### What Is Algebra?

Consider the following arithmetic calculations

(53)(5+3)=2×8=16=25–9=5232

(81)(8+1)=7×9=63=64–1=8212

(123)(12+3)=9×15=135=144–9=12232

Here we see a pattern. To achieve generality, we may use letters to represent unspecified numbers, and write

$\dpi{120}&space;\large&space;({\color{Blue}&space;a}-{\color{Blue}&space;b})({\color{Blue}&space;a}+{\color{Blue}&space;b})={\color{Blue}&space;a}^2-{\color{Blue}&space;b}^2$

While arithmetic deals with calculations of specified numbers, in algebra and calculus to express universal facts, we often use letters to denote numbers in general, not particular numbers.

### Constants and Variables

If the letter represents a specific number that does not change during a problem, it is called a constant. But if the letter is allowed to represent different numbers during a single problem, it is called a variable. Of course, numbers like $2,-7,\pi,\sqrt{3}$ are also constants.

In equations, the unknown is also called the variable. For example, in the equation $3x+2=5$, the letter $x$ is called the variable although the equation implies that $x$ can take only one value $x=1$. But in an equation like $y=2x+3$, $x$ and $y$ represent infinitely many numbers. It is easier to always call the unknown a variable although it sometimes represents a single value.

If in a single discussion, both constants and variables appear, constants are usually denoted by the first letters of the alphabet as $a,b,c,…$ and variables often by the last letters of the alphabet as $x,y,z$. But this is not a hard-and-fast rule, and for example, the following statements have the same meaning:

$(a-b)(a+b)=a^{2}-b^{2},\quad\text{and}\quad (x-y)(x+y)=x^2-y^2.$

The fact that a letter is a constant or a variable should be easily understood from the context of the problem.