Complex number

A complex number is defined as a pair of real numbers ${\displaystyle z=(x,y)}$ where the following operations are defined:

• Addition: ${\displaystyle z_1+z_2=(x_1+x_2,y_1+y_2)}$
• Multiplication: ${\displaystyle z_1{z}_2=(x_1{y}_1-x_2{y}_2,x_2{y}_1+x_2{y}_1)}$

When the second element equals zero the complex numbers behaves as real numbers. That's why the first element of the complex number is known as the real part and the second element as the imaginary part.

Multiplying ${\displaystyle (0,1)(0,1)}$ we get ${\displaystyle (-1,0)}$. Since no real number is the square root of -1, we can now understand why the second element is the imaginary part.

An alternate (and more used) notation is ${\displaystyle z=x+iy}$. From the previous paragraph we get: ${\displaystyle i^2=-1}$.

Using this notation and the definitions above we can deduce all basic operations on complex numbers:

• Addition:

${\displaystyle z_1+z_2=(x_1+x_2)+(y_1+y_2)i}$

• Subtraction:

${\displaystyle z_1-z_2=(x_1+x_2)+(-y_1-y_2)i}$

• Multiplication:

${\displaystyle z_1{z}_2=(x_1{y}_1-x_2{y}_2)+(x_1{y}_2+x_2{y}_1)i}$

• Division:

${\displaystyle \frac{z_1}{z_2}=\frac{x_1{y}_1+x_2{y}_2}{x_2^2+y_2^2}+\frac{x_2{y}_1-x_1{y}_2}{x_2^2+y_2^2}i}$

• Square root:

${\displaystyle \sqrt{x+iy}=\sqrt{\frac{|x+iy|+x}{2}}\pm i\sqrt{\frac{|x+iy|-x}{2}}}$

where the sign of the imaginary part of the root is the same as the sign of the imaginary part of the original number and

${\displaystyle |x+iy|=\sqrt{x^2+y^2}}$

External links

• Wikipedia