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
(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....