Science Library - free educational site

Complex Numbers

Complex numbers are a means of dealing with the imaginary concept of 'the square root of minus one'. A negative times a negative results in a positive. In the Real number realm, there is no number which multiplied by itself will result in a negative number.

The equation $x^2 + 1 = 0$ has no solution in traditional mathematics.

However, mathematicians like Leonard Euler in the eighteenth century discovered that by inventing a special set of numbers which uses 'i' to represent $√{-1}$ , a new type of mathematics permitted useful calculations to be made, encompassing an imaginary extension of the three dimensional realm.

Engineering uses complex numbers to describe rotational motion and stresses.

A complex number has two components. One is in the Real Number Set, and one is imaginary: $a + bi$, where $i$ is the representation of $√{-1}$.

Graph of imaginary numbers
Complex numbers consist of two components: the real part and the imaginary.
$$i^2 = -1$$

Example: $4 + √{-25}$ can be expressed as: $4 + 5i$, since $√{-25} = √{25(-1)} = √{(5^2i^2)} = 5i$


The modulus is the absolute value of a real number: |x| = $\{\table x, x≥0; -x, x < 0$

The modulus is the geometric distance of the number x to the origin along the number line.

In the case of complex numbers, the modulus is gien the symbol |z|, and is the distance from the point P(x,y), which represents the complex number z = x + iy, to the origin in the complex plane.

This can be calculated by, yes, you guessed it, Pythagoras:

$|z| = √{(x-0)^2+(y-0)^2} = √{x^2+y^2} = √{Re^2(z)+Im^2(z)}$

|$z$| = |${x+iy}$| = $√{x^2+y^2}$

Operations with complex numbers

To be considered equal, two complex numbers must have equal real parts and equal imaginary parts. Complex numbers cannot be defined as inequalities, since $√{-1}$ cannot be said to be greater than or less than 0.



$λz=λ(a+ib)=(λa)+i(λb)$, where $a$, $b$, $λ$ ∈ ℝ

$z_1⋅z_2 = (a_1+ib_1)⋅(a_2+ib_2) = a_1a_2 + ib_1a_2 + a_1ib_2 + i^2b_1b_2 = (a_1a_2 - b_1b_2) + i(a_1b_2 + a_2b_1)$

$z_1⋅z_2 = (a_1a_2 - b_1b_2) + i(a_1b_2 + a_2b_1)$


Conjugate Complex Numbers

Properties of Conjugates

$(z^{*})^{*} = z$

$(z_1 + z_2)^{*} = z_1^{*}+z_2^{*}$

$(z_1 ⋅ z_2)^{*} = z_1^{*}⋅z_2^{*}$

$z⋅z^{*} = |z|^2$

$(z^n)^{*} = (z^{*})^n$, $n ∈ ℤ$

The conjugate to a complex number has the same real number part and the same magnitude of the imaginary part, but with opposite sign. the conjugate to complex number $z = a + ib$ is $z^{*} = a - ib$:

$z + z^{*} = 2a$

$(z_1 + z_2)^{*} = z_1^{*}+z_2^{*}$

$(z_1 ⋅ z_2)^{*} = z_1^{*}⋅z_2^{*}$

Content © All rights reserved. Created : December 15, 2013 Last updated :February 14, 2016

Latest Item on Science Library:

The most recent article is:


View this item in the topic:

Vectors and Trigonometry

and many more articles in the subject:

Subject of the Week


'Universe' on covers astronomy, cosmology, and space exploration. Learn Science with


Great Scientists

John Herschel

1792 - 1871

John Herschel is the son of William Herschel, and the nephew of Caroline Herschel, two famous astronomers. He continued his father's work, publishing enhanced catalogues of astronomical objects, but was also prolific in many other fields of science and technology, notably as a pioneer of photography.

John Herschel

Quote of the day...

As far as the laws of mathematics refer to reality, they are not certain; and as far as they are certain, they do not refer to reality.

ZumGuy Internet Promotions

ZumGuy Publications and Promotions