Absolute Value Squared Complex Number

An of import concept for numbers, either real or complex is that of absolute value. Recall that the absolute value |x| of a existent number x is itself, if information technology's positive or nada, but if x is negative, so its absolute value |x| is its negation –x, that is, the corresponding positive value. For example, |iii| = 3, only |–4| = iv. The absolute value part strips a real number of its sign.

For a complex number z =x +yi, we ascertain the absolute value |z| as being the distance from z to 0 in the complex plane C. This volition extend the definition of absolute value for real numbers, since the absolute value |10| of a real number x can be interpreted as the distance from x to 0 on the real number line. Nosotros tin can find the distance |z| past using the Pythagorean theorem. Consider the right triangle with one vertex at 0, another at z and the 3rd at ten on the real axis directly below z (or in a higher place z if z happens to be below the existent axis). The horizontal side of the triangle has length |ten|, the vertical side has length |y|, and the diagonal side has length |z|. Therefore,

|z|ⁱⁱ = 10 ⁱⁱ + y ^two.

(Note that for existent numbers like x, we can driblet accented value when squaring, since |10|^two =x ².) That gives us a formula for |z|, namely,

the absolute value of z is the square root of (x^2+y^2)

The unit circumvolve.

Some complex numbers take accented value ane. Of course, ane is the accented value of both 1 and –1, only information technology'due south besides 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 circuitous numbers of absolute value 1, so information technology has the equation |z| = ane.

A complex number z =x +yi will prevarication on the unit of measurement circumvolve when x ² +y ² = 1. Some examples, besides i, –1, i, and –1 are ±√2/2 ±i√two/2, where the pluses and minuses can be taken in whatsoever gild. 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 observe other complex numbers on the unit of measurement circle from Pythagorean triples. A Pythagorean triple consists of iii whole numbers a, b, and c such that a ⁱⁱ +b ² =c ² If you dissever this equation by c ⁱⁱ, then you lot detect that (a/c)² + (b/c)² = 1. That means that a/c +ib/c is a complex number that lies on the unit circumvolve. The best known Pythagorean triple is 3:4:5. That triple gives us the complex number 3/v +i 4/5 on the unit circle. Another Pythagorean triples are 5:12:13, 15:viii:17, 7:24:25, 21:20:29, 9:forty:41, 35:12:27, and 11:threescore:61. Equally you lot might await, there are infinitely many of them. (For a little more on Pythagorean triples, encounter the cease of the page at http://www.clarku.edu/~djoyce/trig/right.html.)

The triangle inequality.

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

the absolute value of z+w is less than or equal to the sum of the absolute values of z and w

You can run across this from the parallelogram dominion for improver. Consider the triangle whose vertices are 0, z, and z +w. One side of the triangle, the one from 0 to z +westward has length |z +due west|. 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 |westward|. Now, in whatsoever 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 to a higher place.