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© Copyright 1999, Jim Loy
Some people might guess that (x+y)^2=x^2+y^2. But, if you are reading this article, I suspect that you know that:
(x+y)^2 = x^2+2xy+y^2
How do we know that? Let's see:
(x+y)^2 = (x+y)(x+y)
= x(x+y) + y(x+y) {distributive law)
= x^2+xy + xy+y^2
= x^2+2xy+y^2
Not bad. You might want to verify, for yourself that the second step is an application of the distributive law, which says a(x+y)=ax+ay.
There is a way to make this more obvious. To the left
is a square. Each side is x+y. What is the area? Well, it is (x+y)^2. But, we
can also find the area, by summing up the areas of the four rectangles (two of
which are squares) which make up our big square:
area = x^2+xy+xy+y^2
= x^2+2xy+y^2
Beginners should find that easy to understand.
Here is a magic trick. Use a calculator with square roots, for this:
The answer may be .999999..., if your number was not an integer, because of round off error.
The trick involved building up x^2+2x+1, which we know is (x+1)^2. So, then we can take the square root, and subtract x, to get 1.
When I was in school, I occasionally (only four or five times) was marked off for writing things like .25, when I should have written 0.25. 0.25 is a little clearer, as you may not notice the decimal point with .25. But, many many math books leave out the leading "0".
I once overheard an argument. One person said that .1 was 1/10. The other person said that .1 was 10%. Of course, they were both right.