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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.

1. 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).
2. Keeping the answer in your calculator, what would happen if you divided by 9 again? Try it and see!
3. 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.

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

Answers

1. 111111111111…
2. 01234567901234567…
3. 0101010101010101…
4. 000102030405060708091011…
5. 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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