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# 6.2 Geometrical Representation

Since , we can identify complex numbers with points in a plane.

Then is identified with the -axis, and points on the -axis are of the form where is real. I will call the -axis the real axis, and I'll call the -axis the imaginary axis. If , then represents the result of reflecting about the real axis. Also represents the result of reflecting through the origin.

If and are two points in , and , then are the vertices of a right triangle having legs of length , and . By the Pythagorean theorem, the distance from to is . Also,

and in particular, for ,

Claim: If , then is the fourth vertex of the parallelogram having consecutive vertices .

To make this look like a geometry proof, I'll denote points by upper case letters, and let denote the distance from to . Let , , , . Then

Hence, since the quadrilateral has opposite sides equal, it is a parallelogram.

We can now give a geometrical interpretation for the triangle inequality (which motivates its name). In the figure above,

says

i.e, the sum of two sides of a triangle is greater than or equal to the third side. This is proposition 20 of book 1 of Euclid [19]  In any triangle, two sides taken together in any manner are greater than the remaining one." (Euclid did not consider triangles in which all three vertices lie on a line.)

It was the habit of the Epicureans, says Proclus ...to ridicule this theorem as being evident even to an ass, and requiring no proof, and their allegation that the theorem was  known" even to an ass was based on the fact that, if fodder is placed at one angular point and the ass at another, he does not, in order to get his food, traverse the two sides of the triangle but only the one side separating them [19, vol. I page 287].

6.8   Definition (Circle, disc.) Let . The circle with center and radius is

The open disc with center and radius is

and the closed disc with center and radius is

is called the unit circle, and is called the unit disc. A complex number is in the unit circle if and only if .

6.9   Warning. The word circle'' is sometimes used to mean disc'', although the word disc'' is never used to mean circle''. When you see the word circle'' used in a mathematical statement, you should determine from the context which of the two words is meant. For example, in the statement the area of the unit circle is '', the word circle'' means disc'', since the unit circle is, in fact, a zero-area set. In these notes the word circle'' always means circle'' except in warning 5.2.

6.10   Theorem. The product of two numbers in the unit circle is in the unit circle.

Proof: Let ; i.e., . Then , so .

We can also give a geometrical interpretation to the product of two complex numbers. Let and be complex numbers and let . Let and let .

Then is similar to . The proof consists in showing that

 (6.11)

6.12   Exercise. Prove the equalities listed in (6.11). Assume and .

From the similarity of and , we have . In particular, if we take , we get the picture

where

angle(--) angle (--),
which indicates that lies on the line through that passes through . Also

so the length of is obtained by multiplying the length of by .

The figure below shows the powers of a complex number a.

Powers of a: , a, a, a, a.

In each case the four triangles , , , and are all similar. In the third figure, where a is in the unit circle, the triangles and are in fact congruent.

If are points on the unit circle, then

angle(--) angle(--),

which indicates that is the point in the unit circle such that
angle(--) angle(--) angle(--).

From trigonometry, you know that the point on the unit circle making angle with the segment is .

The previous geometrical argument suggests that

 (6.13)

6.14   Exercise. Using standard trigonometric identities, prove (6.13), and show that .

6.15   Exercise. Let . Let . Prove that
 (6.16)

Then show that formula (6.16) is in fact valid for all . (Formula (6.16) is called De Moivre's Formula.)

Next: 6.3 Roots of Complex Up: 6. The Complex Numbers Previous: 6.1 Absolute Value and   Index