WIP draft of complex-numbers concept

Author: Colin Leach

concepts.wip/complex-numbers/.meta/config.json

@@ -0,0 +1,7 @@
+{
+  "authors": [
+    "colinleach"
+  ],
+  "contributors": [],
+  "blurb": "Complex numbers are very widely used in engineering and the physical sciences. They are a standard numerical type in Julia."
+}

concepts.wip/complex-numbers/about.md

@@ -0,0 +1,248 @@
+# About
+
+`Complex numbers` are not complicated.
+They just need a less alarming name.
+
+They are so useful, especially in engineering and science, that Julia includes [complex numbers][complex] as standard numeric types alongside integers and floating-point numbers.
+
+## Basics
+
+A `complex` value in Julia is essentially a pair of numbers: usually but not always floating-point.
+These are called the "real" and "imaginary" parts, for unfortunate historical reasons.
+Again, it is best to focus on the underlying simplicity and not the strange names.
+
+To create complex numbers from two real numbers, just add the suffix `im` to the imaginary part.
+
+```julia-repl
+julia> z = 1.2 + 3.4im
+1.2 + 3.4im
+
+julia> typeof(z)
+ComplexF64 (alias for Complex{Float64})
+
+julia> zi = 1 + 2im
+1 + 2im
+
+julia> typeof(zi)
+Complex{Int64}
+```
+
+Thus there are various `Complex` types, derived from the corresponding integer or floating-point type.
+
+To create a complex number from real variables, the above syntax will not work.
+Writing `a + bim` confuses the parser into thinking `bim` is a (non-existent) variable name.
+
+Writing `b*im` is possible, but the preferred method uses the `complex()` function.
+
+```julia-repl
+julia> a = 1.2; b = 3.4; complex(a, b)
+1.2 + 3.4im
+```
+
+You may wonder, why `im` rather than `i` or `j`?
+
+Most engineers are happy with `j`.
+Most scientists and mathematicians prefer the mathematical notation `i` for _imaginary_, but that notation conflicts with the use of `i` to mean _current_ in Electrical Engineering.
+
+The [manual][complex] states that "Using mathematicians' i or engineers' j for this global constant was rejected since they are such popular index variable names".
+
+By happy coincidence, this also avoids long arguments between engineers and everyone else.
+Feel free to form your own judgement about whether this influenced the decision.
+
+To access the parts of a complex number individually:
+
+```julia-repl
+julia> z = 1.2 + 3.4im
+1.2 + 3.4im
+
+julia> real(z)
+1.2
+
+julia> imag(z)
+3.4
+```
+
+Either part can be zero and mathematicians may then talk of the number being "wholly real" or "wholly imaginary".
+However, it is still a complex number in Julia.
+
+```julia-repl
+julia> zr = 1.2 + 0im
+1.2 + 0.0im
+
+julia> typeof(zr)
+ComplexF64 (alias for Complex{Float64})
+
+julia> zi = 3.4im
+0.0 + 3.4im
+
+julia> typeof(zi)
+ComplexF64 (alias for Complex{Float64})
+```
+
+You may have heard that "`i` (or `j`) is the square root of -1".
+
+For now, all this means is that the imaginary part _by definition_ satisfies the following equality:
+
+```julia-repl
+julia> 1im * 1im == -1
+true
+```
+
+This is a simple idea, but it leads to interesting consequences.
+
+## Arithmetic
+
+Most of the [`operators`][operators] used with floats and integers also work with complex numbers:
+
+```julia-repl
+julia> z1 = 1.5 + 2im
+1.5 + 2.0im
+
+julia> z2 = 2 + 1.5im
+2.0 + 1.5im
+
+julia> z1 + z2  # addition
+3.5 + 3.5im
+
+julia> z1 * z2  # multiplication
+0.0 + 6.25im
+
+julia> z1 / z2  # division
+0.96 + 0.28im
+
+julia> z1^2  # exponentiation
+-1.75 + 6.0im
+
+julia> 2^z1  # another exponentiation
+0.5188946835878313 + 2.7804223253571183im
+```
+
+Explaining the rules for complex number multiplication and division is out of scope for this concept (_and you are unlikely to have to perform those operations "by hand" very often_).
+
+Any [mathematical][math-complex] or [electrical engineering][engineering-complex] introduction to complex numbers will cover this, should you want to dig into the topic.
+
+Alternatively, Exercism has a `Complex Numbers` practice exercise where you can implement a complex number class with these operations from first principles.
+
+Integer division is (mostly) ___not___ possible on complex numbers, so the `//`, `÷` and `%` operators will fail for the complex number type.
+An exception is using `//` with complex _integers_ to get complex rational numbers, but this is probably not often useful.
+
+## Functions
+
+Most mathematical functions will work with complex inputs.
+
+However, providing real inputs and expecting a complex output will not usually work.
+This is a performance optimization that should make more sense after looking at the Multiple Dispatch concept.
+
+```julia-repl
+julia> sin(1.5 + 2.0im)
+3.752771340479298 + 0.25655395609048176im
+
+julia> sqrt(-1)
+ERROR: DomainError with -1.0:
+sqrt was called with a negative real argument but will only return a complex result if called with a complex argument. Try sqrt(Complex(x)).
+
+julia> sqrt(-1 + 0im)
+0.0 + 1.0im
+```
+
+There are several functions, in addition to `real()` and `imag()`, with particular relevance for complex numbers.
