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 $√{-1}$.

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:

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

|$z$| = |${x+iy}$| = $√{x^2+y^2}$

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.

$z_1+z_2=(a_1+ib_1)+(a_2+ib_2)=(a_1+a_2)+i(b_1+b_2)$

$z_1-z_2=(a_1+ib_1)-(a_2+ib_2)=(a_1-a_2)+i(b_1-b_2)$

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

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

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

${z_1}/{z_2}={z_1⋅z_2^{*}}/{|z_2|^2}$

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

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

$(z_1 ⋅ z_2)^{*} = z_1^{*}⋅z_2^{*}$

$z⋅z^{*} = |z|^2$

$(z^n)^{*} = (z^{*})^n$, $n ∈ ℤ$

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

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Mathematics is the most important tool of science. The quest to understand the world and the universe using mathematics is as old as civilisation, and has led to the science and technology of today. Learn about the techniques and history of mathematics on ScienceLibrary.info.

1791 - 1871

Charles Babbage was a polymath, who is most famous for his development of mechanical computational machines.

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