Class 10 Maths Chapter 1 Notes
Real Numbers (CBSE)
What are Real Numbers?
Real numbers include all the numbers we generally use: positive numbers, negative numbers,
fractions, decimals, and irrational numbers.
The only numbers not in this category are imaginary numbers (like numbers involving i).
Examples:
● Integers: -8, 0, 5
● Fractions: 3/4, -7/2
● Decimals: 2.71, -4.5
● Irrational numbers: √3, π
Important points:
● Real numbers are made up of rational and irrational numbers together.
● Any real number can be shown on the number line.
Euclid’s Division Lemma
Statement:
Given two positive integers a and b (where a > b), there exist unique integers q and r such that
a = b × q + r, where 0 ≤ r < b.
Example:
Take a = 45 and b = 7.
, Then
45 = 7 × 6 + 3
So, q = 6 and r = 3.
Euclid’s Division Algorithm
This is a process based on Euclid’s Lemma to find the HCF (Highest Common Factor) of two
numbers.
Steps:
1. Divide the larger number by the smaller number and get the remainder.
2. If the remainder is zero, the divisor is the HCF.
3. If not, take the divisor and remainder as the new pair, and repeat the process until you
get a remainder of zero.
Example:
Find HCF of 84 and 32.
84 = 32 × 2 + 20
32 = 20 × 1 + 12
20 = 12 × 1 + 8
12 = 8 × 1 + 4
8 = 4 × 2 + 0
So, HCF = 4.
The Fundamental Theorem of Arithmetic
This theorem says that every composite number can be written uniquely (apart from the order)
as a product of prime numbers.
Example:
90 = 2 × 3 × 3 × 5
If written differently, like 3 × 2 × 5 × 3, it still has the same prime factors.
Real Numbers (CBSE)
What are Real Numbers?
Real numbers include all the numbers we generally use: positive numbers, negative numbers,
fractions, decimals, and irrational numbers.
The only numbers not in this category are imaginary numbers (like numbers involving i).
Examples:
● Integers: -8, 0, 5
● Fractions: 3/4, -7/2
● Decimals: 2.71, -4.5
● Irrational numbers: √3, π
Important points:
● Real numbers are made up of rational and irrational numbers together.
● Any real number can be shown on the number line.
Euclid’s Division Lemma
Statement:
Given two positive integers a and b (where a > b), there exist unique integers q and r such that
a = b × q + r, where 0 ≤ r < b.
Example:
Take a = 45 and b = 7.
, Then
45 = 7 × 6 + 3
So, q = 6 and r = 3.
Euclid’s Division Algorithm
This is a process based on Euclid’s Lemma to find the HCF (Highest Common Factor) of two
numbers.
Steps:
1. Divide the larger number by the smaller number and get the remainder.
2. If the remainder is zero, the divisor is the HCF.
3. If not, take the divisor and remainder as the new pair, and repeat the process until you
get a remainder of zero.
Example:
Find HCF of 84 and 32.
84 = 32 × 2 + 20
32 = 20 × 1 + 12
20 = 12 × 1 + 8
12 = 8 × 1 + 4
8 = 4 × 2 + 0
So, HCF = 4.
The Fundamental Theorem of Arithmetic
This theorem says that every composite number can be written uniquely (apart from the order)
as a product of prime numbers.
Example:
90 = 2 × 3 × 3 × 5
If written differently, like 3 × 2 × 5 × 3, it still has the same prime factors.
