Mentally multiply by π

This post will give three ways to multiply by π taken from [1].

Simplest approach

Here’s a very simple observation about π :

π ≈ 3 + 0.14 + 0.0014.

So if you need to multiply by π, you need to multiply by 3 and by 14. Once you’ve multiplied by 14 once, you can reuse your work.

For example, to compute 4π, you’d compute 4 × 3 = 12 and 4 × 14 = 56. Then

4π ≈ 12 + 0.56 + 0.0056 = 12.5656.

The correct value is 12.56637… and so the error is .00077.

First refinement

Now of course π = 3.14159… and so the approximation above is wrong in the fourth decimal place. But you can squeeze out a little more accuracy with the observation

π ≈ 3 + 0.14 + 0.0014 + 0.00014 = 3.14154.

Now if we redo our calculation of 4π we get

4π ≈ 12 + 0.56 + 0.0056 + 0.00056 = 12.56616.

Now now our error is .00021, which is 3.6 times smaller.

Second refinement

The approximation above is based on an underestimate of π. We can improve it a bit by adding half of our last term, based on

π ≈ 3 + 0.14 + 0.0014 + 0.00014 + 0.00014/2 = 3.14157

So in our running example,

4π ≈ 12 + 0.56 + 0.0056 + 0.00056 + 00028 = 12.5656 = 12.56654.

which has an error of 0.00007, which is three times smaller than above.

Related posts

• Mentally compute common functions
• Mentally approximating factorials
• Mentally calculating the day of the week

[1] Trevor Lipscombe. Mental mathematics for multiples of π. The Mathematical Gazette, Vol. 97, No. 538 (March 2013), pp. 167–169

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