COMPLEX NUMBER

(ℝ^{2},+,⋅) is the field of Complex Numbers, identified by the symbol ℂ.

ℝ^{2} = {(a,b) | a,b ∈ ℝ}

(a,b)+(c,d) = (a+c,b+d)

(a,b)⋅(c,d) = (ac-bd,ad+bc)

(0,0) is the neutral element of addition; (a,b)+(0,0) = (a+0,b+0) = (a,b)

∀ (a,b) ∈ ℝ^{2}, ∃ the additive inverse (the opposite number) (-a,-b) | (a+b)+(-a,-b) = (a-a,b-b) = (0,0)

(1,0) is the neutral element of multiplication; (a,b)⋅(1,0) = (a⋅1-b⋅0,a⋅0+b⋅1) = (a,b)

∀ (a,b) ∈ ℝ^{2}, (a,b) ≠ (0,0), ∃ the multiplicative inverse (the inverse number) (a/(a^{2}+b^{2}),-b/(a^{2}+b^{2})) | (a,b)⋅(a/(a^{2}+b^{2}),-b/(a^{2}+b^{2})) = (1,0)

ℂ_{0} = {(a,0) | a ∈ ℝ} ⊆ ℂ

(a,0)+(b,0) = (a+b,0)

(a,0)⋅(b,0) = (a⋅b-0⋅0,a⋅0+0⋅b) = (ab,0)

(a,0) ∈ ℂ_{0} → a ∈ ℝ

ℂ_{0} = ℝ ⊆ ℂ

(a,b) = (a,0)+(0,1)⋅(b,0) = a + i⋅b

(a,0) = a; (0,1) = i; (b,0) = b

Imaginary unit: i = (0,1)

Algebraic form of complex numbers: (a,b) = a+i⋅b

i⋅i = i^{2} = (0,1)⋅(0,1) = (0⋅0-1⋅1,0⋅1+1⋅0) = (-1,0) = -1 ⇔ i^{2} = -1

(a+i⋅b)+(c+i⋅d) = a+c+i(b+d) = (a+c,b+d)

(a+i⋅b)⋅(c+i⋅d) = a⋅c+i⋅a⋅d+i⋅b⋅c+i^{2}⋅b⋅d = a⋅c+i⋅a⋅d+i⋅b⋅c-b⋅d = a⋅c-b⋅d+i(a⋅d+b⋅c) = (a⋅c-b⋅d,a⋅d+b⋅c)

(3+i)+(6+2i) = (3+6)+i(1+2) = 9+3i

(4+2i)+3i = 4+i(2+3) = 4+5i

(1+i)⋅(2+3i) = 2+3i+2i+3i^{2} = 2+3i+2i-3 = 2-3+i(3+2) = -1+5i

2i⋅(3+6i) = 6i+12i^{2} = -12+6i

Algebraic form of complex numbers: z = a+ib; real part of z: a = Re(z); imaginary part of z: b = Im(z)

z = 4-7i; Re(z) = 4; Im(z) = -7

z = 2i; Re(z) = 0; Im(z) = 2

A real number a can be regarded as a complex number a+0i, whose imaginary part is 0.

A purely imaginary number bi is a complex number 0+bi, whose real part is zero.

Purely imaginary: z ∈ ℂ | Re(z) = 0

z = (a,b) = a+ib

1 = (1,0); i = (0,1); 1+i = (1,1)

(a+ib)+(c+id) = a+c+i(b+d)

The complex conjugate of the complex number z = x+yi is given by x−yi. It is denoted by either z or z*.

Geometrically, z is the "reflection" of z about the real axis.

Considering z ∈ ℂ, z = a+ib, the complex conjugate of z is z = a-ib; Re(z) = Re(z), Im(z) = -Im(z)

z = 2+6i ⇒ z = 2-6i

z = -2i ⇒ z = 2i

z = 4 ⇒ z = 4

The conjugate of a sum is the sum of the conjugates: z+w = z+w, ∀ z,w ∈ ℂ

The conjugate of a product is the product of the conjugates: z⋅w = z⋅w, ∀ z,w ∈ ℂ

The conjugate of a ratio is the ratio of the conjugates: (z/w) = z/w, ∀ z,w ∈ ℂ, w ≠ 0

z⋅z = a^{2}+b^{2} if z = a+ib

(1+3i)(1-3i) = 1-3i+3i-9i^{2} = 1+9 = 10

Conjugating twice gives the original complex number.

