Explore the complete number system from integers to irrational numbers, and discover how all real numbers can be represented on the number line.
Understanding the hierarchy of numbers
1, 2, 3, 4, 5, ... (counting numbers)
0, 1, 2, 3, 4, 5, ... (natural numbers + 0)
..., -3, -2, -1, 0, 1, 2, 3, ... (positive, negative, and zero)
Fractions, terminating decimals, repeating decimals
Non-repeating, non-terminating decimals (√2, π, e)
All rational and irrational numbers combined
Every point on the number line represents a real number
Numbers that can be expressed as fractions
Definition: A rational number can be expressed as a fraction a/b where a and b are integers and b ≠ 0.
Examples:
Key Properties:
Terminating Decimals:
• 1/4 = 0.25
• 3/8 = 0.375
• 7/20 = 0.35
Repeating Decimals:
• 1/3 = 0.333... = 0.3̄
• 2/7 = 0.285714285714... = 0.285714̄
• 5/6 = 0.8333... = 0.83̄
Converting Decimals to Fractions:
• 0.75 = 75/100 = 3/4
• 0.6̄ = 6/9 = 2/3
• 0.125 = 125/1000 = 1/8
Numbers that cannot be expressed as fractions
Definition: An irrational number cannot be expressed as a fraction a/b where a and b are integers.
Examples:
Key Properties:
√2 (Square Root of 2):
The diagonal of a unit square. Discovered by the ancient Greeks, it was the first known irrational number.
π (Pi):
The ratio of a circle's circumference to its diameter. Approximately 3.14159...
e (Euler's Number):
The base of natural logarithms. Approximately 2.71828... Used in calculus and exponential growth.
φ (Golden Ratio):
(1 + √5)/2 ≈ 1.618... Appears in art, architecture, and nature.
How to represent different types of numbers on the number line
Exact Placement:
Rational numbers can be located exactly on the number line using fractions or decimals.
Approximate Placement:
Irrational numbers can only be approximated on the number line, never located exactly.
Classify numbers and understand their properties
Given: Classify each number as rational or irrational:
0.75, √9, π, 2/3, √5, 0.123123123...
Solution:
• 0.75 = 3/4 → Rational
• √9 = 3 = 3/1 → Rational
• π ≈ 3.14159... → Irrational
• 2/3 = 0.666... → Rational
• √5 ≈ 2.236... → Irrational
• 0.123123123... → Rational (repeating)
Given: Place these numbers on a number line:
1.5, √2, 2.5, π, 3
Solution:
• 1.5 (exact placement)
• √2 ≈ 1.414 (approximate)
• 2.5 (exact placement)
• π ≈ 3.142 (approximate)
• 3 (exact placement)