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Free Printable Math Worksheets for 10th Grade

Math just got more exciting for Grade 10 students! Discover a vast collection of free printable math worksheets by Quizizz, designed to help teachers enhance their students' learning experience and master essential skills.

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Math worksheets for Grade 10 are essential tools for teachers looking to help their students master the challenging concepts in high school mathematics. These worksheets cover a wide range of topics, including algebra, geometry, trigonometry, and probability, ensuring that students have ample practice in each area. Teachers can use these resources to supplement their lesson plans, provide extra practice for struggling students, or even as a basis for group activities in the classroom. By incorporating Grade 10 math worksheets into their curriculum, teachers can ensure that their students are well-prepared for the rigors of high school mathematics and beyond.

Quizizz is an excellent platform for teachers seeking to enhance their Grade 10 math curriculum with engaging and interactive content. In addition to offering a vast collection of Math worksheets for Grade 10, Quizizz also provides teachers with access to a wide variety of quizzes, games, and other educational resources. These materials can be easily integrated into the classroom, allowing teachers to create a dynamic and engaging learning environment for their students. Furthermore, Quizizz offers real-time feedback and analytics, enabling teachers to track student progress and identify areas where additional support may be needed. With Quizizz, teachers have a powerful tool at their disposal to help their Grade 10 students excel in mathematics.

NCERT Solutions for Class 10 Maths Chapter 1 Real Numbers

Class 10 Maths Chapter 1 Solutions Class 10 Maths Chapter 1 Main Topics Class 10 Maths Chapter 1 Practice Tests 10th Maths Chapter 1 Question wise Solutions Class 10 Maths Chapter 1 MCQ Class 10 Maths Chapter 1 Assignments

NCERT Solutions for Class 10 Maths Chapter 1

Learning Class 10 Maths Chapter 1, Real Numbers, effectively requires a systematic approach and a clear understanding of the concepts involved. Here are some steps and strategies to help you learn this chapter.

Get here the NCERT Solutions for Class 10 Maths Chapter 1 Real Numbers all exercises in Hindi and English medium revised and modified for session 2024-25. The solution of chapter 1 class 10th mathematics is updated according to NCERT textbooks published for 2024-25 CBSE exams.

Euclid’s Division Lemma states that for any two positive integers a and b, there exist unique integers q and r such that: a = bq + r, where 0 ≤ r < b. The Fundamental Theorem of Arithmetic states that every positive integer greater than 1 can be expressed as a product of prime numbers in a unique way, up to the order of factors.

Methods for proving the irrationality of certain numbers, such as the square root of a prime number, are discussed. Understanding decimal expansions of real numbers. Terminating decimals and recurring decimals. Rationalization is the process of converting an irrational number into a rational one by multiplying or dividing by a suitable number.

According to rationalised syllabus for academic session 2024-25, only two exercises are in chapter 1. The exercise 1.1 deleted and exercise 1.2 become the exercise 1.1. Similarly exercise 1.3 become exercise 1.2 and exercises 1.4 is also deleted. The course structure of class 10 mathematics chapter 1 is given below: Fundamental Theorem of Arithmetic – statements after reviewing work done earlier and after illustrating and motivating through examples, Proofs of irrationality of √2, √3, √5.

>> Total Number of Exercises: 2 >> Total Marks in Board Exam: 6 >> Number of Periods needed: 15

Class 10 Maths Chapter 1, titled Real Numbers, is an important chapter that lays the foundation for understanding various mathematical concepts. The main points covered in this chapter include introduction to Real Numbers. Real numbers are a set of numbers that includes all rational and irrational numbers. Real numbers can be represented on the number line.

Utilize online resources like video tutorials, educational websites like Tiwari Academy, and interactive quizzes to reinforce your understanding of the chapter. As you progress, practice solving sample papers or past year question papers. This will help you become familiar with the exam pattern and time management. Periodically test your knowledge by attempting self-assessment quizzes or problems from the chapter. This will help you gauge your progress.

In class 10 Maths Chapter we will learn about understanding square numbers and their properties. Finding the square root of a number using prime factorization. Real numbers are used in various mathematical applications, such as finding the distance between two points in coordinate geometry.

