Geometrically demonstration for

algebraic identity

By: Ahmad Ghandehari

In the first place I have to define algebraic identity.

Definition:

If in two algebraic expressions which contain letters, for all real numbers which settle instead letters, they are equal to each other then, this equality called algebraic identity.

In this essay I am going to prove all algebraic identity by geometrically demonstration.

1.      Geometrically demonstration for.

We draw a square with length  as the figure.

The area of original square is, and the sum of inside area are  so

2.      Geometrically demonstration for .

We draw a square with length a as below.

The area of original square is , and the sum of inside area are  so

.

3.      Geometrically demonstration for .

We draw a rectangle with length of  and width of .

The area of original rectangle is , and the sum of inside area are  so

4.      Geometrically demonstration for .

We draw a square with side of .

The area of original square is A , and the sum of inside area are

 so

5.      Geometrically demonstration for.

We draw a rectangle with dimensions of  and, as figure.

The area of original rectangle is A , and the sum of inside area are  so

.

6.      Geometrically demonstration for .

We make a cube with edge of  .

In the inside of this cube there are a cube with edge “a” and a cube with edge “b” and there cuboids with dimensions of  , a and b.

We know that it is hard to imagine, for this purpose in the end of this proof we will give you a model that you will be able to make it and understand the proof .

With considerate that the volume of a cube with edge of x, equal to , and the volume of a cuboids with dimensions of x, y and z is x.y.z .

Therefore the volume of original cube is and the sum of inside volumes is  so .

 The model of this case is below.

7.      Geometrically demonstration for .

We make a cube with edge of  “a”  and make a cube with edge of  in inside of original cube.

In the inside of original cube there are a cube with edge  and a cube with edge b and there cuboids with dimensions of  , a and b.

With considerate to the explanations in article (6) we can say that the volume of original cube is , and the sum of volumes of inside is , therefore  or

.

The model of this case is the next page.