As originally posted on Edvie.com  

REAL NUMBERS 

 

Class 10 Notes 

 

 

 

Introduction: All the numbers including whole numbers, integers, fractions and decimals can be written in the form of

 

 

Rational numbers: A rational number is a number which can be written in the form

of where both p and q are integers and

.  They are a bigger collection than integers as there can be many rational numbers between two integers. All rational numbers can be written either in the form of terminating decimals or non-terminating repeating decimals.  

Fundamental theorem of Arithmetic: Every composite number can be expressed (factorized) as a product of primes, and this factorization is unique, apart from the order in which the prime factors occur.  

1. In general, given a composite

number x , written in product of

primes as where

are primes and written

in ascending order,

 

Once we have decided that the order will be ascending, then the way the number is factorized, is unique. 

Example:  

 

Example:  

Decimal representation of Rational numbers: Every rational number when written in decimal form is either a terminating decimal or a non-terminating repeating decimal. Terminating decimal: A terminating

decimal is a decimal that contains

finite number of digits. 

Example:

 

In the above examples, we can observe

that the denominators of the rational

numbers don’t have any other prime

factors except 2 or 5 or both. Hence at

some stage a division of numerator by

2 or 5 the remainder is zero and we get

a terminating decimal. 

If a Rational number, in its standard form has no other prime

 

 

i.e., If we use the

same primes, we will get powers of

primes.

factor except 2 or 5 it can be expressed as a terminating decimal. 

Non-terminating repeating decimals: If a rational number in its standard form has

prime factors other than 2 or 5 or in addition to 2 and 5, the division does not end.

During the process of division we get a digit or a group of digits recurring in the same

order. Such decimals are called non terminating repeating decimals.  

We draw a line segment over the recurring part to indicate the non-terminating

nature. 

Example: 

   

∴ = 1.3636……=  

Similarly   Observe the following decimals as rationals. 

Now, (i)  

(ii)  

(iii)  

 

 

(iv)  

It is observed that the decimal expression expressed in simplest rational form, the

denominator is having only powers of 2 or 5, or both 2 and 5.  

From above examples, the conclusion is: 

Let x be a rational number whose decimal expansion terminates. Then x can be

expressed in the form of where p and q are coprimes and the prime factorization

of q is of the form where n , m are non-negative integers. 

Let be a rational number, such that the prime factorization of q is of

the form where n , m are non-negative integers. Then x has a decimal

expansion which terminates. This is known as converse of fundamental theorem of

arithmetic. 

 Observe the following decimals as rationals. 

  

i) The recurring part of the Non – terminating decimal is called period. 

Example: ⇒ Period = 3 

ii) Number of digits in the period is called periodicity. 

Example:  

 

 

⇒ Period = 142857 

⇒ Periodicity = 6.  

From above example, the conclusion is: Let be a rational number, such that the

prime factorization of q is not of the form where n , m are non-negative integers.

Then, x has a decimal expansion which is non-terminating repeating (recurring). 

 Example: Without actual division, state whether the following rational numbers are

terminating or nonterminating repeating decimals. 

(i) (ii) (iii)  

Solution:  

(i) is terminating decimal. 

(ii) is non-terminating, repeating decimal as the denominator is

not of the form  

(iii) is non-terminating, repeating decimal as the denominator is

not of the form  

 Example: Write the decimal expansion of the following rational numbers without

acutal division. 

(i) (ii) (iii)  

Solution: 

(i)  

(ii)  

 

 

(iii)  

Irrational numbers: A number which cannot be written in the form of where p

and q are integers and (or) a decimal number which is neither terminating nor

repeating is called irrational number. 

Example: etc. 

Before proving … etc.. are irrational using fundamental theorem of arithmetic.

There is a need to learn another theorem which is used in the proof. 

 Theorem: Let p be a prime number. If p divides a 2 , where a is a positive integer. Then

p divides a . 

Proof: Let the positive integer be a.  

