Some cool math explained with colors.

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Color Maths

Because it helps to know which part of an equation does what.

Bayes' Theorem

$$Pr(H\mid E)=\frac{Pr(E\mid H)Pr(H)}{Pr(E\mid H)Pr(H)+Pr(E\mid \mathrm{\neg }H)Pr(\mathrm{\neg }H)}$$

The $\text{chance}$ $\text{evidence}$ is real (supports $\text{a hypothesis}$) and is the $\text{chance of a true positive}$ $\text{among}$ all positives ($\text{true}$ or $\text{false}$).

Combination

$$C(n,k)=\frac{P(n,k)}{k!}$$

$\text{To group (combine)}$ $k$ $\text{items from}$ $n$ $\text{choices}$, $\text{count the specific permutations}$ and $\text{consolidate the reorderings within the group}$.

Sines

Sine is a fundamental wave function, appearing in geometry, calculus, physics, and signal processing.

Sine (Unit Circle)

$$\sin (\theta )=\frac{\text{height}}{\text{radius}}$$

Pick an $\text{angle on the unit circle}$: $\text{sine}$ is the $\text{vertical coordinate}$ as a $\text{percent of the maximum}$.

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Sine (Series Definition)

$$\sin (x)=x-\frac{x^3}{3!}+\frac{x^5}{5!}+\dots$$

$\text{Sine is}$ $\text{an initial impulse}$, $\text{with a restoring force}$, $\text{with a restoring force on the restoring force}$, $\text{and so on}$.

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Sine (Geometric Definition)

$$\sin (\theta )=\frac{\text{opposite}}{\text{hypotenuse}}$$

In a right triangle, pick $\text{an angle}$: $\text{sine}$ is the $\text{opposite side}$ as a $\text{percent of the largest side}$.

Radian

$$x^{\circ }$$

$\text{Degrees see}$ $\text{angles}$ $\text{from the viewer’s perspective (0-360 scale)}$

$$\theta =\frac{s}{r}$$

$\text{Radians measure}$ $\text{the mover’s path}$
$\text{in terms of the radius}$.

Pythagorean Theorem

$$c^2=a^2+b^2$$

When an $\text{object}$ spans $\text{perpendicular}$ $\text{directions}$, $\text{its}$ $\text{area}$ is the $\text{combined}$ $\text{area}$ of $\text{each}$ $\text{part}$.

Permutation

$$P(n,k)=\frac{n!}{(n-k)!}$$

$\text{To order (permute)}$ $k$ $\text{items from}$ $n$ $\text{choices}$,
$\text{start counting all orderings}$ and
$\text{stop counting after}$ $k$ $\text{items are found}$.

Imaginary Number

A maybe different way to think about imaginary numbers:

• What's an imaginary number?
  • A number pointing sideways (North/South) instead of the typical East/West number line.
• What does i mean?
  • $i$, by itself, points North. Multiplying by $i$ rotates you $90$ degrees. $2$ rotations points you backwards $(i\cdot i=-1)$, $4$ rotations spins you around fully $(i^4=1)$
    

$$1\cdot i^2=-1$$$$i=\sqrt{-1}$$

$\text{Facing forward}$, $\text{two 90-degree rotations}$ is $\text{backward}$.
$\text{An imaginary number}$ is $\text{halfway}$ $\text{backward}$.

LaPlace Transform

$$e^{-st}=e^{-(a+bi)t}=e^{-at}\cdot e^{-bit}$$

A $\text{complex exponential spiral}$ has an implied $\text{decay and spin rate}$

$$F(s)={\int }_0^{\mathrm{\infty }}f(t)\cdot e^{-st}dt$$

$\text{To measure}$ $\text{a specific decay and spin rate}$ $\text{in a signal}$,
$\text{project onto}$ $\text{a spiral of that}$ $\text{rate}$.

Fourier Transform

$$X_k=\frac{1}{N}\sum _{n=0}^{N-1}{x}_n{e}^{i2\pi k\frac{n}{N}}$$

To find $\text{the energy}$ $\text{at a particular frequency}$,
$\text{spin}$ $\text{your signal}$ $\text{around a circle}$ $\text{at that frequency}$, and
$\text{average a bunch of points along that path}$.

Euler's Identity

$$e^{i\pi }=-1$$

$\text{Growth}$ $\text{pushing sideways}$ $\text{lasting for half a circle}$
$\text{points you}$ $\text{backwards}$.

Eulers Formula

$$e^{ix}=\cos (x)+i\sin (x)$$

$\text{Growth}$ in a $\text{perpendicular direction}$ over $\text{time}$
is circular: $\text{here are the}$ $\text{horizontal}$
and $\text{vertical}$ coordinates

E

The constant $e$ (approximately $2.718$) is the base of natural logarithms and arises naturally in many areas of mathematics, especially those involving growth and change.

• Why's $e$ special?
  • All circles are the unit circle, scaled up. All continuously growing systems are $e^{rt}$, scaled to some rate and time.

E (Compound Interest Definition)

$$e=\underset{n\to \mathrm{\infty }}{lim}{\left(1+\frac{1}{n}\right)}^{1\cdot n}$$

$\text{The base for continuous growth}$ is
$\text{the unit quantity}$ $\text{earning unit interest}$ $\text{for unit time}$, $\text{compounded}$ $\text{as fast as possible}$

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E (Derivative Definition)

$$\frac{d}{dx}{e}^x=100\mathrm{\%}\cdot e^x$$

$e^x$ is the function where the $\text{rate of change}$
$\text{is always}$ $100\mathrm{\%}$ of your $\text{current value}$.

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E (Natural Log Definition)

$$\ln (a)={\int }_1^a\frac{1}{x}dx$$$$\ln (e)=1$$

$\text{The natural log}$ is $\text{the time to grow from 1 to a value}$
using $\text{100\% continuous interest}$.

$e\text{ is the number}$ that takes $\text{the natural logarithm}$
$\text{1 unit of time to reach}$.

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E (Series Definition)

$$e=1+\frac{1}{1!}+\frac{1}{2!}+\frac{1}{3!}+\cdots$$

$\text{The base for continuous growth}$ is
$\text{the unit quantity}$ $\text{earning unit interest}$

plus $\text{the interest on the interest}$

plus $\text{the interest on the interest on the interest}$ and $\text{so on}$

Convolution

$$(f\ast g)(t)\stackrel{\text{def}}{=}{\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}f(\tau )g(t-\tau )d\tau$$

To $\text{convolve}$ $\text{a kernel}$ with an $\text{input signal}$:
$\text{flip the signal}$, $\text{move to the desired time}$,
and $\text{accumulate every interaction}$ $\text{with the kernel}$.

Derivative

$$\frac{df}{dt}=\underset{h\to 0}{lim}\frac{f(t+h)-f(t)}{h}$$

Take the $\text{new value}$ $\text{minus the old value}$ to get the change of value of the quantity,
then divide by the $\text{time interval}$,
and let that interval $\text{tend to 0}$ to get the
$\text{rate of change}$ $\text{of quantity}$ $\text{with respect to time}$.

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_{Loosely based on Kalid Azad's ColorizedMath - but cleaned up and extended.}