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Imaginary Concepts -- Equality and Inequality in Complex Numbers

Module by: Kenny Felder

Summary: This module introduces the concept of equality and inequality in complex numbers.

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What does it mean for two complex numbers to be equal? As always, equality asserts that two things are exactly the same. 7+3i7+3i is not equal to 7, or to 3i3i, or to 7-3i7-3i, or to 3+7i3+7i. It is not equal to anything except 7+3i7+3i.

Definition of Equality

Two complex numbers are equal to each other only if their real parts are equal, and their imaginary parts are equal.

So if we say that two complex numbers equal each other, we are actually making two separate, independent statements. We can use this, for instance, to solve for two separate variables.

Table 1
Example: Complex Equality
If 3x+4yi+7=4x+8i3x+4yi+7=4x+8i, what are xx and yy?
Normally, it is impossible to solve one equation for two unknowns. But this is really two separate equations!
Real part on the left = real part on the right: 3x+7=4x3x+7=4x
Imaginary part on the left = imaginary part on the right: 4y=84y=8
We can now solve both of these equations trivially. x=7x=7, y=2y=2

And what about inequalities? The answer may surprise you: there are no inequalities with complex numbers, at least not in the form we’re seeing.

The real numbers have the property that for any two real numbers aa and bb, exactly one of the following three statements must be true: a=ba=b, a>ba>b, or a<ba<b. This is one of those properties that seems almost too obvious to bother with. But it becomes more interesting when you realize that the complex numbers do not have that property. Consider two simple numbers, ii and 1. Which of the following is true?

  • ii = 1 graphics1.png
  • ii > 1 graphics2.png
  • ii < 1 graphics3.png

None of them is true. It is not generally possible to describe two complex numbers as being “greater than” or “less than” each other.

Visually, this corresponds to the fact that all the real numbers can be laid out on a number line: “greater than” means “to the right of” and so on. The complex numbers cannot be laid out on a number line. They are sometimes pictured on a 2-dimensional graph, where the real part is the xx coordinate and the imaginary part is the yy coordinate. But one point on a graph is neither greater than, nor less than, another point!

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