📌 What is the Number System?
The Number System is a way to classify, represent, and operate with numbers. It forms the foundation of all mathematical operations and concepts.
Different types of numbers serve different purposes, from simple counting to solving complex equations.
🧱 Types of Numbers
1. Natural Numbers (ℕ)
Also known as counting numbers. These are the numbers starting from 1, 2, 3, 4, and so on.
Notation: ℕ = {1, 2, 3, 4, ...}
2. Whole Numbers
Natural numbers plus 0.
Notation: {0, 1, 2, 3, ...}
3. Integers (ℤ)
Includes all positive and negative whole numbers.
Notation: ℤ = {..., -3, -2, -1, 0, 1, 2, 3, ...}
4. Rational Numbers (ℚ)
Any number that can be written as a fraction of two integers (a/b where b ≠ 0).
Includes integers, terminating decimals, and repeating decimals.
Examples: 1/2, -3/4, 0.75, 2, -1
5. Irrational Numbers
Numbers that cannot be written as a fraction. Their decimal expansions are non-repeating and non-terminating.
Examples: √2, π, e
6. Real Numbers (ℝ)
All rational and irrational numbers together form the real numbers.
Includes: All numbers that can be placed on a number line.
7. Imaginary Numbers
Numbers that are multiples of the square root of -1, denoted as i.
Example: i, 2i, -5i
8. Complex Numbers (ℂ)
Numbers that have a real part and an imaginary part.
Form: a + bi where a and b are real numbers, and i is the imaginary unit.
Examples: 3 + 4i, -1 + 0i (which is just -1)
📏 Properties of the Number System
1. Closure Property
The result of an operation (like addition or multiplication) on numbers in a set is also in that set.
Example: Integers are closed under addition.
2. Commutative Property
a + b = b + a and a × b = b × a
3. Associative Property
(a + b) + c = a + (b + c) and (a × b) × c = a × (b × c)
4. Distributive Property
a × (b + c) = a × b + a × c
5. Identity Elements
Additive identity: 0 → a + 0 = a
Multiplicative identity: 1 → a × 1 = a
6. Inverse Elements
Additive inverse: a + (-a) = 0
Multiplicative inverse: a × (1/a) = 1, for a ≠ 0
📊 Comparing and Ordering Numbers
- Use a number line to compare values visually
- Negative numbers are smaller than positive numbers
- Fractions and decimals can be converted to compare
- Irrational numbers like √2 can be approximated for comparison
🔢 Number Bases
The decimal system (base 10) is most common, but other systems include:
- Binary (base 2): Used in computers (0 and 1 only)
- Octal (base 8)
- Hexadecimal (base 16): Used in computing (0-9 and A-F)
🧠 Prime Numbers
A prime number is a number greater than 1 that has only two factors: 1 and itself.
Examples: 2, 3, 5, 7, 11, 13, 17...
Used in cryptography, number theory, and factoring.
🎯 Perfect, Composite, and Even/Odd Numbers
Composite Numbers
Have more than two factors. Example: 4, 6, 8, 9
Perfect Numbers
Equal to the sum of their proper divisors. Example: 6 → 1 + 2 + 3 = 6
Even Numbers
Divisible by 2
Odd Numbers
Not divisible by 2
🧮 Practice Questions
- Classify the number √2
- Is -5 a rational number?
- Convert 1011 (binary) to decimal
- Is 13 a prime number?
- Find the multiplicative inverse of 4
- List all rational numbers between 0 and 1