Number Systems
The Number System Hierarchy
Numbers are organised into nested sets, each building on the previous. Understanding this hierarchy helps you see how different number types relate to each other.
Real Numbers (โ)
โโโ Rational Numbers (โ)
โ โโโ Integers (โค)
โ โ โโโ Whole Numbers (W)
โ โ โ โโโ Natural Numbers (โ)
โโโ Irrational Numbers
Every natural number is also a whole number, integer, rational, and real number. The sets nest inside each other like Russian dolls.
Key Terms
Natural Numbers (โ)
Natural numbers are the counting numbers โ the ones you first learn as a child: 1, 2, 3, 4, 5, 6, โฆ
They are always positive and never include zero, fractions, or decimals. They are used for counting discrete objects (e.g. 3 apples, 12 students).
Whole Numbers (W)
Whole numbers are natural numbers with the addition of zero: 0, 1, 2, 3, 4, 5, โฆ
Zero was a significant mathematical invention โ it represents "none" and acts as a placeholder in our number system.
Integers (โค)
Integers extend whole numbers to include negative numbers: โฆ-3, -2, -1, 0, 1, 2, 3โฆ
The symbol โค comes from the German Zahlen (numbers). Integers are used for temperatures, debts, altitudes below sea level, and other quantities that can go below zero.
Rational Numbers (โ)
A rational number is any number that can be expressed as a fraction p/q, where p and q are integers and q โ 0.
This includes:
- All integers (e.g. -3 = -3/1)
- Proper fractions (3/4, -2/5)
- Terminating decimals (0.75 = 3/4)
- Repeating decimals (0.333โฆ = 1/3, 0.666โฆ = 2/3)
Worked Example: Is 0.666โฆ rational? Yes โ it equals 2/3, which is a fraction of two integers with a non-zero denominator.
Irrational Numbers
Irrational numbers cannot be expressed as a fraction p/q. Their decimal expansions:
- Never terminate (go on forever)
- Never repeat in a pattern
Common irrational numbers:
- โ2 โ 1.41421356โฆ (the square root of any non-perfect square)
- โ5 โ 2.23606797โฆ
- ฯ โ 3.14159265โฆ (ratio of a circle's circumference to its diameter)
- e โ 2.71828182โฆ (base of natural logarithm)
Worked Example: Is โ9 rational or irrational? โ9 = 3, which is an integer โ so it is rational, not irrational. But โ7 โ 2.6457513โฆ never terminates or repeats โ irrational.
Real Numbers (โ)
Real numbers are all rational and irrational numbers together. They correspond to every point on the number line. Every number you encounter in everyday mathematics is a real number.
The only numbers that are NOT real numbers are imaginary numbers (such as โ(-1)), which are studied in advanced mathematics.
Classifying Numbers
| Number | Natural | Whole | Integer | Rational | Real |
|---|---|---|---|---|---|
| 5 | โ | โ | โ | โ | โ |
| 0 | โ | โ | โ | โ | โ |
| -4 | โ | โ | โ | โ | โ |
| 3/4 | โ | โ | โ | โ | โ |
| โ2 | โ | โ | โ | โ | โ |
| ฯ | โ | โ | โ | โ | โ |
Common Mistakes
- Mistake: thinking 0.333โฆ is irrational because it goes on forever. Fix: it repeats (3 repeating), so it equals 1/3 โ that makes it rational.
- Mistake: thinking โ9 is irrational because it has a root sign. Fix: โ9 = 3, a plain integer โ always simplify first.
- Mistake: assuming every fraction is less than 1. Fix: 7/3 is a fraction and it is greater than 2.
- Mistake: thinking natural numbers include 0. Fix: natural numbers start at 1. Whole numbers start at 0.
Tips and Tricks
- A quick test for rational: can you write it as a fraction? If yes, it is rational.
- Terminating decimals always end (0.5, 0.75, 0.125) โ always rational.
- Repeating decimals have a block of digits that cycle forever (0.142857142857โฆ) โ always rational.
- Non-repeating, non-terminating decimals (like ฯ or โ2) are always irrational.
- Every integer n is rational because it can be written as n/1.