Binomial approximation

The binomial approximation is useful for approximately calculating <a href="/facts/Exponentiation/pac6QY2g">powers</a> of sums of 1 and a small number x. It states that

(
        1
        +
        x
        
          )
          
            α
          
        
        ≈
        1
        +
        α
        x
        .
      
    
    {\displaystyle (1+x)^{\alpha }\approx 1+\alpha x.}

It is valid when 
 
 
 
 
 |
 
 x
 
 |
 
 <
 1
 
 
 {\displaystyle |x|<1}
 
 and 
 
 
 
 
 |
 
 α
 x
 
 |
 
 ≪
 1
 
 
 {\displaystyle |\alpha x|\ll 1}
 
 where 
 
 
 
 x
 
 
 {\displaystyle x}
 
 and 
 
 
 
 α
 
 
 {\displaystyle \alpha }
 
 may be <a href="/facts/Real_number/R02gw5Pb">real</a> or <a href="/facts/Complex_number/2w7mqI7e">complex numbers</a>.
The benefit of this approximation is that 
 
 
 
 α
 
 
 {\displaystyle \alpha }
 
 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 <a href="/facts/Binomial_theorem/hd4SmkUd">binomial theorem</a>. By <a href="/facts/Bernoulli%27s_inequality/HrVesN8o">Bernoulli's inequality</a>, the left-hand side of the approximation is greater than or equal to the right-hand side whenever 
 
 
 
 x
 >
 −
 1
 
 
 {\displaystyle x>-1}
 
 and 
 
 
 
 α
 ≥
 1
 
 
 {\displaystyle \alpha \geq 1}
 
.

Binomial approximation open-in-new

Binomial approximation