Complex numbers     ℂ

A superset of the Real numbers.
To accommodate square root of negative number. So to fully solve and factor polynomial (incl. quadratic) equations.

Imaginary unit i = √-1        Just like with all math, "you get used to it".
i² = -1      √a√a=a [√9√9=9]       i² = i·i = √-1√-1 = -1

Simplify: √-a = √a√-1 = √a i
√-4 = √4√-1 = 2i
√-18 = √9√-2 = 3√2√-1 = 3√2 i    [= i3√2    i either first or last]

NB. √a√b=√(ab) only if a and b are real numbers. not if a or b complex:
Ex. √-4√-9≠√(-4·-9)=√36=6      √-4√-9 = i√4·i√9 = 2i·3i = 6i² = -6 (Also: i√4·i√9 = i²√36 = 6i² = -6 )
√-4√9 = 2i·3 = 6i

Complex number: a + bi, a and b real numbers. a is the real part, bi the imaginary part.
5+2i    4-i    -5+2i    -3-3√2i
Every real number is a complex number with b=0.
Pure imaginary number: a=0.     2i    -3i    3√2i

The set of complex numbers, ℂ:

z is often used as the generic complex number: z = a+bi
Just like with fractions, decimal numbers, negative numbers, and irrational numbers, "you get used to them".

Complex addition/subtraction: add/subtract the real part, add/subtract the imaginary part
(5+2i) + (3+4i) = 8+6i
(5+2i) - (3+4i) = 2-2i      Parenthesize each complex number in an expression.

Complex multiplication: FOIL the two complex numbers as if they were binomials. NB i²=-1
(5+2i) · (3+4i) = 15+20i+6i+8i² = 15+26i-8 = 7+26i

Complex conjugate: of a+bi is a-bi
Each is each other's conjugate.
Ex. 2-3i, 2+3i      -2+3i, -2-3i      3i, -3i
If z is your complex number, then z̄ or z* is its conjugate.

Multiplying a pair of complex conjugates yields a positive real number:
(a+bi) · (a-bi) = a²-abi+abi-b²i² = a²+b²
(5+2i) · (5-2i) = 25-10i+10i-2i² = 25+4 = 29
(3+4i)(3-4i) = 9+16 = 25
zz* = a²+b²

Complex division: first multiply numerator and denominator by denominator's conjugate to clear the denominator of i's, then simplify to standard form by "splitting" the numerator.
(5+2i)/(3+4i) = (5+2i)(3-4i) / (3+4i)(3-4i) = (15-14i-8i²) / 9+16 = (23-14i)/25 = 23/25-14/25i

Multiplying or dividing two pure imaginary numbers yields a real number.
bi·ci = bci² = -bc
bi/ci = b/c

MP understands i and can do complex arithmetic. but cannot solve equations having complex solutions.

Powers of i:
i⁰ = 1
i¹ = i
i² = -1
i³ = -i
All higher powers are one of the four above, cycling through them. Remainder after integer dividing by 4, i.e. MOD 4.
i⁷³ [73 MOD 4 = 1] → i¹ = i
i⁷⁴ [74 MOD 4 = 2] → i² = -1
MP can do these.

(MP: factors only over integers, e.g. x²-9 → (x+3)(x-3) and Solve =0.
not over rationals, e.g. x²-9/4 → "" but does Solve =0.
nor over irrationals, e.g. x²-3 → "Doesn't factor" but does Solve =0.
Does not Solve over complexs, e.g. x²+1=0 → "No real solutions")
MP can partially calculate with x and i: (x+1)(x-i)(x+i) but Wolfram does.

A quadratic equation that has no real number solutions is called irreducible.
The discriminant b²-4ac (the part under the radical in the quadratic formula) is negative.
The quadratic expression can not be factored over the Real numbers.
The graph of the corresponding quadratic function has no x-intercepts.

BUT now with complex numbers we can solve ALL quadratic equations and factor them.
Ex.   x²+1=0 has no real solutions: x²=-1.   But go complex: x= ±√-1 = ±i
   x²+1 = (x-i)(x--i)    fully factored. 2 solutions: a pair of complex conjugates: i and -i
Ex.   x²-4x+8=0 has no real solutions: QF: x= (4±√-16)/2.   But go complex: x= (4±4i)/2 = 2±2i
   x²-4x+8 = (x-(2+2i))(x-(2-2i))    fully factored. 2 solutions: a pair of complex conjugates: 2+2i and 2-2i
Ex.   x²+x+1=0 has no real solutions but does have a pair of complex conjugates solutions:
   Using quadratic formula, instead of stopping at negative discriminant x=(-1±√-3)/2 go complex:
   x = (-1±√3i)/2 = -1/2 + √3i/2, -1/2 - √3i/2
   x²+x+1 = (x-(-1/2 + √3i/2))(x-(-1/2 - √3i/2))    fully factored. 2 solutions, a pair of complex conjugates.

Higher-degree polynomial functions/equations factor into linear factors (x-c) and irreducible quadratic factors each of which can be factored into a pair of complex conjugates. For a total of n factors and n solutions for an n-degree polynomial.
Ex.   x³+x²+x+1 has one x-intercept. MP factors it: (x+1)(x²+1).
   We know from above about complex factors of x²+1, so: x³+x²+x+1 = (x+1)(x+i)(x-i)    fully factored. 3 solutions: one a real number (-1) and a pair of complex conjugates.
   Wolfram: does all

The complex plane
Analogous to the real number line for real numbers:
To visualize numbers and their relationships to other numbers.
(Is NOT the Cartesian XY rectangular coordinate system plane.) but is essentially the same kind of "geometry".

Note that conjugates are reflections across the Real axis.      Exs. 3+2i and 3-2i, 2i and -2i

Distance ("magnitude", "modulus", "norm") of a complex number z from the origin is |z| = √(a²+b²). Pythagorean theorem.
|i| = 1
Ex. 3+2i is √(9+4)=√13 from origin
Ex. -4-3i is √(16+9)=5 from origin
Ex. -4i is √(0+16)=4 from origin
|z|² = a²+b² = zz*

Distance between two complex numbers is analogous to distance between two points on XY Cartesian plane (viz. Euclidean distance): the real parts (the a's) are the "X's" and the imaginary parts' b's are the "Y's". Pythagorean theorem.
d = √((a₁-a₂)² + (b₁-b₂)²)
Ex. distance between 3+2i and -4-3i = √((3--4)²+(2--3)²) = √(49+25) = √74

Euler's identity: 1 (multiplicative identity, 100%, "whole","all"), 0 (additive identity, "nothing"), e Euler's number, imaginary unit i, π.
"reaches down into the very depths of existence", "of exquisite beauty", "the most famous formula in all mathematics", "most beautiful theorem in mathematics", "greatest equation ever".

i is in the equations of quantum mechanics: Dirac's equation and QED which describe the fundamental stuff and activity of the universe: light, electrons (atoms, chemistry, biology, electronics)