## Definition and types of real numbers

**Real numbe**r -:The number that can be represented on the number line is called a real number. It Includes

- Whole numbers (0,1,2,3,4,5…..)
- Natural Numbers (1,2,3,4,5,…)
- Integers (….,-2,-1,0,1,2,3,…..)

OR

The numbers which we use in real life are called Real numbers. The numbers we used for the calculation, measurement, etc are Real numbers. (like 5 foot,10 Kg,100 meters, etc )

### Types of the real number

There are basically two types of real numbers

- Rational number
- Irrational number

**Rational number -:** Those Real numbers that can be expressed in the form of p/q where p and q are integers and q≠0 is called rational numbers. For Example: 2,\frac{3}{2},0 etc

**Point to Remember **

**✔** The set of rational numbers is represented by the letter “Q”.

**✔** Rational numbers consist of **Whole numbers,** **natural numbers**, **integers**, **terminating decimals**, and **non-terminating** but repeating decimals like (0.333,0.232323) are also Rational numbers.

**✔** o is a rational number because it can be expressed as p/q like \frac{0}{1}.

**Irrational number -: **The Real numbers which cannot be expressed in the form of p/q are called Irrational numbers. For Example √2,√3,π(pie), etc

Points to Remember

**✔ **Irrational number consists of square root and cube root of certain natural numbers (like √2,3√13 etc. ) and Non-terminating numbers (3.141592…,1.41…. etc.)

**✔**Some of the popular irrational numbers are **π(pie)**, **Φ(Golden phi)**,**e (Euler’s number).**

### Important terms definition

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

For Example : 2,3,5,…..etc are prime numbers .

Note: 2 is the only even prime number.

**Composite numbe**r-: A number that has **more than 2-factors or 2 prime factors** is called a composite number

For Example : 4,6,8 …..etc are examples of composite numbers

Note: 4 is the smallest composite number .This is why ,

It has 2 prime factors or 3 factors which make it smallest composite number .

**Co-prime number** -: Two numbers are said to be co-prime if their HCF is 1.

For Example 3 and 5 are coprime because their HCF is 1, (5,7) is pair of other co-prime numbers.

**Terminating decimal number** -: Rational number whose value terminate after fixed decimal places is called Terminating decimal number.

For Example : \frac{25}{100} will terminate after two decimal places i.e 0.25 .

**Pure recurring decimals** -: A decimal representation in which digits repeats periodically after the decimal point is called pure Recurring decimals.

For Example 0.33333…,0.525252….are pure recurring decimals. because its digits are repeating after the decimal point. These types of numbers also fall in the category of rational numbers.

**Non-terminating decimal number -: **The numbers whose value doesn’t terminate after the decimal point is called non-terminating decimal numbers.

For Example -3.1415……, 0.3333… values never terminate, it keep going forever.**These types of numbers are also Non-recurring decimal number .**

**Lemma -: **A lemma is a proven statement that is used to prove other statements. In class 10 real number, we will learn about **“Euclid’s division lemma”** for finding HCF of numbers.

** Euclid’s division lemma**

Let a and b are two positive integers then, there exist unique integers q and r such that a=bq+r where 0≤r≤b.

OR

Dividend=(Divisor × Quotient ) + Remainder

Where

a is divided,

b is the divisor,

q is the quotient, and

r is the remainder. Let’s understand it with an example

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 a multiple of 8 but less than 45 so that values don’t exceed 45

When we divide it, we get** two more positive integers as result,** namely 5 which is Quotient, and 3 which is the remainder.

** 45=8×5 + 5 **

This is Euclid’s division lemma, we can express any division in the form of Euclid’s division lemma.

**Algorithm-: **An algorithm means a series of well-defined steps that provide a procedure for solving certain types of problems.

**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. This means we will learn the step-by-step procedures to compute HCF using Euclid’s Division lemma.

Procedure :

Let’s say, you’ve two positive integers a and b and you want to compute its HCF using Euclid’s Division Algorithm

**Step:01** Apply Euclid’s Division Lemma to a and b and obtain Quotient and remainder q and r such that a=bq+r where 0≤r<b

**Step:02 **

- If remainder r=0 ,then HCF(a,b)=b (divisor )
- If remainder r≠ 0 ,then again apply euclid’s division lemma to b(Divisor) and r (Remainder) .Here Remainder acts as Divisor and Divisor acts as Divided

**Step:0**3 Continue till procedure until remainder = 0

Let’s take an example and understand it

Example:01 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 , remainder ≠ 0. So we will again apply Euclid’s division lemma for 55 and 45

55= 45 × 1 + 10

Again ,remainder ≠ o, apply Same procedure for 55 and 45 until remainder becomes 0

45= 10 × 4 + 5

10= 5×2 + 0

Here remainder is 0, so HCF (210 , 55) = 5 Divisor

** Fundamental theorem of arithmetic **

According to Fundamental Theorem of arithmetic ,

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

For Example We know,

4 is the smallest composite number, and we can express it as the product of prime factors.

4= 2×2

Similarly, we can express every composite number as the product of prime factors using the prime factorization method.

#### HCF and LCM

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

For Example: Let we have two numbers 16 and 32,so 32 will be the least possible number which is exactly divisible by 16 and 32.

Similarly,

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

So, Largest number which will divide 16 and 32 completely is 16 .

Relation between LCM,HCF, and Numbers

Check these stuff as well

- Real number class 10 fill in the blanks
- Real number class 10 online test with answer
- Real number important question with solution 2021 board

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