Calculus/Complex numbers

Complex Numbers

In mathematics, a complex number is a number of the form

$a+bi$

where $a,b$ are real numbers, and $i$ is the imaginary unit, with the property $i^{2}=-1$ . The real number $a$ is called the real part of the complex number, and the real number $b$ is the imaginary part. Real numbers may be considered to be complex numbers with an imaginary part of zero; that is, the real number $a$ is equivalent to the complex number $a+0i$ .

For example, $3+2i$ is a complex number, with real part 3 and imaginary part 2. If $z=a+bi$ , the real part $a$ is denoted ${\text{Re}}(z)$ or $\Re (z)$ , and the imaginary part $b$ is denoted ${\text{Im}}(z)$ or $\Im (z)$ .

Complex numbers can be added, subtracted, multiplied, and divided like real numbers and have other elegant properties. For example, real numbers alone do not provide a solution for every polynomial algebraic equation with real coefficients, while complex numbers do (this is the fundamental theorem of algebra).

Equality

Two complex numbers are equal if and only if their real parts are equal and their imaginary parts are equal. That is, $a+bi=c+di$ if and only if $a=c$ and $b=d$ .

Notation and operations

The set of all complex numbers is usually denoted by ${\rm {C}}$ , or in blackboard bold by $\mathbb {C}$ (Unicode ℂ). The real numbers $\mathbb {R}$ may be regarded as "lying in" $\mathbb {C}$ by considering every real number as a complex: $a=a+0i$ .

Complex numbers are added, subtracted, and multiplied by formally applying the associative, commutative and distributive laws of algebra, together with the equation $i^{2}=-1$

$(a+bi)+(c+di)=(a+c)+(b+d)i$

$(a+bi)-(c+di)=(a-c)+(b-d)i$

$(a+bi)(c+di)=ac+bci+adi+bdi^{2}=(ac-bd)+(bc+ad)i$

Division of complex numbers can also be defined (see below). Thus, the set of complex numbers forms a field which, in contrast to the real numbers, is algebraically closed.

In mathematics, the adjective "complex" means that the field of complex numbers is the underlying number field considered, for example complex analysis, complex matrix, complex polynomial and complex Lie algebra.

The field of complex numbers

Formally, the complex numbers can be defined as ordered pairs of real numbers $(a,b)$ together with the operations:

$(a,b)+(c,d)=(a+c,b+d)$

$(a,b)\cdot (c,d)=(ac-bd,bc+ad)$

So defined, the complex numbers form a field, the complex number field, denoted by $\mathbb {C}$ (a field is an algebraic structure in which addition, subtraction, multiplication, and division are defined and satisfy certain algebraic laws. For example, the real numbers form a field).

The real number $a$ is identified with the complex number $(a,0)$ , and in this way the field of real numbers $\mathbb {R}$ becomes a subfield of $\mathbb {C}$ . The imaginary unit $i$ can then be defined as the complex number $(0,1)$ , which verifies

$(a,b)=a\cdot (1,0)+b\cdot (0,1)=a+bi\quad {\text{and}}\quad i^{2}=(0,1)\cdot (0,1)=(-1,0)=-1$

In $\mathbb {C}$ , we have:

• additive identity ("zero"): $(0,0)$
• multiplicative identity ("one"): $(1,0)$
• additive inverse of $(a,b)$ : $(-a,-b)$
• multiplicative inverse (reciprocal) of non-zero $(a,b)$ : $\left({\frac {a}{a^{2}+b^{2}}},-{\frac {b}{a^{2}+b^{2}}}\right)$

Since a complex number $a+bi$ is uniquely specified by an ordered pair $(a,b)$ of real numbers, the complex numbers are in one-to-one correspondence with points on a plane, called 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

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

$x=r\cos(\varphi )$

$y=r\sin(\varphi )$

Conversion from the Cartesian form to the polar form

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

$\varphi ={\begin{cases}\arctan \left({\frac {y}{x}}\right)&{\text{if }}x>0\\\arctan \left({\frac {y}{x}}\right)+\pi &{\text{if }}x<0,y\geq 0\\\arctan \left({\frac {y}{x}}\right)-\pi &{\text{if }}x<0,y<0\\{\frac {\pi }{2}}&{\text{if }}x=0,y>0\\-{\frac {\pi }{2}}&{\text{if }}x=0,y<0\\{\text{undefined}}&{\text{if }}x=0,y=0\end{cases}}$

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:

$\varphi ={\begin{cases}\arccos \left({\frac {x}{r}}\right)&{\text{if }}y\geq 0,r\neq 0\\-\arccos \left({\frac {x}{r}}\right)&{\text{if }}y<0\\{\text{undefined}}&{\text{if }}r=0\end{cases}}$

