![]() ![]() A second side of the triangle, the one from 0 to z, has length | z|. One side of the triangle, the one from 0 to z + w has length | z + w|. Consider the triangle whose vertices are 0, z, and z + w. You can see this from the parallelogram rule for addition. If z and w are any two complex numbers, then There's an important property of complex numbers relating addition to absolute value called the triangle inequality. ![]() ![]() Little more on Pythagorean triples, see the end of the page at. As you might expect, there are infinitely many of them. That triple gives us the complex number 3/5 + i 4/5 on the unit circle. The best known Pythagorean triple is 3:4:5. That means that a/c + i b/c is a complex number that lies on the unit circle. 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 ![]() You can find other complex numbers on the unit circle from Pythagorean triples. We'll see them later as square roots of i and i. They are the four points at the intersections of the diagonal lines y = x and y = x with the unit circle. Some examples, besides 1, ≡, i, and 1 are ±√2/2 ± i√2/2, where the pluses and minuses can be taken in any order. It include all complex numbers of absolute value 1, so it has the equation | z| = 1.Ī complex number z = x + yi will lie on the unit circle when x 2 + y 2 = 1. The unit circle is the circle of radius 1 centered at 0. Of course, 1 is the absolute value of both 1 and ≡, 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. Some complex numbers have absolute value 1. (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, The horizontal side of the triangle has length | x|, the vertical side has length | y|, and the diagonal side has length | z|. 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). We can find the distance | z| by using the Pythagorean theorem. 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. The absolute value function strips a real number of its sign.įor 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. 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. Notation for the (principal) square root of x. For other uses, see Square Roots (disambiguation). ![]()
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