Types of Numbers

whole numbers

natural numbers

positive, whole numbers

0 1 2 3

integers

positive or nevative whole numbers

-2 -1 0 1 2

real numbers

rational numbers

division of integers are not necessarily whole numbers.

1.5, 2.3

irrational numbers

divisions with an infinite number of decimal places (pi, square-root of 2, ...)

3.14...

multiples

prime numbers

A natural number greater than 1 that is not a product of two smaller natural numbers
See https://en.wikipedia.org/wiki/List_of_prime_numbers

2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59 ...
# 1 is not a prime number

def is_prime(num):
    if num < 2:
        return False
    return all(num % i != 0 for i in range(2, num - 1))

def earliest_divisible_prime(num):
    for i in range(2, num - 1):
        if num % i == 0:
            return i

def prime_numbers_until(num):
    for i in range(2, num - 1):
        if all(i % ii != 0 for ii in range(2, i - 1)):
            print(i)

composite numbers

numbers that can be made by multiplying two prime numbers together.
(All positive whole numbers that are not prime numbers)

4 6

complex numbers

You may use a single-number to represent a 2D point. While normally you would use it's cartesian coordinates, (1, 2), you can also describe it as 1 + 2i. The 2i is entirely unecessary, but a convenience for more complicated equations.

# (y)
# |
# 3
# 2 o
# 1
# 0 1 2 3 -- (x)

o = (1 + 2i)

The value of i is an imaginary number (one that cannot possibly exist), whose only defining attribute is i^2 == -1. This is a convenience used in electrical engineering, and calculating rotations.

• multiplication by i indicates a rotation of 90*
• i^2 is a rotation of 180*
• i^4 is a rotation of 360*
• the concept of i was created to describe how a squared real number is always positive.

Quaternions

Expands upon the idea of a complex number (within a 2D space) so that it is represented in 3D. 3x new variables are introduced - i, j, and k.

quaternion = w + xi + yj + zk

Properties of the new variables

i^2 = -1
j^2 = -1
k^2 = -1

# NOTE:
# the order in which these variables are multiplied
# affects whether the outcome is postive or negative.
ij = k     ji = -k
jk = i     kj = -i
ki = j     ik = -j

• When no rotation has been performed, w is 1.0.

1.0 + 0.0i + 0.0j + 0.0k

• Until i is 1, w is inversely proportional

1.0 + 0.0i + 0.0j + 0.0k
0.5 + 0.5i + 0.0j + 0.0k
0.0 + 1.0i + 0.0j + 0.0k

• Beyond i as 1 (or -1), the relationship between i and w becomes more complicated. Think of it as a projection, and seriously watch this video: https://www.youtube.com/watch?v=d4EgbgTm0Bg
  • 26:00 overview of axes
  • 28.00 visual representation of relationship between i and w (where the numbers really come from).
• Note that j/k are not affected whatsoever.

The values multiplied against j/k are inversely proportional.

0.0 + 0.0i - 1.0j + 0.0k
0.0 + 0.0i + 1.0j + 0.0k
0.0 + 0.0i + 0.0j + 1.0k
0.0 + 0.0i + 0.0j - 1.0k

https://www.youtube.com/watch?v=a_lb0MRkuz0 https://www.youtube.com/watch?v=jlskQDR8-bY https://www.youtube.com/watch?v=d4EgbgTm0Bg