7.3 Imaginary concepts -- equality and inequality in complex numbers

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

Definition of equality

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.

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 $a$ and $b$ , exactly one of the following three statements must be true: $a=b$ , $a>b$ , or $a<b$ . 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, $i$ and 1. Which of the following is true?

• $i$ = 1
• $i$ >1
• $i$ <1

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 $x$ coordinate and the imaginary part is the $y$ coordinate. But one point on a graph is neither greater than, nor less than, another point!