algorithm/javascript-algorithms
1
0
Fork 0
mirror of https://github.com/trekhleb/javascript-algorithms.git synced 2026-03-13 08:51:02 +08:00
Code Issues Packages Projects Releases Wiki Activity

Chore(math-translation-FR-fr): a pack of translations for the math section (#558)

Browse Source 
* chore(factorial): translation fr-FR

* feat(math-translation-fr-FR): fast powering

* feat(math-translation-fr-FR): fibonacci numbers

* chore(math-translation-fr-FR): bits

* chore(math-translation-fr-FR): complex number

* chore(math-translation-fr-FR): euclidean algorithm

* chore(math-translation-fr-FR): fibonacci number

* chore(math-translation-fr-FR): fourier transform

* chore(math-translation-fr-FR): fourier transform WIP

* chore(math-translation-fr-FR): fourier transform done

* chore(math-translation-fr-FR): fourier transform in menu
This commit is contained in:
Loïc TRUCHOT
2020-10-05 21:13:47 +02:00
committed by GitHub
parent 07bc4a4b97
commit d6b8dd394a
15 changed files with 1200 additions and 328 deletions
Download Patch File Download Diff File Expand all files Collapse all files

73
src/algorithms/math/complex-number/README.md
View File

@@ -1,11 +1,14 @@
# Complex Number
A **complex number** is a number that can be expressed in the
_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*.
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_.
![Complex Number](https://www.mathsisfun.com/numbers/images/complex-example.svg)
@@ -13,56 +16,56 @@ A Complex Number is a combination of a Real Number and an Imaginary Number:
![Complex Number](https://www.mathsisfun.com/numbers/images/complex-number.svg)
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.
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
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
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 |
| 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*.
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 Number](https://upload.wikimedia.org/wikipedia/commons/a/af/Complex_number_illustration.svg)
> 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).
> 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
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
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.
![Polar Form](https://upload.wikimedia.org/wikipedia/commons/7/7a/Complex_number_illustration_modarg.svg)
The *absolute value* (or modulus or magnitude) of a complex number `z = x + yi` is:
The _absolute value_ (or modulus or magnitude) of a complex number `z = x + yi` is:
![Radius](https://wikimedia.org/api/rest_v1/media/math/render/svg/b59629c801aa0ddcdf17ee489e028fb9f8d4ea75)
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
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`:
![Phase](https://wikimedia.org/api/rest_v1/media/math/render/svg/7cbbdd9bb1dd5df86dd2b820b20f82995023e566)
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
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:
![Polar Form](https://wikimedia.org/api/rest_v1/media/math/render/svg/b03de1e1b7b049880b5e4870b68a57bc180ff6ce)
@@ -107,7 +110,7 @@ To subtract two complex numbers we subtract each part separately:
### Multiplying
To multiply complex numbers each part of the first complex number gets multiplied
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" (
@@ -138,7 +141,7 @@ Use this rule:
**Example**
```text
(3 + 2i)(1 + 7i)
(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)
@@ -164,7 +167,7 @@ ______
5 − 3i = 5 + 3i
```
On the complex plane the conjugate number will be mirrored against real axes.
On the complex plane the conjugate number will be mirrored against real axes.
![Complex Conjugate](https://upload.wikimedia.org/wikipedia/commons/6/69/Complex_conjugate_picture.svg)
@@ -172,7 +175,7 @@ On the complex plane the conjugate number will be mirrored against real axes.
The conjugate is used to help complex division.
The trick is to *multiply both top and bottom by the conjugate of the bottom*.
The trick is to _multiply both top and bottom by the conjugate of the bottom_.
**Example**
@@ -207,7 +210,7 @@ In the previous example, what happened on the bottom was interesting:
(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:
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