The imaginary number i

In real numbers, we notice that √3, √5, √7 etc. are irrational numbers because 3,5,7 are not perfect square numbers like 4, 9, 16.

What is the square root of -9?

Since the square of every real number must be non-negative there is no real number where its square is negative.

We call the square root of -1, i where ‘i’ is called an imaginary number.

then -1 = i2

With this new convention

√-9 = ∓ 3i

We now constitute a new set of numbers called complex numbers. The set is denoted by C and we write

We can now solve equations of the form

x2+ 9 = 0

x2 + 7 = 0

x2+ 15 = 0 etc.

The complex numbers are written is the form a + ib

Example

In general, we write z = a + ib.

Complex numbers in depth

Definition

Any expression of the form a+ ib where a, b represent real numbers is called a complex number. The number ‘a’ is called the real part of the complex number, and the number ‘b’ is called the imaginary part of the complex number.

Any complex number can be written as an ordered pair. So

z = a +ib = (a,b)

If in a complex number

a = 0, b ≠ 0 then

z = ib = (0,b)

This is a purely imaginary number.

If in a complex number

a ≠ 0, b = 0 then

z = a = (a, 0)

then the number is purely real.

We can write b = Im (z) in the first case

and a = Re (z) in the second case.

where Imz is imaginary z and Re(z) is real z.

Definition

Two complex numbers are equal provided the real parts are equal and the imaginary parts are equal.

If z₁ = a₁ + ib₁ and

z₂= az₂ + ibz₂

Then z₁ = z₂ if and only if

a₁ = a₂

b₁ = b₂

Definition

Two complex numbers differing only in the signs of the imaginary parts are called complex conjugates. Each is the conjugate of the other.

That is if z = a + ib is a complex number its complex conjugate is

 = a - ib.

Examples

1. z = 6 + 2i, = 6 - 2i
2. z = -6 - 2i, = -6 + 2i

Try these questions

1. z = -5i + 8
   Answer: = 5i + 8
2. z = 12i -6/7
   Answer: = -12i - 6/7

Which is a perfect square?

1. 144
2. 133
3. 100
4. 47

Answer:   1. The square root of 144 is 12.