Numbers and the Real Number System
FreeGrade 9
1. Classifying Numbers in the Real Number System
The real number system includes several types of numbers. Natural numbers are counting numbers (1, 2, 3 ...). Whole numbers include 0 and the natural numbers. Integers include negative numbers too. Rational numbers can be written as a fraction (including all integers, fractions, and terminating or repeating decimals). Irrational numbers cannot be written as exact fractions — their decimals go on forever without repeating, like π or √2.
Key terms
Natural Numbers
Counting numbers starting from 1: {1, 2, 3, 4, ...}
Whole Numbers
Natural numbers plus zero: {0, 1, 2, 3, ...}
Integers
All whole numbers and their negatives: {..., −2, −1, 0, 1, 2, ...}
Rational Numbers
Any number that can be written as a fraction pq where q ≠ 0. Includes all integers, fractions, and terminating or repeating decimals.
Irrational Numbers
Numbers whose decimal expansion is non-terminating and non-repeating. Cannot be written as a fraction. Examples: π, √2, √3.
Remember: the sets are nested
Every natural number is also a whole number, an integer, and a rational number. The irrational numbers are separate — they are real numbers that are not rational.
Worked Examples
- 1−7 is a negative whole number, so it is an integer.
- 2It is also rational, since it can be written as the fraction −71.
- 3It is not a natural number or a whole number, since those sets exclude negatives.
- 4Classification: integer and rational number ✓
- 1Calculate: √16 = 4 (because 4 × 4 = 16).
- 24 is a natural number, a whole number, an integer, and therefore also rational.
- 3Sipho is incorrect. Not all square roots are irrational — only square roots of non-perfect squares are irrational. ✓
Diagram
Nested diagram showing natural numbers inside whole numbers, inside integers, inside rational numbers, with irrational numbers as a separate set, all inside the real numbers
2. Identifying Rational and Irrational Numbers
To determine if a number is rational, check if it can be written as a fraction, or if its decimal terminates or repeats in a pattern. Irrational numbers have decimals that go on forever with no repeating pattern. Square roots of non-perfect squares (like √2, √3, √5) are irrational.
How to test a number
Terminates — the decimal ends after a finite number of digits (e.g. 0,75 = 34). Rational.
Repeats — the decimal has a block of digits that repeats indefinitely (e.g. 0,333... = 13). Rational.
Non-repeating and non-terminating — the decimal goes on forever with no pattern (e.g. π, √2). Irrational.
Square roots: the key question
Is the number under the square root sign a perfect square? If yes, the result is rational. If no, the result is irrational. Perfect squares: 1, 4, 9, 16, 25, 36, 49, 64, 81, 100, ...
Turning a repeating decimal into a fraction
Any repeating decimal can be written as a fraction, which is the proof that it is rational:
Write 0,272727… as a common fraction.
Let x = 0,272727…
Two digits repeat, so multiply by 100: 100x = 27,272727…
Subtract: 100x − x = 27, so 99x = 27
x = 2799 = 311
Estimating an irrational number
Even though √11 cannot be written exactly, you can trap it between two whole numbers: 3² = 9 and 4² = 16, so √11 lies between 3 and 4, and closer to 3 since 11 is nearer 9. Being able to place a surd between consecutive integers — and on a number line — is a Grade 9 skill that comes straight back in Grade 10 surds and in Pythagoras answers left in surd form.
Simplify before you decide
√(1625) looks irrational but equals 45, which is rational. ³√27 = 3. And 227 is rational, even though it is often used as an approximation for the irrational π — the two are not the same number.
Common mistake
Deciding from a calculator display. A calculator shows √2 as 1,414213562 and stops — that does not mean the decimal terminates. It also rounds 13 to 0,3333333333. Decide from the definition: can it be written as a fraction of two integers?
Worked Examples
- 1The decimal 0,333... has a repeating pattern — the digit 3 repeats indefinitely.
- 2A repeating decimal can always be expressed as a fraction: 0,333... = 13.
- 3Since it can be written as a fraction, it is rational. ✓
- 13,14 is only an approximation of π, used for convenience in calculations.
- 2The actual decimal value of π continues forever without any repeating pattern: 3,14159265358979...
- 3Because π is non-terminating and non-repeating, it cannot be written as an exact fraction. It is irrational.
- 4Lerato is incorrect. ✓
3. Properties and Operations Across the Real Number System
We apply our knowledge of number types when performing calculations, recognising that operations between certain number types produce predictable results — for example, the sum of two rational numbers is always rational, but combinations involving irrational numbers can be irrational or sometimes rational.
Rules for operations
Rational + Rational
Always rational. Example: 12 + 13 = 56.
Rational + Irrational
Always irrational. Example: 3 + √2 is irrational.
Irrational + Irrational
Usually irrational, but can be rational. Example: √5 − √5 = 0.
Irrational × Irrational
Can be rational or irrational. Example: √3 × √3 = 3 (rational); √2 × √3 = √6 (irrational).
Always simplify first
Before classifying the result, simplify the expression completely. A square root that simplifies to a whole number is rational, even if it did not look rational at first.
Why rational plus irrational is always irrational
Suppose 3 + √2 were rational. Then subtracting the rational number 3 from it would leave a rational number — but that leaves √2, which is irrational. The assumption leads to a contradiction, so 3 + √2 cannot be rational. The same argument works for any rational added to any irrational, which is why that row of the table has no exceptions, while the irrational-with-irrational rows do.
Classify each result as rational or irrational.
√8 × √2 = √16 = 4 → rational
√8 + √2 = 2√2 + √2 = 3√2 → irrational
(2 + √3) + (2 − √3) = 4 → rational
π ÷ 2 → irrational — dividing an irrational by a non-zero rational never fixes it
The properties you are allowed to use
Commutative
a + b = b + a and ab = ba. Not true for − or ÷.
Associative
(a + b) + c = a + (b + c). Regroup freely when adding or multiplying.
Distributive
a(b + c) = ab + ac. The rule behind every expansion in algebra.
Common mistake
Writing √a + √b = √(a + b). It is false: √9 + √16 = 3 + 4 = 7, but √25 = 5. Square roots distribute over multiplication and division, never over addition and subtraction.
Worked Examples
- 1Identify the types: 3 is a rational number. √2 is an irrational number (2 is not a perfect square).
- 2Apply the rule: Adding a rational number to an irrational number always gives an irrational result.
- 3The non-repeating decimal pattern of √2 is preserved — no amount of adding a rational number can make it terminate or repeat.
- 4Conclusion: √2 + 3 is irrational ✓
- 1Calculate: √3 × √3 = (√3)² = 3.
- 2Classify: 3 is a whole number, therefore rational.
- 3Multiplying an irrational number by itself (squaring a square root) cancels the radical and produces a rational result.
- 4Conclusion: √3 × √3 = 3, which is rational ✓