Class 10 Maths Chapter 1 Real Numbers

The chapter provides numerous examples and practice problems to reinforce the concepts discussed. These are the main points covered in Class 10 Maths Chapter 1, Real Numbers. It serves as the building block for further mathematical concepts and is essential for a solid foundation in mathematics. Get here topic wise NCERT Solutions for 10th Mathematics chapter 1 all exercises in Hindi and English medium. Students can take help of videos for solving questions of each exercise.

Class 10 Maths Chapter 1 Solutions for CBSE Board

• Class 10 Maths Chapter 1 Exercise 1.1
• Class 10 Maths Chapter 1 Exercise 1.2

Class 10 Maths Chapter 1 Solutions for State Boards

• Class 10 Maths Chapter 1 Exercise 1.3
• Class 10 Maths Chapter 1 Exercise 1.4
• Class 10 Maths Chapter 1 NCERT Book
• Class 10 Mathematics NCERT Solutions
• Class 10 all Subjects NCERT Solutions

10th Maths Chapter 1 Exercise 1.1 Solutions

Class 10 Maths Chapter 1 Exercise 1.1 Question 1 Express each number as a product of its prime factors: (i) 140 (ii) 156 (iii) 3825 (iv) 5005 (v) 7429 The step by step solutions of each part is given here with practice questions>. You can discuss with us Part (i) , Part (ii) , Part (iii) , Part (iv) , and Part (v) to learn more about the solution.

Class 10 Maths Chapter 1 Exercise 1.1 Question 2 Find the LCM and HCF of the following pairs of integers and verify that LCM × HCF = product of the two numbers. (i) 26 and 91 (ii) 510 and 92 (iii) 336 and 54. Get the solution of question 2 and discuss Part (i) , Part (ii) and Part (iii) with us along with assignments and other doubts for more clarification.

Class 10 Maths Chapter 1 Exercise 1.1 Question 3 Find the LCM and HCF of the following integers by applying the prime factorisation method. (i) 12, 15 and 21 (ii) 17, 23 and 29 (iii) 8, 9 and 25. Discuss with part one , two and three in detail and get worksheets to understand the solution in simple way.

Class 10 Maths Chapter 1 Exercise 1.1 Question 4 Given that HCF (306, 657) = 9, find LCM (306, 657). The solution of this question is very simple and usually asked in MCQ type questions. For more revision , please write to us .

Class 10 Maths Chapter 1 Exercise 1.1 Question 5 Check whether 6ⁿ can end with the digit 0 for any natural number n. The solution of question 5 is similar to the example given in textbook. Discuss with us and Practice here more questions based on question 5.

Class 10 Maths Chapter 1 Exercise 1.1 Question 6 Explain why 7 × 11 × 13 + 13 and 7 × 6 × 5 × 4 × 3 × 2 × 1 + 5 are composite numbers. Extra questions for practice based on the solution of question 6 are given here to download. Put your view to discuss with us.

Class 10 Maths Chapter 1 Exercise 1.1 Question 7 There is a circular path around a sports field. Sonia takes 18 minutes to drive one round of the field, while Ravi takes 12 minutes for the same. Suppose they both start at the same point and at the same time, and go in the same direction. After how many minutes will they meet again at the starting point? Question number 7 is important for exams. Get here the solution and practice assignments with answers for revision.

10th Maths Chapter 1 Exercise 1.2 Solutions

Class 10 Maths Chapter 1 Exercise 1.2 Question 1 Prove that √5 is irrational. To understand the solution of question 1, student need to practice assignments and discuss the explanation in depth.

Class 10 Maths Chapter 1 Exercise 1.2 Question 2 Prove that 3 + 2√5 is irrational. Get here the assignments and practice questions based on the solution of question 2. If still there is any doubt, please visit to discussion forum.

Class 10 Maths Chapter 1 Exercise 1.2 Question 3 Prove that the following are irrationals: (i) 1/√2 (ii) 7√5 (iii) 6 + √2 The solution of question 3 is similar to question 1 and 2. Get more practice to learn the concepts in better way. Discuss in detail here.

Use number lines and diagrams to visualize concepts like decimal expansions and the placement of real numbers on the number line. Visual aids can help make abstract concepts more concrete. Don’t hesitate to ask your teacher or a classmate for help if you encounter difficulties. Sometimes, discussing problems with others can provide valuable insights. Maintain a study schedule and allocate specific time slots for learning and practicing this chapter. Consistency is key.

