Complex number

What is a Complex number?

• Complex numbers allow solutions to certain equations that have no solutions in real numbers.
• A set of complex numbers is denoted by C

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Example :

(x – 1)² = – 9

x = 1 +√-9

x = 1 + 3i

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• A complex number is a number that can be expressed in the form of a + bi,
• where a and b are real numbers
• i represents the imaginary unit ; i² = −1

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Complex number Classification

• Based on the nature of the real part and imaginary part, any complex number can be classified into four types:
  • • imaginary number
    • zero complex number
    • purely imaginary number
    • purely real number

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a ≠ 0	b ≠ 0	a + ib	Imaginary number
a = 0	b = 0	0 + i0	Zero complex number
a = 0	b ≠ 0	ib	Purely imaginary number
a ≠ 0	b = 0	a	Purely real number

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Complex number Properties

• Every real number is a complex number, but every complex number is not necessarily a real number
• The set of all complex numbers is denoted by Z ∈ C
• The set of all imaginary numbers is denoted as  Z ∈ C−R
• important properties of i

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i² =  -1

i³ = i² x i = (-1) x i = -i

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i⁴ = i² x i²

= (-1) x (-1)

= 1

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• The Sum of four consecutive powers of i is equal to 0

i^k + i^k+1 + i^k+2 + i^k+3 = 0

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• Addition of Complex Numbers:

(a + bi) + (c +di) = (a+c) + (b+d)i

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