The complex plane

The complex plane

A complex number $z$ can be viewed as a point or a position vector in a two-dimensional Cartesian coordinate system called the complex plane or Argand diagram. The point and hence the complex number $z$ can be specified by Cartesian (rectangular) coordinates. The Cartesian coordinates of the complex number are the real part $x={\text{Re}}(z)$ and the imaginary part $y={\text{Im}}(z)$ . The representation of a complex number by its Cartesian coordinates is called the Cartesian form or rectangular form or algebraic form of that complex number.

Polar form

Complex number in polar coordinates

Alternatively, the complex number $z$ can be specified by polar coordinates. The polar coordinates are $r=|z|\geq 0$ , called the absolute value or modulus, and $\phi =\arg(z)$ , called the argument of $z$ . For $r=0$ any value of $\varphi$ describes the same number. To get a unique representation, a conventional choice is to set $\arg(0)=0$ . For $r>0$ the argument $\varphi$ is unique modulo $2\pi$ ; that is, if any two values of the complex argument differ by an exact integer multiple of $2\pi$ , they are considered equivalent. To get a unique representation, a conventional choice is to limit $\varphi$ to the interval $(-\pi ,\pi ]$ i.e. $-\pi <\varphi \leq \pi$ . The representation of a complex number by its polar coordinates is called the polar form of the complex number.

Conversion from the polar form to the Cartesian form

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x=r\cos(\varphi)}

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle y=r\sin(\varphi)}

Conversion from the Cartesian form to the polar form

$r={\sqrt {x^{2}+y^{2}}}$

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \varphi= \begin{cases} \arctan\left(\frac{y}{x}\right)&\text{if }x>0\\ \arctan\left(\frac{y}{x}\right)+\pi&\text{if }x<0,y\ge0\\ \arctan\left(\frac{y}{x}\right)-\pi&\text{if }x<0,y<0\\ \frac{\pi}{2}&\text{if }x=0,y>0\\ -\frac{\pi}{2}&\text{if }x=0,y<0\\ \text{undefined}&\text{if }x=0,y=0 \end{cases}}

The previous formula requires rather laborious case differentiations. However, many programming languages provide a variant of the arctangent function. A formula that uses the arccos function requires fewer case differentiations:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \varphi= \begin{cases} \arccos\left(\frac{x}{r}\right)&\text{if }y\ge0,r\ne0\\ -\arccos\left(\frac{x}{r}\right)&\text{if }y<0\\ \text{undefined}&\text{if }r=0 \end{cases}}

Notation of the polar form

The notation of the polar form as

$z=r{\big (}\cos(\varphi )+i\sin(\varphi ){\big )}$

is called trigonometric form. The notation Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \text{cis}(\varphi)} is sometimes used as an abbreviation for Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \cos(\varphi)+i\sin(\varphi)} . Using Euler's formula it can also be written as

$z=re^{i\varphi }$

which is called exponential form.

Multiplication, division, exponentiation, and root extraction in the polar form

Multiplication, division, exponentiation, and root extraction are much easier in the polar form than in the Cartesian form.

Using sum and difference identities its possible to obtain that

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle r_1e^{i\varphi_1}\cdot r_2e^{i\varphi_2}=r_1r_2e^{i(\varphi_1+\varphi_2)}}

and that

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{r_1e^{i\varphi_1}}{r_2e^{i\varphi_2}}=\frac{r_1}{r_2}\cdot e^{i(\varphi_1-\varphi_2)}}

Exponentiation with integer exponents; according to de Moivre's formula,

${\big (}re^{i\varphi }{\big )}^{n}=r^{n}e^{ni\varphi }$

Exponentiation with arbitrary complex exponents is discussed in the article on exponentiation.

The addition of two complex numbers is just the addition of two vectors, and multiplication by a fixed complex number can be seen as a simultaneous rotation and stretching.

Multiplication by Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle i} corresponds to a counter-clockwise rotation by 90° or Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{\pi}{2}} radians. The geometric content of the equation $i^{2}=-1$ is that a sequence of two 90° rotations results in a 180° (Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \pi} radians) rotation. Even the fact $(-1)\cdot (-1)=1$ from arithmetic can be understood geometrically as the combination of two 180° turns.

All the roots of any number, real or complex, may be found with a simple algorithm. The $n$-th roots are given by

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \sqrt[n]{re^{i\varphi}}=\sqrt[n]{r}\,e^{i\left(\frac{\varphi+2k\pi}{n}\right)}}

for $k=0,1,2,\ldots ,n-1$ , where Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \sqrt[n]{r}} represents the principal Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle n} -th root of $r$ .

Absolute value, conjugation and distance

The absolute value (or modulus or magnitude) of a complex number Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle z=re^{i\varphi}} is defined as Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle |z|=r} .

Algebraically, if $z=a+bi$ then Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle |z|=\sqrt{a^2+b^2}} .

One can check readily that the absolute value has three important properties:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle |z|=0} if and only if $z=0$

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle |z+w|\le|z|+|w|} (triangle inequality)

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle |z\cdot w|=|z|\cdot|w|}

for all complex numbers $z,w$ . It then follows, for example, that $|1|=1$ and $\left|{\frac {z}{w}}\right|={\frac {|z|}{|w|}}$ . By defining the distance function $d(z,w)=|z-w|$ we turn the set of complex numbers into a metric space and we can therefore talk about limits and continuity.

The complex conjugate of the complex number $z=a+bi$ is defined to be $a-bi$ , written as ${\bar {z}}$ or $z^{*}$ . As seen in the figure, ${\bar {z}}$ is the "reflection" of $z$ about the real axis. The following can be checked:

${\overline {z+w}}={\bar {z}}+{\bar {w}}$

${\overline {z\cdot w}}={\bar {z}}\cdot {\bar {w}}$

${\overline {\left({\frac {z}{w}}\right)}}={\frac {\bar {z}}{\bar {w}}}$

${\bar {\bar {z}}}=z$

${\bar {z}}=z$ if and only if $z$ is real

$|z|=|{\bar {z}}|$

$|z|^{2}=z\cdot {\bar {z}}$

$z^{-1}={\bar {z}}\cdot |z|^{-2}$ if $z$ is non-zero.

The latter formula is the method of choice to compute the inverse of a complex number if it is given in rectangular coordinates.

That conjugation commutes with all the algebraic operations (and many functions; e.g. $\sin({\bar {z}})={\overline {\sin(z)}}$) is rooted in the ambiguity in choice of $i$ (−1 has two square roots). It is important to note, however, that the function $f(z)={\bar {z}}$ is not complex-differentiable.

Resources

• Complex Numbers, Wikibooks: Calculus
• The complex plane. Written notes created by Professor Esparza, UTSA.