[This is the seventh in a series of posts on the decadic numbers (previous posts: A curiosity, An invitation to a funny number system, What does "close to" mean?, The decadic metric, Infinite decadic numbers, More fun with infinite decadic numbers). I know it's been a while since I've written on this topic, so if you've been following along, you might want to go back and refresh your memory.]
Finally, as promised, I can show you the strange number u which is its own square (but which isn’t zero or one!). Up until now all the decadic numbers we’ve considered have been equivalent to familiar rational numbers, but zero and one are the only rational numbers which are their own square; clearly u must be something quite different!
Assuming that such a u could exist—and assuming it’s a decadic integer, that is, has no digits to the right of the decimal point—let’s think for a minute about what u could possibly be. For example, what could its last digit be?
When we multiply two integers, the last digit of the result depends only on the last digits of the integers being multiplied, since all the other digits contribute some power of 10. So we can narrow down the possibilities for the last digit of any self-square decadic integer by seeing which digits have squares that end in the same digit:
I’ve highlighted the digits with the desired property: of course, and , but also ends in and ends in . We already know we don’t want to consider and . So for now, let’s suppose that ends with .
What about the last two digits of ? Again, the last two digits of depend only on the last two digits of . (If this isn’t obvious to you, you should try a few examples to convince yourself. For example, what is ? What information did you need to compute the last two digits of the answer?) So whatever the last two digits are, their square, when considered on their own as a two-digit number, must be some number that ends in the same two digits. Assuming the last digit is , we can turn this requirement into a modular equation which we can use to solve for the second-to-last digit:
Sure enough, which ends with .
Can we take this further? What about the last three digits?
Check: , which indeed ends with . Continuing in a similar fashion (I’ll let you work out the details on your own), we find that the fourth digit must be , and the fifth digit must be .
Are you seeing a pattern? Let’s make a table of the results so far:
Why did I put some numbers in bold? Well, hopefully you’ve noticed by now that each number in the left-hand column always seems to be a suffix of the square of the previous number. So perhaps the next digit will be 8? Sure enough, ends in .
Will this always work? Yes, in fact, it will, and here’s why. Let’s let denote the last digits of (so , , and so on). Once we have found , we can set up a modular equation to find the next digit (this is just a generalization of what we did earlier):
Now, is clearly divisible by so that term goes away. But what about ? It seems that we only know it is divisible by , not necessarily by . Ah, but wait! We know that ends with , and hence is divisible by ; combining this with the gives us another factor of ! So this term goes away too, and we are left with
Now, (by definition), so subtracting from both sides leaves a multiple of in the place of (namely, with the rightmost digits set to zero). But we can also get rid of all the digits to the left of the st because we are working mod . Dividing by we find that must be equal to that st digit of .
So here’s the procedure: starting with , define
That is, square the current number of length and take the last digits to get the next number. The above proof shows that
- At every step we will have a number whose square ends with the digits of ;
- this procedure will always work; and
- this procedure gives the unique sequence of with this property when starting with !
So we have
and so on.
So what is ? It is simply the limit of carrying out this procedure to infinity!
We know that any suffix of , when squared, yields something ending with itself. So it makes sense (although it takes a bit of imagination!) that squaring itself yields again.