Complex numbers: absolute value

Dave's Short Course on

Absolute value

An important concept for numbers, either real or complex is that of absolute value. Recall that the absolute value |x| of a real number x is itself, if it's positive or zero, but if x is negative, then its absolute value |x| is its negation –x, that is, the corresponding positive value. For example, |3| = 3, but |–4| = 4. The absolute value function strips a real number of its sign.

For a complex number z = x + yi, we define the absolute value |z| as being the distance from z to 0 in the complex plane C. This will extend the definition of absolute value for real numbers, since the absolute value |x| of a real number x can be interpreted as the distance from x to 0 on the real number line.

We can find the distance |z| by using the Pythagorean theorem. Consider the right triangle with one vertex at 0, another at z and the third at x on the real axis directly below z (or above z if z happens to be below the real axis). The horizontal side of the triangle has length |x|, the vertical side has length |y|, and the diagonal side has length |z|. Therefore,

|z|² = x² + y².

(Note that for real numbers like x, we can drop absolute value when squaring, since |x|² = x².) That gives us a formula for |z|, namely,

|z| = √(x² + y²)

The unit circle. Some complex numbers have absolute value 1. Of course, 1 is the absolute value of both 1 and –1, but it's also the absolute value of both i and –i since they're both one unit away from 0 on the imaginary axis. The unit circle is the circle of radius 1 centered at 0. It include all complex numbers of absolute value 1, so it has the equation |z| = 1.

A complex number z = x + yi will lie on the unit circle when x² + y² = 1. Some examples, besides 1, –1, i, and –1 are ±√2/2 ± i√2/2, where the pluses and minuses can be taken in any order. They are the four points at the intersections of the diagonal lines y = x and y = x with the unit circle. We'll see them later as square roots of i and –i.

You can find other complex numbers on the unit circle from Pythagorean triples. A Pythagorean triple consists of three whole numbers a, b, and c such that a² + b² = c² If you divide this equation by c², then you find that (a/c)² + (b/c)² = 1. That means that a/c + i b/c is a complex number that lies on the unit circle. The best known Pythagorean triple is 3:4:5. That triple gives us the complex number 3/5 + i 4/5 on the unit circle. Some other Pythagorean triples are 5:12:13, 15:8:17, 7:24:25, 21:20:29, 9:40:41, 35:12:27, and 11:60:61. As you might expect, there are infinitely many of them.

The triangle inequality. There's an important property of complex numbers relating addition to absolute value called the triangle inequality. If z and w are any two complex numbers, then

|z + w| ≤ |z| + |w|

You can see this from the parallelogram rule for addition. Consider the triangle whose vertices are 0, z, and z + w. One side of the triangle, the one from 0 to z + w has length |z + w|. A second side of the triangle, the one from 0 to z, has length |z|. And the third side of the triangle, the one from z to z + w, is parallel and equal to the line from 0 to w, and therefore has length |w|. Now, in any triangle, any one side is less than or equal to the sum of the other two sides, and, therefore, we have the triangle inequality displayed above.

Next section: Multiplication

Previous section: The complex plane, addition and subtraction

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Email: djoyce@clarku.edu

These pages are located at http://aleph0.clarku.edu/~djoyce/complex/

Dave's Short Course on Absolute value

Dave's Short Course on

Absolute value