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The complete chapter of 10th Maths chapter 1 is divided in four main parts. The explanation of each part with video explanation is given below. Start by reading the chapter thoroughly and grasp the basic definitions and concepts. Ensure that you understand the definitions of real numbers, Euclid’s Division Lemma, and the Fundamental Theorem of Arithmetic. Take concise notes while reading the chapter. Write down key formulas, definitions, and important points. These notes will be helpful for quick revision.

Learn here how to find the prime factorization of numbers. This skill is essential for various aspects of number theory. Practice finding the prime factorization of different numbers. Practice working with terminating and recurring decimals. Solve problems that involve decimal expansions of real numbers. Familiarize yourself with methods for rationalizing irrational numbers, especially square roots. Practice these techniques with different examples. Remember that learning mathematics often requires patience and persistence. Take your time to understand each concept thoroughly before moving on to the next. By following these steps and strategies, you can effectively learn Class 10 Maths Chapter 1 Real Numbers.

Class 10 Maths Chapter 1 Main Topics Mathematics is best learned through practice. Solve a variety of problems related to real numbers, including examples from your textbook and additional practice questions. Pay attention to different types of questions, such as finding HCF, LCM, prime factorization, and working with decimal expansions.

10th Maths Chapter 1 Practice Test Papers

Euclid’s Division Lemma is a fundamental concept in this chapter. Make sure you understand how it works and can apply it to solve problems related to divisibility and remainders. There are four exercises in chapter 1 covering all the four topics on Real Numbers. Students can practice well using the 6 question papers given below. It will help to prepare the chapter for CBSE exams also.

Class 10 Maths Chapter 1 Practice Tests with Answers

• Class 10 Maths Chapter 1 Practice Test 1
• Class 10 Maths Chapter 1 Practice Test 2
• Class 10 Maths Chapter 1 Practice Test 3
• Class 10 Maths Chapter 1 Practice Test 4
• Class 10 Maths Chapter 1 Practice Test 5
• Class 10 Maths Chapter 1 Practice Test 6
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The word algorithm come from the name of the name of the 9th century Persian mathematician al-Khwarizmi. The word ‘algebra’ is derived from a book, he wrote, called Hisab al-jabr w’al-muqabala.

An equivalent version of Fundamental theorem of Arithmetic was probably first recorded as Proposition 14 of Book IX in Euclid’s Elements, before it came to be known as the Fundamental Theorem of Arithmetic.

However, the first correct proof was given by Carl Friedrich Gauss in his al-Khwarizmi. Carl Friedrich Gauss is often referred to as the ‘Prince of Mathematicians’ and is considered one of the three greatest mathematicians of all time, along with Archimedes and Newton. He has made fundamental contributions to both mathematics and science. Learn here how to make class 10 Maths chapter 1 easy to learn.

How many exercises are there in class 10 Maths chapter 1 Real numbers?

Now there are only 2 exercises in class 10 chapter 1 (Real numbers): In first exercise (Ex 1.1), there are 5 questions. In second exercise (Ex 1.2), there are 7 questions. So, there are in all 18 questions in class 10 chapter 1 (Real numbers). There are in all 11 examples in class 10 chapter 1 (Real numbers).

• Examples 1,2,3,4 are based on Ex 1.1.
• Examples 5,6,7,8 are based on Ex 1.2.

Are there any Theorems or Algorithm in 10th Maths chapter 1?

Yes, there are 7 Theorems and 1 Algorithm in class 10 chapter 1 (Real numbers). First exercise (Ex 1.1) is based on Theorem 1.1 (Euclid’s Division Lemma) and Euclid’s Division algorithm. Theorem 1.2 (Fundamental Theorem of Arithmetic) is used in second exercise (Ex 1.2).

What are the Important questions from Maths class 10 chapter 1 Real numbers?

Important questions from Maths class 10 chapter 1:

• In first exercise (Ex 1.1), all questions are important.
• In second exercise (Ex 1.2), Q2, 4, 5, 7 are important.

Which topics students should recall before starting class 10 Maths chapter 1?

