At Winsight Academy I provide you Real Number Class 10 Notes for your last day revision. If you are a student heading to your 1st term or you are preparing for the Final Exams, these notes clubbed with my Real Numbers Class 10 Important Questions will surely help you to excel in the exam.

What does my Real Number Class 10 Notes cover?
In these notes I have covered how to identify Natural Numbers, Whole Numbers, Integers, Rational Numbers, Complex Numbers, Factors, Multiples, Prime and Composite Numbers. After covering all the types of rational numbers, I introduce you to the whole new world of Irrational Numbers and Euclid’s Division Lemma. In Euclid’s division Lemma, I show you how to find the HCF using Euclid’s Lemma and also how to verify if the HCF you received in the result is correct or not. In the end, I cover The Fundamental Theorem of Arithmetic (One of the most important key theorem in mathematics)

CBSE Real Numbers Class 10 Maths Notes Chapter 1 are prepared according to the new CBSE Exam Pattern, you can also download Real Number Class 10 Notes PDF by clicking the button down below:

Are you an online reader? No worries, I have arranged all the topics of Real Number Class 10 Notes in a properly indexed manner.

List of Topics

• Types of Numbers
• Revisiting Irrational Numbers
• Euclid’s Division Lemma or Algorithm
• Find HCF using Euclid’s Division Lemma
• The Fundamental Theorem of Arithmetic

Real Number Class 10 Notes

Natural Numbers :

The simplest numbers are 1, 2, 3, 4……. the numbers being used in counting. These are called natural numbers.

Whole numbers : 

The natural numbers along with the zero form the set of whole numbers i.e. numbers 0, 1, 2, 3, 4 are whole numbers. W = {0, 1, 2, 3, 4….}

Integers :

The natural numbers, their negatives and zero make up the integers.

Z = {….–4, –3, –2, –1, 0, 1, 2, 3, 4,….}

The set of integers contains positive numbers, negative numbers and zero. 

Rational Number :

(i) A rational number is a number which can be put in the form , where p and q are both integers and q ≠ 0.

(ii) A rational number is either a terminating or non-terminating and recurring (repeating) decimal.

(iii) A rational number may be positive, negative or zero.

Complex numbers :

Complex numbers are imaginary numbers of the form a + ib, where a and b are real numbers and
i = ,  which is an imaginary number.

Factors :

A number is a factor of another, if the former exactly divides the latter without leaving a remainder (remainder is zero) 3 and 5 are factors of 12 and 25 respectively.

Multiples :

A multiple is a number which is exactly divisible by another, 36 is a multiple of 2, 3, 4, 9 and 12.

Even Numbers : 

Integers which are multiples of 2 are even number (i.e.) 2,4, 6, 8…………… are even numbers.

Odd numbers : 

Integers that are not multiples of 2 are odd numbers.

Prime and Composite Numbers :

All natural number which cannot be divided by any number other than 1 and itself is called a prime number. By convention, 1 is not a prime number. 2, 3, 5, 7, 11, 13, 17 …………. are prime numbers. Numbers which are not prime are called composite numbers.

The Absolute Value (or modulus) of a real Number :

If a is a real number, modulus a is written as |a| ; |a| is always positive or zero.It means positive value of ‘a’ whether a is positive or negative

|3| = 3 and |0| = 0, Hence |a| = a ; if a = 0 or  a > 0 (i.e.) a ≥ 0

|–3| = 3 = – (–3) . Hence |a| = – a when a < 0

Hence, |a| = a, if a > 0 ;  |a| = – a, if a < 0

Irrational number : 

(i) A number is irrational if and only if its decimal representation is non-terminating and non-repeating. e.g., , π……………. etc. 

(ii) Rational number and irrational number taken together form the set of real numbers.

(iii) If a and b are two real numbers, then either
(i) a > b or (ii) a =  b or (iii)  a < b

(iv) Negative of an irrational number is an irrational number.

(v) The sum of a rational number with an irrational number is always irrational.

(vi) The product of a non-zero rational number with an irrational number is always an irrational number.

(vii) The sum of two irrational numbers is not always an irrational number.

(viii) The product of two irrational numbers is not always an irrational number.

