An Interesting Thing About Square Numbers

I was driving along this morning thinking about square numbers and how they look. And then I thought of an interesting "thing" about square numbers. You've probably seen this before but I found it quite nifty.

Anyway. pick any 2 numbers that are 2 digits apart, say 4 and 6. Set them out as an array:

4 x 6 = 24

Now if you take the number in between the 2 numbers you used in the array and square it, you need an extra counter/egg/chess piece to make the array:

5 x 5 = 25

Doesn't matter what you use - could be chess pieces or coloured squares or anything.

Or if you want to get symbolic:

X² = (x-1)(x+1) + 1

That's because...

(x-1)(x+1) = X² - 1

So I start thinking, what if I throw in a few decimals?

Like 4.9 x 5.1? They're both 0.1 from 5 which we are squaring.

Hmmm, it equals 24.99

Or how about 3.9 x 6.1? They are both 0.1 from those original numbers we multiplied, 4 and 6.

Well, that equals 23.79

And what about 4.999999999 x 5.000000001?

And that one equals 24.99999999 - is that a calculator error because I only have a 10 digit display? Or is that the "real" answer? And what does it mean?

And what if I use numbers that are 2 integers higher and lower than the square number?

Like 3 x 7 = 21.

But 3 x 8 = 24

Why is that? And will it work for every square number?

Too many questions, too little time.

Just wait till I get back into the classroom next week. Look out kids....