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 ${\text{cis}}(\varphi )$ is sometimes used as an abbreviation for $\cos(\varphi )+i\sin(\varphi )$ . 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

$r_{1}e^{i\varphi _{1}}\cdot r_{2}e^{i\varphi _{2}}=r_{1}r_{2}e^{i(\varphi _{1}+\varphi _{2})}$

and that

${\frac {r_{1}e^{i\varphi _{1}}}{r_{2}e^{i\varphi _{2}}}}={\frac {r_{1}}{r_{2}}}\cdot e^{i(\varphi _{1}-\varphi _{2})}$

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 $i$ corresponds to a counter-clockwise rotation by 90° or ${\frac {\pi }{2}}$ radians. The geometric content of the equation $i^{2}=-1$ is that a sequence of two 90° rotations results in a 180° ($\pi$ 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

${\sqrt[{n}]{re^{i\varphi }}}={\sqrt[{n}]{r}}\,e^{i\left({\frac {\varphi +2k\pi }{n}}\right)}$

for $k=0,1,2,\ldots ,n-1$ , where ${\sqrt[{n}]{r}}$ represents the principal $n$-th root of $r$ .

Absolute value, conjugation and distance

The absolute value (or modulus or magnitude) of a complex number $z=re^{i\varphi }$ is defined as $|z|=r$ .

Algebraically, if $z=a+bi$ then $|z|={\sqrt {a^{2}+b^{2}}}.$

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

$|z|=0$ if and only if $z=0$

$|z+w|\leq |z|+|w|$ (triangle inequality)

$|z\cdot w|=|z|\cdot |w|$

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.

Euler's formula

Let's review the MacLaurin expansion of a given function $y=f(x)$ :

$y=f(x)=f(0)+f'(0)\cdot x+{\frac {f''(0)\cdot x^{2}}{2!}}+{\frac {f'''(0)\cdot x^{3}}{3!}}+\cdots$

Here, $n!$ is the factorial of $n$ .

To write the Maclaurin's expansion, we are supposed to know the first derivative, second derivative, third derivative, ect. of the given function. The higher derivative we know, the more accurate the expansion is. Therefore, ideally, if we are able to know every derivative, then the expansion will be absolutely accurate. Fortunately, there are some functions that their every derivative is known: sine, cosine, and the exponential function $y=e^{x}$ are three examples of such a function.

The derivative of $e^{x}$ is itself, therefore every derivative of $e^{x}$ is $e^{x}$.

The MacLaurin expansion of $e^{x}$ is:

$e^{x}=1+x+{\frac {x^{2}}{2!}}+{\frac {x^{3}}{3!}}+{\frac {x^{4}}{4!}}+\cdots$

which is valid for all real numbers, because it is always convergent.

The derivatives of $\sin(x)$ are:

${\begin{aligned}&y=\sin(x)\\&y'=\cos(x)\\&y''=-\sin(x)\\&y'''=-\cos(x)\\&y^{(4)}=\sin(x)\\&y^{(5)}=\cos(x)\end{aligned}}$

The 5th derivation is the same as the 1st derivation, therefore the 6th derivation is the same as the 2nd derivation, and so on.

The same is for $\cos(x)$ . The 1st derivative is $-\sin(x)$ , the second derivative is $-\cos(x)$ , and so on.

The MacLaurin expansion of $\sin(x)$ is:

$\sin(x)=x-{\frac {x^{3}}{3!}}+{\frac {x^{5}}{5!}}-{\frac {x^{7}}{7!}}+\cdots$

The MacLaurin expansion of $\cos(x)$ is:

$\cos(x)=1-{\frac {x^{2}}{2!}}+{\frac {x^{4}}{4!}}-{\frac {x^{6}}{6!}}+\cdots$

which are valid for all real numbers, because they are always convergent.

But what if someone plugs in an imaginary number and calculates $e^{i\cdot x}$ , where $x$ is real? That may sound ridiculous and unimaginable because an imaginary number as the exponent is not yet defined. However, if we really do this, we will get an interesting result:

${\begin{aligned}e^{ix}&=1+i\cdot x+{\frac {(i\cdot x)^{2}}{2!}}+{\frac {(i\cdot x)^{3}}{3!}}+{\frac {(i\cdot x)^{4}}{4!}}+{\frac {(i\cdot x)^{5}}{5!}}+{\frac {(i\cdot x)^{6}}{6!}}+{\frac {(i\cdot x)^{7}}{7!}}+\cdots \\&=1+i\cdot x-{\frac {x^{2}}{2!}}-{\frac {i\cdot x^{3}}{3!}}+{\frac {x^{4}}{4!}}+{\frac {i\cdot x^{5}}{5!}}-{\frac {x^{6}}{6!}}-{\frac {i\cdot x^{7}}{7!}}+\cdots \\&=\left(1-{\frac {x^{2}}{2!}}+{\frac {x^{4}}{4!}}-{\frac {x^{6}}{6!}}+\cdots \right)+i\left(x-{\frac {x^{3}}{3!}}+{\frac {x^{5}}{5!}}-{\frac {x^{7}}{7!}}+\cdots \right)\\&=\cos(x)+i\sin(x)\end{aligned}}$

The result is $e^{ix}=\cos(x)+i\sin(x)$ . This equation is called Euler's formula.

