According to the quadratic formula ,

if b

For example : x

x

∴ x = $\pm$ √ (-4)

∴ x = $\pm$ 2 √ (-1)

We assume that the square root of (-1) is denoted by the symbol ‘i’, called imaginary unit. The symbol ‘i’ was first introduced by the famous Swiss mathematician, Leonhard Euler in 1748, possibly because ‘i’ is the first letter of the Latin word ‘imaginarius’. Thus , for any two real numbers ‘a’ and ‘b’, we can form a new number

The complex-numbers is of the form ‘a + ib’, where a and b are real numbers and ‘i’ is the imaginary unit which has the property of

i

In the 1st example 4 is a real part and 7 is an imaginary part.

In the 2nd example -1/2 is a real part and 1 is an imaginary part.

A complex-number is denoted by single letter such as z, w etc.

Two complex-numbers

z

A complex-number ‘z’ is said to be zero if its both real and imaginary parts are zero. In other words

z= a + ib = 0 if and only if a = 0 and b = 0

If z = a + ib, the number a – ib is called the complex conjugate(or simply conjugate) of a + ib and is denoted by

i) √(-27) ⇒ √(-1 x 27) = √(-1) √27 = i√27

ii) 4 - √(-5) ⇒ 4 - √(-1 x 5) = 4 - √5 √(-1) = 4 - i√5

2) Write the real and imaginary part of the following numbers.

i) 2 + i √2

Re z = 2 and Im z = √2

ii) √3/ 5 i

Re z = 0 and Im z √3/5

3) Find a and b such that 2a + i 4b and 2i represents the same complex-numbers.

2a + i 4b = 0 + 2i

By the definition of equality,

2a = 0 ⇒ a = 0

And 4b = 2 ⇒ b = 2/4 = ½

∴ a = 0 and b = ½ 4) Find the conjugates of 2 + 5i and √5

2 – 5i and √5 (as there is no imaginary part)

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