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Binomial theorem

Resources

The binomial theorem provides a simple method for determining the coefficients of each term in the series expansion of a binomial with the general form (A + B)n. A series expansion or Taylor series is a sum of terms, possibly an infinite number of terms, that equals a simpler function. The expansion of (A + B)ngiven by the binomial theorem contains only n terms. Its generalized form (where n may be a complex number) was discovered by Isaac Newton.

The binomial theorem has been used extensively in the areas of probability and statistics. The main argument in this theorem is the use of the combination formula to calculate the desired coefficients.

The question of expanding an equation with two unknown variables called a binomial was posed early in the history of mathematics. One solution (for real n), known as Pascals triangle, was determined in China as early as the thirteenth century by the mathematician Yang Hui. His solution was independently discovered in Europe 300 years later by Blaise Pascal (16231662), whose name has been permanently associated with it since. The binomial theorem, a simpler and more efficient solution to the problem, was first suggested by Isaac Newton (16421727). He developed the theorem as an undergraduate at Cambridge and first published it in a letter written for Gottfried Leibniz (16461716), a German mathematician.

Expanding an expression like (A + B)n just means multiplying it out. By using standard algebra, the equation (A + B)2, for example, can be expanded into the form A2 + 2AB + B2. Similarly, (A + B)4can be written A4 + 4A3B + 6A2B2 + 4AB3 + B4. Notice that the terms for A and B follow the general pattern AnB0, An-1B1, An-2B2, An-3B3, A1Bn-1, A0Bn. Also observe that as the value of n increases, the number of terms increases. This makes finding the coefficients for individual terms in an equation with a large n value tedious. For instance, it would be cumbersome to find the coefficient for the term A4B3 in the expansion of (A + B)7 if we used this algebraic approach. The inconvenience of this method led to the development of other solutions for the problem of expanding a binomial.

One solution, known as Pascals triangle, uses an array of numbers (shown below) to determine the coefficients of each term.

This triangle of numbers is created by following a simple rule of addition. Numbers in one row are equal to the sum of two numbers in the row directly above it. In the fifth row, the second term, 4, is equal to the sum of the two numbers above it, namely 3+1. Each row represents the terms for the expansion of the binomial on the left. For example, the terms for (A+B)3 are A3+3A2B+3AB2+B3. Obviously, the coefficient for the terms A3 and B3 is 1. Pascals triangle works more efficiently than the algebraic approach, however, it also becomes tedious to create this triangle for binomials with a large n value.

The binomial theorem provides an easier and more efficient method for expanding binomials that have large n. Using this theorem, the coefficients for each term are found with the combination formula. The combination formula is

The notation n! is read n factorial and means multiplying n by every positive whole integer that is smaller than itself. Thus, 4! is equal to 4 × 3 × 2 × 1=24. Applying the combination formula to a binomial expansion (A + B)n, n is the power to which the formula is expanded, and r is the power of B in each term. For example, for the term A4B3 in the expansion of (A + B)7, n is 7 and r is 3. By substituting these values

KEY TERMS

Binomial An equation consisting of two unknowns such as (A + B).

Coefficient A number that is multiplied by terms in an algebraic equation.

Expansion Multiplying out terms in an equation.

Factorial An operation represented by the symbol !. The term n! is equal to multiplying n by all of the positive whole number integers that are less than it.

Pascals triangle An array of integers that represents the expansion of a binomial equation.

into the combination formula we get 7!/(3! × 4!) = 35, which is the coefficient for this term. The complete binomial theorem can be stated as follows:

(A + B)n = ΣnCrAnrBr

Resources

BOOKS

Larson, Ron. Precalculus. 7th ed. New York: Houghton Mifflin College, 2006.

Mendenhall, William, et al. Introduction to Probability and Statistics. Pacific Grove, CA: Duxbury Press, 2005.