This is the most wonderful formula in mathematics, because the exponential function and the trigonometric functions are connected in such a way via the imaginary unit $i$ . As mentioned before, $e^{ix}$ is not defined, but why not define it like this? It doesn't violate any mathematical rule, and it may show some properties of deep maths. If we plug in $x=\pi$, the equation will become:

$e^{\pi i}=-1$

which is called Euler's identity.

Many people like to write Euler's identity as

$e^{\pi i}+1=0$

because in this formulation, we have five of the most important mathematical constants, $0,1,i,e,\pi$ all in one equation, and nothing else.

By the same reasoning, we get the following identities:

$e^{0i}=e^{0}=1$

$e^{{\frac {\pi }{2}}i}=i$

$e^{\pi i}=-1$

$e^{{\frac {3\pi }{2}}i}=-i$

$e^{2\pi i}=1$

$e^{{\frac {5\pi }{2}}i}=i$

and so on.

Complex fractions

We can divide a complex number $a+bi$ by another complex number $c+di\neq 0$ in two ways. The first way has already been implied: to convert both complex numbers into exponential form, from which their quotient is easily derived. The second way is to express the division as a fraction, then to multiply both numerator and denominator by the complex conjugate of the denominator. The new denominator is a real number.

${\begin{aligned}{\frac {a+bi}{c+di}}&={\frac {(a+bi)(c-di)}{(c+di)(c-di)}}={\frac {(ac+bd)+(bc-ad)i}{c^{2}+d^{2}}}\\&=\left({\frac {ac+bd}{c^{2}+d^{2}}}\right)+\left({\frac {bc-ad}{c^{2}+d^{2}}}\right)i\end{aligned}}$

Matrix representation of complex numbers

While usually not useful, alternative representations of the complex field can give some insight into its nature. One particularly elegant representation interprets each complex number as a 2×2 matrix with real entries which stretches and rotates the points of the plane. Every such matrix has the form

${\begin{bmatrix}a&-b\\b&a\end{bmatrix}}$

where $a,b$ are real numbers. The sum and product of two such matrices is again of this form. Every non-zero matrix of this form is invertible, and its inverse is again of this form. Therefore, the matrices of this form are a field. In fact, this is exactly the field of complex numbers. Every such matrix can be written as

${\begin{bmatrix}a&-b\\b&a\end{bmatrix}}=a{\begin{bmatrix}1&0\\0&1\end{bmatrix}}+b{\begin{bmatrix}0&-1\\1&0\end{bmatrix}}$

which suggests that we should identify the real number 1 with the identity matrix

${\begin{bmatrix}1&0\\0&1\end{bmatrix}}$

and the imaginary unit $i$ with

${\begin{bmatrix}0&-1\\1&0\end{bmatrix}}$

a counter-clockwise rotation by 90 degrees. Note that the square of this latter matrix is indeed equal to the 2×2 matrix that represents −1.

The square of the absolute value of a complex number expressed as a matrix is equal to the determinant of that matrix.

$|z|^{2}={\begin{vmatrix}a&-b\\b&a\end{vmatrix}}=(a^{2})-((-b)(b))=a^{2}+b^{2}$

If the matrix is viewed as a transformation of the plane, then the transformation rotates points through an angle equal to the argument of the complex number and scales by a factor equal to the complex number's absolute value. The conjugate of the complex number $z$ corresponds to the transformation which rotates through the same angle as $z$ but in the opposite direction, and scales in the same manner as $z$ ; this can be represented by the transpose of the matrix corresponding to $z$ .

If the matrix elements are themselves complex numbers, the resulting algebra is that of the quaternions. In other words, this matrix representation is one way of expressing the Cayley-Dickson construction of algebras.

Exercises

1) If $z=\exp(i\theta )\ \ {\text{prove then that}}\ \ z+{\bar {z}}=2\cos \theta .$

2) The line through 0 and z is perpendicular to the line through 0 and w when  $z{\bar {w}}+w{\bar {z}}=0.$

3) Consider the rotation transformation $T(z)\ =\ e^{i\theta }z.$ Show that perpendicular lines are mapped by T to perpendicular lines.

4) Calculate the addition, subtraction, and multiplication of two complex numbers using matrix notation with the following: $z_{1}=3+5i,z_{2}=7-12i$