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,

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

A complex number *z* = *x* + *yi* will lie on the unit circle when *x*^{2} + *y*^{2} = 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 *a*^{2} + *b*^{2} = *c*^{2} If you divide this equation by *c*^{2}, 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. (For a
little more on Pythagorean triples, see the end of the page at http://www.clarku.edu/~djoyce/trig/right.html.)

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

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.