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Categories of Symbols

Let's examine one of the intermediate equations above:

$$2x-7=15$$

There are two kinds of symbols here.

1. Quantities. A quantity is a number or a variable that represents a number. 2, x, 7, and 15 are all quantities.
2. Operations, which are symbols that represent a mathematical operation. The only one in this expression is +; symbols like -, $\times$ (for multiplication), and $\div$ are also in this category.^3.1

Operations have an interesting property: they need quantities in order to exist. We can have expressions like

$$4x+9$$

but not expressions like

$$4x+\,$$

If the 9 is ``taken away'' because, say, we subtract 9 from both sides of an equation, the expression simply becomes 4x.

Symbol motions can be classified as being in the addition/subtraction class or the multiplication/division class. The classes are named this way because of what their symbol motions do to the equation. Symbol motions in the addition/subtraction class have the same effect on an equation as if you add or subtract a term from each side; and those in the multiplication/division class have the same effect as if you multiplied or divided each side by something. Symbol motions within each category are very similar to each other. Let's look each class in detail.

Footnotes

...^3.1

The equal sign would be in a third category, called relations. For now, we are just concerned with the first two categories.