Real Numbers Class 10 Notes

Real Numbers Class 10 Notes

Real Numbers Class 10 Notes contain concept-by-concept explanations. These notes were created by experts and adhere to CBSE guidelines.The Class 10th Math curriculum is designed to provide students with the core fundamentals of this discipline. Amongst the most intriguing of these concepts are Real Numbers which form the basis of several crucial Mathematical topics. You must have studied different kinds of numbers in earlier classes, such as Integers, Natural, Whole, Rational and Irrational. These kinds of numbers combine to produce a set of real numbers.

Here, we are going to understand real number concepts. Read the Real Numbers class 10 maths chapter 1 notes for your exam preparation to score higher marks.

Real Numbers

Real numbers have all irrational and rational numbers; any real numbers can be represented by plotting on the number line.

Necessary Points about Real Numbers

  • All irrational and rational numbers combine to form a collection of real numbers. It’s denoted with the letter R.
  • The number line can represent real numbers, and the square root of each positive real integer is always present and may be seen using a number line.
  • The sum or difference between a rational and an irrational number must be an irrational number.
  • The product or division of a rational number with an irrational number returns an irrational number.
  • The method of successive magnification is used to represent a decimal expansion on a number line.

Real Number Presentation

All sorts of real numbers can be precisely placed on the Number line. Real numbers such as 4 are represented as 4.0, -5 as -5.0, fractions such as 2/5 as 0.40, 1/2 as 0.50, and so forth.

Euclid’s Division Lemma

Let’s understand Euclid’s division lemma in detail:

  • The Division of Euclid Given two numbers a and b, the lemma asserts that there exists a distinct pair of integers q and r such that a=b×q+r and 0≤r<b.
  • Dividend = divisor* quotient remainder is identical to the lemma.
  • The resulting quotient and remainder will always be unique for every given pair of dividends and divisor.

Euclid’s Division Algorithm

Here’s how to use Euclid’s division algorithm:

  • Consider two integers a and b where a > b to determine the highest common factor of two numbers.
  • We use Euclid’s Division Lemma to identify two integers q and r such that a = b × q r and 0 ≤ r < b.
  • If the remainder is 0, the highest common factor is b; otherwise, we can use Euclid’s Division Lemma on b and r to obtain a new pair of quotient and remainder.
  • This approach is followed until the remainder is zero, at which point the divisor is the highest common factor of the provided set of integers.

Prime Factorization

  • This methodology is used to represent a natural number as a prime number product.
  • 12=2 *2* 3 is an example of prime factorization.

Fundamental Theorem of Arithmetic

  • According to this theorem, the prime factorization of a given integer is always distinct if the arrangement of the prime factors is neglected.
  • For instance: 36 = 2*2* 3* 3 or 36 = 2* 3* 2* 3.
  • In the above example, 36 is presented as a product of prime factors, where the factor arrangement does not matter.

Proof of the Arithmetic Fundamental Theorem

A composite number is represented in number theory as the product of primes, and this factorization is distinct aside from the order in which the prime factor comes.

This theorem also shows that not only can a composite number be factorized as the product of its prime components, but also the factorization is unique for each composite number, regardless of the sequence of occurrence of the prime factors.

It means that there is only one way to express a natural number by the product of prime factors. 

Finding the Lowest Common Multiple 

The smallest common multiple of two or more numbers is also defined as the lowest common multiple (LCM).

Method of Finding the Highest Common Factor

We know that the biggest number that divides each of the supplied numbers without leaving any remainder is the highest common factor of two or more given integers.

H.C.F can be calculated using two methods: prime factorization and Euclid’s division methodology.

Prime Factorization:

  • Given two numbers that can both be represented as products of the prime factors, we afterward pick the prime factors that are common to both numbers.
  • For example, find the highest common factor of the integers 20 and 24.
  • 20 = 2 × 2 × 5 and 24 = 2 × 2 × 2 × 3.
  • The factor common by 20 and 24 is 2 2, which equals 4, which is the highest common factor of 20 and 24.

Euclid’s Division Algorithm:

  • Euclid’s Division Algorithm is the repetitive use of Euclid’s division lemma to determine the highest common factor of two integers.

Steps to Apply:

  • Apply Euclid’s division lemma to positive integers, assuming a is greater than b, to obtain whole numbers q and r such that a = bq+r, 0 ≤ r < b.
  • If r = 0, the highest common factor of a and b equals b. If r equals zero, apply the division lemma to b and r and repeat the method until the remainder equals zero. The divisor discovered is the needed highest common factor.

For example, the number 45 can be expressed as its prime factors:

  • 45 = 9 × 5
  • The prime factors of 45 are 9 and 5.

Product of Two Numbers 

  • Every two positive integers a and b,

a × b = H.C.F × L.C.M.

Consider 36 and 56, where the highest common factor is 4, and the L.C.M is 504.

  • 36 × 56 = 2016
  • 4 × 504 = 2016

As a result of solving both, you obtain the answer: 36* 56 = 4* 504.

  • This relation does not hold for three or more numbers.

