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CBSE Guide Real Numbers class 10 Notes

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10 Mathematics notes Chapter 1 Real Numbers

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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.

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