In linear algebra, better theorems and more insight will emerge if complex nimbers are investigated along with real numbers. Thus today we weill begin by introducing the complex numbers and their basic properties. We will generalize the examples of a plane and prginary space to Rn\mathbb{R}^n and Cn\mathbb{C}^n, which we then will generalize to the notion of a vector space. With the help of complex numbers and additional knowledge of calcus, we will confirm Euler’s formula:

eix=cosx+i sinxe^{ix}=\text{cos}x+i\,\text{sin}x

If x=πx=\pi , the formula will be:

eiπ+1=0e^{i\pi}+1=0

It seems absoluytely magical that such a neat equation combines ee(Euler’s number), ii(the unit imaginary number), π\pi(ratio of a circle’s circumference to its diameter), 1(first counting number), 0(zero). And also has the basic operations of add, multiply, and an exponent too! The physicist Richard Feynman called the equation “our jewel” and “the most remarkable formula in mathematics”.

You should already be familar with the basic properties of the set of real numbers, namely R\mathbb{R} . The extension of numeric set always accompanies by the failure of display of the result of some numbers, say 2\sqrt{2} , which failed to be expressed by nothing but rational numbers and thus rational numbers was then extended to real numbers(with the addition of irrational number). The idea is to assume that we have the result of square root of −1-1 , denoted ii , that obey the usual rules of arithmetic. Here we give the formal definations.

Basic Definations

Defination Complex numbers, complex plane, real part, imaginary part, addition, multiplication

A complex number is an ordered pair (x,y)(x,y) , where x,y∈Rx,y \in \mathbb{R}. They can be interpreted as points in the complex plane, with rectangular coordinates xx and yy, just as real numbers xx are thought of as points on the real line.

When real numbers xx are displayed as points (x,0)(x,0) on the real axis, we write x=(x,0)x=(x,0); Complex numbers of the form (0,y)(0,y) correspond to points on the yy axis and are called pure imaginary numbers when y≠0y\neq 0 . The yy axis is then referred to as the imaginary axis.

complex_plane
complex plane

The set of all complex numbvers is denoted by C\mathbb{C} :

C={(x,y):x,y∈R}\mathbb{C} = \left\{(x,y) : x,y \in \mathbb{R} \right\}

The real numbers xx and yy are, morever, known as real and imaginary parts of zz, respectively, and we write Re z=x,  Im z=b\text{Re}\,z=x,\,\,\text{Im}\,z=b .

Then addition z1+z2z_1+z_2 and multiplicationz1z2z_1z_2 of two complex numbers on C\mathbb{C}

z1=(x1,y1),z2=(x2,y2)z_1=(x_1,y_1), z_2=(x_2,y_2)

are defined by:

z1+z2=(x1+x2,y1+y2)z1z2=(x1x2−y1y2,x1y2+x2y1)z_1+z_2=(x_1+x_2,y_1+y_2) \\ z_1z_2=(x_1x_2-y_1y_2, x_1y_2+x_2y_1)

The operation defined by the means above is compatible with real numbers, since:

(x1,0)+(x2,0)=(x1+x2,0)(x1,0)(x2,0)=(x1x2,0)(x_1,0)+(x_2,0)=(x_1+x_2,0)\\ (x_1,0)(x_2,0)=(x_1x_2,0)

The complex number system is, therefore, a natural extension of the real number system.

Any complex number z=(x,y)z=(x,y) can be written as z=(x,0)+(0,y)z=(x,0)+(0,y), and it is easy to see that (0,y)=(1,0)(y,0)(0,y)=(1,0)(y,0). Hence:

z=(x,0)+(1,0)(y,0)=Re z+(Im z)i\begin{aligned} z&=(x,0)+(1,0)(y,0) \\ &=\text{Re}\,z+(\text{Im}\,z)i \end{aligned}

And if we think of a real number as eigher xx or (x,0)(x,0) and let ii denote the pure imaginary number (0,1)(0,1), it is clear that:

z=x+iyz=x+iy

Also, with the convention that z2=zz,z3=z2zz^2=zz, z^3 = z^2z , etc., we gain:

i2=(0,1)(0,1)=(−1,0)=−1i^2=(0,1)(0,1)=(-1,0)=-1

Also notice any complex number times zero is zero, since:

z 0=(x+iy)(0+0 i)=0+0 i=0z\,0=(x+iy)(0+0\,i)=0+0\,i=0

Suppose z=x+yi,x,y∈Rz=x+yi, x,y \in \mathbb{R}. The real part zz, denoted Re z\text{Re}\,z , is defined by Re z=x\text{Re}\,z=x. The imaginary part zz, denoted Im z\text{Im}\,z , is defined by Im z=y\text{Im}\,z=y. Thus for every complex number zz we have

z=Re z+(Im z)iz=\text{Re}\,z+(\text{Im}\,z)i

For any real number x∈Rx \in \mathbb{R} , we could consider xx as x+0ix+0i in the form of complex numbers. Thus we could think of R\mathbb{R} as a subset of C\mathbb{C} .

Defination complex conjugate, absolute value

Suppose z∈Cz \in \mathbb{C} .

  • The complex conjugate of z∈Cz \in \mathbb{C} , denoted zˉ\bar{z} , is defined by

zˉ=Re z−(Im z)i\bar{z}=\text{Re}\,z-(\text{Im}\,z)i

  • The modulus of a complex number zz, denoted ∣z∣\lvert z \rvert , is defined by

∣z∣=(Re z)2+(Im z)2\lvert z \rvert = \sqrt{\left(\text{Re\,z}\right)^2+\left(\text{Im}\,z\right)^2}

Note that:

  • ∣z∣\lvert z \rvert is a nonnegative number for every z∈Cz \in \mathbb{C}
  • z=∣z∣z=\lvert z \rvert if and only if zz is a real number

It is easy to find:

zz‾=∣z∣2z\overline{z} = \lvert z \rvert^2

Basic Properties

Properties of complex arithmetic

  • commutativity
  • associativity
  • distributivity
  • additive identity
  • additive inverse
  • multiplicative inverse

F\mathbb{F} stands for either R\mathbb{R} or C\mathbb{C} (as they are examples of fields). Elements of F\mathbb{F} are called scalars, which is a fancy word for “number” often used when an object is emphasized as a number, as opposed to a vector.

Further Properties

Expoenential Form