Consider the following divisions:
- 15 ÷ 7 = 2 1/7 = 2.142857…
- 50 ÷ 7 = 7 1/7 = 7.142857…
- 162 ÷ 7 = 23 1/7 = 23.142857…
In each case, the fractional part is the same: 1/7, and therefore the decimals will also be the same […142857…]
Some of these fractions (like 1/7, 2/5 and 5/6) appear frequently, so it is worthwhile to memorize their decimals. This is easier than it might sound!
Our goal will be to memorize the decimals for all fractions up to 11/12. This allows us to divide by any number up to 12, with perfect precision. (You could also extend this skill later to divide by other numbers such as 14, 15 and 22, without having to memorize anything else!)
This page explains some of the Mathematics that will help you notice the patterns in these decimals, and memorize them more easily.
Division by 2, 3, 4, 5 and 10
Many people know these already.
Division by 9
I’m starting with division by 9, because it’s the easiest. I’ll also show you the Mathematics:
First, notice that 0.999999… = 1 because the difference 1 – 0.999999… would be 0.000000…, which is zero. if there is no difference between two quantities, then they are equal.
Secondly, notice that 9/9 = 1, and combine this with the previous fact to conclude: 9/9 = 0.999999…
By dividing each digit by 9, we get: 1/9 = 0.111111…
We can then multiply by any number 2–8 to get e.g. 4/9 = 0.444444…
So, to divide a digit from 1–8 by 9, you just repeat that digit forever!
Notice that 3/9 = 0.333333…, because it is the same as 1/3. Does 6/9 also seem familiar…?
Division by 11
Division by 11 is similar to division by 9.
Similar to before, notice that 11/11 = 1 = 0.99 99 99…
By dividing each pair of digits by 11, we get: 1/11 = 0.09 09 09…
We can then multiply the 09 by any number 2–10 to get e.g. 4/11 = 0.36 36 36…
So, to divide anything from 1–10 by 11, you multiply it by 9 (e.g. 4 × 9 = 36) and repeat the resulting 2-digit number forever.
Division by 8
Division by 8 is related to division by 4.
Division by 6 and by 12
I’m grouping these together because they are both related to division by 3.
Division by 7
Finally, the most complex one. But if you remember the magic string of digits “142857”, then you can memorize these very easily.