📚 Number Systems & Exponents
80-Mark Question Paper with Complete Answer Key
Every integer is a whole number.
3/8
"The product of a rational and an irrational number is always irrational."
(a) √23
(b) √225
(c) 0.3796
(d) 7.478478...
B. Give three examples of each type and explain why √7 is irrational while √49 is rational.
C. Using the number line, locate √2 and √3 geometrically.
B. Simplify the following expressions using laws of exponents:
(i) (2^(2/3)) × (2^(1/3))
(ii) (5^2)^(7)
(iii) 7^(1/5) / 7^(1/3)
(iv) 13^(1/5) × 17^(1/5)
📖 Explanation:
A rational number is defined as any number that can be expressed in the form p/q where p and q are integers and q ≠ 0.
Zero can be written as: 0 = 0/1, 0/2, 0/5, etc.
Since 0 and 1 are integers and 1 ≠ 0, zero satisfies the definition of a rational number.
📖 Explanation:
Whole Numbers (W): {0, 1, 2, 3, 4, ...}
Integers (Z): {..., -3, -2, -1, 0, 1, 2, 3, ...}
While every whole number is an integer, not every integer is a whole number.
Counterexample: -5 is an integer but NOT a whole number (whole numbers are non-negative)
📝 Division:
📖 Explanation:
- √2: Cannot be written as p/q → IRRATIONAL ✓
- π: Cannot be written as p/q → IRRATIONAL ✓
- 22/7: Already in p/q form → RATIONAL ✗
- 0.3333...: Repeating decimal, equals 1/3 → RATIONAL ✗
📝 Division:
📝 Step-by-Step Solution:
📝 Solution:
📖 Explanation:
The decimal expansion 0.101001000100001... is:
- ✓ Non-terminating (never ends)
- ✓ Non-recurring (pattern never repeats - 1 zero, 2 zeros, 3 zeros, ...)
Since it's non-terminating AND non-recurring, it is IRRATIONAL.
📖 Explanation:
-125 is an integer, and every integer is a rational number.
We can write it as: -125/1
Here, p = -125 and q = 1 (both integers, q ≠ 0) ✓
Note: We could also write it as -250/2, -375/3, etc., but -125/1 is the simplest form.
📝 Solution:
📝 Step-by-Step Solution:
📝 Using Law of Exponents:
📖 Explanation:
General Rule: If r is rational (non-zero) and s is irrational, then r × s is irrational.
Example: 2 × √3 = 2√3 (irrational) ✓
Exception: 0 × √3 = 0 (rational)
So the statement is TRUE for non-zero rational numbers.
📖 Solution Method:
Method 1 - Decimal Expansion: Any decimal between 3 and 4 is rational
Examples: 3.1, 3.2, 3.3, 3.4, 3.5
Method 2 - Using Fractions:
Write 3 = 18/6 and 4 = 24/6
Between them: 19/6, 20/6, 21/6, 22/6, 23/6
Simplify: 19/6, 10/3, 7/2, 11/3, 23/6
📖 Explanation:
Since √225 = 15, which is an integer, it can be written as 15/1.
Therefore, √225 is RATIONAL ✓
📝 Step-by-Step Solution:
📝 Step-by-Step Solution:
📝 Using Conjugate Method:
📝 Step-by-Step Solution:
📝 Step-by-Step Solution:
📖 Solution:
First, find the decimal range:
Any non-terminating non-recurring decimal between them is irrational.
Example: 0.150150015000150000... (pattern exists but doesn't repeat)
This lies between 1/7 and 2/7, and is irrational ✓
📝 Using Law of Exponents:
📖 Explanation:
(a) √23: 23 is not a perfect square → √23 = 4.795... (non-terminating, non-recurring) → IRRATIONAL ✗
(b) √225: √225 = 15 → RATIONAL ✓
(c) 0.3796: Terminating decimal → Can be written as 3796/10000 → RATIONAL ✓
(d) 7.478478...: Repeating decimal (period 478) → Can be converted to p/q → RATIONAL ✓
📖 Key Differences:
| Feature | Rational Numbers | Irrational Numbers |
|---|---|---|
| Definition | Can be written as p/q (p, q ∈ Z, q ≠ 0) | Cannot be written as p/q |
| Decimal Form | Terminating or repeating | Non-terminating, non-repeating |
| Examples | 1/2, 3/4, -5/2, 0, 5 | √2, √3, π, e |
📝 Examples of Rational Numbers:
1. 1/2 = 0.5 (Terminating)
2. 1/3 = 0.333... (Repeating)
3. 5 = 5/1 (Integer is rational)
📝 Examples of Irrational Numbers:
1. √2 = 1.41421356... (Non-terminating, non-repeating)
2. π = 3.14159265... (Non-terminating, non-repeating)
3. 0.10110111011110... (Pattern visible but doesn't repeat)
📖 Proof:
Assume: √2 is rational
Then: √2 = p/q where p and q are co-prime integers (no common factors)
Step 1: Square both sides: 2 = p²/q²
Step 2: Therefore: p² = 2q²
Step 3: This means p² is even, so p must be even
Let p = 2k for some integer k
Step 4: Substitute: (2k)² = 2q²
Step 5: 4k² = 2q²
Step 6: 2k² = q²
Step 7: This means q² is even, so q is even
Step 8: But if both p and q are even, they have a common factor 2
This contradicts our assumption that p and q are co-prime!
