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 I’ll 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 don’t 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.

Let’s 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 don’t 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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