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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 $√{-25} = √{25(-1)} = √{(5^2i^2)} = 5i$

Modulus

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.

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

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

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

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

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

${z_1}/{z_2}={z_1⋅z_2^{*}}/{|z_2|^2}$

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

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