Complex Numbers 
With quadratic equations, there is not always a real solution. For example, the equation
x^{2} + 1 = 0 or just x^{2} = 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

$The\; imaginary\; unitihas\; the\; property$ 
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 xaxis of the argand plane. The axis is called the real axis. 
The imaginary part of the complex number is displayed on the yaxis 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 
z_{1} + z_{2} = x_{1} + x_{2} + i (y_{1}+ y_{2})
z_{1} + z_{2} = x_{1}  x_{2} + i (y_{1} y_{2})

Excamples 
(1+2i) + (4+3i) = (1+4) + i·(2+3) = 5+5i
(1+2i) + 8i = 1+10i
(12i) + (4+2i) = 5


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

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Multiplication of complex numbers 
The multiplication is done by multiplying the parentheses. 
z_{1} · z_{2} = (x_{1} + y_{1} i) · (x_{2} + y_{2} i)
= x_{1} · x_{2}  y_{1} · y_{2} + i (x_{1} · y_{2} + y_{1} · x_{2})

Example 
(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. 

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: (32i) / (4+5i) 

 The real part is:

 The imaginary part is:


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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) 
Value 
z = √x² + y² 
Magnitude (z) 
Reciprocal 

1 / z 
Eexponential function 
e^{z} = e^{x} · cos y + (e^{x} sin y) · i 
e^{z} 
Root 

√z 
Logarithm 
ln z= 1/2 ln (x² + y²) + atan (y / x) · i 
Ln (z) 
Sine 
sin z = sin x · cosh y + (cos x · sinh y · i) 
Sin (z) 
Cosine 
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) 
Tangent 

Tan (z) 





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