 # 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}\$. 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}\$

### 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^{*}\$

## Site Index

### Latest Item on Science Library:

The most recent article is:

Air Resistance and Terminal Velocity

View this item in the topic:

Mechanics

and many more articles in the subject:

### Physics

Physics is the science of the very small and the very large. Learn about Isaac Newton, who gave us the laws of motion and optics, and Albert Einstein, who explained the relativity of all things, as well as catch up on all the latest news about Physics, on ScienceLibrary.info. ### Great Scientists

#### David Hilbert

1862 - 1943

David Hilbert, 1862 - 1943, was German, and is considered one of the greatest mathematicians ever, leaving a broad legacy in mathematics, physics and philosophy. 