User:Aleksander Stos/ComplexNumberAdvanced

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Definition

Complex numbers are defined as ordered pairs of reals:

$\mathbb {C} =\{(a,b)\colon a,b\in \mathbb {R} \}.$

Such pairs can be added and multiplied as follows

• addition: $(a,b)+(c,d)=(a+c,b+d)$
• multiplication: $(a,b)(c,d)=(ac-bd,bc+ad)$

$\scriptstyle \mathbb {C}$ with the addition and multiplication is the field of complex numbers. From another of view, $\scriptstyle \mathbb {C}$ with complex additions and multiplication by real numbers is a 2-dimesional vector space.

To perform basic computations it is convenient to identify numbers of the form $\{(a,0):\;\;a\in \mathbb {R} \}$ and to introduce the imaginary unit, i=(0,1). The imaginary unit has the property $\scriptstyle i^{2}=-1.$Indeed, $(0,1)\cdot(0,1) = (-1,0) = -1.</math> Any complex number $z=(a,b)$ can be written as $z=a+bi$ (this is often called the algebraic form) and vice-versa. The numbers a and b are called the real part and the imaginary part of z, respectively. We denote $a=\Re (z)$ and $b=\Im (z).$ Remark that two complex numbers are equal if and only if their real and complex part are equal, respectively. Notice that i makes the multiplication quite natural:

$(a+bi)(c+di)=(ac-bd)+(bc+ad)i.$

The square root of number in the denominator in the above formula is called the modulus of z and denoted by $|z|$,

$|z|={\sqrt {a^{2}+b^{2}}}.$

We have for any two complex numbers $z_{1}$ and $z_{2}$

• $|{\bar {z}}|=|z|;$
• $|z_{1}\cdot z_{2}|=|z_{1}|\cdot |z_{2}|;$
• $|{\frac {z_{1}}{z_{2}}}|={\frac {|z_{1}|}{|z_{2}|}},$ provided $z_{2}\not =0.$
• ${\big |}|z_{1}|-|z_{2}|{\big |}\leq |z_{1}+z_{2}|\leq |z_{1}|+|z_{2}|$

For $z=a+bi$ we define also ${\bar {z}}$, the conjugate, by ${\bar {z}}=a-bi.$ Then we have

• ${\bar {({\bar {z}})}}=z$
• ${\bar {z}}_{1}\pm {\bar {z}}_{2}={\overline {(z_{1}\pm z_{2})}};$
• ${\bar {z}}_{1}\cdot {\bar {z}}_{2}={\overline {(z_{1}\cdot z_{2})}};$
• ${\overline {\left({\frac {z_{1}}{z_{2}}}\right)}}={\frac {{\bar {z}}_{1}}{{\bar {z}}_{2}}},$ provided $z_{2}\not =0;$
• $z{\bar {z}}=|z|^{2}.$

Geometric interpretation

Complex numbers may be naturally represented on the complex plane, where $z=x+iy$ corresponds to the point (x,y), see the fig. 1.

Fig. 1. Graphical representation of a complex number and its conjugate

The modulus is just the distance from the point $z=(x,y)$ and the origin. More generally, $|z_{1}-z_{2}|$ is the distance between the two given points. Furthermore, the conjugation is just the symmetry with respect to the x-axis. The sum of two given complex numbers $z_{1}$ and $z_{2}$ can be geometrically determined as a vertex of a parallelogram determined by the points 0, $z_{1}$ and $z_{2}$ (i.e. the fourth vertex given these three, see the fig. 2).

Fig. 2. Graphical representation of a complex number and its conjugate

Trigonometric and exponential form

As the graphical representation suggests, any complex number z=a+bi of modulus 1 (i.e. a point from the unit circle) can be written as $z=\cos \theta +i\sin \theta$ for some $\theta \in [0,2\pi ).$ So actually any (non-null) $z\in \mathbb {C}$ can be represented as

$z=r(\cos \theta +i\sin \theta ),$ where r traditionally stands for |z|.

