The binomial approximation is useful for approximately calculating powers of sums of 1 and a small number x. It states that

may be real or complex numbers.

is converted from an exponent to a multiplicative factor. This can greatly simplify mathematical expressions (as in the example below) and is a common tool in physics.

The approximation can be proven several ways, and is closely related to the binomial theorem. By Bernoulli's inequality.

Derivations

Using linear approximation

The function

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

and so

Thus

By Taylor's theorem that lies between 0 and . In little o notation.

Using Taylor series

The function

may be real or complex can be expressed as a Taylor series about the point zero.

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

starts to approach one, or when evaluating a more complex expression where the first two terms in the Taylor series cancel (see example).

, a better approximation is:

Example

The binomial approximation for the square root, can be applied for the following expression,

.

and recalling that a square root is the same as a power of one half.

which is otherwise not obvious from the original expression.

Generalization

While the binomial approximation is linear, it can be generalized to a quadratic approximation keeping the second term in the Taylor series:

Applied to the square root, it results in:

Quadratic example

Consider the expression:

then the expression unhelpfully simplifies to zero

While the expression is small, it is not exactly zero. So now, keeping the quadratic term:

were kept.

References