Complex Number

A complex number is a number written in the form z = a + bi, where a and b are real numbers, and i² = −1.

Here, a is the real part and b is the imaginary part.

For a complex number z = a + bi:

• Re(z) is the real part of z: Re(z) = a.
• Im(z) is the imaginary part of z: Im(z) = b.

Theorem: C is a field and a vector space

Every complex number has the form z = a + bi, where a, b ∈ R and i² = −1.

1. C is a field

Let z₁ = a + bi and z₂ = c + di.

Closure:

z₁ + z₂ = (a + c) + (b + d)i ∈ C

z₁z₂ = (ac − bd) + (ad + bc)i ∈ C

Additive identity and inverse:

The additive identity is 0 = 0 + 0i. The additive inverse of z = a + bi is −z = −a − bi.

Multiplicative identity:

The multiplicative identity is 1 = 1 + 0i.

Multiplicative inverse:

If z = a + bi ≠ 0, then a² + b² > 0, and

z⁻¹ = 1/(a + bi) = (a − bi)/(a² + b²).

Indeed,

(a + bi)(a − bi)/(a² + b²) = (a² + b²)/(a² + b²) = 1.

Associativity, commutativity, and distributivity follow from the corresponding properties of real-number arithmetic. Therefore, C is a field.

2. C is a vector space over R

Take z₁, z₂ ∈ C and λ, μ ∈ R. Scalar multiplication is defined by ordinary multiplication:

λ(a + bi) = (λa) + (λb)i ∈ C.

The vector-space axioms hold:

λ(z₁ + z₂) = λz₁ + λz₂

(λ + μ)z = λz + μz

λ(μz) = (λμ)z

1z = z

Hence, C is a vector space over R.

Every z = a + bi can be written as z = a·1 + b·i. Therefore, {1, i} is a basis of C over R, and dim_R(C) = 2.

Also, C is a one-dimensional vector space over itself, with basis {1}.

Theorem: (C, +) is an Abelian Group

Let z₁ = a + bi, z₂ = c + di, and z₃ = e + fi, where a, b, c, d, e, f ∈ R.

1. Closure

z₁ + z₂ = (a + c) + (b + d)i.

Since a + c ∈ R and b + d ∈ R, we have z₁ + z₂ ∈ C. Thus, addition is closed on C.

2. Associativity

(z₁ + z₂) + z₃ = [(a + c) + (b + d)i] + (e + fi)

= (a + c + e) + (b + d + f)i.

Also,

z₁ + (z₂ + z₃) = (a + bi) + [(c + e) + (d + f)i]

= (a + c + e) + (b + d + f)i.

Therefore,

(z₁ + z₂) + z₃ = z₁ + (z₂ + z₃).

3. Additive identity

The complex number 0 = 0 + 0i is the additive identity because

z₁ + 0 = (a + bi) + (0 + 0i) = a + bi = z₁.

4. Additive inverse

For every z₁ = a + bi, the number −z₁ = −a − bi belongs to C, and

z₁ + (−z₁) = (a + bi) + (−a − bi) = 0.

5. Commutativity

z₁ + z₂ = (a + c) + (b + d)i,

and

z₂ + z₁ = (c + a) + (d + b)i.

Since addition in R is commutative, a + c = c + a and b + d = d + b. Hence,

z₁ + z₂ = z₂ + z₁.

Therefore, all group axioms are satisfied. Consequently,

(C, +) is an Abelian group.

Complex Numbers as Vectors

​A complex number z = a + bi can be represented by the vector Oz = (a, b) in the complex plane. The real part a gives the horizontal component, while the imaginary part b gives the vertical component. Therefore, the number determines both a magnitude, |z| = √(a² + b²), and a direction, arg(z). This makes the vector representation natural: it allows a complex number to be viewed geometrically as a directed segment from the origin to the point (a, b).