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Edited By Ramraj Saini | Updated on Jan 30, 2024 11:53 AM IST

**Real Numbers Class 10th Notes **are discussed here. These notes are created by expert team at Careers360 considering the latest syllabus and pattern of CBSE also keeping the need of the students in mind. These notes comprehensively covers all the topics sub topics, short tricks, formulae, and other important key points. NCERT Class 10 Maths chapter 1 notes based on real numbers, in which fundamental theorem of arithmetic, rational and irrational numbers are included. The chapter real numbers Class 10 notes also includes Euclid's division lemma and Euclid’s division algorithm.

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This Story also Contains

- NCERT Class 10 Maths chapter 1 notes
- NCERT Class 10 Chapter 1 Notes-
- Real Numbers Class 10 Notes - Topic 1:
- Numbers:
- Real Numbers Class 10 Notes- Topic 1:
- Euclid’s Division Lemma:
- Fundamental Theorem of Arithmetic:
- Significance of NCERT Class 10 Maths Chapter 1 Notes-
- Class 10 Chapter Wise Notes
- NCERT Solutions of Cass 10 Subject Wise
- NCERT Class 10 Exemplar Solutions for Other Subjects:

The introductory part may include: The NCERT Class 10 chapter 1 notes revisiting the rational and irrational number and rational number expansion are the highlighting key feature of Real Numbers. CBSE Class 10 Maths chapter 1 notes contain systematic explanations of topics using examples and exercises. CBSE Class 10 Maths chapter 1 notes include FAQ or frequently asked questions about the chapter. These topics can be easily downloaded from Class 10 Maths chapter 1 notes pdf download.

**Also, students can refer,**

- NCERT Solutions for Class 10 Maths Chapter 1 Real Numbers
- NCERT Exemplar Solutions for Class 10 Maths Chapter 1 Real Numbers

1. **Natural Numbers:** These are the numbers that are started from the count of one.

The natural numbers can be allocated as “N”.

Example: 1, 2, 3, 4, 5, 6……. And so on.

2.** Whole Numbers: **These are the numbers that are started from the count of zero.

The natural numbers are denoted by the letter “W”.

Counting must be started from “0”.

Example: 0, 1, 2, 3, 4, 5, 6……. And so on.

3.** Integers:** These are the numbers that are positive and negative natural numbers

Zero is also included.

The natural numbers are denoted by the letter “Z”. Z is Zahlen count.

Example: ……-5, -4, -3, -2, -1, 0, 1, 2, 3, 4, 5……. And so on.

4. **Rational numbers: **These are the numbers that are any real number and they can be expressed in the form p/q of R where denominator q is not equivalent to 0.

This can be denoted by the letter “Q”.

It is noted that every integer number is a rational number.

Example: 1/2, 3/4, etc.

**Decimal expansion:** The decimal expansion of rational numbers is terminating and non-terminating. Terminating if the numbers are finite and non-terminating if the numbers are repeating decimals.

5. **Irrational numbers:** These are the numbers that can not be expressed in the numerator by denominator form. The numbers are real but are not in the ratio of integers.

Example: √2,√3, π, 0.34343………..

The numbers neither can express in terms of terminating numbers nor in non-terminating numbers.

6. **Real numbers:** These are the numbers that include all the numbers such as rational, integers, fractions, and irrational numbers are included.

7. **Algorithm**: This can be defined as solving the problems we must require a number of steps in infinite numbers.

**8. Lemma:** Lemma is defined as another proven statement for providing another statement.

According to Euclid’s division Lemma, for two positive integers, a and b there exist a unique integer such that a=bq+r, where 0≤r<b. Where a is a dividend, b is a divisor, q is a quotient, and r is remainder.

Use: Euclid’s division Lemma can be used in finding the highest common factor of any two positive integers.

Steps are included to find HCF with the help of Euclid’s Division Lemma.

Let the two positive integers will be a and b where “a” is greater than “b”.

Now, applying Euclid’s division to these numbers a and b, also find the two numbers q and r.

In this step checking the value of r if it is found to be 0 then b is the HCF of these numbers.

Continuing this process until we get the value 0. Reaching the value of 0 makes the divisor b as the HCF of a and b.

Prime Number: The numbers that can be divisible by themselves and by 1 are called prime numbers.

The number contains only natural numbers.

Example: 2, 3, 5, 7, 11, 13, 19, 23…………and so on.

**Composite numbers:** The numbers that have more than two integral divisors are called composite numbers.

Example: 4, 6, 8, 10, 12, 14, 16… etc.

Co-prime numbers: These are the numbers that have no common factor other than one are termed co-prime numbers.

**Note to remember:** All the prime numbers are co-prime numbers.

Consecutive numbers: These are the numbers that are always co-prime numbers.

Statement: Each and every composite number is a product of primes and this factorisation is different in which prime factors occur.

**Theorems:**

Let x be a rational number whose decimal expansion terminates. Then x can be expressed in the form p/q, where p and q are co-prime, and the prime factorization of q is of form 2

^{m}5^{n}, where m, n are non-negative integers.- Let x = p/q be a rational number, such that the prime factorization of q is of form 2
^{m}5^{n}, where m, n are non-negative integers. Then x has terminating decimal expansion which terminates after k places of decimals, where k is the larger of m and n. - Let x = p/q be a rational number, such that prime factorization of q is not of form 2
^{m}5^{n}, where m, n are non-negative integers. Then x has a non-terminating repeating decimal expansion.

Real Numbers Class 10 notes, will be helpful in understanding the formulas, statements, rules in detail. Also, it contains the previous year’s questions and NCERT textbook pdf. CBSE Class 10 Maths chapter 1 notes it contains FAQ or frequently asked questions along with a topic-wise explanation. These topics can also be downloaded from Real Numbers Class 10 notes pdf download.

1. Solve the products of its prime numbers 98.

98=2×7^{2}

2. HCF and LCM of 13 and 17 with the help of prime factorisation is

13=1×13;17=1×17

HCF=1 and LCM=13×17=221

3. As per NCERT Class 10 Chemistry chapter 1 notes and class 10 Real Numbers, calculate the largest number that can divide the 398, 436, and 542 which leaves the remainder 7, 11, and 15 respectively.

Algorithm: 398-7=391

436-11=425

542-15=527

HCF- 391, 425, 527 will be 17.

4. calculate the value HCF when it is expressed in terms like,1032×2+408 calculate p?

HCF of 408 and 1032 →24

1032×2+408×p=24

408p=24-2064

P=-5

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