Katie, one of our editors, has been contacted by Brendan with a question about some maths he’s been investigating. Read on to find out what he’s discovered, and read Katie’s response.

Dear The Aperiodical,

I’ve noticed an interesting property of numbers, and I wondered if you could tell me if this is something which is already known to mathematicians? I’ve been calling them Dual Inversal Numbers, but I’d love to know if they have an existing name, and if there’s anything else you can tell me about them.

Certain pairs of numbers have the property that if you divide one by each number you get a decimal which contains the other. For example, $\frac{1}{2} = 0.5$, and $\frac{1}{5} = 0.2$. I’ve found several examples of such number pairs, including $\frac{1}{25} = 0.4$ which pairs with $\frac{1}{4} = 0.25$, and  $\frac{1}{16} = 0.0625$ with $\frac{1}{625} = 0.0016$.

I’ve also noticed that some of these pairs involve a decimal which repeats, like $\frac{1}{27} = 0.\overline{037}$ and  $\frac{1}{37} = 0.\overline{027}$. There’s also the exciting $\frac{1}{7} = 0.\overline{142857}$ with $\frac{1}{142857} = 0.\overline{000007}$. There’s even a number which is dual inversal with itself: $\frac{1}{3} = 0.\overline{3}$. For this reason, I call 3 a ‘cannibal’ dual inversal number.

I’ve noticed a couple of interesting facts about these numbers:

• If A & B and C & D are dual inversal pairs, then AC & BD and AD & BC are dual inversal pairs.
• This doesn’t work for all numbers – 6 doesn’t seem to have a pair, and nor do 12, 15 and 18. These are all multiples of three – although I don’t think that’s a rule, as some do work – e.g. $\frac{1}{21} = 0.\overline{047619}$ and $\frac{1}{47619} = 0.\overline{000021}$.
• $\frac{1}{6} = 0.1\overline{6}$, but 16 is already paired with 625.
• The length of the recurring sequence is always the same in a pair – for example, $\frac{1}{17} = 0.\overline{588235294117647}$ and $\frac{1}{588235294117647 } = 0.\overline{0000000000000017}$ – you need all the zeroes to make it the same length.

I’d be grateful if you could tell me if what I’ve found is interesting, or if it already exists as a concept in mathematics. I look forward to hearing from you!
Brendan Bethlehem, aged 9

Dear Brendan,

Thanks very much for your email! I think I’ve spotted what might be going on here, and also have an explanation for why some numbers work and some don’t.

If you look at all the pairs which work without a recurring decimal, such as 2 and 5, or 04 and 25, the product of these numbers is always 10, or 100, or 1000 – a power of ten. Since we use powers of ten as the basis of the way we write down numbers (we have columns for units, tens, hundreds etc) then these numbers behave in this way with fractions and decimals. You can even prove this quite easily: if you call your two numbers a and b, you can write down $\frac{1}{a} = \frac{b}{10}$ (if you work out $\frac{b}{10}$, it is “0.b”) – but you can put any power of 10 here depending on how many digits are in the numbers a and b. Then you can rearrange this equation (multiply both sides by a, then by 10) to get $a \times b = 10$.

The pairs of numbers which work but with a repeated set of digits, like 27 and 37, are more interesting – if you multiply them together you get 9, or 99, or 999, or 999999 (in the case of 1/7). This shows they’re almost two numbers which multiply to a power of ten, but not quite. The reason the fractions work is because the decimal is that number repeated forever – each time you add on another repeat of those digits you get closer to the power of ten, and with infinitely many 9s on the end you find that e.g. 9.999999… (recurring) equals ten, or 99.99999… = 100. This is exactly the same as the way in which 0.9 recurring equals one, as explained in this Numberphile video by James Grime.

This means that the numbers which don’t work are ones which don’t multiply to give powers of ten, or one less than powers of ten. In this case, you won’t get a nice repeating decimal – the difference between something like 1/27 which is 0.037037037… (the repeated section starts from the very beginning) and something like 1/12 which is 0.0833333…, is that (in this case) there’s an 08 before the threes, which never gets repeated as it’s all threes from there. Unless the decimal starts with the thing that repeats, you won’t be able to invert it this way. You’ve noticed that 1/6 gives 0.16666… – but this isn’t the same as 0.16161616…, and that’s why 16 is paired with something else.

If you want to work out which numbers this will work for, you can easily find which numbers divide certain other numbers by looking at their prime factors. The nice thing about powers of 10 is that ten is the product of two primes – 2 and 5. This means that $10 = 2 \times 5$, $100 = 10 \times 10 = 2 \times 5 \times 2 \times 5$, $1000 = 10 \times 10 \times 10 = 2 \times 5 \times 2 \times 5 \times 2 \times 5$ and so on. To find pairs of divisors, you just have to split the twos and fives into two piles. So, for 100, I have two twos and two fives, which I can split as $2 \times 2=4$, $5 \times 5=25$, which is a pair; or I can split it as $2 \times 5=10$ and $2 \times 5=10$, which is also technically a working pair ($\frac{1}{10} = 0.10$) and I can also split it into $2$ and $2 \times 5 \times 5=50$, which is a pair because $\frac{1}{50} = 0.02$ and $\frac{1}{2} = 0.50$ (as you said, you have to look at the right number of decimal places). Does that make sense? How many ways are there to split the six prime factors of 1000 into two groups?

This is a really nice bit of maths you’ve found – I showed it to some mathematicians I know, and they all agreed it’s quite pretty. I’ve not been able to find an existing name for this idea, but I’m sure if any of the Aperiodical’s readers have seen it they’ll get in touch! I also find your first interesting point to be very insightful – can you see why it works, based on what I’ve said above?

Thanks again for getting in touch, and we’d love to hear from you if you have any other mathematical adventures!