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An Introduction to Algebra

Algebra is about working with numbers; adding them, subtracting them,
multiplying them and dividing them. Usually algebra deals with things
that are true whatever the numbers may be.

For example, suppose be have two numbers. Call them a and b. We can
find a line that is a units long and another line that is b units long. If we
join the line for b to the end of the line for a the resulting line is a+b
units long, as is shown below with the lines expanded into rectangles for
visibility.

Now consider two numbers; call them a and c. The product of a and c,
axc, is the area of a rectangle of length a and height c, as shown below.

It is clear from flipping the rectangle that axc is equal to cxa. If side
c is the same as side a then the rectangle is a square. Therefore axa is
called a-squared and it is shown as a^{2}.

Suppose we have two numbers a and b and two numbers
c and d. Now consider what happens if a and b together and c and d together
and then multiply (a+b) times (c+d). The result is shown below.

The diagram shows visually that (a+b)x(c+d) is equal to axc+axd+bxc+bxd.
This can also be shown by algebraic manipulation. The quantity (a+b)x(c+d)
is equal to ax(c+d) + bx(c+d). The quantity ax(c+d) is equal to axc+axd.
Likewise bx(c+d) is equal to bxc+bxd. Therefore

For the special case of (a+b)x(a+b), which is written also written
as (a+b)^{2}, the formula reduces to

(a+b)^{2} = a^{2} + 2axb + b^{2}

This shown below:

Take the special case when a=7 and b=2. In this case (a+b)=9. Nine squared,
9^{2}, is 81. The above algebra says that 81 is the same as
7^{2} + 2x(7x2) + 2^{2}, which is 49 + 2x14 + 4 or
49+28+4 which is, in fact, equal to 81.

A special case is when b=1. Let us use n as the length of the side of the square.
Then the formula says that

(n+1)^{2} = n^{2} + 2xn + 1^{2} since 1^{2} is just 1 and 2xn can be written as just 2n
this reduces to
(n+1)^{2} = n^{2} + 2n + 1.

Suppose n=4. The formula says that 5^{2} should be 4^{2}=16 plus 2 times 4
plus 1. And indeed 16+8+1 is 25, which is 5^{2}. The diagram below shows the
formula visually.