Number Systems Solved Examples (Class 9 Maths)
Number systems include natural numbers, whole numbers, integers, rational numbers, and irrational numbers. Understanding their properties and operations is
TL;DR: Number systems include natural numbers, whole numbers, integers, rational numbers, and irrational numbers. Understanding their properties and operatio…
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Number systems include natural numbers, whole numbers, integers, rational numbers, and irrational numbers. Understanding their properties and operations is
Number Systems — Solved Numerical Examples (Step by Step)
Example 1: Classify the following numbers: 5, -3, 0, 2/3, sqrt(2).
Solution: 5: Natural number, whole number, integer, rational number
-3: Integer, rational number
0: Whole number, integer, rational number
2/3: Rational number (ratio of two integers)
sqrt(2): Irrational number (cannot be expressed as a ratio of two integers)
Example 2: Express 0.333... (repeating) as a fraction.
Solution: Let x = 0.333...
Multiply both sides by 10: 10x = 3.333...
Subtract the first equation from the second:
10x - x = 3.333... - 0.333...
9x = 3
x = 3/9 = 1/3
Example 3: Simplify sqrt(50).
Solution: sqrt(50) = sqrt(25 × 2)
sqrt(50) = sqrt(25) × sqrt(2)
sqrt(50) = 5sqrt(2)
Example 4: Rationalize the denominator: 1 / sqrt(3).
Solution: To rationalize, multiply both numerator and denominator by sqrt(3):
1 / sqrt(3) = 1 / sqrt(3) × sqrt(3) / sqrt(3)
1 / sqrt(3) = sqrt(3) / (sqrt(3))^2
1 / sqrt(3) = sqrt(3) / 3
Example 5: Find (2 + sqrt(3))(2 - sqrt(3)).
Solution: This is in the form (a + b)(a - b) = a^2 - b^2
(2 + sqrt(3))(2 - sqrt(3)) = (2)^2 - (sqrt(3))^2
(2 + sqrt(3))(2 - sqrt(3)) = 4 - 3
(2 + sqrt(3))(2 - sqrt(3)) = 1
Example 6: Express 1.25 as a fraction in simplest form.
Solution: 1.25 = 125 / 100
Find the GCD of 125 and 100:
125 = 5 × 25 = 5 × 5 × 5
100 = 4 × 25 = 2 × 2 × 5 × 5
GCD = 25
125 / 100 = (125 ÷ 25) / (100 ÷ 25) = 5 / 4
Example 7: Is sqrt(2) a rational or irrational number? Justify.
Solution: sqrt(2) is an irrational number.
Proof by contradiction:
Assume sqrt(2) is rational, so sqrt(2) = p/q where p and q are coprime integers.
Squaring: 2 = p^2 / q^2, so p^2 = 2q^2
This means p^2 is even, so p is even. Let p = 2k.
Then (2k)^2 = 2q^2, so 4k^2 = 2q^2, so q^2 = 2k^2.
This means q^2 is even, so q is even.
But if both p and q are even, they are not coprime, which is a contradiction.
Therefore, sqrt(2) is irrational.
Tips
- Remember the hierarchy: Natural numbers are a subset of whole numbers, which are a subset of integers, which are a subset of rational numbers. Rational and irrational numbers together form real numbers.
- To convert a repeating decimal to a fraction, multiply by an appropriate power of 10 to shift the decimal, then subtract to eliminate the repeating part.
- When simplifying radicals, factor out perfect squares. For example, sqrt(50) = sqrt(25 × 2) = 5sqrt(2). Always simplify radicals in the denominator by rationalizing.
Frequently Asked Questions
What is the difference between rational and irrational numbers?
Rational numbers can be expressed as the ratio of two integers (p/q where q is not zero). Examples: 3/4, -2, 0.5. Irrational numbers cannot be expressed as such a ratio. Their decimal representation never terminates and never repeats. Examples: pi, e, sqrt(2).
How do I rationalize a denominator with a binomial like 1/(2 + sqrt(3))?
Multiply both numerator and denominator by the conjugate of the denominator. The conjugate of (2 + sqrt(3)) is (2 - sqrt(3)). So: 1/(2 + sqrt(3)) × (2 - sqrt(3))/(2 - sqrt(3)) = (2 - sqrt(3))/(4 - 3) = 2 - sqrt(3).
More Maths Solved Examples
- Quadratic Equations
- Trigonometry
- Arithmetic Progressions
- Surface Areas and Volumes
- Coordinate Geometry
- Probability
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