# 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 EqualityOfComplexNumbers 2015-05-09 17:11:34 2015-05-09 17:11:34 pahio (2872) pahio (2872) 7 pahio (2872) Topic msc 30-00 ModulusOfAComplexNumber ArgumentOfProductAndQuotient ComplexLogarithm