01. Real Numbers
All numbers that can be represented on the number line
are called Real Numbers.
Real Numbers include:
• Rational Numbers
• Irrational Numbers
✏ Example
5, −3, 0,
3
4
, √2, √5 and π are all real numbers.
02. Rational Numbers
A number that can be expressed in the form
is called a
Rational Number.
✏ Example
3 =
3
1
0.75 =
75
100
=
3
4
Therefore, both are rational numbers.
03. Irrational Numbers
Numbers which cannot be expressed in the form
p
q
are called Irrational Numbers.
Their decimal expansions are
non-terminating and non-repeating.
✏ Example
√2 = 1.4142135...
√3 = 1.7320508...
√5 = 2.2360679...
These are irrational numbers.
04. Prime Numbers
A natural number greater than 1 having exactly two positive factors,
1 and itself, is called a Prime Number.
✏ Example
Factors of 7:
1, 7
Therefore, 7 is prime.
⚠️ 1 is neither prime nor composite.
05. Composite Numbers
A natural number greater than 1 having more than two positive factors
is called a Composite Number.
✏ Example
Factors of 12:
1, 2, 3, 4, 6, 12
Therefore, 12 is composite.
06. Prime Factorisation ⭐⭐⭐
Expressing a composite number as a product of prime numbers
is called Prime Factorisation.
✏ Example
60 = 2 × 30
= 2 × 2 × 15
= 2 × 2 × 3 × 5
Therefore,
07. Fundamental Theorem of Arithmetic ⭐⭐⭐
Every composite number can be expressed as a product of primes,
and this prime factorisation is unique apart from the order
of the prime factors.
✏ Example
72 = 2 × 36
= 2 × 2 × 18
= 2 × 2 × 2 × 9
Therefore,
72 = 2³ × 3²
08. HCF using Prime Factorisation
For HCF, take the common prime factors with the smallest powers.
✏ Example
36 = 2² × 3²
48 = 2⁴ × 3
HCF = 2² × 3
= 4 × 3
= 12
09. LCM using Prime Factorisation
For LCM, take all prime factors with the highest powers.
✏ Example
36 = 2² × 3²
48 = 2⁴ × 3
LCM = 2⁴ × 3²
= 16 × 9
= 144
10. HCF × LCM Relation ⭐⭐⭐
✏ Example
For 12 and 18:
HCF = 6
LCM = 36
6 × 36 = 216
12 × 18 = 216
Hence verified.
11. Prove √2 is Irrational ⭐⭐⭐
We use the proof by contradiction method.
Step 1:
Assume that √2 is rational.
Step 2:
Therefore,
√2 =
p
q
where p and q are coprime integers and q ≠ 0.
Step 3:
Squaring both sides:
2 =
p²
q²
Therefore,
p² = 2q²
Step 4:
Since p² is divisible by 2,
p is also divisible by 2.
Let p = 2k.
Step 5:
Substituting p = 2k:
(2k)² = 2q²
4k² = 2q²
q² = 2k²
Therefore, q is also divisible by 2.
Step 6:
Thus, both p and q are divisible by 2.
This contradicts the fact that p and q are coprime.
Therefore, our assumption is false.
Hence, √2 is irrational. ✓
12. Prove √3 is Irrational ⭐⭐⭐
Step 1:
Assume that √3 is rational.
Step 2:
√3 =
p
q
where p and q are coprime integers and q ≠ 0.
Step 3:
Squaring both sides:
3 =
p²
q²
Therefore,
p² = 3q²
Step 4:
Since p² is divisible by 3,
p is also divisible by 3.
Let p = 3k.
Step 5:
(3k)² = 3q²
9k² = 3q²
q² = 3k²
Therefore, q is also divisible by 3.
Step 6:
Thus, both p and q are divisible by 3.
This contradicts the fact that p and q are coprime.
Therefore, our assumption is false.
Hence, √3 is irrational. ✓
13. Prove √5 is Irrational ⭐⭐⭐
Step 1:
Assume that √5 is rational.
Step 2:
√5 =
p
q
where p and q are coprime integers and q ≠ 0.
Step 3:
Squaring both sides:
5 =
p²
q²
Therefore,
p² = 5q²
Step 4:
Since p² is divisible by 5,
p is also divisible by 5.
Let p = 5k.
Step 5:
(5k)² = 5q²
25k² = 5q²
q² = 5k²
Therefore, q is also divisible by 5.
Step 6:
Thus, both p and q are divisible by 5.
This contradicts the fact that p and q are coprime.
Therefore, our assumption is false.
Hence, √5 is irrational. ✓
14. Prove 3 + 2√5 is Irrational ⭐⭐⭐
Step 1:
Assume that 3 + 2√5 is rational.
Step 2:
2√5 = (3 + 2√5) − 3
Since rational − rational is rational,
2√5 would be rational.
Step 3:
√5 =
2√5
2
Therefore, √5 would be rational.
Step 4:
But √5 is irrational.
This is a contradiction.
Therefore, our assumption is false.
