## Square minus square

Sometimes the connection between the mathematics and the real
world is hard to find. In schools we study formulas and equations
which are proved by other formulas and equations, and lemmas, and
theorems. We try to remember them but with less success. Students
often ask themselves Why are we learning this, there is no use of
it. After many years, some people say Oh, I see now why we have
studied that in school. Other people never say it. So, here Ill prove
2 equations in both the mathematics and the real world way.

**How is in mathematics**

In mathematics we can meet these equations a^2 b^2 = (a b)
(a + b) or a^2 1 = (a 1) (a + 1) and these formulas are very easy
to prove with simple mathematic operations.

a^2 b^2 <=>

a^2 b^2 + ab ab <=>

a(a b) + b(a b) <=>

(a + b) (a b) <=>

**How is in real world**

This is what we use to learn in school but somehow the
mathematic hides the real world behind. This can cause some trouble
when we have a real world situation to solve. We know what we have to
solve, but we dont know that this equation will help us in that case.

To see how this works with real world objects we can just use
real world objects, instead of the formulas.

Lets try it with apples.

We can put apples in 4 rows and 4 columns so they will form a
square.

Then we can remove one apple from the upper right corner.

So, the upper row will be with 3 apples and the most right
column will be with 3 apples too.

We can take these 3 apples from the upper row and put them at
right one under the other. In these case we will have one row less and
one column more. But also we will have all apples forming rectangle.

**Real world situation**

So, if you make an apple party and you have 16 apples in
beautiful square. And you want to eat one apple very desperately
before the others come. You dont have to worry about the decoration.
You can eat one apple from the upper right corner and than move the
apples to the right. And you will have a nice rectangle and nice
party, and the first apple in your stomach.

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