This is the trigonometric form of the complex number z. If we adopt convention that $\theta \in [0,2\pi )$ then such $\theta$ is unique and called the argument of z. The equality of two complex numbers $z_{1}=r_{1}e^{i\theta _{1}}$ and $z_{2}=r_{2}e^{i\theta _{2}}$ is equivalent to $r_{1}=r_{2}$ and $\theta _{1}=\theta _{2}+2k\pi$ for certain integer k. Graphically, the number $\theta$ is the (oriented) angle between the x-axis and the interval containing 0 and z. Closely related is the exponential notation. If we define complex exponential as

$e^{z}=\sum _{0}^{\infty }{\frac {z^{n}}{n!}},$

then it may be shown that

$e^{i\theta }=\cos \theta +i\sin \theta ,\quad \quad \theta \in \mathbb {R} .$

Consequently, any (non-zero) $z\in \mathbb {C}$ can be written as

$z=re^{i\theta }$ with the same r and $theta$ as above.

This is called the exponential form of the complex number z. It is well-adapted to perform multiplications. Indeed, for any $z_{1}=r_{1}e^{i\theta _{1}}$ and $z_{2}=r_{2}e^{i\theta _{2}}$ we have

• $z_{1}z_{2}=r_{1}r_{2}e^{i(\theta _{1}+\theta _{2})}$
• ${\frac {z_{1}}{z_{2}}}={\frac {r_{1}}{r_{2}}}e^{i(\theta _{1}-\theta _{2})},$ provided $z_{2}\not =0.$

The following particular case of complex multiplication is well-know as the de Moivre's formula

$(cos\theta +i\sin \theta )^{n}=\cos(n\theta )+i\sin(n\theta )$

Fig 2. Multiplication by $i$ amounts to rotation by 90 degrees.

Graphically, multiplication by a constant complex number $z=re^{i\theta }$ amounts to the rotation by $\theta$ and the homothety of ratio r. In particular, the multiplication by i amounts to the rotation by the right angle (counter-clockwise), see Fig. 2.

Complex roots

Any non-constant polynomial with complex coefficients has a complex root. This result is known as the Fundamental Theorem of Algebra. Consequently, any complex polynomial of degree n has exactly n roots (counted with multiplicities). In particular, the equation

$z^{n}=a$,

where z is the variable and a a non-zero constant has exactly n solutions. They are called n^th (complex) roots of a. If a is written in the exponential form, $a=re^{i\theta },$ then the n roots of a, denoted as $z_{0},z_{2},\ldots ,z_{n-1}$, are given by

$z_{k}={\sqrt[{n}]{r}}\cdot \exp \left(i\left({\frac {\theta +2k\pi }{n}}\right)\right),\quad k=0,1,\ldots ,n-1.$

One may observe that

$z_{k}=z_{k-1}e^{i\theta /n};$ or, equivalently, $z_{k}=z_{0}e^{ik\theta /n},\quad k=1,2,\ldots ,n-1.$

Geometrically it means that the roots form a regular n-sided polygon centred at the origin; the vertices of the polygon belong to the circle of radius ${\sqrt[{n}]{r}}.$

Particularly important are the roots of unity, i.e. solutions of $z^{n}=1$. The cubic roots of 1 (with n=3) are

$1,\,-{\frac {1}{2}}+i{\frac {\sqrt {3}}{2}},\,-{\frac {1}{2}}-i{\frac {\sqrt {3}}{2}}$

and for n=4 we have

${\frac {1}{2}}+{\frac {\sqrt {2}}{2}},\,{\frac {1}{2}}-{\frac {\sqrt {2}}{2}},\,{\frac {1}{2}}+{\frac {\sqrt {2}}{2}},\,-{\frac {1}{2}}-{\frac {\sqrt {2}}{2}}.$