TOPICS

Introduction

Complex Numbers And Quadratic Equations

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.

(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

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).

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

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/(a^{2} + b^{2}) + i (-b)/(a^{2} + b^{2}) (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

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.

In general, for any integer k, i^{4k} = 1, i^{4k + 1} = i, i^{4k + 2} = –1, i^{4k + 3} = – i

Now, i^{3} = -i, i^{4} = 1, i^{5} = i etc.

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

(i) (z₁ + z₂)^{2} = z₁^{2} + 2 z₁.z₂ + z₂^{2}

(ii) (z₁ - z₂)^{2} = z₁^{2} - 2 z₁.z₂ + z₂^{2}

(iii) (z₁ + z₂)^{3} = z₁^{3} + z₂^{3} + 3 z₁^{2}.z₂ + 3 z₁.z₂^{2}

(iv) (z₁ - z₂)^{3} = z₁^{3} - z₂^{3} - 3 z₁^{2}.z₂ + 3 z₁.z₂^{2}

(v) z₁^{2} - z₂^{2} = (z₁ + z₂)(z₁ - z₂)

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

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.

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