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Algorithms and data-structures implemented on JavaScript
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# Complex Number
_Read this in other languages:_
[français](README.fr-FR.md).
A **complex number** is a number that can be expressed in the
form `a + b * i`, where `a` and `b` are real numbers, and `i` is a solution of
the equation `x^2 = −1`. Because no _real number_ satisfies this
equation, `i` is called an _imaginary number_. For the complex
number `a + b * i`, `a` is called the _real part_, and `b` is called
the _imaginary part_.
A Complex Number is a combination of a Real Number and an Imaginary Number:
Geometrically, complex numbers extend the concept of the one-dimensional number
line to the _two-dimensional complex plane_ by using the horizontal axis for the
real part and the vertical axis for the imaginary part. The complex
number `a + b * i` can be identified with the point `(a, b)` in the complex plane.
A complex number whose real part is zero is said to be _purely imaginary_; the
points for these numbers lie on the vertical axis of the complex plane. A complex
number whose imaginary part is zero can be viewed as a _real number_; its point
lies on the horizontal axis of the complex plane.
| Complex Number | Real Part | Imaginary Part | |
| :------------- | :-------: | :------------: | ---------------- |
| 3 + 2i | 3 | 2 | |
| 5 | 5 | **0** | Purely Real |
| −6i | **0** | -6 | Purely Imaginary |
A complex number can be visually represented as a pair of numbers `(a, b)` forming
a vector on a diagram called an _Argand diagram_, representing the _complex plane_.
`Re` is the real axis, `Im` is the imaginary axis, and `i` satisfies `i^2 = −1`.
> Complex does not mean complicated. It means the two types of numbers, real and
> imaginary, together form a complex, just like a building complex (buildings
> joined together).
## Polar Form
An alternative way of defining a point `P` in the complex plane, other than using
the x- and y-coordinates, is to use the distance of the point from `O`, the point
whose coordinates are `(0, 0)` (the origin), together with the angle subtended
between the positive real axis and the line segment `OP` in a counterclockwise
direction. This idea leads to the polar form of complex numbers.
The _absolute value_ (or modulus or magnitude) of a complex number `z = x + yi` is:
The argument of `z` (in many applications referred to as the "phase") is the angle
of the radius `OP` with the positive real axis, and is written as `arg(z)`. As
with the modulus, the argument can be found from the rectangular form `x+yi`:
Together, `r` and `φ` give another way of representing complex numbers, the
polar form, as the combination of modulus and argument fully specify the
position of a point on the plane. Recovering the original rectangular
co-ordinates from the polar form is done by the formula called trigonometric
form:
Using Euler's formula this can be written as:
## Basic Operations
### Adding
To add two complex numbers we add each part separately:
```text
(a + b * i) + (c + d * i) = (a + c) + (b + d) * i
```
**Example**
```text
(3 + 5i) + (4 − 3i) = (3 + 4) + (5 − 3)i = 7 + 2i
```
On complex plane the adding operation will look like the following:
### Subtracting
To subtract two complex numbers we subtract each part separately:
```text
(a + b * i) - (c + d * i) = (a - c) + (b - d) * i
```
**Example**
```text
(3 + 5i) - (4 − 3i) = (3 - 4) + (5 + 3)i = -1 + 8i
```
### Multiplying
To multiply complex numbers each part of the first complex number gets multiplied
by each part of the second complex number:
Just use "FOIL", which stands for "**F**irsts, **O**uters, **I**nners, **L**asts" (
see [Binomial Multiplication](ttps://www.mathsisfun.com/algebra/polynomials-multiplying.html) for
more details):
- Firsts: `a × c`
- Outers: `a × di`
- Inners: `bi × c`
- Lasts: `bi × di`
In general it looks like this:
```text
(a + bi)(c + di) = ac + adi + bci + bdi^2
```
But there is also a quicker way!
Use this rule:
```text
(a + bi)(c + di) = (ac − bd) + (ad + bc)i
```
**Example**
```text
(3 + 2i)(1 + 7i)
= 3×1 + 3×7i + 2i×1+ 2i×7i
= 3 + 21i + 2i + 14i^2
= 3 + 21i + 2i − 14 (because i^2 = −1)
= −11 + 23i
```
```text
(3 + 2i)(1 + 7i) = (3×1 − 2×7) + (3×7 + 2×1)i = −11 + 23i
```
### Conjugates
We will need to know about conjugates in a minute!
A conjugate is where we change the sign in the middle like this:
A conjugate is often written with a bar over it:
```text
______
5 − 3i = 5 + 3i
```
On the complex plane the conjugate number will be mirrored against real axes.
### Dividing
The conjugate is used to help complex division.
The trick is to _multiply both top and bottom by the conjugate of the bottom_.
**Example**
```text
2 + 3i
------
4 − 5i
```
Multiply top and bottom by the conjugate of `4 − 5i`:
```text
(2 + 3i) * (4 + 5i) 8 + 10i + 12i + 15i^2
= ------------------- = ----------------------
(4 − 5i) * (4 + 5i) 16 + 20i − 20i − 25i^2
```
Now remember that `i^2 = −1`, so:
```text
8 + 10i + 12i − 15 −7 + 22i −7 22
= ------------------- = -------- = -- + -- * i
16 + 20i − 20i + 25 41 41 41
```
There is a faster way though.
In the previous example, what happened on the bottom was interesting:
```text
(4 − 5i)(4 + 5i) = 16 + 20i − 20i − 25i
```
The middle terms `(20i − 20i)` cancel out! Also `i^2 = −1` so we end up with this:
```text
(4 − 5i)(4 + 5i) = 4^2 + 5^2
```
Which is really quite a simple result. The general rule is:
```text
(a + bi)(a − bi) = a^2 + b^2
```
## References
- [Wikipedia](https://en.wikipedia.org/wiki/Complex_number)
- [Math is Fun](https://www.mathsisfun.com/numbers/complex-numbers.html)