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[parent] equality of complex numbers (Topic)

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 $ 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 chain 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

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



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See Also: modulus of a complex number, argument of product and quotient, complex logarithm


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Cross-references: vanish, argument, modulus, polar form, squares, implies, equations, multiple, integer, angle, number, positive, implication, properties, field, sums, real numbers, ordered pairs, quotient ring, complex numbers, consequence, equality relation
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This is version 3 of equality of complex numbers, born on 2008-01-24, modified 2008-01-24.
Object id is 10210, canonical name is EqualityOfComplexNumbers.
Accessed 407 times total.

Classification:
AMS MSC: 30-00 (Functions of a complex variable :: General reference works )
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