## REAL NUMBERS Class 10th

All national and inrational number are **Real numbers.**

**OR **Rational numbers and irrational numbers taken together are called Real Numbers.

- The set of real numbers is denoted by R.
- Every real number is either a rational number or an irrational number.
- A real number which is not rational is called an irrational number.
- The
**sum, difference**or**product**of a rational and irrational number is**irrational.**

### TYPES OF REAL NUMBERS

**Natural Numbers :**

Counting number 1,2,3... are called are **Natural Numbers. **

- i.e ; N = {1, 2, 3,........}
- The set of Natutral number represented by
**N.**

**Whole Numbers : **

Natural number including zero.

- i.e; W = { 0,1,2,3,4.......}
- The set of Whole number represented by
**W.**

**Even Numbers :**

A number that is exactly divisible by 2 or multiple of 2 called **even numbers. **

Eg : 2, 4, 6........

**Odd Number : **

A number that is not divisible by 2 or not multiple of 2 are called odd number.

Eg : 1, 3, 5......

**Integers :**

All **positive** and **negative natural** number are integers including **zero**.
i

- .e ; I = {.....-3, -2, -1 , 0, 1, 2, 3......}
- The set of Integer represented by
**I.**

**Rational Numbers :**

Those number which can be written in the form of p/q where q is not equal to 0.

- The set of all rational numbers is denoted by
**Q.** - Every natural number a can be written as a / 1 , so is a rational number.
- Every rational number has a peculiar characteristic that when expressed in decimal form is expressible either
in
**terminating decimals**or**non-terminating repeating decimals.**

**Irrational Numbers :**

**NOT Rational.**

**OR,**An irrational number can not be written in the form p/q where p and q are integer and q is not equal to 0.

**OR,**Those numbers which when expressed in decimal form are

**neither terminating nor repeating decimals**are known as

**irrational numbers.**

- The exact value of pie is not 22/7. 22/7 is rational while pie is irrational numbers. 22/7 is approximate value of pie. Similarly, 3.14 is not an exact value of it.

**Prime Numbers :**

- 1 is not a prime number
- The only even prime number is 2.
- The smallest prime number is 2.
- All prime numbers are odd except 2.

**Co-prime Numbers :**

Two positive numbers are said to be co-prime if they have no common factor other than 1. HCF (2,3) = 1

**Composite Numbers :**

- 1 is not a composite number.
- 4 is the smallest composite number.
- 1 is neither prime nor composite.

Consecutive Numbers: A series of natural numbers each differing by one is called consecutive numbers.

3, 4, 5, 5, 6 are consecutive numbers.

### FUNDAMENTAL THEOREM OF ARITHMETIC:

Every positive integer greater than 1 can be the uniquely written as a prime or as the product of two or more primes.

**Examples ;**

(i) 60 = 2 × 2 × 3 × 5

(ii) 35 = 60 = 5 × 7

__Check Your Understanding :__

**Question :**

**1. Express each number as a product of its prime
factors: **

a. 140

b. 156

c. 3825

d. 5005

e. 7429

**NOTE : **

- If p is a prime and p divides a², then p divides a, where a is a positive integer.

**Example 01 :**

**Prove that √5 is irrational.**

**Explanation :**

⇒ (√5)2 = (a/b)2

⇒ 5 = a2/b2

⇒ a2 = 5b2

⇒ a2 is multiple of 5.

So, a is a multiple of 5.

From this equation , we can see that 'a2' is a multiple of 5. This means that 'a' must be a multiple of 5 as well since squaring an integer will result in another integer. Let's represent 'a' as '5k' where 'k' is an integer:

⇒ a = 5k for some integer k.

⇒ a2 = 25k2

⇒ 5b2 = 25k2

⇒ b2 = 5k2

⇒ b2 is a multiple of 5

So, b is a multiple of 5

Now, we can see that 'b2' is also a multiple of 5. This implies that 'b' must be a multiple of 5 as well since squaring an integer will result in another integer.

**false**, and

**√5 is irrational.**

**Example 02 :**

**Prove that 6 + √5 is irrational.**

**Explanation :**

**6 + √5 is rational.**

**√5 is irrational.**So, our assumption is incorrect. Hence,

**6 + √5 is irrational.**

__Check Your Understanding :__

**Question : **

**Prove that the following are irrationals:**

a. **√**2

b. 3 + **√**2

c. 1 / **√**7

d. 2 - 3**√**5

e. **√**3 - √5

- L.C.M × H.C.F = Product of two numbers.
- L.C.M of two or more prime numbers is equal to their product.
- H.C.F of two or more prime numbers is always 1.

