As originally posted on Edvie.com  

COMPLEX NUMBERS 

Class 10 Notes 

 Introduction: If a , b are natural numbers

such that a > b, then the equation

is not solvable in N, the set of natural

numbers i.e., there is no natural number

satisfying the equation . So, the

set of natural numbers is extended to

form the set I of integers in which every

equation of the form ; a , b N is

solvable.

But equations of the form , where

are not solvable in I.

Therefore, the set I of integers is extended

to obtain the set Q of all rational numbers

in which every equation of the form

is uniquely solvable.  

The equation of the form etc.

are not solvable in Q because there is no

rational number whose square is 2. Such

numbers are known as irrational

numbers. The set Q of all rational

numbers is extended to obtain the set R

which includes both rational and

irrational numbers. This set is known as

the set of real numbers. The equation of

the form etc. are not

solvable in R i.e., there is no real number

whose square is a negative real number.  

 

 Euler was the first mathematician to

introduce the symbol (iota) for the

square root of –1 with the property

He also called this symbol as

the imaginary unit. 

So, the necessity to study of COMPLEX

NUMBERS arose. 

 Integral powers of iota (i): For positive

integral powers of i: 

We have  

 

In order to compute for n > 4, we

divide n by 4 and obtain the remainder

r . Let m be the quotient when n is

divided by 4. Then, 

 

 

Thus, the value of for n > 4 is ,

where r is the remainder when n is

divided by 4. 

 

 

 

Negative integral powers of i : 

By the law of indices, we have 

 

If  

where r is the remainder when n is divided by 4. 

is defined as 1.  Example: Evaluate the following: 

(i) (ii) (iii) (iv)  

Solution:

(i) 135 leaves remainder as 3 when it is divided by 4. Therefore, 

 

(ii) The remainder is 3 when 19 is divided by 4. Therefore,  

 

(iii) We have,  

 

On dividing 999 by 4, we obtain 3 as the remainder. Therefore,  

So,  

(iv) We have, 

 

 

 

Example: Show that  

(i)  

(ii)  

(iii)  

Solution: (i) We have, 

 (ii) We have, 

 (iii) We have, 

  Imaginary quantities: The square root of a negative real number is called an

imaginary quantity or an imaginary number. 

For example, etc. are imaginary quantities. 

A useful result: If a, b are positive real numbers, then 

 Proof: We have,  

 

 

And  

Therefore,  

 

For any two real numbers is true only when at least one of a

and b is either positive or zero. 

 

In other words, is not valid if a and b both are negative.  

For any positive real number a, we have .  Example: Compute the following: 

(i) (ii) (iii)  Solution: (i) We have, 

 

(ii) We have,  

(iii) We have, 

 

Example: A student writes the formula . Then he substitutes a = –1 and

b = –1 and finds 1 = –1. Explain where is he wrong? 

Solution: Since a and b both are negative. Therefore, cannot be written as

. In fact for a and b both negative, we have 

 

Example: Is the following computation correct? If not give the correct computation: 

 

 

Solution: The said computation is not correct, because –2 and –3 both are negative

and is true when at least one of a and b is positive or zero. The correct

computation is as given below: 

 

Complex number: If a, b are two real numbers, then the number of the form a + ib is

called a complex number. 

For example 7 + 2i, –1 + i, 3 – 2i, 0 + 2i, 1 + 0i etc. are complex numbers. 

Real and imaginary parts of a complex number: If z = a + ib is a complex number,

then ‘ a ’ is called the real part of z and ‘ b ’ is known as the imaginary part of z . The real

part of z is denoted by Re ( z ) and the imaginary part by Im (z). 

For example, if z = 3 – 4 i , then Re ( z ) = 3 and Im ( z ) = – 4 

 Purely real and purely imaginary complex numbers: A complex number z is purely

real if its imaginary part is zero i.e., Im ( z ) = 0 and purely imaginary if its real part is

zero i.e., Re (z) = 0. 

E.g.: purely real = 3; purely imaginary = - 4i 

 Set of complex numbers: The set of all complex numbers is denoted by C. 

i.e.,  

since a real number ‘a’ can be written as a + 0i, therefore every real number is a

complex number. Hence, R ⊂ C where R is the set of all real numbers. 

 

Equality of complex numbers: Two complex numbers and are

equal if i.e., and . 

Thus, and . 

Example: If are equal, then find x and y. 

Solution: We have, 

 

Example: If find . 

 

Solution: We have, 

Solving these eqautions, we get 

Square root of a complex number: 

The square root of a negative number is called an imaginary

number

Example:

Solution:

 

to simplify  , divide n  by 4 and use the remainder to simplify it further. 

Change the denominator to power with multiple of 4. 

  

GET MORE FREE STUDY MATERIAL & PRACTICE TESTS ON 

Edvie.com 

 

Follow us on Facebook