Students should recall some topics of class 9 chapter 1 (Number systems). Topics :

• Types of numbers (Natural number, Whole number, Integers, Rational numbers, Irrational numbers and Real numbers).
• Decimal expansion of Rational numbers (Terminating and Non-terminating recurring) and Irrational numbers (Non-terminating non-recurring).
• HCF (Highest Common Factor) and LCM (Lowest/Least Common Multiple).
• And some other topics like Meaning of Dividend, Divisor, Quotient and Remainder, Prime numbers, Composite numbers, Factors.

Chapter 2: Polynomials »

Shikhar Tiwari

Having graduated from Electronics and Communication Engineering from AKTU – Noida, India, in 2021, working for Tiwari Academy as a content writer and reviewer. My main focus is to provide an easy to understand methods in all subjects specially mathematics and making study material with step by step explanation.

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Real Numbers Sample Question A number, when divided by 30 gives 55 as the quotient and 18 as the remainder. Find the number. A. 1678 B. 1650 C. 1668 D. None of these

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Polynomials Sample Question If two zeros of polynomial x 3 + bx 2 + cx + d are 1+√2 and 1-√2, find its third zero. A. c - 2 B. -b - 2 C. b - 2 D. - c - 2

Linear equations in two variables sample question find the values of ^@p^@ and ^@q^@ for which the following pair of linear equations have an infinite number of solutions:@^6x + y = 3, (p + q)x + (p - q)y = (2p - 24)@^ a. ^@\begin{aligned}p = 21 \text{ and } & q = 16\end{aligned}^@ b. ^@\begin{aligned}p = 21 \text{ and } & q = 15\end{aligned}^@ c. ^@\begin{aligned}p = 26 \text{ and } & q = 15\end{aligned}^@ d. ^@\begin{aligned}p = 20 \text{ and } & q = 13\end{aligned}^@, quadratic equations sample question the sum of the n consecutive natural numbers starting from 5 is 68. find the value of n . a. 8 b. 10 c. 7 d. 9, arithmetic progressions sample question which term in the arithmetic progression 8, 17, 26, 35 will be 261 more than the 30 th term a. 59 b. 56 c. 58 d. 62, triangles sample question two poles of heights ^@ 20 \space metres^@ and ^@40 \space metres^@ stand vertically on a plane ground. if the distance between their feet is ^@21 \space metres^@, find the distance between their tops. a. ^@29 \space m^@ b. ^@36 \space m^@ c. ^@31 \space m^@ d. ^@28 \space m^@, coordinate geometry sample question if ^@a(2,8)^@, ^@b(6 , y)^@, ^@c(x, 4)^@ and ^@d(0, 4)^@ are the vertices of a parallelogram ^@abcd^@, find the values of ^@x^@ and ^@y^@. a. ^@ x = 4, y = 8 ^@ b. ^@ x = 8 , y = 12 ^@ c. ^@ x = 6 , y = 10 ^@ d. ^@ x = 7 , y = 11 ^@, trigonometry and its applications sample question an equilateral triangle with a side of length 14√ 3 cm is inscribed in a circle. find the radius of the circle. a. 14√ 2 cm b. 11.2 cm c. 7 cm d. 14 cm, circles sample question if the lengths of tangents drawn from an external point ^@a^@ to a point ^@p^@ and ^@q^@ on the circle are equal then ^@ap^@ is equal to: a. ^@ op ^@ b. ^@ aq ^@ c. ^@ oq ^@ d. ^@ oa ^@, constructions sample question a. b. c. d., area related to circles sample question a chord of a circle of radius ^@ 28 \space cm ^@ subtends a right angle at the center. find the area of the corresponding major segment.^@ \bigg[\pi = \dfrac { 22 } { 7 }\bigg] ^@ o a b 90° a. ^@ 224 \space cm^2 ^@ b. ^@ 2240 \space cm^2 ^@ c. ^@ 474 \space cm^2 ^@ d. ^@ \text { none of these } ^@, surface area and volume sample question there is a cylindrical scale holder of height ^@ 12 \space cm^@ and diameter ^@ 14 \space cm^@. now the scale holder is cut into two pieces such that the height of the new scale holder is reduced to ^@ \left(\dfrac{ 5 }{ 6 } \right)^{th} ^@ of the height. the diameter of the new scale holder remains the same. if the new scale holder is filled with water up to the brim, then find the amount of water held in liters. (where ^@ \pi = 3.14 ^@) a. ^@ 1846.32 \space l ^@ b. ^@ 184.632 \space l ^@ c. ^@ 1.84632 \space l ^@ d. ^@ 18.4632 \space l ^@, statistics sample question calculate the median for the following data: ^@ \text{ marks obtained }^@ ^@ \text{ number of students }^@ ^@ \text{ less than } 10 ^@ ^@ 16 ^@ ^@ \text{ less than } 20 ^@ ^@ 44 ^@ ^@ \text{ less than } 30 ^@ ^@ 58 ^@ ^@ \text{ less than } 40 ^@ ^@ 80 ^@ ^@ \text{ less than } 50 ^@ ^@ 101 ^@ ^@ \text{ less than } 60 ^@ ^@ 126 ^@ ^@ \text{ less than } 70 ^@ ^@ 153 ^@ ^@ \text{ less than } 80 ^@ ^@ 174 ^@ a. ^@45.333^@ b. ^@44.333^@ c. ^@41.333^@ d. ^@43.333^@, probability sample question two dice are rolled. what is the probability that the two numbers add up to a prime number a.   12 36   b.   19 36   c.   17 36   d.   15 36  , full year 10th grade review sample question if the zeros of the polynomial ^@ x^3 - 10 x^2 + 27 x - 18 ^@ are ^@(a - b), a, (a + b)^@, find ^@ a ^@ and ^@ b ^@. a. ^@ a = \dfrac { 10 } { 3 } \space \text{and} \space b = \pm \sqrt{\dfrac { 19 } { 2 }} ^@ b. ^@ a = \dfrac { 10 } { 3 } \space \text{and} \space b = \pm \sqrt{\dfrac { 19 } { 3 }} ^@ c. ^@ a = \dfrac { 3 } { 10 } \space \text{and} \space b = \pm \sqrt{\dfrac { 19 } { 3 }} ^@ d. ^@ \text{ none of these } ^@, dice and cubes sample question faces of a cube are marked with a,b,c,d,e and f. if two views of a cube are as shown below, what will be there on the face opposite to face e a. f b. b c. d d. cannot be determined, venn diagrams sample question following chart shows number of students in a school who play one of three sports. find the number of students who play exactly two sports. a. 45 b. 38 c. 42 d. 44, number series sample question find the missing number 8 4 11 52 32 a. 55 b. 67 c. 79 d. 71, alphabet problems sample question which is the odd one out in this set: xci, hmt, glr, jou a. hmt b. glr c. xci d. jou, ntse_mat sample question which is the odd one out in this set: â â â â â  nails, lanis, slain, snail a. nails b. lanis c. snail d. slain.

