eVidyarthi

Main Menu
  • eVidyarthi
  • School
    • Class 6th
      • Maths Class 6
      • Science Class 6
      • Hindi Class 6
      • व्याकरण
      • English Class 6
      • English Grammar
      • Sanskrit Class 6
      • Geography
      • Civics
      • History
    • Class 7th
      • Maths Class 7
      • Science Class 7
      • Hindi Class 7
      • व्याकरण
      • English Class 7
      • English Grammar
      • Sanskrit Class 7
      • Geography
      • Civics
      • History
    • Class 8th
      • Maths Class 8
      • Science Class 8
      • Hindi Class 8
      • व्याकरण
      • English Class 8
      • English Grammar
      • Sanskrit Class 8
      • Geography
      • Civics
      • History
    • Class 9th
      • Maths Class 9
      • Science Class 9
      • Hindi Class 9
      • English Class 9
      • English Grammar
      • व्याकरण
      • Economics Class 9
      • Geography Class 9
      • Civics Class 9
      • History Class 9
    • Class 10th
      • Maths Class 10
      • Science Class 10
      • Hindi Class 10
      • English Class 10
      • English Grammar
      • व्याकरण
      • Economics Class 10
      • History Class 10
      • Civics Class 10
      • Geography Class 10
    • Class 11th
      • Maths Class 11
      • Accounts Class 11
      • English Class 11
      • English Grammar
      • व्याकरण
      • Chemistry Class 11
      • Physics Class 11
      • Biology Class 11
    • Class 12th
      • Maths Class 12
      • Accounts Class 12
      • Chemistry Class 12
      • Physics Class 12
      • Biology Class 12
      • English Class 12
      • English Grammar
      • व्याकरण
    • Close
  • English
    • Basic English Grammar
    • Basic English Speaking
    • English Vocabulary
    • English Idioms & Phrases
    • Personality Enhancement
    • Interview Skills
    • Close
  • Softwares
    • Microsoft Word
    • PhotoShop
    • Excel
    • Illustrator
    • PowerPoint
    • Close
Class 10th Maths || Menu
  • Important
    • Previous Year Papers 2019
    • CBSE Board Paper 2018
    • CBSE Board Paper 2017
    • Sample Papers
    • Sample Paper – I
    • MCQs
    • Important Formulas
    • Revision Notes
    • Mathematics Book
    • Marking Scheme
    • Mathematics Syllabus
    • Close
  • Real Numbers
    • Ex. 1.1 & Intro
    • Ex. 1.2 NCERT
    • Ex. 1.3 & Ex. 1.4
    • R.D Sharma Solutions
    • Important Formulas
    • NCERT Solutions
    • Close
  • Polynomials
    • Ex. 2.1 & Intro
    • Ex. 2.2 NCERT
    • Ex. 2.3 & Ex. 2.4 NCERT
    • R. D Sharma Solutions
    • NCERT Solutions
    • Close
  • Pair of Linear Equations in 2 Variables
    • Ex. 3.1 & Intro
    • Ex. 3.2 NCERT
    • Ex. 3.3 NCERT
    • Ex 3.4 NCERT
    • Ex 3.5 NCERT
    • Ex. 3.6 & Ex. 3.7 NCERT
    • R. D Sharma Solutions
    • NCERT Solutions
    • Close
  • Quadratic Equations
    • Ex 4.1 & Intro
    • Ex 4.2 NCERT
    • Ex. 4.3 NCERT
    • Ex. 4.4 NCERT
    • R. D Sharma Solutions
    • NCERT Solutions
    • Close
  • Arithmetic Progression
    • Ex 5.1 & Intro
    • Ex 5.2 NCERT
    • Ex 5.3 NCERT
    • Ex 5.4 NCERT
    • R. D Sharma Solutions
    • NCERT Solutions
    • Close
  • Triangles
    • Intro & Theorem
    • Ex 6.2 NCERT
    • Ex. 6.3 NCERT
    • Ex 6.4 NCERT
    • Ex 6.5 NCERT
    • Ex. 6.6 NCERT
    • R D Sharma Solutions
    • NCERT Solutions
    • Close
  • Coordinate Geometry
    • Ex. 7.1 & Intro
    • Ex. 7.2 NCERT
    • Ex. 7.3 NCERT
    • Ex 7.4 NCERT
    • R D Sharma Solutions
    • NCERT Solutions
    • Close
  • Introduction to Trigonometry
    • Ex 8.1 & Intro
    • Ex. 8.2 NCERT
    • Ex 8.3 NCERT
    • Ex 8.4 NCERT
    • Ex. 6.1 with Examples – R.D Sharma
    • R D Sharma Solutions
    • NCERT Solutions
    • Close
  • Some Applications of Trigonometry
    • Ex. 9.1 & Intro
    • R D Sharma Solutions
    • NCERT Solutions
    • Close
  • Circles
    • Intro & Theorem
    • Ex. 10.1 & Ex. 10.2 NCERT
    • R D Sharma Solutions
    • NCERT Solutions
    • Close
  • Constructions
    • Construction of Similar Figures
    • Construction of Tangents
    • NCERT Solutions
    • Close
  • Areas Related to Circles
    • Ex 12.1 & Intro
    • Ex 12.2 NCERT
    • Ex. 12.3 NCERT
    • R D Sharma Solutions
    • Important Formulas
    • NCERT Solutions
    • Close
  • Surface Areas and Volumes
    • Ex. 13.1 NCERT
    • Ex. 13.2 NCERT
    • Ex. 13.3 NCERT
    • Ex. 13.4 & Ex. 13.5 NCERT
    • R. D Sharma Solutions
    • NCERT Solutions
    • Close
  • Statistics
    • Ex. 14.1 NCERT
    • Ex. 14.2 NCERT
    • Ex. 14.3 NCERT
    • Ex. 14.4 NCERT
    • R D Sharma Solutions
    • NCERT Solutions
    • Close
  • Probability
    • Ex. 15.1 & Intro
    • Ex. 15.2 NCERT
    • R D Sharma Solutions
    • NCERT Solutions
    • Close

