RedCrab Math Tutorial

Complex Numbers

With quadratic equations, there is not always a real solution. For example, the equation
x2 + 1 = 0 or just x2 = -1
In order to be able to count on solutions of such equations, the mathematician Leonard Euler introduced a new imaginary number and designated it with the letter i.
A complex number z consists of a real part a and an imaginary part b. The imaginary part is marked with the letter i

z = a + b i

The imaginary unit i has the property

i² = -1

The value of a complex number corresponds to the length of the vector z in the Argand plane.

Graphical interpretation of complex numbers

For the graphical interpretation of complex numbers the Argand plane is used. The Argand plane is a special form of a normal Cartesian coordinate system. The difference is in the name of the axles.
The real part of the complex number is displayed on the x-axis of the argand plane. The axis is called the real axis.
The imaginary part of the complex number is displayed on the y-axis of the argand plane. The axis is called the imaginary axis.

The following figure shows a graphical representation of a complex number


Addition and subtraction of complex numbers

The addition and subtraction of complex numbers corresponds to the addition and subtraction of the vectors. The real and imaginary components are added or subtracted

z1 + z2 = x1 + x2 + i (y1+ y2)

z1 + z2 = x1 - x2 + i (y1- y2)


(1+2i) + (4+3i) = (1+4) + i·(2+3) = 5+5i

(1+2i) + 8i = 1+10i

(1-2i) + (4+2i) = 5

The following figure shows an addition and graphical display in the RedCrab Calculator

Multiplication of complex numbers

The multiplication is done by multiplying the parentheses.

z1 · z2 = (x1 + y1 i) · (x2 + y2 i)

= x1 · x2 - y1 · y2 + i (x1 · y2 + y1 · x2)


(1+2i) · (4+3i) = (1·4 - 2·3) + i·(1·3 + 2·4) = -2+11i

The following figure shows the multiplication and graphic display in the RedCrab Calculator

Conjugate a complex number

To divide a complex number, you need the conjugate of a complex number.
  • The conjugate to z = a + bi is written z = a - bi

  • Property of the operation:  and  

In the following example we search the sum of    and    that is
  • Sum:

  • Conjugate:


Division of complex numbers

Complex numbers are divided by multiplying the numerator and denominator by the complex conjugate of the denominator.
Example for calculating the quotient: (3-2i) / (4+5i)
  • The real part is:
  • The imaginary part is:
Elementary complex functions RedCrab Calculator
    Calculator Syntax

Complex number

z = x + y ·i z = x + yi

Real part

Re (z) = x Re (z)

Imaginary part

Im (z) = y Im (z)

Conjugate complex number 

z = x - y ·i Conjugate (z)


|z| = √x² + y² Magnitude (z)


1 / z
Eexponential function ez = ex · cos y + (ex sin y) · i ez




ln z= 1/2 ln (x² + y²) + atan (y / x) · i Ln (z)


sin z = sin x · cosh y + (cos x · sinh y · i) Sin (z)


con z = cos x · cosh y + (sin x · sinh y · i) Cos (z)

Sinus hyperbolic

sinh z = sinh x · cos y + (cosh x · sin y · i) Sinh (z)

Cosine hyperbolic

cosh z = cosh x · cos y - (sinh x · sin y · i) Cosh (z)


Tan (z)




Integer and Real Numbers
Complex Numbers
Roots and Power
Percentage Calculation
Interest Calculation
Absolute Value of a Number
Euclidean division
Modulo - Remainder of a division

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