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Assignment For Class 10 Mathematics

Assignment for Class 10 Mathematics have been developed for Standard 10 students based on the latest syllabus and textbooks applicable in CBSE, NCERT and KVS schools. Parents and students can download the full collection of class assignments for class 10 Mathematics from our website as we have provided all topic wise assignments free in PDF format which can be downloaded easily. Students are recommended to do these assignments daily by taking printouts and going through the questions and answers for Grade 10 Mathematics. You should try to do these test assignments on a daily basis so that you are able to understand the concepts and details of each chapter in your Mathematics book and get good marks in class 10 exams.

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Assignments for Class 10 Mathematics as per CBSE NCERT pattern

All students studying in Grade 10 Mathematics should download the assignments provided here and use them for their daily routine practice. This will help them to get better grades in Mathematics exam for standard 10. We have made sure that all topics given in your textbook for Mathematics which is suggested in Class 10 have been covered ad we have made assignments and test papers for all topics which your teacher has been teaching in your class. All chapter wise assignments have been made by our teachers after full research of each important topic in the textbooks so that you have enough questions and their solutions to help them practice so that they are able to get full practice and understanding of all important topics. Our teachers at https://www.assignmentsbag.com have made sure that all test papers have been designed as per CBSE, NCERT and KVS syllabus and examination pattern. These question banks have been recommended in various schools and have supported many students to practice and further enhance their scores in school and have also assisted them to appear in other school level tests and examinations. Its easy to take print of thee assignments as all are available in PDF format.