Prime factorization of where are prime

numbers. 

Therefore  

 

Given p divides a2 ⇒ p is one of the prime factors of a 2 . 

The only prime factor of a 2 is  

∴ If p divides , then p is one of  

Since p is one of ⇒ p divides a  

 Example: Verify the statement above for p = 2 and a2 = 64 

Clearly 64 is even number which is divisible by 2 

 

8 is also divisible by 2 which is even number. 

∴ The theorem above is verified. 

 

 

Now, by using the above theorem it is easy to prove are irrational. For

this contradiction technique is used. 

 Example: Prove that is irrational. 

Solution: Let is not an irrational  

∴  

∴  

∴ 5 divides a 2 ⇒ 5 divides a ……..(2) (Theorem)  

∴  

From (1) , (3)  

 

∴ 5 divides b 2 ⇒ 5 divides b …… (4) (Theorem) 

From (2), (4) 

5 divides a and b . 

But a and b are co – primes (by assumption) 

∴ It is contradiction to our assumption. 

∴ is not a rational  

∴ is an irrational.  

Example: Show that is irrational. 

Solution: Let is not an irrational  

∴  

 Here a, b are rational then  

 

 

 ∴ a rational is not equal to irrational  

∴ our assumption is wrong. 

∴ is an irrational.  

 i) The sum of the two irrational numbers need not be irrational.

Example: If , then both x and y are irrational, but

which is rational.

ii) The product of two irrational numbers need not be irrational.

Example: , then both x and y are irrational, but

which is rational.

 Surd: An irrational root of a rational number is called a surd. 

General form of a surd: is called a surd of order n , where a is positive rational

number, n is a positive integer greater than 1 and is not a rational number. 

Example : i) are surds of order 2. 

ii) , are surds of order 3. 

Operations on surds:  

Addition and subtraction: Similar surds can be added or subtracted. Addition and

subtraction can be done using distributive law. i.e.,  

Example: i)  

ii)  

Multiplication of surds: 

i) Surds of the same order can be multiplied as  

 

 

ii) Surds of different order can be multiplied by reducing them to the same

order. 

Example: 

i)

ii)  Division of surds: Apply same procedure as in case of multiplication of surds. 

Example: 

i)

ii)

 

iii)  

The order of radicals   are 4, 6 and 6 respectively we 

note that   and the order of  is 2. Thus the order of surd is not a property 

of the surd itself, but of the way in which it is expressed. 

 

 

i) If is a surd, the ‘ a ’ is called radicand and the symbol is called

radical sign. 

ii) are not surds as they are not irrational numbers. 

 

 

iii) In a surd the radicand should always be a rational number. So

and are roots of an irrational number, hence cannot called surds.  

iv) All surds are irrational numbers, whereas all irrational numbers are

not surds.

Example: is irrational but not a surd. 

 

Real numbers: A number whose square is non-negative, is called a real number. 

In fact, all rational and irrational numbers form the collection of real numbers. 

Every real number is either rational or irrational. 

 

Consider a real number. 

i) If it is an integer or it has a terminating or repeating decimal

representation then it is rational.

ii) If it has a either-terminating nor-repeating decimal representation then

it is irrational.

Rational and irrational number together form the collection of all real numbers     

 In the given number line, 4/10 is to the left of 5/10, 6/10,7/10 etc. Thus. 

 The above implies that 5/10 lines between 4/10 and 6/10. Consider two rational numbers 5/10 

and 6/10. We can find a rational number between them. For example, 

 

 

 Again, between 5/10 and 11/20, we can find another rational number. For example, 

 and the process continues. Thus, we can find a rational number between any two rational 

numbers however close they may be end, hence, infinite rational numbers lies between two 

rational numbers. This property of rational numbers explains that rational numbers and 

present ever,  y  where on the number line. 

 

 

 

 

 

GET MORE FREE STUDY MATERIAL & PRACTICE TESTS ON 

Edvie.com 

 

Follow us on Facebook