(1+i)/(1+2i) = (1+i)/(1+2i) ⋅ (1-2i)/(1-2i) = (1+i)(1-2i)/(1+2i)(1-2i) = (1-2i+i-2i^{2})/(1-2i+2i-4i^{2}) = (1-2i+i+2)/(1-2i+2i+4) = (3-i)/5 = 3/5 - (1/5)i

Modulus of a complex number: z ∈ ℂ, z = a+ib, the modulus is the real number |z| = √(a^{2}+b^{2})

z = 1+i; |z| = √(1^{2}+1^{2}) = √2

z = a, |z| = √(a^{2}) = |a|

|z| ≥ 0, the modulus of a complex number is always ≥ 0.

|z| = 0 ⇔ z = 0, the modulus of z is 0 when a and b are 0, therefore z is 0.

|w⋅z| = |w|⋅|z|, the modulus of the product is the product of the moduli.

|z+w| ≤ |z|+|w|, triangle inequality.

z⋅z = |z|^{2}

ρ (rho): length of the vector connecting the point to the origin; ρ ∈ [0,+∞].

θ (theta): angle that the vector forms with the x axis; θ ∈ [0,2π).

ρ and θ are polar coordinates.

From Polar coordinates to Cartesian coordinates: a = ρ⋅cos(θ); b = ρ⋅sin(θ)

Fron Cartesian coordinates to Polar coordinates: ρ = √(a^{2}+b^{2}); tan(θ) = b/a

z ∈ ℂ, z = a+ib = ρ⋅cos(θ)+i⋅ρ⋅sin(θ) = ρ(cos(θ)+i⋅sin(θ))

Algebraic form of complex numbers: z = a+ib

Trigonometric form of complex numbers: z = ρ(cos(θ)+i⋅sin(θ))

ρ = √(a^{2}+b^{2}) = |z|

θ = arg(z)

z = 1+i; ρ = |z| = √(a^{2}+b^{2}) = √(1^{2}+1^{2}) = √2; tan(θ) = 1 ⇔ θ = arg(z) = π/4; z = ρ(cos(θ)+i⋅sin(θ)) = √2(cos(π/4)+i⋅sin(π/4)) = √2(√2/2+i√2/2) = 1+i

z = -5; ρ = |z| = √(a^{2}+b^{2}) = √(-5^{2}+0^{2}) = √25 = 5; z = ρ(cos(θ)+i⋅sin(θ)) = 5(cos(π)+i⋅sin(π)) = 5(-1+i⋅0) = -5

z = i; ρ = |z| = √(a^{2}+b^{2}) = √(0^{2}+1^{2}) = 1; z = ρ(cos(θ)+i⋅sin(θ)) = 1(cos(π/2)+i⋅sin(π/2)) = 1(0+i⋅1) = 1(i) = i

z = a+ib, algebraic form of a complex number; a,b ∈ ℝ; i^{2} = -1

z = ρ(cos(θ)+i⋅sin(θ)), trigonometric form of a complex number; a = ρ⋅cos(θ), b = ρ⋅sin(θ), ρ ∈ [0,+∞), θ ∈ [0,2π)

z_{1},z_{2} ∈ ℂ, z_{i} = ρ_{i}(cos(θ_{i})+i⋅sin(θ_{i})), i = 1,2

z_{1}⋅z_{2} = ρ_{1}⋅ρ_{2}(cos(θ_{1}+θ_{2})+i⋅sin(θ_{1}+θ_{2}))

ρ = √(a^{2}+b^{2}) = |z|; θ = arg(z)

|z_{1}⋅z_{2}| = |z_{1}|⋅|z_{2}|; arg(z_{1}⋅z_{2}) = arg(z_{1})+arg(z_{2})

z_{1}⋅z_{2} = ρ_{1}(cos(θ_{1})+i⋅sin(θ_{1}))⋅ρ_{2}(cos(θ_{2})+i⋅sin(θ_{2})) = ρ_{1}⋅ρ_{2}[cos(θ_{1})cos(θ_{2})-sin(θ_{1})sin(θ_{2})+i(sin(θ_{1})cos(θ_{2})+cos(θ_{1})sin(θ_{2}))] = ρ_{1}⋅ρ_{2}(cos(θ_{1}+θ_{2})+i⋅sin(θ_{1}+θ_{2}))

z_{1}/z_{2} = ρ_{1}/ρ_{2}(cos(θ_{1}-θ_{2})+i⋅sin(θ_{1}-θ_{2}))

|z_{1}/z_{2}| = |z_{1}|/|z_{2}|; arg(z_{1}/z_{2}) = arg(z_{1})-arg(z_{2})

