Binomial theorem
The Binomial Theorem is a fundamental principle in algebra that describes the expansion of powers of binomials, which are expressions consisting of two terms, such as \(x + y\). It states that for any real numbers \(x\) and \(y\), and a whole number \(n\), the expression \((x + y)^n\) can be expanded into a sum involving binomial coefficients, denoted as \(c_k^n\). These coefficients correspond to the entries in Pascal's Triangle, a geometric representation of these values. The theorem has historical significance, with roots tracing back to ancient mathematicians across various cultures, including contributions from Greek, Indian, Chinese, and Islamic scholars, culminating in a more formalized approach by European mathematicians in the 16th and 17th centuries.
The theorem is not only taught in middle and high school mathematics but also finds applications in combinatorics, calculus, and number theory. For example, it aids in solving counting problems and proving foundational results like Fermat's Little Theorem. Generalizations of the binomial theorem extend its applicability to infinite series and complex numbers, allowing for broader mathematical exploration. Notably, the theorem has connections to various fields, including graph theory and fractals, making it a versatile tool in both theoretical and applied mathematics.
Subject Terms
Binomial theorem
Summary: The binomial theorem is the basis of Pascal’s Triangle and is used to solve a variety of problems.
A binomial is an algebraic expression with two terms, like x+y. When binomials are multiplied together, they produce higher powers of the individual terms that are called “binomial coefficients.” The binomial theorem states that for any real numbers x and y, and whole number n:
where ck is the binomial coefficient
and n! is the product of the numbers 1 through n. These coefficients are the entries in what is referred to as Pascal’s Triangle, named for mathematician Blaise Pascal. Students typically encounter this theorem in middle school or high school algebra, and in high school or college calculus. It also has uses in other areas of mathematics, such as in combinatorics, where it helps in calculations for certain counting problems.
The binomial theorem is found across ages and cultures. It appears in the ancient world in the work of Greek mathematician Euclid of Alexandria. His formula was for the square (n=2) of a binomial, but it was described geometrically rather than algebraically. There is also evidence that the Hindu scholar Aryabhata knew the theorem for cubes in the sixth century. At least as early as the eleventh century, Chinese mathematicians such as Jia Xian and later Zhu Shijie knew the binomial coefficients in the form of Pascal’s Triangle. They used the binomial theorem to find square and cube roots, and evidence suggests they knew of the binomial theorem for large values of n. Around the fifteenth century, the binomial theorem and binomial coefficients to at least the seventh power were found in the writings of Islamic scholars including Omar Khayyam, Abu Bekr ibn Muhammad ibn al-Husayn Al-Karaji, Abu Ali al-Hasan ibn al-Haytham (Alhazen), and Ibn Yahya al-Maghribi al-Samaw’al.
In the sixteenth century, European mathematicians began using the binomial theorem and binomial coefficients. For example, mathematician Michael Stifel’s 1544 work Arithmetica integra contained the binomial coefficients. Other contributors include Françis Viète, Blaise Pascal, James Gregory, Sir Isaac Newton, and Niels Abel. John Wallace’s seventeenth century book De Algebra Tractatus is cited as the first published account of Newton’s binomial work. While Pascal was not the first to study the binomial coefficients, he is credited with linking algebraic and combinatorial interpretations of the coefficients.
Pascal’s Triangle is a triangular representation of the binomial coefficients, which may be attributed to him because of his 1653 work Traité du Triangle Arithmétique in which he compiled and expounded on much of what was known about binomial coefficients. The related Pascal matrix is a symmetric, positive definite matrix with the Pascal triangle represented on its antidiagonals.
Generalizations and Extensions
James Gregory and Sir Isaac Newton generalized the binomial theorem to allow first fractional, and then real powers, which requires replacing the finite sum with an infinite series and extending the definition of the binomial coefficients. When generalizing to an infinite series, another issue that must be considered is convergence, which imposes restrictions on the numbers x and y for which the series converges to the binomial. Newton came to this generalization indirectly while trying to calculate areas under certain curves.
Another way to generalize the binomial theorem is to broaden the types of values that x and y can take. One such generalization allows x, y, and n to be complex numbers. The definition of the binomial coefficients has to be generalized to complex numbers, and certain restrictions on the variables are required for convergence of the resulting infinite series. Alternatively, one can allow x and y to be commuting elements of a Banach algebra, a normed algebra studied in such fields as complex analysis, real analysis, and functional analysis. Banach algebra is named for twentieth century mathematician Stefan Banach.
One more type of generalization considers not just the sum of two numbers x and y, but sums with more terms. Such a sum would be called a multinomial, and the multinomial coefficients would be appropriate generalizations of the binomial coefficients. Pascal’s Pyramid or Pascal’s Simplex are extensions of Pascal’s Triangle for three or more dimensions.
Applications
The binomial theorem gives a quick way of expanding a power of the form (x+y)n, making the formula useful for basic algebraic calculations. The binomial theorem, along with De Moivre’s formula, can be used to prove the trigonometric double-angle identities, as well as more general formulas for cos(nx and sin(nx). The mathematical constant e also can be written as the infinite limit of
Mathematical induction and the binomial theorem, or the multinomial theorem, can be used to prove what is known as “Fermat’s little theorem,” named for mathematician Pierre de Fermat. This result in number theory states that if p is a prime number and n is an integer not divisible by p, then np-n. is divisible by p. Fermat’s little theorem is itself used in cryptography, providing an indirect application of the binomial theorem. One theorem in graph theory states that a graph with n vertices and adjacency matrix A is connected if and only if all the entries in the matrix (1+A)n-1 are positive. This theorem is proved using the binomial theorem, generalized to certain matrices, and some basic graph theory results. Certain colorings of Pascal’s Triangle produce fractal figures like Sierpinski’s Triangle, named for mathematician Waclaw Sierpinski. In set theory, the regions of a Venn diagram for n distinct sets are in one-to-one correspondence with the binomial coefficients ck for k ranging from 0 to n. Venn diagrams are named for John Venn.
Bibliography
“The Binomial Theorem.” In Math. http://www.intmath.com/series-binomial-theorem/4-binomial-theorem.php.
Chauvenet, William. Binomial Theorem and Logarithms. Self-published: Biblio Bazaar, 2008.
Coolridge, J. L. “The Story of the Binomial Theorem.” American Mathematical Monthly 56, no. 3 (March 1949).
Friedberg, Stephen H. “Applications of the Binomial Theorem.” International Journal of Mathematical Education in Science and Technology 29, no. 3 (1998).
Fulton, C. M. “Classroom Notes: A Simple Proof of the Binomial Theorem.” American Mathematical Monthly 59, no. 4 (1952).