Why is it so good to know that $(1+x)^n \approx 1+nx$ for $nx \ll 1$?

Question

I’ve often heard that it’s good to memorize the fact that $(1+x)^n \approx 1+nx$ for $nx \ll 1$ (most recently here), especially for mental arithmetic or making quick approximations. But why?

Could you post an example or two that illustrate its typical use, and/or an explanation of why applications of this fact occur frequently?

Answer 1

Many of us know $\sqrt 2 \approx 1.414$. This allows you to find square roots of numbers near $2$, so $\sqrt {2.05}=\sqrt{2(1.025)}=\sqrt 2 \cdot \sqrt{1+0.025}\approx 1.414(1+\frac {0.025}2)= 1.414+\frac {1.414}{80}\approx 1.414+.018=1.432$. Maybe you know $9^3=729$ and want $9.1^3=9^3(1+\frac 1{90})^3\approx 9^3(1+\frac 1{30})\approx 729+24.3\approx 753$ It allows you to make small corrections for many facts you know.

I would do the first by saying to myself $2.05$ is $2.5\%$ bigger than $2$, so the square root is $1.25\%$ bigger, which is $\frac 1{80}$ to get to the final calculation.

Answer 2

You use it every day when you say that a yearly pay increase of 2% will give you a 10% increase in 5 years. Inaccurate but not too far from the truth.

Answer 3

People use it for approximating square roots, for example.

$\sqrt{101} = 10\sqrt{1+1/100}$ which is approximately $10 \times (1+1/200) = 10.05 $