Irrational Numbers

Any number that cannot be represented in the form of p/q is irrational; examples include √2, π, e, and more.

Number Theory:

  • If a prime integer p divides a2, then p divides a. 
  • The total or difference of two irrational numbers is also irrational.
  • Irrational numbers are the product and quotient of a non-zero rational value with an irrational value.
  • √p is irrational when p is a prime integer. For example, √3 is irrational; this statement can be shown using the “Proof by contradiction” approach.

Uses of Number Theory

Here are some of the most important applications of number theory. Some of the most important divisibility tests are used to determine if a given number m divides the integer n. Numerous applications can be found in mathematics as well as in practical applications like as

  • Security systems similar to the ones used in banking securities
  • Barcodes
  • Memory management system
  • Designing modular structures
  • Websites for selling products online
  • Authentication system 

Rational Numbers and Their Decimal Expansions

Rational Numbers

Real numbers can be expressed as p/q, where p and q are integers and q is not equal to zero. A rational number is -1/2, 4/5, 1, 0, -3, and more.

Terminating and Nonterminating Decimals

  • Terminating decimals are decimals with a specific endpoint, such as 0.3, 2.76, and so on.
  • Non-terminating decimals are decimals that have no end, such as 0.33333333333333….., 0.13135243…

Non-terminating decimals can be:

a) recurring – the same component of the decimal continues to repeat

b) Non-Recurring – No recurrence of a decimal component occurs, as in = 3.1415926535…

How to know if the rational number is terminating or not?

  • Consider that if a/b is a rational number, then its decimal expansion would end if the following criteria are met:
  • The highest common factor of two integers a and b is 1.
  • b can be expressed as a prime factorization of 2 and 5, i.e., b = 2m × 5n where either m or n, or both can = 0.
  • If the prime factorization of ‘b’ has any number other than 2 or 5, then the decimal expansion of that number will be recurring.

Things to Keep in Mind

  • Real numbers are both rational and irrational numbers.
  • They can be positive or negative.
  • Real numbers have three properties: 
  1. Commutative, 
  2. Associative, 
  3. Distributive.
  • According to Euclid’s division lemma, a positive integer can be divided by another positive integer to provide a residue that is less than the second positive integer.
  • Arithmetic Fundamental Theorem – Any integer bigger than one is either a prime number or a prime number form.
  • All natural numbers can be expressed as the product of their prime factors.

Benefits of class 10th real numbers notes 

  • Real Numbers Notes are prepared in such a manner that students can go through all of the topics in a short period while not missing any crucial chapter topics.
  • Chapter Class 10 Math’s real numbers Notes can benefit learners to improve their problem-solving skills.
  • Class 10th real numbers notes can assist you in preparing for the crucial and scoring topics of real numbers.
  • Class 10 maths chapter 1 notes are written in a very exemplifying and informative manner. We keep every key factor in mind when writing these notes, such as the CBSE pattern, important concepts that need to be emphasized, and all the topics relevant to Real Number Class 10 Notes.
  • Students can understand and revise all of the formulae and topics using Chapter 1 Math’s Class 10 real numbers Notes. Last-minute revision can be highly advantageous to students as it can speed up their examination preparations and assist them in answering long questions.
  • Studying from the class 10 real numbers notes in such a manner that you take less time studying any chapter, these Maths 10th Class Notes can enable you to save time to practice more questions.

General Tips

  1. Go through every point and topic related to the chapter attentively in class 10 real numbers notes Mathematics chapter because some crucial topics will get missed if  considered easy.
  2. Revise the real numbers class 10 maths chapter 1 notes and the main topics because some of the basic principles from this chapter get repeated in coming classes.


Real Numbers Class 10 Notes is one of the excellent study resources for the CBSE final exam preparation examination. Students can go through class 10th real numbers notes to improve their learning and score good marks in the exam.

Frequently Asked Questions

         Q. What is the division algorithm of Euclid?

Euclid’s division procedure is a method for calculating the greatest common factor of two positive numbers.

         Q. What is explained in chapter 1 real numbers for Class 10?

We will study Euclid’s division algorithm, the fundamental theorem of arithmetic, techniques for calculating the lowest common multiple and highest common factors, and a thorough explanation of rational and irrational numbers in chapter 1 real numbers for Class 10.

         Q. Is the number 10i real?

No, 10i is an imaginary number, not a real number.

         Q. How difficult is Chapter 1 real numbers of Class 10 Math?

Nothing is difficult when there is patience, practice, and constant effort. To get high scores in CBSE, students must identify their mistakes and weaknesses and work extra hard on them. When students understand the fundamentals, learning becomes simple and more enjoyable. With further practice, the student will gain fluency and clarity, allowing them to answer the CBSE question paper confidently. The class 10 real numbers notes will assist the student in accessing and revising the relevant sections and becoming well-prepared.

Is zero a real number?

Zero is regarded as both a real and an imaginary number. Imaginary numbers are the square root of non-positive real numbers. And, because 0 is also a non-positive number, it meets the conditions for an imaginary number. 0 is also a rational number that can be represented on a number line and, therefore, a real number.