Some advantages of Free Assignments for Class 10 Mathematics

• Solving Assignments for Mathematics Class 10 helps to further enhance understanding of the topics given in your text book which will help you to get better marks
• By solving one assignments given in your class by Mathematics teacher for class 10 will help you to keep in touch with the topic thus reducing dependence on last minute studies
• You will be able to understand the type of questions which are expected in your Mathematics class test
• You will be able to revise all topics given in the ebook for Class 10 Mathematics as all questions have been provided in the question banks
• NCERT Class 10 Mathematics Workbooks will surely help you to make your concepts stronger and better than anyone else in your class.
• Parents will be able to take print out of the assignments and give to their child easily.

All free Printable practice assignments are in PDF single lick download format and have been prepared by Class 10 Mathematics teachers after full study of all topics which have been given in each chapter so that the students are able to take complete benefit from the worksheets. The Chapter wise question bank and revision assignments can be accessed free and anywhere. Go ahead and click on the links above to download free CBSE Class 10 Mathematics Assignments PDF.

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You can get free PDF downloadable assignments for Grade 10 Mathematics from our website which has been developed by teachers after doing extensive research in each topic.

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Class 10 Mathematics Real Numbers Assignments

We have provided below free printable Class 10 Mathematics Real Numbers Assignments for Download in PDF. The Assignments have been designed based on the latest NCERT Book for Class 10 Mathematics Real Numbers . These Assignments for Grade 10 Mathematics Real Numbers cover all important topics which can come in your standard 10 tests and examinations. Free printable Assignments for CBSE Class 10 Mathematics Real Numbers , school and class assignments, and practice test papers have been designed by our highly experienced class 10 faculty. You can free download CBSE NCERT printable Assignments for Mathematics Real Numbers Class 10 with solutions and answers. All Assignments and test sheets have been prepared by expert teachers as per the latest Syllabus in Mathematics Real Numbers Class 10. Students can click on the links below and download all Pdf Assignments for Mathematics Real Numbers class 10 for free. All latest Kendriya Vidyalaya Class 10 Mathematics Real Numbers Assignments with Answers and test papers are given below.

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We have provided below the biggest collection of free CBSE NCERT KVS Assignments for Class 10 Mathematics Real Numbers . Students and teachers can download and save all free Mathematics Real Numbers assignments in Pdf for grade 10th. Our expert faculty have covered Class 10 important questions and answers for Mathematics Real Numbers as per the latest syllabus for the current academic year. All test papers and question banks for Class 10 Mathematics Real Numbers and CBSE Assignments for Mathematics Real Numbers Class 10 will be really helpful for standard 10th students to prepare for the class tests and school examinations. Class 10th students can easily free download in Pdf all printable practice worksheets given below.

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• Real Numbers For Class 10

Real Numbers Class 10 Notes: Chapter 1

Cbse real numbers class 10 notes:- download pdf here, class 10 maths chapter 1 real number notes.

CBSE Class 10 Maths Chapter 1 Real Numbers Notes are provided here in detail. As we all know, any number, excluding complex numbers, is a real number. Positive and negative integers, irrational numbers, and fractions are all examples of real numbers. To put it another way, any number found in the real world is a real number. Numbers can be found all around us. Natural numbers are being used to count objects, integers are used to measure temperature, rational numbers are used to represent fractions, irrational numbers are used to calculate the square root of a number, etc. These various types of numbers form a collection of real numbers. Here, we are going to learn what a real number is, Euclid’s division algorithm, the fundamental theorem of arithmetic, methods of finding LCM and HCF and the complete explanation of rational and irrational numbers with examples.

Real Numbers

Positive integers, negative integers, irrational numbers, and fractions are all examples of real numbers. In other words, we can say that any number is a real number, except for complex numbers. Examples of real numbers include -1, ½, 1.75, √2, and so on. In general,

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

To know more about real numbers, visit here .

Students can refer to the short notes and MCQ questions along with a separate solution pdf of this chapter for quick revision from the links below.

• Real Numbers Short Notes
• Real Numbers MCQ Practice Questions
• Real Numbers MCQ Practice Solutions

Euclid’s Division Lemma

• Euclid’s Division Lemma states that given two integers a and b , there exists a unique pair of integers q and r such that a = b × q + r   a n d   0 ≤ r < b .
• This lemma is essentially equivalent to :  dividend = divisor × quotient + remainder
• In other words, for a given pair of dividend and divisor, the quotient and remainder obtained are going to be unique.

