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COMPLEX NUMBER


(ℝ2,+,⋅) is the field of Complex Numbers, identified by the symbol ℂ.

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) ∈ ℝ2, ∃ 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) ∈ ℝ2, (a,b) ≠ (0,0), ∃ the multiplicative inverse (the inverse number) (a/(a2+b2),-b/(a2+b2)) | (a,b)⋅(a/(a2+b2),-b/(a2+b2)) = (1,0)

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) ∈ ℂ0 → a ∈ ℝ

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 = i2 = (0,1)⋅(0,1) = (0⋅0-1⋅1,0⋅1+1⋅0) = (-1,0) = -1 ⇔ i2 = -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+i2⋅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+3i2 = 2+3i+2i-3 = 2-3+i(3+2) = -1+5i

2i⋅(3+6i) = 6i+12i2 = -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 = zw, ∀ 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 = a2+b2 if z = a+ib

(1+3i)(1-3i) = 1-3i+3i-9i2 = 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-2i2)/(1-2i+2i-4i2) = (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| = √(a2+b2)

z = 1+i; |z| = √(12+12) = √2

z = a, |z| = √(a2) = |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 = √(a2+b2); 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 = √(a2+b2) = |z|

θ = arg(z)

z = 1+i; ρ = |z| = √(a2+b2) = √(12+12) = √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| = √(a2+b2) = √(-52+02) = √25 = 5; z = ρ(cos(θ)+i⋅sin(θ)) = 5(cos(π)+i⋅sin(π)) = 5(-1+i⋅0) = -5

z = i; ρ = |z| = √(a2+b2) = √(02+12) = 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; i2 = -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)

z1,z2 ∈ ℂ, zi = ρi(cos(θi)+i⋅sin(θi)), i = 1,2

z1⋅z2 = ρ1⋅ρ2(cos(θ12)+i⋅sin(θ12))

ρ = √(a2+b2) = |z|; θ = arg(z)

|z1⋅z2| = |z1|⋅|z2|; arg(z1⋅z2) = arg(z1)+arg(z2)

z1⋅z2 = ρ1(cos(θ1)+i⋅sin(θ1))⋅ρ2(cos(θ2)+i⋅sin(θ2)) = ρ1⋅ρ2[cos(θ1)cos(θ2)-sin(θ1)sin(θ2)+i(sin(θ1)cos(θ2)+cos(θ1)sin(θ2))] = ρ1⋅ρ2(cos(θ12)+i⋅sin(θ12))

z1/z2 = ρ12(cos(θ12)+i⋅sin(θ12))

|z1/z2| = |z1|/|z2|; arg(z1/z2) = arg(z1)-arg(z2)

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

(1+i)5; a+ib, a = 1, b = 1; ρ = √(a2+b2) = √2; θ = π/4; z = ρ(cos(θ)+i⋅sin(θ)) = √2(cos(π/4)+i⋅sin(π/4)); z5 = 25/2(cos(5π/4)+i⋅sin(5π/4)) = 25/2(√2/2+i√2/2) = 25/2(1/√2+i/√2) = 25/2(-1/21/2-i/21/2) = -24/2-24/2i = -22-22i = -4-4i ⇔ (1+i)5 = -4-4i

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

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

e = cos(π)+i⋅sin(π) = -1; e = -1 ⇔ e + 1 = 0

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

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

zk = r(cos(θk)+i⋅sin(θk)); r = ρ1/n = n√ρ; θk = (θ+2kπ)/n, k = 1, ..., n

w = ρ(cos(θ)+i⋅sin(θ)), z = r(cos(φ)+i⋅sin(φ)) | zn = w; using the De Moivre's formula zn = rn(cos(nφ)+i⋅sin(nφ)); zn = w, rn(cos(nφ)+i⋅sin(nφ)) = ρ(cos(θ)+i⋅sin(θ)) ⇔ rn = ρ ⇔ r = ρ1/n, nφ = θ+2kπ ∀ k ∈ ℤ ⇔ φ = (θ+2kπ)/n ∀ k ∈ ℤ; k = 1, φ1 = (θ+2π)/n; k = n+1, φn+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)); zk = ρ1/2(cos((θ+2kπ)/2)+i⋅sin((θ+2kπ)/2) k = 0, 1; z0 = cos(π/2)+i⋅sin(π/2) = 0+i⋅1 = i; z1 = 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; i2 = -i2 = -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 ρ1/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 ⇒ z2 ≥ 0 and z ∈ ℂ, z ≥ 0 ⇔ -z ≤ 0; i2 = -1 ≥ 0 and 1 ≥ 0 ⇒ -1 ≤ 0 and it is an absurd result, a contradiction.