Math, Better Explained: Learn to Unlock Your Math Intuition

by Kalid Azad

With natural laws, we’re an observer describing the motion of others. Radians are about them, not us.

Degrees are arbitrary because they’re based on the sun (365 days ~ 360 degrees), but they are backwards because they are from the observer’s perspective.

Degrees

Even Euler, the genius who discovered e and much more, didn’t understand negatives as we do today.

Till 1700 negatives didn't make much sense

i is a “new imaginary dimension” to measure a number i (or -i) is what numbers “become” when rotated Multiplying i is a rotation by 90 degrees counter-clockwise Multiplying by -i is a rotation of 90 degrees clockwise Two rotations in either direction is -1: it brings us back into the “regular” dimensions of positive and negative numbers.

Imaginary numbers

Numbers are 2-dimensional.

Like negative numbers modeling flipping, imaginary numbers can model anything that rotates between two dimensions “X” and “Y”.

complex number is the fancy name for numbers with both real and imaginary parts. They’re written a + bi, where a is the real part b is the imaginary part

Complex number

Size of -x = √(-x)2 = |x|

Size of a negative number

Size of a + bi = √(a2 + b2)

Size of complex numbers

Multiplying by a complex number rotates by its angle

Original heading: 3 units East, 4 units North = 3 + 4i Rotate counter-clockwise by 45 degrees = multiply by 1 + i If we multiply them together we get: (3 + 4i) · (1 + i) = 3 + 4i + 3i + 4i2 = 3−4 + 7i = −1 + 7i So our new orientation is 1 unit West (-1 East), and 7 units North,

Example calculation of complex numbers

complex numbers can make ugly calculations simple (like calculating cosine(a + b)).

Imaginary numbers have an intuitive explanation: they “rotate” numbers, just like negatives make a “mirror image” of a number.

Helpful insight

Complex numbers are similar: they have their real and imaginary parts “contained” in a single variable (shorthand is often Re and Im).

number z = 3 + 4i would have a magnitude of 5. The shorthand for “magnitude of z” is this: |z|

Size of complex numbers

Didya know that 1/i = -i? Just multiply both sides by i and see for yourself!

Instead of z = a + bi, think about a number z* = a−bi, called the “complex conjugate”. It has the same real part, but is the “mirror image” in the imaginary dimension. The conjugate or “imaginary reflection” has the same magnitude, but the opposite angle!

Complex conjugates

Complex conjugates are indicated by a star (z*) or bar () above the number

z = a + bi has the complex conjugate z* = a − bi

Example of a complex conjugate

The number should be |z|2 since we scaled by the size twice. Now let’s do an example: (3 + 4i) · (3−4i) = 9 − 16i2 = 25

Multiplying conjugates ~> z * z*

Rotate by opposite angle: multiply by (1 – i) instead of (1 + i) Divide by magnitude squared: divide by |√2|2 = 2

Division

The more traditional “plug and chug” method is to multiply top and bottom by the complex conjugate:

Division - other approach

there’s a few properties to consider: (x + y)* = x* + y* (x · y)* = x* · y*

The conjugate is a way to “undo” a rotation.

I deposited $3, $10, $15.75 and $23.50 into my account. What transaction will cancel these out? To find the opposite: add them up, and multiply by -1.

Example of conjugate

See the conjugate z* as a way to “cancel” the rotation effects of z, just like a negative number “cancels” the effects of addition. One caveat: with conjugates, you need to divide by |z| · |z| to remove the scaling effects as well.

The mathematical constant e is the base of the natural logarithm.

e

The natural logarithm, formerly known as the hyperbolic logarithm, is the logarithm to the base e, where e is an irrational constant approximately equal to 2.718281828459.

Natural log

Pi is the ratio between circumference and diameter shared by all circles.

Pi

Pi is important and shows all circles are related, not to mention the trigonometric functions derived from circles (sin, cos, tan).

Importance of Pi

e is the base amount of growth shared by all continually growing processes.

e definition from the author

e shows up whenever systems grow exponentially and continuously: population, radioactive decay, interest calculations,

Just like every number can be considered a “scaled” version of 1 (the base unit), every circle can be considered a “scaled” version of the unit circle (radius 1), and every rate of growth can be considered a “scaled” version of e (the “unit” rate of growth).

Best way to understand e

growth = 2x Said another way, doubling is 100% growth. We can rewrite our formula like this: growth = (1 + 100%)x It’s the same equation, but we separate “2” into what it really is: the original value (1) plus 100%. Clever, eh?

Growth

the general formula for x periods of return is: growth = (1 + return)x

The growth of two half-periods of 50% is: growth = (1 + 100%/2)2 = 2.25

e is defined to be that rate of growth if we continually compound 100% return on smaller and smaller time periods:

When n is close to 1000, rate of growth converges to 2.71

Sure, you start out expecting to grow from 1 to 2. But with each tiny step forward you create a little “dividend” that starts growing on its own. When all is said and done, you end up with e (2.718...) at the end of 1 time period, not 2.

Why e is not = 2

If we start with $2.00, we get 2e. If we start with $11.79, we get 11.79e.

e is like a speed limit (like c, the speed of light) saying how fast you can possibly grow using a continuous process.

Well, it’s just half the number of changes: (1+.01)50 = (1 + .01)100/2 = ((1 + .01)100)1/2 = e1/2

Case of 50% annual grow

Even though growth can look like addition (+1%), we need to remember that it’s really a multiplication (times 1.01).

The key is that for any rate we pick, it’s just a new exponent on e: growth = erate

Suppose we have 300% growth for 2 years. We’d multiply one year’s growth (e3) by itself two times: growth=(e3)2 = e6 And in general: growth = (erate)time = erate · time

Different times

When we write: ex the variable x is a combination of rate and time. x = rate · time

10 years of 3% growth means 30 changes of 1%.

growth = ex = ert If we have a return of r for t time periods, our net compound growth is ert

Our rate is 100% every 24 hours, so after 10 days we get: 300 · e1 · 10 = 6.6 million kg

Example of growth calculation

Using the natural log we can deduce the initial rate of 100%.

constantly slowing decay is the reverse of constantly compounding growth.