TOPICS
Introduction
Introduction of Complex Numbers

A number of the form a + ib, where a and b are real numbers, is called a complex number, a is called the real part and b is called the imaginary part of the complex number.
For example, 3 + i5, (– 2) + i√3 etc.

Algebra of Complex Numbers

(i) Addition of two complex numbers
(ii) Difference of two complex numbers
(iii) Multiplication of two complex numbers
(iv) Division of two complex numbers
(v) Power of i
(vi) The square roots of a negative real number (vii) Identities

Addition of two complex numbers

Let z₁ = a + ib and z₂ = c + id be any two complex numbers. Then, the sum z₁ + z₂ is defined as follows: z₁ + z₂ = (a + c) + i (b + d), which is again a complex number.
For example, (3 + i5) + (– 7 + i9) = (3 – 7) + i (5 + 9) = – 4 + i 48
The addition of complex numbers satisfy the following properties:
(i) The closure law The sum of two complex numbers is a complex number, i.e., z₁ + z₂ is a complex number for all complex numbers z₁ and z₂.
(ii) The commutative law For any two complex numbers z₁ and z₂, z₁ + z₂ = z₂ + z₁
(iii) The associative law For any three complex numbers z₁, z₂, z₃, (z₁ + z₂) + z₃ = z₁ + (z₂ + z₃).
(iv) The existence of additive identity There exists the complex number 0 + i 0 (denoted as 0), called the additive identity or the zero complex number, such that, for every complex number z, z + 0 = z.
(v) The existence of additive inverse To every complex number z = a + ib, we have the complex number – a + i(– b) (denoted as – z), called the additive inverse or negative of z. We observe that z + (–z) = 0 (the additive identity).

Difference of two complex numbers

Given any two complex numbers z₁ and z₂, the difference z₁ – z₂ is defined as follows: z₁ – z₂ = z₁ + (– z₂).
For example, (8 + 2i) – (3 – 7i) = 5 + 9i

Multiplication of two complex numbers

Let z₁ = a + ib and z₂ = c + id be any two complex numbers. Then, the product z₁ z₂ is defined as follows: z₁ z₂ = (ac – bd) + i(ad + bc)
For example, (3 + i5) (2 + i6) = (3 × 2 – 5 × 6) + i(3 × 6 + 5 × 2) = – 24 + i28 The multiplication of complex numbers possesses the following properties, which we state without proofs.
(i) The closure law The product of two complex numbers is a complex number, the product z₁ z₂ is a complex number for all complex numbers z₁ and z₂.
(ii) The commutative law For any two complex numbers z₁ and z₂, z₁ z₂ = z₂ z₁.
(iii) The associative law For any three complex numbers z₁, z₂, z₃, (z₁ z₂) z₃ = z₁ (z₂ z₃).
(iv) The existence of multiplicative identity There exists the complex number 1 + i 0 (denoted as 1), called the multiplicative identity such that z.1 = z, for every complex number z. (v) The existence of multiplicative inverse For every non-zero complex number z = a + ib or a + bi(a ≠ 0, b ≠ 0), we have the complex number a/(a2 + b2) + i (-b)/(a2 + b2) (denoted by 1/z or z-1 ), called the multiplicative inverse of z such that z by 1/z = 1 (the multiplicative identity).
(vi) The distributive law For any three complex numbers z₁, z₂, z₃,
(a) z₁ (z₂ + z₃) = z₁ z₂ + z₁ z₃
(b) (z₁ + z₂) z₃ = z₁ z₃ + z₂ z₃

Division of two complex numbers

Given any two complex numbers z₁ and z₂, where 2 0 z ≠ , the quotient z₁/z₂ is defined by z₁/z₂ = z₁ By 1/z₂.

Power of i

In general, for any integer k, i4k = 1, i4k + 1 = i, i4k + 2 = –1, i4k + 3 = – i
Now, i3 = -i, i4 = 1, i5 = i etc.

The square roots of a negative real number

The square roots of – 1 are i, – i. However, by the symbol √-1 , we would mean i only. Generally, if a is a positive real number, √-a = √a √-1 = √a i

Identities

(i) (z₁ + z₂)2 = z₁2 + 2 z₁.z₂ + z₂2
(ii) (z₁ - z₂)2 = z₁2 - 2 z₁.z₂ + z₂2
(iii) (z₁ + z₂)3 = z₁3 + z₂3 + 3 z₁2.z₂ + 3 z₁.z₂2
(iv) (z₁ - z₂)3 = z₁3 - z₂3 - 3 z₁2.z₂ + 3 z₁.z₂2
(v) z₁2 - z₂2 = (z₁ + z₂)(z₁ - z₂)

Conjugate

The conjugate of the complex number z = a + ib, denoted by z , is given by z = a – ib

Polar Form

The polar form of the complex number z = x + iy is r (cosθ + i sinθ), where r = √x² + y² (the modulus of z) and cosθ = x/r , sinθ = y/r . (θ is known as the argument of z. The value of θ, such that – π < θ ≤ π, is called the principal argument of z. polar form

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