Doubling Square Numbers

As we saw before, if we multiply two square numbers, their product is a square number.  There has also been some interest in doubling square numbers, especially whether it is possible to find one square number which is double another.

If you are at all unclear about what a square number or its square root is, or what it means for a number to be irrational, this article on square numbers should clear things up for you.  If you have further questions about the vocabulary, or for that matter anything else in this article, please leave them in the comments and I’ll do my best to clear them up for you.

Let’s start with an example.  16 is a square number.  Doubled, it makes 32.  32 doesn’t seem to be a square number because 5 squared would be 25, 6 squared would be 36, and 32 falls into the hole in between them.  You can find this out another way by using a calculator; the square root of 32 is one of those long, messy irrational numbers: 5.656854249492380195206754896838… .

Maybe we got unlucky with this example.  Let’s try another square number, 144 (12×12).  If you double 144, you get 288.  I don’t happen to know right off what the square numbers closest to 288 are, but my calculator tells me that its square root is 16.97056274847714058562026469051… .  Yuck.  (Note: Now that I know the square root is between 16 and 17, I can tell that the nearest squares are 16×16, or 256, and 17×17, or 289.)  Looks like 288 isn’t a square number either.

Looks to me like doubling a square number is not a good way to find another square number.  In fact, you can never double a square number to get another square number.  Here’s why.

Let’s name our square number n².  Remember that this means n·n, so n is n²’s square root.  If you double n², you get 2n², which is the same as 2·n·n.

You might remember the trick that let us split the factors, toward the end of the article on products of square numbers, in order to show that they made a square number.  It would really be nice to be able to do that to 2·n·n.  The problem with trying to use that trick for this is that we only have one 2.

Let’s try some possibilities.  If we say that the square root might be 2·n, we should square that to check and see if it squares to 2n².  Squaring, we get (2·n)² which is 2·n·2·n which is 2·2·n·n, or 4n².  Not what we were looking for.

Maybe leaving the 2 out would work better.  We could say the square root might be n.  Squaring, we would get n², which is still not the 2n² we’re looking for.

Using the whole 2 or leaving it out entirely isn’t working.  Maybe using half would work better; let’s give ½·n a try.  (½·n)²=½·n·½·n=¼·n².  Definitely still not 2n².

You might be able to see what’s going on here.  When we threw in the 2, it got squared to 4.  When we threw in ½, it got squared to ¼.  What we really need is something that would get squared to 2 — in other words, the square root of 2.  But 2 is not a square number, so its square root is something irrational and messy that we call 1.414213562373095048801688724209… for short.  That means the square root of 2n² is 1.414213562373095048801688724209…·n.

Since square numbers are numbers that have a whole number for a square root, and the square root of 2n² is definitely not a whole number, we can now be sure that it’s impossible to double one square number and get another.