equality of complex numbers

The equality relation “=” among the 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 of the complex numbers as the ordered pairs $(a,\,b)$ of real numbers and also as the sums $a\!+\!ib$ where $i^{2}=-1$ .

\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 in the reverse direction is apparent.

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

\displaystyle a=r\cos\varphi,\qquad b=r\sin\varphi,

(2)

where $r$ 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

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}}},\qquad\sin\varphi=\frac{% b}{\sqrt{a^{2}+b^{2}}}.

(4)

The equations (4) are , since the sum of the squares of their 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

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

where $r$ 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 $2\pi$ .

Title	equality of complex numbers
Canonical name	EqualityOfComplexNumbers
Date of creation	2015-05-09 17:11:34
Last modified on	2015-05-09 17:11:34
Owner	pahio (2872)
Last modified by	pahio (2872)
Numerical id	7
Author	pahio (2872)
Entry type	Topic
Classification	msc 30-00
Related topic	ModulusOfAComplexNumber
Related topic	ArgumentOfProductAndQuotient
Related topic	ComplexLogarithm