Notes of Ch 1 Real Numbers| Class 10th Math

Revision Notes of Ch 1 Real Numbers Class 10th Math

Topics in the Chapter

• Euclid’s Division Lemma
• Euclid’s Division Algorithm
• The Fundamental Theorem of Arithmetic
• Revisiting Irrational Numbers
• Revisiting Rational Numbers and Their Decimal Expansions

Euclid’s Division Lemma

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

     a = bq + r, 0  ≤ r < b

Here,  a, b, q and r are called dividend, divisor, quotient and remainder respectively.

→ Example: Dividing 34 by 6, we get quotient as 6 and remainder as 4.

So we can write it as, 34 = 6×5 + 4

Here, a = 34, b = 6, q = 5 and r = 4

∴ a = bq + r, 0  ≤ r < b

Euclid’s Division Algorithm 

→ Euclid’s Division Algorithm is application of Euclid’s Division Lemma means it is technique to compute HCF of two natural numbers.

→ To obtain the HCF of two positive integers, say c and d, with c > d, we follow these steps:
Step 1: Apply Euclid’s division lemma to c and d to obtain two whole numbers q and r.
            Thus, c = dq + r, 0  ≤ r < d

Step 2 : If r = 0, d is the HCF of c and d. If r ≠ 0, again apply the division lemma to d and r.

Step 3 : Continue the process till the remainder is zero. The last divisor when remainder becomes zero is the HCF of c and d

→ Example: Use Euclid’s division algorithm to find the HCF of 420 and 130.
Here, 420 > 130 so we can write,
Step 1: 420 = 130 × 3 + 30 (remainder is 30 so we will apply division lemma again on 130 and 30)
Step 2: 130 = 30 × 4 + 10 (remainder is 10 so we will apply division lemma again on 30 and 10)
Step 3: 30 = 10 × 3 + 0
Now, the remainder is 0 and the last divisor is 10. So, 10 is the HCF of 420 and 130.


The Fundamental Theorem of Arithmetic

→ Every composite number can be expressed (factorised) as a product of primes, and this factorisation is unique, apart from the order in which the prime factors occur.

→ Let x be composite number which can be factorised as,
x =  p₁p₂ ..... p_n , where  p₁p₂ ..... p_n are primes and written in ascending order, i.e., p₁ ≤ p₂ ≤ ..... ≤ p_n. If we combine the same primes, we will get powers of primes. This is called prime factorization.
For example: 32760 = 2 × 2 × 2 × 3 × 3 × 5 × 7 × 13 = 2³ × 3² × 5 × 7 × 13

By expressing any two numbers as their prime factors, their highest common factor (HCF) and lowest common multiple (LCM) can be calculated.