+Please skip over these if they make no sense to you!
+
+- `conj()` simply flips the sign of the imaginary part of a complex number (_from + to - or vice-versa_).
+    - Because of the way complex multiplication works, this is more useful than you might think.
+- `abs(<complex number>)` is guaranteed to return a real number with no imaginary part.
+- `abs2(<complex number>)` returns the square of `abs(<complex number>)`: quicker to calculate than `abs()`, and often what a calculation needs.
+- `angle(<complex number>)` returns the phase angle in radians.
+
+```julia-repl
+julia> z1
+1.5 + 2.0im
+
+julia> conj(z1)
+1.5 - 2.0im
+
+julia> abs(z1)
+2.5
+
+julia> abs2(z1)
+6.25
+
+julia> angle(z1)
+0.9272952180016122
+```
+A partial explanation, for the mathematically-minded _(again, feel free to skip)_:
+
+- The `(real, imag)` representation of `z1` in effect uses Cartesian coordinates on the complex plane.
+- The same complex number can be represented in `(r, θ)` notation, using polar coordinates.
+- Here, `r` and `θ` are given by `abs(z1)` and `angle(z1)` respectively.
+
+
+
+Here is an example using some constants:
+
+```julia-repl
+julia> euler = exp(1im * π)
+-1.0 + 1.2246467991473532e-16im
+
+julia> real(euler)
+-1.0
+
+julia> round(imag(euler), digits=15)  # round to 15 decimal places
+0.0
+```
+
+So a simple expression with three of the most important constants in nature `e`, `i` (or `j`) and `pi` gives the result `-1`.
+Some people believe this is the most beautiful result in all of mathematics.
+It dates back to around 1740.
+
+-----
+
+## Optional section: a Complex Numbers FAQ
+
+This part can be skipped, unless you are interested.
+
+### Isn't this some strange new piece of pure mathematics?
+
+It was strange and new in the 16th century.
+
+500 years later, it is central to most of engineering and the physical sciences.
+
+### Why would anyone use these?
+
+It turns out that complex numbers are the simplest way to describe anything that rotates or anything with a wave-like property.
+So they are used widely in electrical engineering, audio processing, physics, computer gaming, and navigation - to name only a few applications.
+
+You can see things rotate.
+Complex numbers may not make the world go round, but they are great for explaining _what happens_ as a result of the world going round: look at any satellite image of a major storm.
+
+Less obviously, sound is wave-like, light is wave-like, radio signals are wave-like, and even the economy of your home country is at least partly wave-like.
+
+A lot of this wave processing can be done with trig functions (`sin()` and `cos()`) but that gets messy quite quickly.
+
+Complex exponentials are ___much___ easier to work with.
+
+### But I don't need complex numbers!
+
+Only true if you are living in a cave and foraging for your food.
+
+If you are reading this on any sort of screen, you are utterly dependent on some useful 20th-Century advances made through the use of complex numbers.
+
+1. __Semiconductor chips__.
+    - These make no sense in classical physics and can only be explained (and designed) by quantum mechanics (QM).
+    - In QM, everything is complex-valued by definition. (_it's waveforms all the way down_)
+
+2. __The Fast Fourier Transform algorithm__.
+    - FFT is an application of complex numbers, and it is in _everything_ connected to sound transmission, audio processing, photos, and video.
+
+    - MP3 and other audio formats use FFT for compression, ensuring more audio can fit within a smaller storage space.
+    - JPEG compression and MP4 video, among many other image and video formats, also use FFT for compression.
+
+    - FFT is also deployed in the digital filters that allow cellphone towers to separate your personal cell signal from everyone else's.
+
+So, you are probably using technology that relies on complex number calculations thousands of times per second.
+
+
+[complex]: https://docs.julialang.org/en/v1/manual/complex-and-rational-numbers/#Complex-Numbers
+[math-complex]: https://www.nagwa.com/en/videos/143121736364/
+[engineering-complex]: https://www.khanacademy.org/science/electrical-engineering/ee-circuit-analysis-topic/ee-ac-analysis/v/ee-complex-numbers
+[operators]: https://docs.python.org/3/library/stdtypes.html#numeric-types-int-float-complex

concepts.wip/complex-numbers/introduction.md

@@ -0,0 +1 @@
+# Introduction

concepts.wip/complex-numbers/links.json

@@ -0,0 +1,18 @@
+[
+  {
+    "url": "https://docs.julialang.org/en/v1/manual/complex-and-rational-numbers/#Complex-Numbers",
+    "description": "Complex number section in the Julia manual."
+  },
+  {
+    "url": "https://en.wikipedia.org/wiki/Complex_number",
+    "description": "Wikipedia: Complex Number."
+  },
+  {
+    "url": "https://www.nagwa.com/en/videos/143121736364/",
+    "description": "Mathematical view of complex numbers (video)."
+  },
+  {
+    "url": "https://www.khanacademy.org/science/electrical-engineering/ee-circuit-analysis-topic/ee-ac-analysis/v/ee-complex-numbers",
+    "description": "Engineering view of complex numbers, from the Khan Academy."
+  }
+]