# Real Numbers class 10 Notes Mathematics

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## CBSE Guide Real Numbers class 10 Notes

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## 10 Mathematics notes Chapter 1 Real Numbers

Download CBSE class 10th revision notes for chapter 1 Real Numbers in PDF format for free. Download revision notes for Real Numbers class 10 Notes and score high in exams. These are the Real Numbers class 10 Notes prepared by team of expert teachers. The revision notes help you revise the whole chapter in minutes. Revising notes in exam days is on of the the best tips recommended by teachers during exam days.

CBSE Class–10 Mathematics
Revision Notes
CHAPTER 01
REAL NUMBERS

• Natural numbers: Counting numbers are called Natural numbers. These numbers are denoted by N = {1, 2, 3, ………}
• Whole numbers: The collection of natural numbers along with 0 is the collection of Whole number and is denoted by W.
• Integers: The collection of natural numbers, their negatives along with the number zero are called Integers. This collection is denoted by Z.
• Rational number: The numbers, which are obtained by dividing two integers, are called Rational numbers. Division by zero is not defined.
• Coprime: If HCF of two numbers is 1, then the two numbers area called relatively prime or coprime.

1. Euclid’s division lemma :

For given positive integers ‘a’ and ‘b’ there exist unique whole numbers ‘q’ and ‘r’ satisfying the relation ${\text{a }} = \;{\text{bq }} + \;{\text{r}},{\text{ }}0\; \leqslant \;{\text{r }} < \;{\text{b}}.$.

Theorem: If $a$ and $b$ are non-zero integers, the least positive integer which is expressible as a linear combination of $a$ and $b$ is the HCF of $a$ and $b$, i.e., if $d$ is the HCF of $a$ and $b$, then these exist integers $x_1$ and $y_1$, such that $d = a{x_1} + b{y_1}$ and $d$ is the smallest positive integer which is expressible in this form.

The HCF of $a$ and $b$ is denoted by HCF$\left( {a,b} \right)$.

2. Euclid’s division algorithms :

HCF of any two positive integers a and b. With a > b is obtained as follows:

Step 1 : Apply Euclid’s division lemma to a and b to find q and r such that

${\text{a }} = \;{\text{bq }} + \;{\text{r}}\;,{\text{ }}0\; \leqslant \;{\text{r }} < \;{\text{b}}.$

b = Divisor

q = Quotient

r =  Remainder

Step II: If r  = 0, HCF (a,b)=b if $r \ne 0$, apply Euclid’s lemma to b and r.

Step III: Continue the process till the remainder is zero. The divisor at this stage will be the required HCF.

3. The Fundamental Theorem of Arithmetic :

Every composite number can be expressed (factorized) as a product of primes and this factorization is unique, apart from the order in which the prime factors occur.

$Ex:\;\;24 = 2 \times 2 \times 2 \times 3 = 3 \times 2 \times 2 \times 2$

4. Let $x = {p \over q},$ $q \ne 0$  to be a rational number, such that the prime factorization of ‘q’ is of the form 2m+5n, where m, n are non-negative integers. Then x has a decimal expansion which is terminating.

5. Let $x = \frac{p}{q},\;\;q \ne 0$ be a rational number, such that the prime factorization of q is not of the form 2m+5n, where m, n are non-negative integers. Then x has a decimal expansion which is non-terminating repeating.

6. $\sqrt p$ is irrational, which p is a prime. A number is called irrational if it cannot be written in the form $\frac{P}{q}$ where p and q are integers and ${\text{q}}\;\,\, \ne {\text{ }}0.$

8. If a and b are two positive integers, then HCF(a, b) x LCM(a, b) = a x b

i.e., (HCF x LCM) of two intergers = Product of intergers.

9. A rational number which when expressed in the lowest term has factors 2 or 5 in the denominator can be written as terminating decimal otherwise a non-terminating recurring decimal. In other words, if the rational number ${a \over b}$ is, such that the prime factorization of b is of form ${2^m}{.5^n},$ where m and n are natural numbers, then ${a \over b}$ has a terminating decimal expansion.

10. We conclude that every rational number can be represented in the form of terminating or non-terminating recurring decimal.

## Real Numbers class 10 Notes

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## CBSE Class-10 Revision Notes and Key Points

Real Numbers class 10 Notes. CBSE quick revision note for Class-10 Mathematics, Chemistry, Maths, Biology and other subject are very helpful to revise the whole syllabus during exam days. The revision notes covers all important formulas and concepts given in the chapter. Even if you wish to have an overview of a chapter, quick revision notes are here to do if for you. These notes will certainly save your time during stressful exam days.

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