https://en.wikipedia.org/wiki/Number
N = {1, 2, 3, ...}
Operations: n + m, n * m, n // m, n^m.
Mathematical induction is often used as a proof technique.
A binary relation n < m (less then) is present.
Z = {..., -2, -1, 0, 1, 2, ...}
Z = {(a, b): a and b are natural numbers from N}
A pair (a, b) denotes 'a-b' (negative numbers appear).
Some pairs are considered equal:
(a1, b1) ~ (a2, b2) only if a1 + b2 = a2 + b1.
Addition (a1, b1) + (a2, b2) = (a1 + a2, b1 + b2).
Substraction (a1, b1) - (a2, b2) = (a1 + b2, a2 + b1).
Multiplication (a1, b1) * (a2, b2) = (a1 * a2 + b1 * b2, a1 * b2 + a2 * b1).
Natural numbers can be identified with pairs (a, b), where b < a.
Zero is identified with pairs (a, a).
Q = {(a, b): a and b are integer numbers from Z, b is not zero}
A pair (a, b) denotes a fraction 'a / b' [numerator / denumerator].
Some pairs are considered equal:
(a1, b1) ~ (a2, b2) only if a1 * b2 = a2 * b1 [2 / 4 ~ 1 / 2].
Addition (a1, b1) + (a2, b2) = (a1 * b2 + a2 * b1, b1 * b2).
Substraction (a1, b1) - (a2, b2) = (a1 * b2 - a2 * b1, b1 * b2).
Multiplication (a1, b1) * (a2, b2) = (a1 * a2, b1 * b2).
Division (a1, b1) / (a2, b2) = (a1, b1) * (b2, a2), where a2 is not zero.
Integer numbers can be identified with pairs (a, 1).
Problem: x^2 = 2, x can not be rational.
R - there are several definitions ... Every real number corresponds to a point on the number line. All rational numbers are also real numbers but the reverse is not true. Most real numbers can only be 'approximated' by decimal numerals: 1 / 2 = 0.5 (rational) 1 / 3 = 0.333... = 0.(3) (rational) 0.(12) = 0.1212... = 12 * [10^(-2) + 10^(-4) + ...] = 12 / 99 sqrt(2) = 1.41421... (irrational) pi = 3.14159... (irrational) e = 2.71828... (irrational) Problem: x^2 = -1, x can not be real.
https://en.wikipedia.org/wiki/Complex_number
C = {(x, y): x and y are real numbers from R}
x is called the real part and y is called the imaginary part.
Notation: z = (x, y) = x + y i, where i^2 = -1.
Every complex number corresponds to a point in the plane.
Real numbers can be identified with pairs (x, 0).
Addition (x1, y1) + (x2, y2) = (x1 + x2, y1 + y2).
Substraction (x1, y1) - (x2, y2) = (x1 - x2, y1 - y2).
Multiplication (x1, y1) * (x2, y2) = (x1 * x2 - y1 * y2, x1 * y2 + y1 * x2).
The complex conjugate of z is z^(*) = (x, -y).
Norm |z|^2 = |x|^2 + |y|^2 = z * z^(*) = z^(*) * z.
1 / z = z^(*) / (z * z^(*)) = z^(*) / |z|^2 = z^(*) * (1/|z|^2).
Division z1 / z2 = z1 * (1 / z2) = (1 / z2) * z1.
Every polynomial with complex coefficients has a root in the complex numbers.
The complex numbers lack a total order that is compatible with field operations.
https://en.wikipedia.org/wiki/Quaternion
H = {(x, y, z, t): x, y, z, t are real numbers from R}
Notation: q = (x, y, z, t) = x + y i + z j + t k,
where i^2 = j^2 = k^2 = ijk = -1,
ij = -ji = k, jk = -kj = i, ki = -ik = j.
x is the scalar part of q.
y i + z j + t k is the vector part of q.
The conjugate of q is q^(*) = (x, -y, -z, -t).
Norm |q|^2 = |x|^2 + |y|^2 + |z|^2 + |t|^2 = q * q^(*) = q^(*) * q.
Warning: in general q1 * q2 != q2 * q1.
The reciprocal of q is q^(-1) = q^(*) / |q|^2.
Warning: the notation q1 / q2 is ambiguous because in general
q1 * q2^(-1) != q2^(-1) * q1.
A unit quaternion u is a quaternion of norm one, |u| = 1.
If q is a non-zero quaternion, then u = q / |q|.
Unit quaternions can be identified with rotations in R^3.