Unit 1: Real And Complex Numbers - Mathematics For Class IX (Science Group) - Explanation Of Exercise 1.5

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Unit 1: Real And Complex Numbers
Explanation Of Exercise 1.5

Definition of Complex Number

A number which consists of an imaginary part is called complex number. it is denoted by z.
e.g. z = a + ib
Where a and b are real numbers and i is an imaginary unit or part indication which is called iota and its value is, i = √-1. So i² = -1.
Thus, in a complex number we have both components (a + ib) so:

• a = real part and
• b = imaginary part

The complex number a+ ib can be written in order pair form (a, b) such as:

• 5+ 8i = (5, 8)
• 3 + 4i = (3, 4)

Explanation:
Imaginary Number:
We know that the square of real number is non-negative. So the solution of the equation x² + 1 = 0 does not exist in R. To overcome this inadequacy of real number, mathematicians introduced a new number √-1, imaginary unit and denoted it by the letter i (iota) having the property that i² = -1. Obviously i is not real number. It is a new mathematical entity that enables us to find the solution of every algebraic equation of the type x² + a = 0 where a > 0. Numbers like √-1 = i, √-5 = √5i, √-49 = 7i are called pure imaginary number.

Recognize 'a' as real part and 'b' as imaginary part of z = a + ib

In the complex number z = a + ib, "a" is the real part of complex number and "b" is the imaginary part of complex number.

• The real part number is denoted by Re(z)
• Its imaginary part is denote Im(z)

Conjugate Of A Complex Number

If we have a complex number (x + iy) then (x - iy) is called the conjugate of the given complex number .Conjugate of z is denoted by z i.e., 

• If z = a + ib,
  then z = a - ib

or

• If z = (a, b)
  then z = (a, - b)

or

• if z = a - ib,
  then z = a + ib

or

• If z = (a, - b),
  then z =(a, b)

In conjugate we just change the sign of imaginary part.

The Condition Of Equality Of Complex Numbers:

Two complex numbers are said to be equal if and only if, they have same real parts and same imaginary parts. i.e.
∀a, b, c, d ∈R, such that:
a + ib = c + id, iff a = c and b = d.