Welcome to the journey through number systems! Click any section below to begin your exploration.
Welcome to the Number Systems Journey!
Numbers are the foundation of mathematics. Throughout history, humans have developed different types of numbers to solve various problems. Let's explore these fascinating number systems together!
What You'll Learn:
Natural Numbers (ℕ) - The counting numbers
Whole Numbers (W) - Natural numbers with zero
Integers (ℤ) - Positive and negative whole numbers
Rational Numbers (ℚ) - Numbers that can be expressed as fractions
How these number systems relate to each other
Natural Numbers (ℕ)
Natural numbers are the first numbers humans used for counting. They start from 1 and go on forever!
ℕ = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, ...}
Key Properties:
Start from 1 (not 0)
Only positive numbers
Used for counting objects
Infinite in size
Example: "I have 5 apples" - 5 is a natural number
Whole Numbers (W)
Whole numbers include all natural numbers plus zero. Zero was a revolutionary concept in mathematics!
W = {0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, ...}
Key Properties:
Start from 0 (includes zero)
Natural numbers + zero
Represents "nothing" or absence
Example: "I have 0 oranges" - 0 is a whole number
Note: Every natural number is a whole number, but not vice versa!
Integers (ℤ)
Integers extend whole numbers to include negative numbers. The symbol ℤ comes from the German word "Zahlen" (numbers).
ℤ = {..., -3, -2, -1, 0, 1, 2, 3, ...}
Key Properties:
Include positive, negative, and zero
Extend infinitely in both directions
Represent debt, temperature below zero, etc.
Etymology: ℤ from German "Zahlen"
Example: "-5°C temperature" - negative 5 is an integer
Rational Numbers (ℚ)
Rational numbers can be expressed as a fraction p/q where p and q are integers and q ≠ 0. They include all integers, fractions, and terminating or repeating decimals.
ℚ = {p/q : p, q ∈ ℤ, q ≠ 0}
Key Properties:
Can be written as fractions: 3/4, -5/2, 7/1
All integers are rational: 5 = 5/1
Decimals that terminate: 0.5 = 1/2
Decimals that repeat: 0.333... = 1/3
Dense on number line - infinite rationals between any two numbers
Nested Sets Visualization
Number systems form a beautiful nested structure, like Russian dolls. Each larger set contains all the previous sets!
The Relationship:
ℕ ⊂ W: Natural numbers are inside Whole numbers
W ⊂ ℤ: Whole numbers are inside Integers
ℤ ⊂ ℚ: Integers are inside Rational numbers
ℕ ⊂ W ⊂ ℤ ⊂ ℚ: The complete hierarchy
ℕ ⊂ W ⊂ ℤ ⊂ ℚ
Practice Examples
Example 1: True or False?
Every whole number is a natural number FALSE - Zero is a whole number but not a natural number
Every integer is a rational number TRUE - Any integer m can be written as m/1
Every rational number is an integer FALSE - 3/5 is rational but not an integer
Example 2: Find 5 Rational Numbers Between 1 and 2