMIAL THEOREM : BINOMIAL THEOREM In elementary algebra, the binomial theorem describes the algebraic expansion of powers of a binomial. According to the theorem, it is possible to expand the power (x + y)n into a sum involving terms of the form axbyc, where the coefficient of each term is a positive integer, and the sum of the exponents of x and y in each term is n. For example: The coefficients appearing in the binomial expansion are known as binomial coefficients. They are the same as the entries of Pascal's triangle, and can be determined by a simple formula involving factorials. These numbers also arise in combinatorics, where the coefficient of xn−kyk is equal to the number of different combinations of k elements that can be chosen from an n-element set. HISTORY : HISTORY This formula and the triangular arrangement of the binomial coefficients are often attributed to Blaise Pascal, who described them in the 17th century, but they were known to many mathematicians who preceded him. The 4th century B.C. Greek mathematician Euclid mentioned the special case of the binomial theorem for exponent 2 as did the 3rd century B.C. Indian mathematician Pingala to higher orders. A more general binomial theorem and the so-called "Pascal's triangle" were known in the 10th-century A.D. to Indian mathematician Halayudha and Persian mathematician Al-Karaji, and in the 13th century to Chinese mathematician Yang Hui, who all derived similar results. Al-Karaji also provided a mathematical proof of both the binomial theorem and Pascal's triangle, using mathematical induction. STATEMENT OF THE THEOREM : STATEMENT OF THE THEOREM According to the theorem, it is possible to expand any power of x + y into a sum of the form
where denotes the corresponding binomial coefficient. Using summation notation, the formula above can be written
This formula is sometimes referred to as the binomial formula or the binomial identity.
A variant of the binomial formula is obtained by substituting 1 for x and x for y, so that it involves only a single variable. In this form, the formula reads
or equivalently GEOMETRICAL EXPLANATION : GEOMETRICAL EXPLANATION For positive values of a and b, the binomial theorem with n = 2 is the geometrically evident fact that a square of side a + b can be cut into a square of side a, a square of side b, and two rectangles with sides a and b. With n = 3, the theorem states that a cube of side a + b can be cut into a cube of side a, a cube of side b, three a×a×b rectangular boxes, and three a×b×b rectangular boxes. PASCAL'S TRIANGLE : THE BINOMIAL COEFFICIENTS : THE BINOMIAL COEFFICIENTS The coefficients that appear in the binomial expansion are called binomial coefficients. These are usually written , and pronounced “n choose k”.
The coefficient of xn−kyk is given by the formula
which is defined in terms of the factorial function n!. Equivalently, this formula can be written
with k factors in both the numerator and denominator of the fraction. Note that, although this formula involves a fraction, the binomial coefficient is actually an integer. COMBINATORIAL INTERPRETATION : COMBINATORIAL INTERPRETATION The binomial coefficient can be interpreted as the number of ways to choose k elements from an n-element set. This is related to binomials for the following reason: if we write (x + y)n as a product
then, according to the distributive law, there will be one term in the expansion for each choice of either x or y from each of the binomials of the product. For example, there will only be one term xn, corresponding to choosing 'x from each binomial. However, there will be several terms of the form xn−2y2, one for each way of choosing exactly two binomials to contribute a y. Therefore, after combining like terms, the coefficient of xn−2y2 will be equal to the number of ways to choose exactly 2 elements from an n-element set. PASCAL’S TRIANGLE : PASCAL’S TRIANGLE In mathematics, Pascal's triangle is a triangular array of the binomial coefficients in a triangle. It is named after the French mathematician Blaise Pascal in much of the Western world, although other mathematicians studied it centuries before him in Greece, India, Persia, China, and Italy.
The rows of Pascal's triangle are conventionally enumerated starting with row n = 0 at the top. The entries in each row are numbered from the left beginning with k = 0 and are usually staggered relative to the numbers in the adjacent rows. A simple construction of the triangle proceeds in the following manner. On row 0, write only the number 1. Then, to construct the elements of following rows, add the number directly above and to the left with the number directly above and to the right to find the new value. If either the number to the right or left is not present, substitute a zero in its place. For example, the first number in the first row is 0 + 1 = 1, whereas the numbers 1 and 3 in the third row are added to produce the number 4 in the fourth row. Slide 10: This construction is related to the binomial coefficients by Pascal's rule, which states that if
for any nonnegative integer n and any integer k between 0 and n.
Pascal's triangle has higher dimensional generalizations. The three-dimensional version is called Pascal's pyramid or Pascal's tetrahedron, while the general versions are called Pascal's simplices. HISTORY : HISTORY The set of numbers that form Pascal's triangle were well known before Pascal. But, Pascal developed many applications of it and was the first one to organize all the information together in his treatise, Traité du triangle arithmétique (1653). The numbers originally arose from Hindu studies of combinatorics and binomial numbers and the Greeks' study of figurate numbers.
The earliest explicit depictions of a triangle of binomial coefficients occur in the 10th century in commentaries on the Chandas Shastra, an Ancient Indian book on Sanskrit prosody written by Pingala between the 5th and 2nd century BC. While Pingala's work only survives in fragments, the commentator Halayudha, around 975, used the triangle to explain obscure references to Meru-prastaara, the "Staircase of Mount Meru". It was also realised that the shallow diagonals of the triangle sum to the Fibonacci numbers. Slide 13: At around the same time, it was discussed in Persia (Iran) by the Persian mathematician, Al-Karaji (953–1029). It was later repeated by the Persian poet-astronomer-mathematician Omar Khayyám (1048–1131); thus the triangle is referred to as the Khayyam triangle in Iran. Several theorems related to the triangle were known, including the binomial theorem. Khayyam used a method of finding nth roots based on the binomial expansion, and therefore on the binomial coefficients.
