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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?

Ben Kovitz
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  • I answered a question asking what constants are useful for mental calculation. This was my list. What is useful depends on what calculation you like to do. Another thing I find useful is doing multiplication from the top down. If I multiply $abc \times de$ I start with $a \times d$, then add in $bd, ae$ one digit to the right and so on. I find it easier to keep track of where I am. You can also stop when you have enough accuracy. – Ross Millikan Dec 21 '17 at 22:16
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    You need to make friends with numbers. When I say know the squares up to $1000$, I don't mean that if somebody asks you for $18^2$ you know it is $324$. I mean that if you see $324$ one of the things that pops into your head is $18^2$. If all of these are at your mental tips you will often find a way to get where you want to go. – Ross Millikan Dec 21 '17 at 22:18
  • It's useful in real or numerical analysis, so you will use tricks like that to show things behave the way you want so you can make statements about their size in relation to something else. Inequalities are powerful creatures. – CluelessIndividual Dec 22 '17 at 02:44

3 Answers3

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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.

Ross Millikan
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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.

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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 $

Higurashi
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