Merge branch 'exercism:main' into fft

Author: depial

concepts/complex-numbers/about.md

@@ -193,7 +193,11 @@ So a simple expression with three of the most important constants in nature `e`,
 Some people believe this is the most beautiful result in all of mathematics.
 It dates back to around 1740.
 
-The polar `(r, θ)` notation is so useful, that there are built-in functions `cis` (short for cos(x) + isin(x)) and `cispi` (short for cos(πx) + isin(πx)) which can help in constructing it more efficiently.
+The polar `(r, θ)` notation is so useful, that there are built-in functions `cis` (short for `cos(x) + isin(x)`) and `cispi` (short for `cos(πx) + isin(πx)`) which can help in constructing it more efficiently.
+
+The usefulness of polar notation is found in Euler's elegant formula, `ℯ^(iθ) = cos(θ) + isin(θ) = x + iy`, where `|x + iy| = 1`.
+With `|x + iy| = r`, we have the more general polar form of `r * ℯ^(iθ) = r * (cos(θ) + isin(θ)) = x + iy`.
+Note that the exponential form, in particular, is compact and easy to manipulate.
 
 ```julia-repl
 julia> exp(1im * π) ≈ cis(π) ≈ cispi(1)
@@ -220,6 +224,14 @@ julia> cispi(θ / π)  # θ/π == 1/2
 0.0 + 1.0im
 ```
 
+Incidentally, this makes complex numbers very useful for performing rotations and radial displacements in 2D.
+
+For rotations, the complex number `z = x + iy`, can be rotated an angle `θ` about the origin with a simple multiplication: `z * ℯ^(iθ)`.
+Note that the `x` and `y` here are just the usual coordinates on the real 2D Cartesian plane, and a positive angle results in a *counterclockwise* rotation, while a negative angle results in a *clockwise* one.
+
+Likewise simply, a radial displacement `Δr` can be made by adding it to the magnitude `r` of a complex number in the polar form (eg. `z = r * ℯ^(iθ)` -> `z' = (r + Δr) * ℯ^(iθ)`).
+Note how the angular part stays the same and only the magnitude, `r`, is varied, as expected.
+
 -----
 
 ## Optional section: a Complex Numbers FAQ

concepts/complex-numbers/introduction.md

@@ -159,7 +159,11 @@ julia> round(imag(euler), digits=15)  # round to 15 decimal places
 0.0
 ```
 
-The polar `(r, θ)` notation is so useful, that there are built-in functions `cis` (short for cos(x) + isin(x)) and `cispi` (short for cos(πx) + isin(πx)) which can help in constructing it more efficiently.
+The polar `(r, θ)` notation is so useful, that there are built-in functions `cis` (short for `cos(x) + isin(x)`) and `cispi` (short for `cos(πx) + isin(πx)`) which can help in constructing it more efficiently.
+
+The usefulness of polar notation is found in Euler's elegant formula, `ℯ^(iθ) = cos(θ) + isin(θ) = x + iy`, where `|x + iy| = 1`.
+With `|x + iy| = r`, we have the more general polar form of `r * ℯ^(iθ) = r * (cos(θ) + isin(θ)) = x + iy`.
+Note that the exponential form, in particular, is compact and easy to manipulate.
 
 ```julia-repl
 julia> exp(1im * π) ≈ cis(π) ≈ cispi(1)
@@ -185,3 +189,11 @@ julia> cis(θ)
 julia> cispi(θ / π)  # θ/π == 1/2
 0.0 + 1.0im
 ```
+
+Incidentally, this makes complex numbers very useful for performing rotations and radial displacements in 2D.
+
+For rotations, the complex number `z = x + iy`, can be rotated an angle `θ` about the origin with a simple multiplication: `z * ℯ^(iθ)`.
+Note that the `x` and `y` here are just the usual coordinates on the real 2D Cartesian plane, and a positive angle results in a *counterclockwise* rotation, while a negative angle results in a *clockwise* one.
+
+Likewise simply, a radial displacement `Δr` can be made by adding it to the magnitude `r` of a complex number in the polar form (eg. `z = r * ℯ^(iθ)` -> `z' = (r + Δr) * ℯ^(iθ)`).
+Note how the angular part stays the same and only the magnitude, `r`, is varied, as expected.

config.json

@@ -227,7 +227,7 @@
           "function-composition",
           "functions"
         ],
-        "status": "wip"
+        "status": "beta"
       },
       {
         "slug": "old-annalyns-infiltration",