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Equality Of Complex Numbers

The equality relation “=” among the complex numbers is determined as consequence of the definition of the complex numbers as elements of the quotient ring $\mathbb{R}/(X^2\!+\!1)$ , which enables the interpretation of the complex numbers as the ordered pairs $(a,\,b)$ of real numbers and also as the sums $a\!+\!ib$ where .

$\displaystyle a_1+ib_1 = a_2+ib_2 \quad \Longleftrightarrow \quad a_1 = a_2\; \wedge\; b_1 = b_2$

(1)

This condition may as well be derived by using the field properties of $\mathbb{C}$ and the properties of the real numbers:

$\displaystyle a_1+ib_1 = a_2+ib_2\;$	$\displaystyle \implies\; \;a_2-a_1 = -i(b_2-b_1)$
	$\displaystyle \implies\; (a_2-a_1)^2 = -(b_2-b_1)^2$
	$\displaystyle \implies\; (a_2-a_1)^2+(b_2-b_1)^2 = 0$
	$\displaystyle \implies\; \;a_2-a_1 = 0, \;\; b_2-b_1= 0$
	$\displaystyle \implies\; \;a_1 = a_2, \;\;\; b_1 = b_2$

The implication chain in the reverse direction is apparent.

If $a+ib \neq 0$ , then at least one of the real numbers and differs from 0. We can set

$\displaystyle a = r\cos\varphi, \quad b = r\sin\varphi,$

(2)

where

is a uniquely determined positive number and $\varphi$ is an angle which is uniquely determined up to an integer multiple of $2\pi$ . In fact, the equations (2) yield

$\displaystyle a^2+b^2 = r^2(\cos^2\varphi+\sin^2\varphi) = r^2,$

whence

$\displaystyle r = \sqrt{a^2+b^2}.$

(3)

Thus (2) implies

$\displaystyle \cos\varphi = \frac{a}{\sqrt{a^2+b^2}}, \quad \sin\varphi = \frac{b}{\sqrt{a^2+b^2}}.$

(4)

The equations (4) are compatible, since the sum of the squares of their right sides is 1. So these equations determine the angle $\varphi$ up to a multiple of $2\pi$ . We can write the

Theorem. Every complex number may be represented in the polar form

$\displaystyle r(\cos\varphi+i\sin\varphi),$

where

is the modulus and $\varphi$ the argument of the number. Two complex numbers are equal if and only if they have equal moduli and, if the numbers do not vanish, their arguments differ by a multiple of $2\pi$ .

1	Equality
1	Angle
1	Complex Number
1	Complex
1	Field
1	Real Number
1	Positive
1	Property
1	Summation
1	Integer
1	Number
1	Ordered Pair
1	Logical Implication
1	Implication
1	General Associativity
1	Equation
2	Square Of A Number
2	Zero Of A Function
2	Modulus Of Complex Number
3	Quotient Ring