1.1 What Is Algebra?

What Is Algebra?

Consider the following arithmetic calculations

(5–3)(5+3)=2×8=16=25–9=5² – 3²

(8–1)(8+1)=7×9=63=64–1=8² – 1²

(12–3)(12+3)=9×15=135=144–9=12² – 3²

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

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 $a,b,c,\dots$ 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:

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