In division for all rationals of the form
(q ≠ 0), p & q are integers, two things can happen either the remainder becomes zero or never becomes zero.

These numbers are called irrational numbers.

Eg. :   

0.1279312793	rational	terminating
0.1279312793….	rational	non terminating
	&	recurring
0.32777	rational	terminating
	rational	non terminating
0.32777…….	&	recurring
0.5361279	rational	terminating
0.3712854043….	irrational	non terminating

Euclid’s Division Lemma or Euclid’s Division Algorithm

For any two positive integers a and b, there exist unique integers q and r satisfying a = bq + r, where 0 ≤ r < b. 

For Example 

(i) Consider number 23 and 5, then:

23 = 5 × 4 + 3

Comparing with a = bq + r; we get:

a = 23, b = 5, q = 4, r = 3 

and 0 ≤ r < b (as 0 ≤ 3 < 5).

(ii) Consider positive integers 18 and 4.

18 = 4 × 4 + 2

⇒ For 18 (= a) and 4(= b) we have q = 4, 

r = 2 and  0 ≤ r < b.

In the relation a = bq + r, where 0 ≤ r < b is nothing but a statement of the long division of number a by number b in which q is the quotient obtained and r is the remainder. 

Thus, dividend = divisor × quotient + remainder ⇒ a = bq + r

What is H.C.F (Highest Common Factor)?

 The H.C.F. of two or more positive integers is the largest positive integer that divides each given positive number completely.

i.e., if positive integer d divides two positive integers a and b then the H.C.F. of a and b is d.

For Example

(i) 14 is the largest positive integer that divides 28 and 70 completely; therefore H.C.F. of 28 and 70 is 14.

(ii) H.C.F. of 75, 125 and 200 is 25 as 25 divides each of 75, 125 and 200 completely and so on.

How to Find H.C.F Using Euclid’s Division Lemma?

Consider positive integers 418 and 33.

Step-1

Taking bigger number (418) as a and smaller number (33) as b 

express the numbers as a = bq + r

⇒ 418 = 33 × 12 + 22

Step-2

Now taking the divisor 33 and remainder 22; apply the Euclid’s division algorithm to get:

33 = 22 × 1 + 11   [Expressing as a = bq + r]

Step-3

Again with new divisor 22 and new remainder 11; apply the Euclid’s division algorithm to get:

22 = 11 × 2 + 0 

Step-4

Since, the remainder = 0 so we cannot proceed further.

Step-5

The last divisor is 11 and we say H.C.F. of 418 and 33 = 11

How to verify the H.C.F resulted from Euclid’s Lemma?

(i) Using factor method: 

∴ Factors of 418 = 1, 2, 11, 19, 22, 38, 209 and 418 and, 

Factor of 33 = 1, 3, 11 and 33. 

Common factors = 1 and 11

⇒ Highest common factor = 11 i.e., H.C.F. = 11 

(ii) Using prime factor method: 

Prime factors of 418 = 2, 11 and 19.

Prime factors of 33 = 3 and 11. 

  ∴ H.C.F. = Product of all common prime factors  = 11. For any two positive integers a and b which can be expressed as a = bq + r, where 0 ≤ r < b, the, H.C.F. of (a, b) = H.C.F. of (q, r) and so on. For number 418 and 33

418 = 33 × 12 + 22

33 = 22 × 1 + 11

and 22 = 11 × 2 + 0

⇒ H.C.F. of (418, 33) = H.C.F. of (33, 22)     = H.C.F. of (22, 11) = 11.

The Fundamental Theorem of Arithmetic

Statement : Every composite number can be decomposed as a product prime numbers in a unique way, except for the order in which the prime numbers occur.  

For example : 

(i)  30 = 2 × 3 × 5, 30 = 3 × 2 × 5, 30 = 2 × 5 × 3 and so on.

(ii) 432 = 2 × 2 × 2 × 2 × 3 × 3 × 3 = 2⁴ × 3³    

    or 432 = 3³ × 2⁴. 

(iii) 12600 = 2 × 2 × 2 × 3 × 3 × 5 × 5 × 7 

= 2³ × 3² × 5² × 7

In general, a composite number is expressed as the product of its prime factors written in