Hence, 3 + 2√5 is irrational. ✓
15. Rational + Irrational
The sum of a rational number and an irrational number
is always irrational.
✏ Example
3 + √2
3 is rational and √2 is irrational.
Therefore,
3 + √2 is irrational.
16. Non-zero Rational × Irrational
The product of a non-zero rational number and an irrational number
is always irrational.
✏ Example
5√3
5 is a non-zero rational number
and √3 is irrational.
Therefore,
5√3 is irrational.
⚠️ The condition non-zero is important.
0 × √3 = 0, which is rational.
17. Irrational + Irrational
The sum of two irrational numbers may be rational
or irrational.
✏ Example 1 — Rational
√2 + (−√2) = 0
0 is rational.
✏ Example 2 — Irrational
√2 + √3
is irrational.
18. Irrational × Irrational
The product of two irrational numbers may be rational
or irrational.
✏ Example 1 — Rational
√2 × √2 = 2
✏ Example 2 — Irrational
√2 × √3 = √6
√6 is irrational.
19. Irrational ÷ Irrational
The quotient of two irrational numbers may be rational
or irrational.
✏ Example 1 — Rational
√2
√2
= 1
✏ Example 2 — Irrational
√2
√3
is irrational.
20. Perfect Square and √n
If n is a perfect square, then √n is an integer
and therefore rational.
If n is not a perfect square, then √n is irrational.
✏ Example
√81 = 9
Therefore, √81 is rational.
√10 is irrational because 10 is not a perfect square.
21. HCF–LCM Based Question ⭐⭐⭐
✏ Example
The HCF of two positive integers is 6
and their LCM is 180.
If one number is 30, find the other number.
HCF × LCM = Product of the two numbers
6 × 180 = 30 × x
1080 = 30x
x = 36
22. HCF and LCM Together
✏ Example
Find HCF and LCM of 72 and 120.
72 = 2³ × 3²
120 = 2³ × 3 × 5
HCF
= 2³ × 3
= 24
LCM
= 2³ × 3² × 5
= 360
Check:
24 × 360 = 8640
72 × 120 = 8640 ✓
23. Irrationality Proof — General Pattern ⭐⭐⭐
Step 1:
Assume that the given number is rational.
Step 2:
Express it as p/q,
where p and q are coprime integers and q ≠ 0.
Step 3:
Perform the required algebraic manipulation.
Step 4:
Show that p and q are both divisible by the same prime number.
Step 5:
This contradicts the fact that p and q are coprime.
Step 6:
Therefore, the original assumption is false.
Step 7:
Hence, the given number is irrational.
Exam Pattern:
Assumption → Algebra → Contradiction → Conclusion
24. Important Number Facts
| Number |
Type |
| 0 |
Rational |
| 1 |
Rational |
| −5 |
Rational |
| 3/7 |
Rational |
| √2 |
Irrational |
| √3 |
Irrational |
| √5 |
Irrational |
| π |
Irrational |
25. Identify Rational or Irrational
✏ Example
√49 = 7
Therefore, √49 is rational.
√50 = 5√2
Since √2 is irrational and 5 is non-zero rational,
√50 is irrational.
26. Quick Concept Check
✏ Q1
√2 + √2 = 2√2
Answer: Irrational
✏ Q2
√5 × √5 = 5
Answer: Rational
✏ Q3
√3 − √3 = 0
Answer: Rational
27. Exam Traps ⚠️
❌ 1 is neither prime nor composite.
❌ HCF → highest powers नहीं.
✔ HCF → lowest powers.
❌ LCM → lowest powers नहीं.
✔ LCM → highest powers.
❌ Two irrational numbers do not always have an irrational sum.
❌ Two irrational numbers do not always have an irrational product.
❌ In irrationality proofs, p and q must be taken as coprime.
❌ q = 0 is not allowed.
28. Quick Revision ⭐⭐⭐
Fundamental Theorem of Arithmetic
Every composite number can be expressed as a product of primes,
and the prime factorisation is unique apart from the order
of the prime factors.
Irrationality Proof
Assume rational → Express as p/q
→ Algebraic manipulation → Contradiction
→ Hence irrational.
29. Real Numbers — Exam Strategy 🎯
Prime Factorisation:
Express the number as a product of prime factors.
HCF:
Take common prime factors with the lowest powers.
LCM:
Take all prime factors with the highest powers.
HCF–LCM Relation:
HCF × LCM = Product of two positive integers.
√2, √3, √5:
Use the contradiction method.
3 + 2√5:
Assume rational and isolate √5.
Rational / Irrational:
Use the definition and known irrational numbers.
🏆 Real Numbers — Key Points
Prime Factorisation
→ Unique apart from order of factors.
HCF
→ Lowest powers of common prime factors.
LCM
→ Highest powers of all prime factors.
HCF × LCM
→ Product of two positive integers.
√2, √3, √5
→ Irrational
Rational + Irrational
→ Irrational
Non-zero Rational × Irrational
→ Irrational
Irrational + Irrational
→ May be rational or irrational
Irrational × Irrational
→ May be rational or irrational