**Example 01 :**

**Find the LCM and HCF of the 26 and 91 and verify that LCM × HCF = product
of the two numbers. **

**Explanation : **

To find the LCM (Least Common Multiple) and HCF (Highest Common Factor) of two numbers, 26 and 91, we can use the following steps:

__Step 1: __Find the prime factors of each number.

__Step 2:__ Determine the LCM by taking the product of all the unique prime factors with their highest powers.

__Step 3:__ Determine the HCF by taking the product of all the common prime factors with their lowest powers.

Step 1: Prime factors of 26 and 91

26 = 2 × 13

91 = 7 × 13

Since, **LCM =** 2 × 7 × 13 = 182

** HCF =** 13

**Now, let's verify if LCM × HCF = product of the two numbers:**

LCM × HCF = 182 × 13 = 2366

Product of the two numbers = 26 × 91 = 2366

As we can see, LCM × HCF and the product of the two numbers are both equal to 2366. **Therefore, the verification is successful.**

So, the LCM of 26 and 91 is 182, the HCF is 13, and LCM × HCF is **indeed equal to the product of the two numbers** (26 × 91 = 2366).

**Example 02 :**

**Find the LCM and HCF of 12, 15 and 21 by applying the prime factorization method.**

**Solution : **

To find the LCM (Least Common Multiple) and HCF (Highest Common Factor) of 12, 15, and 21 using the prime factorization method, we need to follow these steps:

__Step 1:__ Find the prime factors of each number.

__Step 2: __Determine the LCM by taking the product of all the unique prime factors with their highest powers.

__Step 3:__ Determine the HCF by taking the product of all the common prime factors with their lowest powers.

Now,

12 = 22 × 3

15 = 3 × 5

21 = 3 × 7

Since,

**LCM =** 22 × 3 × 5 × 7 = 420

**HCF =** 3

So, the LCM of 12, 15, and 21 is 420, and the HCF is 3

**Example 03 :**

**Given that HCF (306, 657) = 9. Find the LCM (306, 657).**

Solutions :

HCF (306, 657) × LCM (306, 657) = 306 × 657

⇒ 9 × LCM (306, 657) = 306 × 657

⇒ LCM (306, 657) = 306 x 657/9 = 34 x 657

⇒ LCM (306, 657) = 22338.

__Check Your Understanding :__

**Question : **

**a. Find the LCM and HCF of 510 and 92 and verify that LCM × HCF = product
of the two numbers.**

**b. Find the LCM and HCF of 8, 9 and 25 by applying the prime factorisation method.**

**c. Given that HCF (96, 404) = 4. Find the LCM (96, 404).**

Frequently Asked Questions on Class 10 Real Numbers

**1. What are Real Numbers, and how are they different from other types of numbers?**

**Answer: **Real numbers include all rational and irrational numbers. They can be represented on the number line and can take any value between two numbers. Rational numbers are those that can be expressed as fractions, whereas irrational numbers cannot be expressed as fractions and have non-repeating, non-terminating decimal expansions.

**2. How can we find the LCM of two or more numbers using the prime factorization method?**

**Answer: **To find the LCM of two or more numbers, we use the prime factorization method. First, factorize each number into its prime factors. Then, the LCM is found by taking the product of all the unique prime factors with their highest powers.

**3. What is the Fundamental Theorem of Arithmetic, and how does it relate to the prime factorization of a number?**

**Answer: **The Fundamental Theorem of Arithmetic states that every positive integer greater than 1 can be represented uniquely as the product of prime numbers, up to the order of the factors. This theorem is fundamental in understanding the concept of prime factorization.

**4. How do we prove that the square root of 2 is irrational?**

**Answer: **The proof that the square root of 2 is irrational is done by assuming the opposite, i.e., assuming that √2 is rational. Then, we derive a contradiction by showing that this assumption leads to an impossibility, thus proving that √2 is irrational.

**5. How can we find the HCF and LCM of more than two numbers using the prime factorization method?**

**Answer: **To find the HCF and LCM of more than two numbers using the prime factorization method, follow these steps:

Step 1: Prime factorize all the given numbers.

Step 2: Determine the HCF by taking the product of all the common prime factors with their lowest powers.

Step 3: Determine the LCM by taking the product of all the unique prime factors with their highest powers.