## Counting by division

By David Shaw, 5 December 2016

A calculator is very handy for this activity

Let’s look at a few interesting patterns you can get by dividing. Some of these patterns need a lot of digits, so get the best calculator you can find.

If you have a computer with the internet handy, there are some good online calculators to use, such as wolframalpha.com. Some mathematicians might prefer to use long division with paper and a pen.

### You will need

- Calculator
- Paper and pen (optional)

### What to do

If you’re using a computer calculator, remember that ‘/’ means division.

- Let’s start with a relatively simple calculation. What do you think you’ll get if you calculate 1÷9? Key 1÷9 into your calculator (or 1/9 on a computer).
- Keeping the answer in your calculator, what would happen if you divided by 9 again? Try it and see!
- Let’s try to extend the pattern. What do you think you’ll get when you calculate 1÷99?
To really understand what’s going on, you can use long division.

- Think long and hard before this one – what do you think you will get when you divide by 99 again?
- Final question, and this one’s tricky. What will you get when you calculate 1÷998001? Can you see why this works?

### Answers

- 111111111111…
- 01234567901234567…
- 0101010101010101…
- 000102030405060708091011…
- 000001002003004005006007008…

### What’s happening?

Amazingly, the division begins to count!

Maths is all about patterns, and this activity shows a really neat one. Dividing by 9 and then 9 again gives an answer that looks a lot like counting! You can extend the pattern by using more 9s – dividing by 99 twice. The final twist is combining the two divisions. 998001 = 999 x 999, which is why the final calculation counts too.

The pattern arises because the number 9 is one less than 10. Or to put it another way, when you divide 10 by 9, you get a remainder of 1. When you move to the next place value, that 1 becomes a 10 and the process repeats. That’s why you get 0.111111…

The second division reveals another trick of the number 9. When you divide 20 by 9, you get a remainder of 2. When you divide 30 by 9, you get a remainder of 3. This process will keep ticking up until eventually it gets to 90, which has no remainder.

It’s a bit hard to understand just by reading. The best way to see the pattern is by doing long division to calculate 1 ÷ 9 and 0.1111… ÷ 9 by hand. The patterns you find will help explain what happens when you divide by 99 999.

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