The story of each individual’s math education can be represented, in stages, by the sets of numbers which he or she has used as a universe of discourse. A universe of discourse is best thought of, in math terms, as the largest set of numbers being talked about — i.e., the set of numbers from which all solutions to problems and equations must be drawn.

In the beginning, we usually have only the counting numbers or the whole numbers, and if a number isn’t in one of those sets it doesn’t exist. That’s why, for example, a 1st-grader will solemnly assure us that, “5 doesn’t go into 19” and, “You can’t take 5 from 3!”

Once the rational numbers, including negative integers, have been revealed, many new problems become solvable. The process continues until the entire set of real numbers is on the table, including the somewhat mysterious irrationals. This is a “map” of the real numbers:

The next, and last, big step is the introduction of the imaginary numbers, when the square root of -1 is defined to be the imaginary unit, i. When the imaginaries are added to the reals, we have the set (or, more properly speaking, the field) of complex numbers. Any number of the form a + bi, where a and b are real numbers, is a complex number. If b happens to be 0, then a + bi is the real number, a. If b is not 0, then a + bi is an imaginary number.

The complete picture looks like this:

Sometimes there is confusion on this last point: Some contend that, for example, 2 + 3i is “complex,” while 3i is not. This is incorrect: 2 + 3i and 3i are imaginary numbers (try finding them on a number line!), but both are complex, because both can be written in the form a + bi.

larry-headshotAbout the Author
A former math teacher in Georgia, Larry Coty is now USATestprep’s Math Content Team Leader. He has two daughters and resides in Tucker, GA.