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In Algebra "A Number" Doesn't Mean "Any Old Number"

In algebra, a number might be either a variable number or a constant number. Conastant numbers are numbers whose value remains constant -- never changes. Variables are numbers whose value varies -- changes. Five, 1, 3, -2, pi, six, and 39 are constants. P, q, r, x, y, and z are variables. But, when the phrase "a number" is used in algebra, the possibility exists that the number is not a constant. So, "a number" means a variable number.

Of course, one might say "the number one" and mean "one." Or one might say "number five" and mean "the fifth item" or "the fifth question," still referring to a specific number. One might also say, "I'm thinking of a number" and go on to describe a specific number, but when in algebra, when one says "a number," one means a variable number.

Numbers may be represented in different ways. They may be named orally so as to be heard or they may be written in words.

onetwoa number

They may be written in symbols.


Markers or tokens or chips may be used to indicated how many are under discussion.

Numbers may be graphed. The y value for the numbers 1 and 2 are constant. The graphs are horizontal lines because the numbers are constants. But, the y value of "a number" changes as the value of the number changes. Its graph is not a horizontal line.

Expressions may be composed of variable numbers or constant numbers or both. Expressions composed only of constant numbers are constant expressions. Expressions which include even one variable are variable expressions. The graphs of the expressions one more than one, 1 + 1, and one more than two, 2 + 1, are constant expressions. The expression one more than a number, x + 1, is a variable expression. Notice, that the variable expression is not a horizontal line.

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