## Introduction

we have already studied about the natural number, whole number, integers, and real number as well in class 9. Now in class 10th, we will study more about the real numbers. we will study **Euclid’s division lemma,** **Euclid’s division algorithm**, and** the fundamental theorem of arithmetic **in real numbers for class 10. Before learning real numbers for class 10, let’s recall the real number of class 9. If you would like to practice the standard problem to strengthen your learning concept, you can further check the important Question of the real numbers for class10

## Definition and types of real numbers

**Real numbe**r -: Real number are those numbers which can be represented on the number line

OR

The numbers which we use in real life are called the Real number.

### The real numbers are of two types

- Rational number
- Irrational number

**Rational number -:** Those numbers which can be expressed in the form of p/q where p and q are integers and q≠0 is called a rational number.

example – 2\5, 1 , 2.5 …………….etc

**Irrational number -: **Those numbers which cannot be expressed in the form of p/q is called Irrational number

example-√2, √5, √7 ……..etc

## some important terms definition for Real number

**Prime number**-: The number is said to be a prime number if it is exactly divisible by 1 and itself only.

Example – 2, 3, 5 ………..etc.

**Composite numbe**r-: The number which has **more than 2-factor or2 prime factor** is called a composite number

Example- 4,6,8,10 ……………etc.

**Coprime number** -: Two numbers are said to be coprime if their HCF is 1.

Example – 3 and 5 are coprime because their HCF is 1.

**Terminating decimal number** -: The rational number in which value gets terminate after fixed decimals place is called Terminating decimal number.

Example -25/100 on the division will terminate after 2 decimal places

**Non-terminating decimal number -: **The number which values do not terminate after the decimal point is called non-terminating decimal number.

Example -3.1415……, 0.3333…., etc

** There are two types of non-terminating decimal representation **

- Pure recurring decimals
- Mixed recurring decimals

**Pure recurring decimals** -: A decimal in which all the digits after the decimal point are repeated is called pure recurring decimals.

Example – 0.33…., 1.515151……. etc

**mixed recurring decimals -: **A decimal in which at least one of digits after the decimal point is not repeated and then some digits or digits repeated is called mixed recurring decimals

Example-2.1666….,0.35555…., 0.78585….. etc

### Important points for real number

**All the terminating and recurring decimal are rational number****Non-terminating decimals are irrational number****2 is only even prime number****4 is the smallest composite number**

## Real numbers for class 10

**Lemma -: **A lemma is a proven statement that is used to prove other statements.

**Algorithm-: **An algorithm means a series of well-defined steps that provide a procedure of calculations repeated successively on the results of earlier steps until the desired result is obtained.

** Euclid’s division lemma**

It is nothing but a statement of the long division.consider a division of two positive integers 45 and 8 .when you ask for division, what you will do?

You will think about the number which is multiple of 8 but does not exceed 45 so that our remainder should always less than 8.

when we divide it we will get the result of** two more positive integers,** namely 5 which is Quotient and 3 which is the remainder. so our results can be express as

** 43= 8* 5 +5 **

This is nothing but Euclid’s division lemma

**Euclid’s division lemma** -: Let a and b any positive integers then, there exist unique integers q and r such that a=bq+r where 0≤r≤b

**Euclid’s division algorithm**-: It is an algorithm (procedure) to compute the highest common factor (HCF) of two positive integers, using Euclid’s division lemma. let’s understand it with an example

Q.1 Use Euclid’s division algorithm to find HCF of 210 and 55.

sol. we have two positive integers 210 and 55, applying Euclid’s division lemma, we get

210= 55 * 3 + 45

since the remainder is not 0. so we will apply same things for 55 and 45, then we get

55= 45 *1 + 10

again the remainder is not o, so we continue the same process until the remainder becomes 0.

45= 10 *4 + 5

10= 5*2 + 0

Here remainder is 0, so HCF of 210 and 55 will be 5

** Fundamental theorem of arithmetic **

Every composite number can be expressed as the product of prime factor and this factorization is unique except for the order in which the prime factor occurs.

Ex – we have already studied about the composite number and prime number above.

we know, 4 is the smallest composite number, we can express it as the product of prime numbers

4= 2* 2

similarly, we can express every composite number as prime factors

#### HCF and LCM

LCM(Least common multiple ) -: LCM of two or more numbers is the least number which is divisible by given number completely.

Ex – 16 and 32, the least number which is divisible by 16 and 32 is 32

HCF(Highest common factor)=HCF of two or more number is the largest number which divides the given number completely

The largest number which divides the given number 16 and 32 completely is 16