For more information on Euclid’s Division Lemma, watch the below video

To know more about Euclid’s Division Lemma, visit here .

Euclid’s Division Algorithm

• Euclid’s Division Algorithm is a method used to find the H.C.F of two numbers, say  a and b where a> b.
• We apply Euclid’s Division Lemma to find two integers q and r such that a = b × q + r   a n d   0 ≤ r < b .
• If r = 0, the H.C.F is b; else, we apply Euclid’s division Lemma to b (the divisor) and r (the remainder) to get another pair of quotient and remainder.
• The above method is repeated until a remainder of zero is obtained. The divisor in that step is the H.C.F. of the given set of numbers.

For more information on Euclid’s Division Algorithm, watch the below video

The Fundamental Theorem of Arithmetic

Prime Factorisation

• Prime Factorisation is the method of expressing a natural number as a product of prime numbers.
• Example: 36 = 2 × 2 × 3 × 3   is the prime factorisation of 36.

Fundamental Theorem of Arithmetic

• The Fundamental Theorem of Arithmetic states that the prime factorisation for a given number is unique if the arrangement of the prime factors is ignored.
• Example: 36 = 2 × 2 × 3 × 3 OR,  36 = 2 × 3 × 2 × 3
• Therefore, 36 is represented as a product of prime factors (Two 2s and two 3s) ignoring the arrangement of the factors.

To know more about the Fundamental Theorem of Arithmetic, visit here .

Method of Finding LCM

As we know, the smallest of the common multiples of two or more numbers is called the lowest common multiple (LCM). Example: To find the Least Common Multiple  ( L.C.M ) of 36 and 56,

•   36 = 2 × 2 × 3 × 3 56 = 2 × 2 × 2 × 7
•  The common prime factors are 2 × 2
•  The uncommon prime factors are  3 × 3 for 36 and 2 × 7 for 56.
• LCM of 36 and 56 = 2 × 2 × 3 × 3 × 2 × 7   which is 504

To know more about LCM, visit here .

Method of Finding HCF

We know that the greatest number that divides each of the given numbers without leaving any remainder is the highest common factor (HCF) of two or more given numbers. H.C.F can be found using two methods – Prime factorisation and Euclid’s division algorithm.

• Given two numbers, we express both of them as products of their respective prime factors. Then, we select the prime factors that are common to both the numbers
• Example – To find the H.C.F of 20 and 24 20 = 2 × 2 × 5   and    24 = 2 × 2 × 2 × 3
• The factor common to 20 and 24 is 2 × 2 , which is 4, which in turn is the H.C.F of 20 and 24.
• It is the repeated use of Euclid’s division lemma to find the H.C.F of two numbers.

• The required HCF is 6 .

To know more about HCF, visit here .

For more information on HCF and LCM, watch the below video

To know more about the Properties of HCF and LCM, visit here .

Product of Two Numbers = HCF X LCM of the Two Numbers

• For any two positive integers a and b, a × b = H . C . F × L . C . M .
• Example – For 36 and 56, the H.C.F is 4 and the L.C.M is 504 36 × 56 = 2016 4 × 504 = 2016 Thus, 36 × 56 = 4 × 504
• Let us consider another example: For 5 and 6, the H.C.F is 1 and the L.C.M is 30 5 × 6 = 30 1 × 30 =30 Thus, 5 × 6 = 1 × 30
• The above relationship, however, doesn’t hold true for 3 or more numbers

To know the Relationship between LCM and HCF, visit here .

Applications of HCF & LCM in Real-World Problems

L.C.M can be used to find the points of common occurrence. For example,ringing of bells that ring with different frequencies, the time at which two persons running at different speeds meet, and so on.

For more information on Applications Of LCM, watch the below video

Revisiting Irrational Numbers

Irrational numbers.

Any number that cannot be expressed in the form of p/q (where p and q are integers and q ≠ 0 .) is an irrational number. Examples √2 , π , e and so on.

To know more about Irrational Numbers, visit here .

Number theory: Interesting results

• If a number p (a prime number) divides a 2 , then p divides a. Example: 3 divides 6 2 i.e 36, which implies that 3 divides 6.
• The sum or difference of a rational and an irrational number is irrational
• The product and quotient of a non-zero rational and irrational number are irrational.
• √p is irrational when ‘p’ is a prime. For example, 7 is a prime number, and √7 is irrational. The above statement can be proved by the method of “Proof by contradiction”.

