Real Numbers Class 10 Notes Maths Chapter 1

CBSE Class 10 Maths Chapter 1 Real Numbers Notes are here to help you understand the basics. Real numbers are the ones we commonly use, like positive and negative whole numbers, fractions, and numbers with decimals. Basically, if you can find a number in the real world, it’s a real number. Think about it: we use numbers every day, whether it’s counting things, measuring temperature with positive or negative whole numbers, or working with fractions for parts of a whole. We’ll cover what real numbers are, how to divide them using Euclid’s division algorithm, the fundamental theorem of arithmetic, ways to find the lowest common multiple (LCM) and highest common factor (HCF), and we’ll explain rational and irrational numbers with examples.

Real Numbers Class 10 Notes

Get your free PDF download of CBSE Class 10 Maths Notes for Chapter 1 on Real Numbers. These notes are designed for quick revision. Included are NCERT Class 10 Maths Notes for Chapter 1, covering the topic of Real Numbers. In the updated CBSE Exam Pattern, expect Multiple Choice Questions (MCQs) in Class 10 Maths, which contribute 20 marks.

R = Real Numbers: All rational and irrational numbers are called real numbers.

I = Integers: All numbers from (…-3, -2, -1, 0, 1, 2, 3…) are called integers.

Q = Rational Numbers: Real numbers of the form \(\frac { p }{ q }\), q ≠ 0, p, q ∈ I are rational numbers.

• All integers can be expressed as rational, for example, 5 = \(\frac { 5 }{ 1 }\)
• Decimal expansion of rational numbers terminating or non-terminating recurring.

Q’ = Irrational Numbers: Real numbers which cannot be expressed in the form \(\frac { p }{ q }\) and whose decimal expansions are non-terminating and non-recurring.

• Roots of primes like √2, √3, √5 etc. are irrational

N = Natural Numbers: Counting numbers are called natural numbers. N = {1, 2, 3, …}

W = Whole Numbers: Zero along with all natural numbers are together called whole numbers. {0, 1, 2, 3,…}

Even Numbers: Natural numbers of the form 2n are called even numbers. (2, 4, 6, …}

Odd Numbers: Natural numbers of the form 2n -1 are called odd numbers. {1, 3, 5, …}

• Why can’t we write the form as 2n+1?

Remember this!

• All Natural Numbers are whole numbers.
• All Whole Numbers are Integers.
• All Integers are Rational Numbers.
• All Rational Numbers are Real Numbers.

Prime Numbers: The natural numbers greater than 1 which are divisible by 1 and the number itself are called prime numbers, Prime numbers have two factors i.e., 1 and the number itself For example, 2, 3, 5, 7 & 11 etc.

• 1 is not a prime number as it has only one factor.

Composite Numbers: The natural numbers which are divisible by 1, itself and any other number or numbers are called composite numbers. For example, 4, 6, 8, 9, 10 etc.
Note: 1 is neither prime nor a composite number.

Since remainder is zero, divisor (8) is HCF.
Although Euclid’s Division lemma is stated for only positive integers, it can be extended for all integers except zero, i.e., b ≠ 0.

1. Algorithm to locate HCF and LCM of two or more positive integers:

Step I: Factorize each of the given positive integers and express them as a product of powers of primes in ascending order of magnitude of primes.
Step II: To find HCF, identify common prime factor and find the least powers and multiply them to get HCF.
Step III: To find LCM, find the greatest exponent and then multiply them to get the LCM.

2. To prove Irrationality of numbers:

• The sum or difference of a rational and an irrational number is irrational.
• The product or quotient of a non-zero rational number and an irrational number is irrational.

3. To determine the nature of the decimal expansion of rational numbers:

• Let x = p/q, p and q are co-primes, be a rational number whose decimal expansion terminates. Then the prime factorization of’q’ is of the form 2^m5ⁿ, m and n are non-negative integers.
• Let x = p/q be a rational number such that the prime factorization of ‘q’ is not of the form 2^m5ⁿ, ‘m’ and ‘n’ being non-negative integers, then x has a non-terminating repeating decimal expansion.

Alert!

• 2³ can be written as: 2³ = 2³5⁰
• 5² can be written as: 5² = 2⁰5²

Euclid’s Division Lemma

Euclid’s Division Lemma” in Class 10 Maths is a fundamental theorem used for proving the existence and uniqueness of the quotient and remainder when one integer is divided by another. It states that for any two positive integers ‘a’ and ‘b’, there exist unique integers ‘q’ (quotient) and ‘r’ (remainder) such that

$𝑎=𝑏𝑞+𝑟$

a=bq+r, where

$0\le 𝑟<𝑏$

0≤r<b.

Euclid’s Division Algorithm

Euclid’s Division Algorithm in Class 10 is a method used to find the highest common factor (HCF) of two numbers by repeatedly dividing the larger number by the smaller one and taking the remainder until we get a zero remainder.

Real Numbers

Real numbers refer to all the numbers that we use in everyday life, including positive and negative whole numbers, fractions, decimals, and irrational numbers like square roots. They are the numbers we can see on a number line and use in various mathematical operations.

• Any real number can be plotted on the number line.
• Real numbers constitute the union of all rational and irrational numbers.

 

 

 

 

 

 

 

 

 

 

 

 

 

Revisiting Irrational Numbers

Revisiting Irrational Numbers” in Class 10 likely refers to a topic in mathematics where students review and deepen their understanding of irrational numbers, which are numbers that cannot be expressed as fractions and have non-repeating, non-terminating decimal representations. This could involve exploring properties of irrational numbers, operations involving them, and their significance in real-world contexts.

FAQs Real Numbers Class 10 Notes

What are real numbers in class 10th?

Real numbers in class 10th are the numbers that include all the positive and negative whole numbers, fractions, and decimals. They're basically numbers we use every day.

What are the types of real numbers?

Real numbers are classified into rational and irrational numbers. Rational numbers can be expressed as fractions, while irrational numbers cannot and include square roots of non-perfect squares.

Is zero a real number?

Yes, zero is a real number. It's a whole number and also considered neither positive nor negative.