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Mathematics

Complex Numbers Operations

Gk Scientist March 22, 2022 No Comments
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Table of Contents

  • What is Complex Numbers?
  • Operations on Complex Numbers:
  • Conjugate of a Complex Number:

What is Complex Numbers?

The numbers of the form a + ib are called complex numbers where a, b are real numbers. Complex numbers are generally denoted by z. Thus complex number is z = a + ib.

A complex number z = a + ib consist of two parts-

  • Real part denoted by Re (z).
  • Imaginary part denoted by Im (z).

Thus z = a + ib then Re (z) = a and Im (z) = b.

Purely Real Complex Number- A complex number is said to be purely real if its imaginary part is zero i.e. z is purely real if and only if the imaginary part of z = 0.

Purely Imaginary Complex Number- A complex number is said to be purely imaginary if its real part is zero i.e. z is purely imaginary if and only if the real part of z = 0.

Equality of Complex Numbers- Two complex numbers z1 = a + ib and z2 = c + id are said to be equal if an only if a = c and b = d i.e. Re (z1) = Re (z2) and Im (z1) = Im (z2).

Operations on Complex Numbers:

(1) Addition of Complex Number- If z1 = a + ib and z2 = c + id with the two complex number, then their addition is denoted by z1 + z2 and is given as-

z1 + z2 = (a + c) + i(b + d)
⇒ z1 + z2 = [Re (z1) + Re (z2)] + i[Im (z1) + Im (z2)]

Properties of Addition of Complex Number:

  • Commutative Property- If z1 and z2 be the two complex numbers then z1 + z2 = z2 + z1.
  • Associative Property- If z1, z2, z3 be the three complex numbers then (z1 + z2) + z3 = z1 + (z2 + z3).
  • Additive Identity-If z be any complex number then z + 0 = z = 0 + z, then O is called the additive identity and O = 0 + i0.
  • Additive Inverse- If z1 and z2 be any two complex numbers such that z1 + z2 = 0 = z2 + z1 then z2 is the additive inverse of z1 and in general z2 = -z1.

(2) Subtraction of Complex Number- If z1 = a + ib and z2 = c + id be two complex numbers, then the subtraction of z2 from z1 is-

z1 – z2 = z1 + (-z2) = (a + ib) + (-c – id) = (a – c) + i(b – d)

(3) Multiplication of Complex Number- If z1 = a + ib and z2 = c + id be the two complex numbers then their multiplication is denoted by z1.z2 and is given as-

z1 . z2 = (a + ib) (c + id)
⇒ z1 . z2 = ac + i(ad + bc) – bd
⇒ z1 . z2 = (ac – bd) + i(ad + bc)
⇒ z1 . z2 = [Re (z1) . Re (z2) – Im (z1) . Im (z2)] + i[Re (z1) . Im (z2) + Im (z1) . Re (z2)]

Properties of Multiplication of Complex Number:

  • Commutative Property- If z1 and z2 be two complex numbers then z1 . z2 = z2 . z1.
  • Associative Property- If z1, z2, z3 be three complex numbers then (z1 . z2) . z3 = z1 (z2 . z3).
  • Multiplicative Identity- If z be any complex number such that z . 1 = z = 1 . z then ‘1’ is called the Multiplicative Identity.
  • Multiplicative Inverse- If z be any complex number such that z . z1 = 1 = z1 . z then z1 is called the multiplicative inverse of z. In general z = 1/z.
Example- Find the Multiplicative Inverse of a + ib.

Solution: Let z = a + ib then multiplicative inverse of z is 1/(a + ib)
⇒ z = 1/(a + ib) x (a – ib)/(a – ib)
⇒ z = (a – ib)/[a2 – (ib)2]
⇒ z = (a – ib)/(a2 + b2)
⇒ z = z̄/| z |2

(4) Division of Complex Number- If z1 = a + ib and z2 = c + id are two complex numbers, then the division of z1 by z2 is defined as-

z1/z2 = (a + ib)/(c + id)
⇒ z1/z2 = (a + ib).(c – id)/(c + id).(c – id)
⇒ z1/z2 = [(ac + bd) + i(bc – ad)]/(c2 + d2)
⇒ z1/z2 = (ac + bd)/(c2 + d2) + i (bc – ad)/(c2 + d2)

Conjugate of a Complex Number:

If z be any complex number such that z = a + ib then its conjugate is denoted by z̄ and is given as z̄ = a – ib.

Properties of Conjugate of a Complex Number:

If z, z1, z2 be any complex number then-

Properties of Conjugate of a Complex Number - Complex Numbers Operations

Heights and Distances in TrigonometryDual Nature of Radiations and Matter
The Cosine FormulaeDerivation of de-Broglie equation
The Sine FormulaHeisenberg Uncertainty Principle
Graphs of Simple Trigonometric FunctionsModern ABC of Chemistry Class-12 Part I & Part II (Set of 2 Books)
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