Real Numbers class 10 Notes

 

Real Numbers class 10 Notes

• Real numbers include both rational numbers and irrational numbers.
• This chapter explains Euclid’s Division Algorithm and the Fundamental Theorem of Arithmetic.
• These concepts are used to find HCF and understand the decimal expansion of rational numbers.

The Fundamental Theorem of Arithmetic

• The Fundamental Theorem of Arithmetic states that "every composite number can be written as a product of prime numbers, and this factorisation is unique (except for the order of the prime factors).

Composite Numbers 

• A composite number is a positive integer greater than 1 that has more than two factors.
• It means the number can be divided exactly by 1, itself, and at least one more number.

Examples

• 4 → factors: 1, 2, 4
• 6 → factors: 1, 2, 3, 6
• 8 → factors: 1, 2, 4, 8
• 9 → factors: 1, 3, 9

- All these numbers have more than two factors, so they are composite numbers.

Difference Between Prime and Composite

1) Prime number: Only 2 factors (1 and itself)

      Example: 2, 3, 5, 7

2) Composite number: More than 2 factors

      Example: 4, 6, 8, 9

Important Points

• 1 is neither prime nor composite.
• 2 is the smallest prime number.
• 4 is the smallest composite number.

Theorem 1.1 (Euclid’s Division Lemma)

Statement

        Given any two positive integers a and b, there exist unique integers q and r such that

                a = bq + r

                    where 0 ≤ r < b.

Meaning

• a = dividend
• b = divisor
• q = quotient
• r = remainder

Example 1) Divide 17 by 5

Solution: 17 = 5 × 3 + 2

So

• a = 17
• b = 5
• q = 3
• r = 2

And 0 ≤ 2 < 5, so the theorem is satisfied.

The remainder r is always greater than or equal to 0 and less than b. 

Simple Example to Find HCF and LCM 

Question: Find the HCF and LCM of: 12 and 18

Step 1: Prime Factorisation

12 = 2 × 2 × 3 
18 = 2 × 3 × 3 

Step 2: Find HCF (Highest Common Factor)

Common factors: 2 and 3

HCF = 2 × 3 = 6

Step 3: Find LCM (Least Common Multiple)

LCM = 2 × 3 × 2 × 3
LCM = 4 × 9 = 36

Formula:

        HCF × LCM = Product of two numbers

Revisiting Irrational Numbers 

• An irrational number is a number that cannot be written in the form $\frac{p}{q}$ where p and q are integers and $q \neq 0$
  
• Their decimal expansion is non-terminating and non-repeating.

Examples

• √2
• √3
• √5
• π

Example of Irrational Number

$$\sqrt{2}=1.4142135\dots$$

• The digits never end and never repeat in a pattern, so it is irrational.

Theorem 1.2 – Revisiting Irrational Numbers

Statement

The number √p is irrational, where p is a prime number.

Proof

Assume that √p is rational.

Then it can be written in the form

$$\sqrt{p} = \frac{a}{b}$$

where a and b are integers, b ≠ 0, and a and b have no common factor (they are in lowest form).

Squaring both sides:

$$p = \frac{a^2}{b^2}$$ $$a^2 = p\,b^2$$

This means a² is divisible by p, so a is also divisible by p.

Let
                                                            a = p × k

Substitute in the equation:

$$(p k)^2 = p\,b^2$$
$$p^2{k}^2=p{b}^2$$

Divide by p:

$$p k^2 = b^2$$

So b² is divisible by p, therefore b is also divisible by p.
This means a and b both have factor p.
But we assumed a and b have no common factor.
This is a contradiction.
Therefore our assumption is wrong.
Hence,

$$\sqrt{p} \text{ is irrational.}$$

Important Result

• The square root of any prime number is irrational.

Examples:

• √2
• √3
• √5
• √7

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