Perry Romanowski

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Binomial (Linnaean System)

Despite the overwhelming diversity of life that exists (and once existed) on this planet, it is clear that some organisms are more similar to each other than to others. Thus, organisms can be assigned to groups based on their overall similarity to other organisms. For example, humans belong to the group "mammals" as do all other organisms that possess mammary glands and hair. The grouping of organisms provides a convenient means of classification; that is, an organism can be described by the groups to which it belongs.

The classification system that is used today is called the Linnaean System after its inventor, the Swedish naturalist Carolus Linnaeus (1707-1778). In his 1758 book, Systema Naturae, Linnaeus categorized all organisms into seven hierarchical groupings arranged from most inclusive to least inclusive. They are kingdom, phylum, class, order, family, genus, and species. Humans belong to the kingdom Animalia, the phylum Chordata, the class Mammalia, the order Primates, and the family Hominidae, and have been given the generic name (genus) Homo and the specific name (species) sapiens. The Linnaean System is hierarchical because there may be many species per genus, many genera (plural of genus) per family, and so on.

Because specific names are not unique (i.e., there may exist a plant with the specific name sapiens), the name of a species always includes both the generic name and the specific name, for example, Homo sapiens. This method of giving every species a unique combination of two names is called "binomial nomenclature," and is part of Linnaeus's classification system. By convention, these scientific names for organisms, as opposed to the common names, are always italicized. Furthermore, the generic name is capitalized while the specific name is not. Biologists prefer scientific names to common names because of their uniqueness, stability, and universality. Common names, on the other hand, often refer to more than one species and vary over time and from place to place. Biologists follow a certain Code of Nomenclature when deciding what to name a newly discovered species.

The practice of naming and classifying organisms is termed " taxonomy . " Linnaeus classified organisms mainly by their physical (morphological) characteristics. He believed that his groups held theological significance, that is, that they revealed God's plan in creating life. However, with the recognition that species evolve, which led to Charles Darwin's On the Origin of Species in 1859, it became apparent that Linnaeus's classification system held biological significance as well. Organisms that are morphologically similar and consequently grouped together are usually similar because they share a common ancestry. The Linnaean System thus reflects evolutionary relationships among organisms. For example, humans are grouped with gorillas and chimpanzees in the order Primates because we are more closely related to gorillas and chimpanzees then we are to other mammals. Likewise, Primates are grouped with Rodentia in the class Mammalia because primates and rodents are more closely related to each other than they are to other organisms in the phylum Chordata, such as reptiles and fish.

Todd A. Schlenke

Bibliography

Darwin, Charles. On the Origin of Species by Means of Natural Selection. London: John Murray, 1859. Facsimile edition reprinted Cambridge, MA: Harvard University Press, 1975.

Jeffrey, Charles. An Introduction to Plant Taxonomy. Cambridge, U.K.: Cambridge University Press, 1982.

Linnaeus, Carolus. Systema Naturae, 10th ed. Stockholm, Sweden: Laurentius Salvius, 1758. Reproduced New York: Stechert-Hafner Service Agency, 1964.

Schuh, Randall T. Biological Systematics: Principles and Applications. Ithaca, NY: Cornell University Press, 2000.

Simpson, George Gaylord. Principles of Animal Taxonomy. New York: Columbia University Press, 1961.

Internet Resources

Maddison, David R., and Wayne P. Maddison. The Tree of Life. <http://phylogeny.arizona.edu/tree/phylogeny.html>.

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Binomial theorem

The binomial theorem provides a simple method for determining the coefficients of each term in the expansion of a binomial with the general equation (A + B)n. Developed by Isaac Newton, this theorem has been used extensively in the areas of probability and statistics . The main argument in this theorem is the use of the combination formula to calculate the desired coefficients.