CBSE Class 10 Maths Chapter 1 Real Numbers Notes

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.

I. Euclid’s Division lemma
Given two positive integers a and b, there exist unique integers q and r satisfying a = bq + r, 0 ≤ r ≤ b.
Notice this. Each time ‘r’ is less than b. Each ‘q’ and ‘r’ is unique.

Real Numbers Class 10 Notes Maths Chapter 1 1

II. Application of lemma
Euclid’s Division lemma is used to find HCF of two positive integers. Example: Find HCF of 56 and 72 ?
Steps:

  • Apply lemma to 56 and 72.
  • Take bigger number and locate ‘b’ and ‘r’. 72 = 56 × 1 + 16
  • Since 16 ≠ 0, consider 56 as the new dividend and 16 as the new divisor. 56 = 16 × 3 + 8
  • Again, 8 ≠ 0, consider 16 as new dividend and 8 as new divisor. 16 = 8 × 2 + 0

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.

III. Constructing a factor tree
Steps

  • Write the number as a product of prime number and a composite number
    Example:
    Factorize 48
  • Repeat the process till all the primes are obtained
    ∴ Prime factorization of 48 = 24 x 3
    Real Numbers Class 10 Notes Maths Chapter 1 2

IV. Fundamental theorem of Arithmetic
Every composite number can be expressed as a product of primes, and this expression is unique, apart from the order in which they appear.
Applications:

  1. To locate HCF and LCM of two or more positive integers.
  2. To prove irrationality of numbers.
  3. To determine the nature of the decimal expansion of rational numbers.

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 2m5n, 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 2m5n, ‘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

Android App

eVidyarthi

Search On eVidyarthi

Evidyarthi on Facebook

Like us on Facebook

Follow Evidyarthi on Youtube

Learn English
Learn English Through हिन्दी
Job Interview Skills
English Grammar
हिंदी व्याकरण - Vyakaran
Mathematics Class 6th
Science Class 6th
हिन्दी Class 6th
Mathematics Class 7th
Science Class 7th
हिन्दी Class 7th
हिन्दी Class 8th
Mathematics Class 8th
Mathematics Class 9th
English Class 9th
Science Class 9th
Mathematics Class 10th
English Class 10th
Mathematics Class XI
Chemistry Class XI
Accountancy Class 11th
Accountancy Class 12th
Mathematics Class 12th
Microsoft Word
Microsoft Excel
Microsoft PowerPoint
Adobe PhotoShop
Adobe Illustrator
Learn German
Learn French
IIT JEE
Privacy Policies, Contact Us
Copyright © 2020 eVidyarthi and its licensors. All Rights Reserved.