Binomial approximation

From Wikipedia, the free encyclopedia
Jump to: navigation, search

The binomial approximation is useful for approximately calculating powers of sums of a small number and 1. It states that if and where and are real or complex numbers, then

If either or are complex then the absolute value denotes the modulus of the complex number.

The benefit of this approximation is that is converted from a power to multiplicative factor. This can greatly simplify mathematical expressions (see example) and is a common tool in physics.[1]

This approximation can be proven several ways including the binomial theorem and ignoring the terms beyond the first two.

By Bernoulli's inequality, the left-hand side of this relation is greater than or equal to the right-hand side whenever and .

Derivation using linear approximation[edit]

The function

is a smooth function for x near 0. Thus, standard linear approximation tools from calculus apply: one has

and so

Thus

Derivation using Taylor Series[edit]

The function

where and may be real or complex can be expressed as a Taylor Series about the point zero.

If and , then the terms in the series become progressively smaller and it can be truncated to

.

This result from the binomial approximation can always be improved by keeping additional terms from the Taylor Series above. This is especially important when starts to approach one, or when evaluating a more complex expression where the first two terms in the Taylor Series cancel (see example).

Sometimes it is wrongly claimed that is a sufficient condition for the binomial approximation. A simple counterexample is to let and . In this case but the binomial approximation yields .

Example simplification[edit]

Consider the following expression where and are real but .

The mathematical form for the binomial approximation can be recovered by factoring out the large term and recalling that a square root is the same as a power of one half.

Evidently the expression is linear in when which is otherwise not obvious from the original expression.

Example keeping the quadratic term[edit]

Consider the expression:

where and . If only the linear term from the binomial approximation is kept then the expression unhelpfully simplifies to zero

.

While the expression is small, it is not exactly zero. It is possible to extract a nonzero approximate solution by keeping the quadratic term in the Taylor Series, i.e. so now,

This result is quadratic in which is why it did not appear when only the linear in terms in were kept.

References[edit]

  1. ^ For example calculating the multipole expansion. Griffiths, D. (1999). Introduction to Electrodynamics (Third ed.). Pearson Education, Inc. pp. 146–148.