The question of expanding an equation with two unknown variables called a binomial was posed early in the history of mathematics . One solution, known as Pascal's triangle , was determined in China as early as the thirteenth century by the mathematician Yang Hui. His solution was independently discovered in Europe 300 years later by Blaise Pascal whose name has been permanently associated with it since. The binomial theorem, a simpler and more efficient solution to the problem, was first suggested by Isaac Newton. He developed the theorem as an undergraduate at Cambridge and first published it in a letter written for Gottfried Leibniz, a German mathematician.

Expanding an equation like (A + B)n just means multiplying it out. By using standard algebra the equation (A + B)2 can be expanded into the form A2 + 2AB + B2. Similarly, (A + B)4can be written A4 + 4A3B + 6A2 B2 + 4AB3 + B4. Notice that the terms for A and B follow the general pattern AnB0,An-1B1,An-2B2,An-3B3,...,A1Bn-1, A0Bn. Also observe that as the value of n increases, the number of terms increases. This makes finding the coefficients for individual terms in an equation with a large n value tedious. For instance, it would be cumbersome to find the coefficient for the term A4B3 in the expansion of (A + B)7 if we used this algebraic approach. The inconvenience of this method led to the development of other solutions for the problem of expanding a binomial.

One solution, known as Pascal's triangle, uses an array of numbers (shown below) to determine the coefficients of each term.

This triangle of numbers is created by following a simple rule of addition . Numbers in one row are equal to the sum of two numbers in the row directly above it. In the fifth row the second term, 4 is equal to the sum of the two numbers above it, namely 3 + 1. Each row represents the terms for the expansion of the binomial on the left. For example, the terms for (A+B)3 are A3 + 3A2B + 3AB2 + B3. Obviously, the coefficient for the terms A3 and B 3 is 1. Pascal's triangle works more efficiently than the algebraic approach, however, it also becomes tedious to create this triangle for binomials with a large n value.

The binomial theorem provides an easier and more efficient method for expanding binomials which have large n values. Using this theorem the coefficients for each term are found with the combination formula. The combination formula is

The notation n! is read "n factorial" and means multiplying n by every positive whole integer which is smaller than it. So, 4! would be equal to 4 × 3 × 2 × 1 = 24. Applying the combination formula to a binomial expansion (A + B)n, n represents the power to which the formula is expanded, and r represents the power of B in each term. For example, for the term A4B3 in the expansion of (A + B)7, n is equal to 7 and r is equal to 3. By substituting these values into the combination formula we get 7! / (3! × 4!) = 35, which is the coefficient for this term. The complete binomial theorem can be stated as the following:

Resources

books

Dunham, William. Journey Through Genius. New York: John Wiley & Sons, 1990.

Eves, Howard Whitley. Foundations and Fundamental Concepts of Mathematics. NewYork: Dover, 1997.

Larson, Ron. Precalculus. 5th ed. New York: Houghton Mifflin College, 2000.

Perry Romanowski

KEY TERMS

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Binomial

—An equation consisting of two unknowns such as (A + B).

Coefficient

—A number that is multiplied by terms in an algebraic equation.

Expansion

—Multiplying out terms in an equation.

Factorial

—An operation represented by the symbol "!". The term n! is equal to multiplying n by all of the positive whole number integers that are less than it.

Pascal's triangle

—An array of integers that represents the expansion of a binomial equation.

binomial

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bi·no·mi·al / bīˈnōmēəl/ • n. 1. Math. an algebraic expression of the sum or the difference of two terms.2. a two-part name, esp. the Latin name of a species of living organism.• adj. 1. Math. consisting of two terms. ∎  of or relating to a binomial or to the binomial theorem.2. having or using two names, used esp. of the Latin name of a species of living organism.

binomial theorem

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binomial theorem Mathematical rule for expanding (as a series) an algebraic expression of the form (x + y)n, where x and y are numerical quantities and n is a positive integer. For n = 2, its expansion is given by (x + y)2 = x2 + 2 xy + y2.

binomial

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binomial (math.) having two terms. XVI. f. F. binôme or modL. binōmius, of uncert. orig. (see BI-, -IAL).

binomial

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