Real and imaginary part

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.

Rules for calculating real and imaginary parts Every set of two complex numbers $z$ , $w$ satisfies the following equalities:

$\Re (z+w)=\Re (z)+\Re (w)$
$\Re (z\cdot w)=\Re (z)\cdot \Re (w)-\Im (z)\cdot\Im (w)$
$\Im (z+w)=\Im (z)+\Im (w)$
$\Im (z\cdot w)=\Re (z)\cdot \Im (w)+\Im (z)\cdot\Re (w)$
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 $\varphi = 2\arctan\left(\tan\left(\frac{\varphi}{2}\right)\right)=2\arctan\left(\frac{\Im(z)}{|z|+\Re(z)}\right)\tiny.$

What is the real part of $-4+11\cdot \ii$ ?

$\Re(-4+11\cdot \ii)={}$ $-4$

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