In 13th century, Yang Hui (1238–98) presented the arithmetic triangle that is the same as Pascal's triangle. Pascal's triangle is called Yang Hui's triangle in China. The "Yang Hui's triangle" was known in China in the upper half of the 11th century by the Chinese mathemtician Jia Xian (1010-1070).
Petrus Apianus (1495–1552) published the triangle on the frontispiece of his book on business calculations in the 16th century. This is the first record of the triangle in Europe. Slide 14: In Italy, it is referred to as Tartaglia's triangle, named for the Italian algebraist Niccolò Fontana Tartaglia (1500–77). Tartaglia is credited with the general formula for solving cubic polynomials, (which may be really from Scipione del Ferro but was published by Gerolamo Cardano 1545).
Traité du triangle arithmétique (Treatise on Arithmetical Triangle) has been published posthumously in 1665. In the Treatise Pascal collected several results then known about the triangle, and employed them to solve problems in probability theory. The triangle was later named after Pascal by Pierre Raymond de Montmort (1708) who called it "Table de M. Pascal pour les combinaisons" (French: Table of Mr. Pascal for combinations) and Abraham de Moivre (1730) who called it "Triangulum Arithmeticum PASCALIANUM" (Latin: Pascal's Arithmetic Triangle), which became the modern Western name. BINOMIAL EXPANSION : BINOMIAL EXPANSION Pascal's triangle determines the coefficients which arise in binomial expansions. For an example, consider the expansion
(x + y)2 = x2 + 2xy + y2 = 1x2y0 + 2x1y1 + 1x0y2.
Notice the coefficients are the numbers in row two of Pascal's triangle: 1, 2, 1. In general, when a binomial like x + y is raised to a positive integer power we have:
(x + y)n = a0xn + a1xn−1y + a2xn−2y2 + ... + an−1xyn−1 + anyn,
where the coefficients ai in this expansion are precisely the numbers on row n of Pascal's triangle. In other words,
This is the binomial theorem.
Notice that the entire right diagonal of Pascal's triangle corresponds to the coefficient of yn in these binomial expansions, while the next diagonal corresponds to the coefficient of xyn−1 and so on. Slide 16: To see how the binomial theorem relates to the simple construction of Pascal's triangle, consider the problem of calculating the coefficients of the expansion of (x + 1)n+1 in terms of the corresponding coefficients of (x + 1)n (setting y = 1 for simplicity). Suppose then that
The two summations can be reorganized as follows: Slide 17: We now have an expression for the polynomial (x + 1)n+1 in terms of the coefficients of (x + 1)n (these are the ais), which is what we need if we want to express a line in terms of the line above it. Recall that all the terms in a diagonal going from the upper-left to the lower-right correspond to the same power of x, and that the a-terms are the coefficients of the polynomial (x + 1)n, and we are determining the coefficients of (x + 1)n+1. Now, for any given i not 0 or n + 1, the coefficient of the xi term in the polynomial (x + 1)n+1 is equal to ai (the figure above and to the left of the figure to be determined, since it is on the same diagonal) + ai−1 (the figure to the immediate right of the first figure). This is indeed the simple rule for constructing Pascal's triangle row-by-row.
It is not difficult to turn this argument into a proof (by mathematical induction) of the binomial theorem. Since (a + b)n = bn(a/b + 1)n, the coefficients are identical in the expansion of the general case.
An interesting consequence of the binomial theorem is obtained by setting both variables x and y equal to one. In this case, we know that (1 + 1)n = 2n, and so
In other words, the sum of the entries in the nth row of Pascal's triangle is the nth power of 2. COMBINATIONS : COMBINATIONS A second useful application of Pascal's triangle is in the calculation of combinations. For example, the number combinations of n things taken k at a time (called n choose k) can be found by the equation
But this is also the formula for a cell of Pascal's triangle. Rather than performing the calculation, one can simply look up the appropriate entry in the triangle. For example, suppose a basketball team has 10 players and wants to know how many ways there are of selecting 8. The answer is entry 8 in row 10 (with the first row and the first entry in a row numbered 0): 45. That is, the solution of 10 choose 8 is 45. OVERALL PATTERNS AND PROPERTIES : OVERALL PATTERNS AND PROPERTIES The pattern obtained by coloring only the odd numbers in Pascal's triangle closely resembles the fractal called the Sierpinski triangle. This resemblance becomes more and more accurate as more rows are considered; in the limit, as the number of rows approaches infinity, the resulting pattern is the Sierpinski triangle, assuming a fixed perimeter. More generally, numbers could be colored differently according to whether or not they are multiples of 3, 4, etc.; this results in other similar patterns. Sierpinski triangle Slide 20: Imagine each number in the triangle is a node in a grid which is connected to the adjacent numbers above and below it. Now for any node in the grid, count the number of paths there are in the grid (without backtracking) which connect this node to the top node (1) of the triangle. The answer is the Pascal number associated to that node. The interpretation of the number in Pascal's Triangle as the number of paths to that number from the tip means that on a Plinko game board shaped like a triangle, the probability of winning prizes nearer the center will be higher than winning prizes on the edges. Slide 21: One property of the triangle is revealed if the rows are left-justified. In the triangle below, the diagonal coloured bands sum to successive Fibonacci numbers. THE END : THE END