De Moivre's formula: z^{n} = ρ^{n}(cos(nθ)+i⋅sin(nθ)), z = ρ(cos(θ)+i⋅sin(θ)), n ∈ ℕ

(1+i)^{5}; a+ib, a = 1, b = 1; ρ = √(a^{2}+b^{2}) = √2; θ = π/4; z = ρ(cos(θ)+i⋅sin(θ)) = √2(cos(π/4)+i⋅sin(π/4)); z^{5} = 2^{5/2}(cos(5π/4)+i⋅sin(5π/4)) = 2^{5/2}(√2/2+i√2/2) = 2^{5/2}(1/√2+i/√2) = 2^{5/2}(-1/2^{1/2}-i/2^{1/2}) = -2^{4/2}-2^{4/2}i = -2^{2}-2^{2}i = -4-4i ⇔ (1+i)^{5} = -4-4i

Euler's formula: e^{iθ} = cos(θ)+i⋅sin(θ), ∀ θ ∈ ℝ; cos(θ) = (e^{iθ}+e^{-iθ})/2, sin(θ) = (e^{iθ}-e^{-iθ})/2i

e^{iθ} = cos(θ)+i⋅sin(θ); e^{-iθ} = cos(-θ)+i⋅sin(-θ) = cos(θ)-i⋅sin(θ); e^{iθ}+e^{-iθ} = cos(θ)+i⋅sin(θ)+cos(θ)-i⋅sin(θ) = 2⋅cos(θ)

e^{iπ} = cos(π)+i⋅sin(π) = -1; e^{iπ} = -1 ⇔ e^{iπ} + 1 = 0

w ∈ ℂ, n ∈ ℕ\{0}, z ∈ ℂ is the n-th root of w if z^{n} = w

w ∈ ℂ, w = ρ(cos(θ)+i⋅sin(θ)), n ∈ ℕ\{0}, there are n different complex roots z_{k}, k = 1, ..., n, of w

z_{k} = r(cos(θ_{k})+i⋅sin(θ_{k})); r = ρ^{1/n} = ^{n}√ρ; θ_{k} = (θ+2kπ)/n, k = 1, ..., n

w = ρ(cos(θ)+i⋅sin(θ)), z = r(cos(φ)+i⋅sin(φ)) | z^{n} = w; using the De Moivre's formula z^{n} = r^{n}(cos(nφ)+i⋅sin(nφ)); z^{n} = w, r^{n}(cos(nφ)+i⋅sin(nφ)) = ρ(cos(θ)+i⋅sin(θ)) ⇔ r^{n} = ρ ⇔ r = ρ^{1/n}, nφ = θ+2kπ ∀ k ∈ ℤ ⇔ φ = (θ+2kπ)/n ∀ k ∈ ℤ; k = 1, φ_{1} = (θ+2π)/n; k = n+1, φ_{n+1} = (θ+2(n+1)π)/n = (θ+2π)/n + 2nπ/n = (θ+2π)/n + 2π

√-1; ρ(cos(θ)+i⋅sin(θ)), ρ = 1, θ = π; -1 = 1(cos(π)+i⋅sin(π)); z_{k} = ρ^{1/2}(cos((θ+2kπ)/2)+i⋅sin((θ+2kπ)/2) k = 0, 1; z_{0} = cos(π/2)+i⋅sin(π/2) = 0+i⋅1 = i; z_{1} = cos((π+2π)/2)+i⋅sin((π+2π)/2) = cos(3π/2)+i⋅sin(3π/2) = 0+i(-1) = -i

The complex roots of degree 2 of -1 are i and -i; i^{2} = -i^{2} = -1

The n-th roots of a complex number are the vertices of a regular polygon of n sides inscribed in the circumference of radius ρ^{1/n} where ρ is the modulus of the number.

(ℂ,+,⋅) is a field containing the field of real numbers (ℝ,+,⋅) and is algebraically closed; algebraically closed means that a polynomial equation can always be solved in ℂ; the field of real numbers is not algebraically closed because many polynomial equations are not solvable in ℝ.

The field of complex numbers ℂ is not orderable.

If ℂ were orderable: z ∈ ℂ ⇒ z^{2} ≥ 0 and z ∈ ℂ, z ≥ 0 ⇔ -z ≤ 0; i^{2} = -1 ≥ 0 and 1 ≥ 0 ⇒ -1 ≤ 0 and it is an absurd result, a contradiction.