Which Complex Number Has an Absolute Value of 5

An of import concept for numbers, either real or circuitous is that of
absolute value.
Recall that the absolute value |x| of a existent number
ten
is itself, if it’s positive or aught, but if
ten
is negative, and so 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 existent number of its sign.

For a complex number
z =ten +yi,
we define the accented value |z| as being the distance from
z
to 0 in the circuitous aeroplane
C. This will extend the definition of absolute value for real numbers, since the accented value |x| of a real number
10
can be interpreted as the altitude from
x
to 0 on the real number line. We can discover the distance |z| by using the Pythagorean theorem. Consider the correct triangle with ane vertex at 0, another at
z
and the 3rd at
x
on the real axis directly beneath
z
(or higher up
z
if
z
happens to be beneath the real centrality). The horizontal side of the triangle has length |10|, the vertical side has length |y|, and the diagonal side has length |z|. Therefore,

|z|²
=
x
^two
+
y
².

(Annotation that for real numbers like
x,
nosotros can drop absolute value when squaring, since |ten|² =10
ⁱⁱ.) That gives us a formula for |z|, namely,

The unit of measurement circle.

Some complex numbers accept absolute value 1. Of course, ane is the accented value of both i and –1, only information technology’due south also the absolute value of both
i
and –i
since they’re both 1 unit of measurement 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 i, then it has the equation |z| = 1.

A complex number
z =ten +yi
volition lie on the unit circle when
ten
² +y
² = 1. Some examples, besides 1, –1,
i,
and –one
are ±√2/2 ±i√2/ii, where the pluses and minuses can be taken in any order. They are the four points at the intersections of the diagonal lines
y =ten
and
y =x
with the unit circle. Nosotros’ll run into them afterwards as square roots of
i
and –i.

You can find other circuitous numbers on the unit circumvolve from Pythagorean triples. A
Pythagorean triple
consists of three whole numbers
a, b,
and
c
such that
a
^two +b
² =c
²
If you lot split up this equation by
c
^two, then you lot find that (a/c)ⁱⁱ + (b/c)^two = one. That means that
a/c +ib/c
is a circuitous number that lies on the unit circle. The best known Pythagorean triple is 3:iv:5. That triple gives us the complex number 3/5 +i 4/5 on the unit circle. Some other Pythagorean triples are v:12:13, 15:8:17, 7:24:25, 21:20:29, nine:40:41, 35:12:27, and 11:60:61. Every bit yous might expect, at that place 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.)

The triangle inequality.

There’s an important property of circuitous numbers relating addition to accented value called the triangle inequality. If
z
and
w
are any 2 complex numbers, then

You can encounter this from the parallelogram dominion for addition. Consider the triangle whose vertices are 0,
z,
and
z +due west.
Ane side of the triangle, the one from 0 to
z +west
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 +westward,
is parallel and equal to the line from 0 to
w,
and therefore has length |west|. Now, in whatever triangle, any 1 side is less than or equal to the sum of the other ii sides, and, therefore, we accept the triangle inequality displayed in a higher place.