Conclusion: √2 cannot be rational, therefore √2 is IRRATIONAL ✓
📝 Step-by-Step Solution:
📝 Step-by-Step Solution:
📖 Construction Steps:
Step 1: Draw a line and mark point O at zero
Step 2: Mark point A at distance 2 from O (so OA = 2)
Step 3: Mark point B at distance 1 from A (so AB = 1), on the same line
Step 4: Now OB = 3. Find the midpoint M of OB
Step 5: OM = MB = 3/2
Step 6: Draw a semicircle with center M and radius MB = 3/2
Step 7: Draw a perpendicular to the line at point A
Step 8: Let this perpendicular meet the semicircle at point P
Step 9: By Pythagoras theorem: AP² = OM² - AM² = (3/2)² - (1/2)² = 9/4 - 1/4 = 8/4 = 2
Wait, let me recalculate for √5:
Correct Method: OA = 2, AB = 1, then OB = 3
AP² = (3/2)² - (1/2)² would give 2, not 5
For √5: Use OA = 2, AB = 1, so we need AP² = 5
Using the right triangle method: construct with sides 1 and 2
📖 Number System Hierarchy:
Real Numbers (R) = Rational (Q) + Irrational (I)
| Type | Symbol | Examples | Characteristics |
|---|---|---|---|
| Natural | N | 1, 2, 3, ... | Counting numbers |
| Whole | W | 0, 1, 2, 3, ... | Natural + zero |
| Integer | Z | ..., -2, -1, 0, 1, 2, ... | Whole + negatives |
| Rational | Q | 1/2, 3/4, -5/2 | p/q form, integers terminating or repeating decimals |
| Irrational | I | √2, π, e | Cannot be p/q, non-terminating non-repeating |
| Real | R | All above | All rational + irrational |
Relationship: N ⊂ W ⊂ Z ⊂ Q ⊂ R
📖 Comparison:
√49:
49 = 7² (perfect square)
√49 = 7 (an integer)
Can be written as 7/1 → RATIONAL ✓
√7:
7 is not a perfect square
√7 = 2.6457513... (non-terminating non-repeating)
Cannot be written as p/q → IRRATIONAL ✗
📖 Laws of Exponents for Real Numbers:
For a > 0, and p, q are rational numbers:
1. Multiplication Rule: a^p · a^q = a^(p+q)
Example: 2^(1/2) · 2^(1/2) = 2^(1/2 + 1/2) = 2^1 = 2
2. Power of a Power: (a^p)^q = a^(pq)
Example: (3^2)^3 = 3^(2×3) = 3^6 = 729
3. Division Rule: a^p / a^q = a^(p-q)
Example: 5^(3/2) / 5^(1/2) = 5^(3/2 - 1/2) = 5^1 = 5
4. Same Exponent Rule: a^p · b^p = (ab)^p
Example: 2^(1/3) · 4^(1/3) = (2×4)^(1/3) = 8^(1/3) = 2
📖 Solutions to (B):
(i) (2^(2/3)) × (2^(1/3)):
(ii) (5^2)^7:
(iii) 7^(1/5) / 7^(1/3):
(iv) 13^(1/5) × 17^(1/5):
📖 Proof of (a^p)^q = a^(pq):
Using a = 2, p = 2/3, q = 3/2:
LHS = RHS = 2 ✓ Therefore (a^p)^q = a^(pq) is proved!
📊 Marks Distribution Summary
🎯 Key Points for Success
- Definition Clarity: Always remember p/q form for rational numbers
- Decimal Recognition: Terminating/Repeating = Rational; Never-ending, never-repeating = Irrational
- Exponent Laws: Master the four main laws - they appear in almost every problem
- Show Working: Partial marks are awarded for correct working even if final answer is wrong
- Rationalizing: Always rationalize denominators in final answers
- Number Line: Understand geometric construction for square roots
- Like Terms: Combine surds with same irrational parts
- Proof Techniques: Practice contradiction method for proving irrationality
