Introduction
⇒First we know about the real number;
π√Real Number includes;
(i) Whole numbers: like 0, 1, 2, 3, 4.................................. etc.
(ii) Rational numbers: like 4/5, 0/6, 0.333........, etc.
(iii) irrational numbers: like π, √3, √2 etc.
We have learnt quadratic equation in previous class. The nature of quadratic equations is
D > 0, {Real and unequal roots}
D = 0, {Real and equal roots}
D < 0, {No Real roots, i.e. Imaginary root}
Look the following example
x2 + 3x + 5 = 0
a = 1, b = 3, c= 5
D =
= 32– 4 ×1 × 5
= 9 – 20
= –11
D < 0, {so equation has no real but imaginary roots}
Now we have to find the roots
Here both the value of x is an imaginary number, which is made by the composition of (i), symbol “i” is called iota. Such number is called complex number.
1. Imaginary Number: A number whose square is negative is known as an imaginary number.
Ex: , , , etc.
2. Complex number: Any number which is of the form of x + iy, where x and y are real number and i = is called a complex number.
Ex : 3 + i5, 2 – i3, 5 + i2 and 4 +i3 etc.
It is denoted by z i.e. z = x +iy, in which Re(z) = x and Im(z) = y
A complex number has two parts;
(I) real part Re(z) {∈ R}
real part : 2, 3, 5, and 4 or may be any real number.
(II) imaginary part Im(z) {Real number with i(iota)}
imaginary part: i, i2, i3, i4, and i5 etc.
Every Real number is a complex number if x∈ R and y ∈ R; such as
z = 3 ⇒ 3 +i0, x = 3, y =0
z = –3⇒ – 3 +i0, x = –3, y =0
z = 7 ⇒ 7 +i0, x = 7, y =0
3. See the following complex numbers
z = 3, z = i3, z = 4, z = i7
z = 3 and z = 4 are purely Real
z = i3 and z = i7 are purely Imaginary