Do you know how to divide by 7?

I discovered this when I was in school and thought I might have discovered some mathematical secret! 🤓 I came across is because of the common approximation for PI as 22/7, accurate to two decimal places.

It turns out this is a rounding effect of 7 in the base 10 (decimal system) and nothing to do with the actual numbers in any objective sense. When we write the number which results from 22/7, we actually break it down into smaller numbers for the purpose of writing it. With a number which has recurring decimals like this, we can never actually correctly represent the number, as to write in shorter form is an approximation.

The effect you see is dependent on the base, as this sets how numbers are represented as smaller numbers when writing. To demonstrate this to yourself, do the same calculations in another base, 16 for example, or even 7. Note that 22 in base 10 is denoted as 16 in base 16, and 31 in base 7. 7 is also represented as 10 in base 7.

Base 10: 22 / 7 = 3.14285714
Base 16: 16 / 7 = 3.24924924
Base 7: 31 / 10 = 3.1

Here's the relevant point. Those numbers are actually the same number (if you discount rounding error) even though the are written differently. The number itself is independent of the representation.

With this in mind the phenomenal you see is pretty interesting but there's nothing more to it than a feature of rounding in base 10. I think of it now like when you see the wheels on a tire appear to rotate slowly backwards on TV. There is a close matching period between the frame rate of the camera and the speed of the wheels turning. The slight difference gives the "pattern" of the wheels appearing to turn. The base is like the camera, and the number which results from the division like rotations of the wheel.

Thanks for reminding me of this! 😄

Do you know how to divide by 7?

Remember:

14 28 57