Complex numbers: Introduction to Complex numbers
Real and imaginary part
In the theory The notion complex number we already came across the following two definitions. Now we will also introduce a notation.
Real and imaginary parts
If #z=a+b\ii# with #a,b\in\mathbb{R}# then
- #a# is called the real part of #z# and it is written as #\Re(z)#, and
- #b# is called the imaginary part of #z# and is written as #\Im (z)#.
#\Re# and #\Im# hence, are real-valued functions on #\mathbb{C}#; These are the Cartesian coordinates of the complex number.
The imaginary part of a complex number is a real number.
- #\Re (z+w)=\Re (z)+\Re (w)#
- #\Re (z\cdot w)=\Re (z)\cdot \Re (w)-\Im (z)\cdot\Im (w)#
- #\Im (z+w)=\Im (z)+\Im (w)#
- #\Im (z\cdot w)=\Re (z)\cdot \Im (w)+\Im (z)\cdot\Re (w)#
- If #z# is real, then we have #\Re (z\cdot w)=z\cdot \Re (w)#
These properties follow directly from the definitions.
The last property is a direct result of the second one, because if #z# is real, we have #\Re(z)=z# and #\Im(z)=0#.
The following concept is useful in the description of the transition between Cartesian and polar coordinates:
Equal modulo a real number
In order to express that two real numbers #a# and #b# differ by an integer multiple of #2\pi#, we can write
\[a=b \mod(2\pi)\tiny.\] We say #a# and #b# are equal modulo #2\pi#.
The numbers #-\frac{\pi}{2}# and #\frac{7\pi}{2}# for example, are equal modulo #2\pi#.
The same definition applies to integers: an even integer is equal to #0# modulo #2#.
Every real number is equal modulo #2\pi# to exactly one number in #\ivoc{-\pi}{\pi}#.
Starting with the absolute value #r=|z|# and an argument #\varphi# of a complex number #z#, the real and imaginary part can be found as follows: \[ \begin{array}{lcl} \Re (z)=r\cdot\cos (\varphi) &,\qquad& \Im (z)=r\cdot\sin (\varphi) \end {array}\tiny.\]
Conversely, the absolute value of #z# can be obtained from the real and imaginary part by means of the formula
\[
|z|= \sqrt{\Re (z)^2+\Im (z)^2}\tiny.
\] If #z\ne0# then the argument #\varphi=\arg(z)# modulo #2\pi# is determined by \[ \begin{array}{rcl}\cos(\varphi)=\frac{\Re(z)}{|z|}&,\qquad&\sin(\varphi)=\frac{\Im(z)}{|z|}\end {array}\tiny.\] In practice, these data are sufficient to determine #\arg(z)#. It is also possible to use the following more complicated formula:\[\arg(z)= \begin{cases}\pi&\text{if } z\text{ lies on the negative real axis}\\ 2\arctan\left(\frac{\Im (z)}{|z|+\Re (z)}\right)&\text{otherwise }\end {cases}\]
Often people think that the principal value of the argument is given by the formula
\[
\arg(z)=\arctan \left( \frac{\Im (z)}{\Re (z)}\right)\tiny.
\]
This is true if #\Re (z)\gt0#, but in general it is incorrect. Check it yourself using the complex number #-1-\ii#.
We now prove the statement that #\arg(z)=2\arctan\left(\frac{\Im (z)}{|z|+\Re (z)}\right)# when #z# is not located on the negative real axis. We do so using the formula \[\tan\left(\frac{\alpha}{2}\right)=\frac{\sin(\alpha)}{1+\cos(\alpha)}\tiny.\]
This last formula follows from the known double angle formulas: \[ \begin{array}{rcl}\tan\left(\frac{\alpha}{2}\right)&=&\frac{\sin\left(\frac{\alpha}{2}\right)}{\cos\left(\frac{\alpha}{2}\right)}\\ &&\phantom{uvwxyz}\color{blue}{\text{definition tangent}}\\ &=&\frac{2\sin\left(\frac{\alpha}{2}\right)\cdot\cos\left(\frac{\alpha}{2}\right)}{2\cos^2\left(\frac{\alpha}{2}\right)}\\ &&\phantom{uvwxyz}\color{blue}{\text{numerator and denominator multiplied by }2\cos\left(\frac{\alpha}{2}\right)}\\&=&\frac{\sin\left({\alpha}\right)}{1+\cos\left(\alpha\right)}\\&&\phantom{uvwxyz}\color{blue}{\cos(2x) =2\cos(x)^2-1\text{ and }\sin(2x)=2\sin(x)\cdot\cos(x)}\\ \end {array}\]
#\varphi=\arg(z)# is determined by the conditions \[ \begin{array}{rcl}\cos(\varphi)&=&\frac{\Re(z)}{|z|}\\ \sin(\varphi)&=&\frac{\Im(z)}{|z|}\\ \varphi&\in&\ivoc{-{\pi}}{{\pi}}\tiny.\end {array}\] From the first two conditions \[\tan\left(\frac{\varphi}{2}\right)=\frac{\sin(\varphi)}{1+\cos(\varphi)}=\frac{ \frac{\Im(z)}{|z|} }{1+\frac{\Re(z)}{|z|}}=\frac{\Im(z)}{|z|+\Re(z)}\tiny\] From the third condition #\frac{\varphi}{2}\in\ivoc{-\frac{\pi}{2}}{\frac{\pi}{2}}# follows.
According to the theory of inverse trigonometric functions, the function #\arctan# is the inverse of #\tan# on #\ivoo{-\frac{\pi}{2}}{\frac{\pi}{2}}#. Therefore, #\varphi\ne\pi#, so for #z# not on the negative real axis, we have \[\varphi = 2\arctan\left(\tan\left(\frac{\varphi}{2}\right)\right)=2\arctan\left(\frac{\Im(z)}{|z|+\Re(z)}\right)\tiny.\]
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