Table of Contents

## Real Numbers Class 10 Notes

**R= Real Numbers**

Real numbers include both irrational and rational numbers.

**I= Integers**

Integers are all numbers starting with (…-3, -2, -1, 0, 1, 2, 3…).

**Q= Rational Numbers**

Rational numbers are real numbers with the pattern p/q, q 0, and p, q I.

- Rational expressions can represent all integers, such as 5 = 5/1.
- Decimal expansion of recurring or non-recurring rational numbers.

**Q’ = Irrational Numbers**

Real numbers cannot be expressed in the form p/q, whose decimal expansions are non-terminating and non-recurring.

- Roots of primes like √2, √3, √5 etc., are irrational.

**N= Natural Numbers**

Natural numbers are those that can be counted. N = {1, 2, 3, …}

**W= Whole Numbers**

Whole numbers are the collective term for zero and all natural numbers. {0, 1, 2, 3,…}

**Even Numbers**

Even numbers are natural numbers with the form 2n. (2, 4, 6, …}

**Odd Numbers**

Odd numbers are natural numbers with the form 2n – 1; examples are 1, 3, 5, etc.

- Why can’t the form be expressed as 2n+1?

**Remember!**

*****Natural numbers are exclusively whole numbers.

*****Every Whole Number is an integer.

*****Rational Numbers are the same as Integers.

*****All Rational Numbers are Real Numbers.

**Prime Numbers**

Prime numbers are all natural numbers bigger than one that can be divided by 1 and the number itself. Examples of prime numbers are 2, 3, 5, 7, and 11.

- Since it only has one factor, 1, it is not a prime number.

**Composite Numbers**

Composite numbers are any natural numbers that may be divided by one, including the number one itself. Four, six, eight, nine, ten, etc.

- 1 is not a prime number nor a composite number.

## Real Numbers Class 10 Notes- Methods

**Euclid’s Division Lemma**

There are distinct integers q and r that satisfy the equation a = bq + r, 0 r b when given two positive integers a and b.

**Observe this: **Every time, “r” is lower than “b.” Each “q” and “r” is distinctive.

**Application Of Lemma**

The HCF of two positive numbers is determined using Euclid’s Division Lemma.

**Example:** How can I find the HCF of 56 and 72?

**Steps:- **

- Lemma applied to 56 and 72.
- Find “b” and “r” by using a larger number. 72 = 56 × 1 + 16
- Consider 56 as the new dividend and 16 as the new divisor since 16 s not equal to 0. 56 = 16 × 3 + 8
- Once more, if 8 is not equal to 0, apply 16 as the new dividend and 8 as the new divisor. 16 = 8 × 2 + 0

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**The remainder being zero, (8) is HCF’s divisor**

Euclid’s Division Lemma can be extended to all integers except zero, i.e., b 0. This is true even if it is only stated for positive integers.

**III. Constructing A Factor Tree**

**Steps for building a factor tree:**

- Put the amount in the form of a prime number plus a composite number.

**Example: 48 factorise**

- Continue until all primes have been achieved.

**∴** Prime factorization of 48 = 24 x 3

**Fundamental Theorem Of Arithmetic**

With the exception of the sequence in which they appear, every composite number can be written as a product of primes, and this expression is singular.

**Applications:**

- In order to find the HCF and LCM of two or more positive integers.
- Demonstrating the irrationality of numbers
- To identify the type of rational number’s decimal expansion.