Natural numbers, integers, rationals, irrationals, and ordering on the number line.
Number sets help you classify values before you operate on them. Knowing whether a number is natural, whole, integer, rational, irrational, or real tells you what kind of answer is possible.
CSEC may ask you to identify numbers in a set, place them on a number line, or compare their sizes. When ordering numbers, convert them to a common form if needed, such as decimals or fractions. The goal is not just to name the set, but to understand how the numbers relate.
Before we can do anything with numbers, we need to understand the different types or sets of numbers. These are like different groups or families of numbers, and each one has specific properties.
Natural Numbers are the numbers you learned to count with as a child: 1, 2, 3, 4, 5, 6, ...
Think of natural numbers as "counting numbers." If someone asked "how many apples are in a basket?" you couldn't answer "0" apples, you'd either say "1 apple" or "5 apples" or some other positive whole number. That's why natural numbers start at 1.
Symbol: We write this set as ℕ or {1, 2, 3, 4, 5, ...}
Real-world examples:
Whole Numbers are natural numbers PLUS zero: 0, 1, 2, 3, 4, 5, ...
The only difference between natural numbers and whole numbers is that whole numbers include 0. This might seem small, but zero is actually revolutionary! It allows us to represent "nothing" as a number.
Symbol: We write this set as W or {0, 1, 2, 3, 4, ...}
Why does 0 matter?
Relationship: Every natural number is a whole number, but not every whole number is a natural number (because 0 is a whole number but not a natural number). We write this as: ℕ ⊂ W (natural numbers are a subset of whole numbers)
Integers are whole numbers PLUS negative numbers: ..., -3, -2, -1, 0, 1, 2, 3, ...
Integers include positive numbers, negative numbers, and zero. Now we can represent things that go in both directions from zero.
Symbol: We write this set as ℤ or {..., -2, -1, 0, 1, 2, ...}
Real-world examples:
Relationship: Every whole number is an integer, but not every integer is a whole number. W ⊂ ℤ
So far we have: ℕ ⊂ W ⊂ ℤ
Rational Numbers are any numbers that can be written as a fraction of two integers.
A rational number is any number of the form where and are integers and (we can't divide by zero).
Symbol: We write this set as ℚ
What counts as a rational number?
Which of these are rational numbers?
Relationship: Every integer is a rational number. ℤ ⊂ ℚ
So far: ℕ ⊂ W ⊂ ℤ ⊂ ℚ
Irrational Numbers are real numbers that CANNOT be written as a fraction .
These are numbers whose decimal representations go on forever WITHOUT repeating in a pattern.
Common irrational numbers:
Why are these "irrational"?
The word "irrational" doesn't mean "crazy", it means "not a ratio." These numbers cannot be expressed as the ratio (fraction) of two integers.
is irrational
If we try to write as a decimal: = 1.41421356237...
This goes on forever without repeating. We can never write it as a simple fraction. Try it on your calculator, keep pressing for more decimal places, and the digits never settle into a repeating pattern.
Real Numbers are ALL the numbers above combined: rational numbers PLUS irrational numbers.
If we can plot it on a number line, it's a real number.
Symbol: We write this set as ℝ
The Complete Relationship:
Within these sets, several special categories of numbers are worth noting:
Square Numbers (Perfect Squares): Numbers that are the result of multiplying an integer by itself.
Even Numbers: Integers divisible by 2: {..., -4, -2, 0, 2, 4, 6, ...}
Odd Numbers: Integers NOT divisible by 2: {..., -3, -1, 1, 3, 5, ...}
Prime Numbers: Natural numbers greater than 1 that have ONLY two factors: 1 and itself. (More on this later!)
Composite Numbers: Natural numbers greater than 1 that have MORE than two factors.
A set of numbers can be arranged in order from smallest to largest (ascending) or from largest to smallest (descending).
The number line is your best friend here. Numbers further to the LEFT are smaller. Numbers further to the RIGHT are larger.
📊 Interactive Number Line - Drag the blue point around!
You can drag the blue point along the number line to see how position relates to value. The red axis shows all the integers, and notice how each tick mark is 1 unit apart.
We use special symbols to compare numbers:
< means "less than" (smaller)> means "greater than" (larger)Order these numbers from smallest to largest: 3, -2, 0.5, -4, 2
Step 1: Look at the interactive number line below, the red points show where each number is:
📊 Ordering Numbers on a Number Line
Highlighted points: -4, -2, 0.5, 2, 3
Numbers to the LEFT are smaller. Numbers to the RIGHT are larger.
Step 2: Write them in order from left to right:
When ordering negative numbers, remember: -10 is smaller than -2 because -10 is further LEFT on the number line. The bigger the negative number looks, the smaller it actually is!