To know more about Number theory, visit here .

Proof by Contradiction

In the method of contradiction, to check whether a statement is TRUE (i)  We assume that the given statement is TRUE. (ii) We arrive at some result which contradicts our assumption, thereby proving the contrary. Eg: Prove that √7 is irrational. Assumption: √7  is rational. Since it is rational √7 can be expressed as √7 = a/b , where a and b are co-prime Integers, b ≠ 0. On squaring, a 2 /b 2 = 7 ⇒ a 2 = 7 b 2 . Hence, 7 divides a. Then, there exists a number c such that a=7c. Then, a 2 = 49 c 2 . Hence, 7 b 2 = 49 c 2 or b 2 = 7 c 2 . Hence 7 divides b. Since 7 is a common factor for both a and b, it contradicts our assumption that a and b are coprime integers. Hence, our initial assumption that √7  is rational is wrong. Therefore, √7 is irrational.

Revisiting Rational Numbers and Their Decimal Expansions

Rational numbers.

Rational numbers are numbers that can be written in the form p/q, where p and q are integers and q ≠ 0 . Examples -1/2 , 4/5,  1 , 0 , − 3   and so on.

To know more about Rational Numbers, visit here .

Terminating and Non-Terminating Decimals

Terminating decimals are decimals that end at a certain point. Example: 0.2, 2.56 and so on. Non-terminating decimals are decimals where the digits after the decimal point don’t terminate. Example: 0.333333….., 0.13135235343… Non-terminating decimals can be : a) Recurring – a part of the decimal repeats indefinitely (0 . 142857 142857 …. ) b) Non-recurring – no part of the decimal repeats indefinitely. Example: π = 3.1415926535…

To know more about terminating and non-terminating decimals, click here .

Check if a given rational number is terminating or not

If a/b is a rational number, then its decimal expansion would terminate if both of the following conditions are satisfied : a) The H.C.F of a and b is 1. b) b can be expressed as a prime factorisation of 2 and 5 i.e b = 2 m × 5 n where either m or n, or both can = 0. If the prime factorisation of b contains any number other  than 2 or 5, then the decimal expansion of that number will be recurring

1/40 = 0.025 is a terminating decimal,  as the H.C.F of 1 and 40 is 1, and the denominator (40) can be expressed as 2 3 × 5 1 .

3/7 = 0.428571 is a recurring decimal as the H.C.F of 3 and 7 is 1 and the denominator (7) is equal to 7 1

Real Numbers for Class 10 Solved Examples

Find the largest number that divides 70 and 125 leaving the remainder 5 and 8 respectively.

First, subtract the remainder from the number.

(i.e) 70-5 = 65

125-8 = 117.

Thus, we need to find the largest number that divides 65 and 117 and leaves the remainder 0.

To find the largest number, take the HCF of 65 and 117.

Finding HCF of 65 and 117.

65 = 5×13

117 = 3×3×13.

Hence, HCF (65, 117) = 13.

Therefore, the largest number that divides 70 and 125 leaving the remainder 5 and 8 respectively is 13.

Find the LCM of 306 and 657, given that HCF (306, 657) = 9.

Given that, HCF (306, 657) = 9.

We know that HCF × LCM = Product of Numbers

Hence, 9×LCM = 306×657

9×LCM = 201042

LCM = 201042/9

LCM = 22338.

Therefore, LCM of 306 and 657 is 22338.

Prove that 1/√2 is an irrational number.

To prove 1/√2 is an irrational number.

Now, let us take the opposite assumption.

(i.e) Take 1/√2 is a rational number.

We know that rational numbers are the numbers that can be written in the form of p/q, where q is not equal to 0. (p and q are two co-prime numbers)

Hence, 1/√2 = p/q.

Now, simplify the above equation by multiplying √2 on both sides.

1 = (p√2)/q

q = p√2

Hence, we get q/p = √2.

Here, p and q are integers, and hence q/p is a rational number.

But, √2 is an irrational number.

Hence, our assumption is wrong.

Therefore, 1/√2 is an irrational number.

Hence, proved.

Frequently Asked Questions on Class 10 Real Numbers

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