What this chapter is about
This chapter explores the vast landscape of numbers that you use in mathematics. Starting from the counting numbers you learned as a child, it builds up to the complete system of real numbers. You will understand how natural numbers, whole numbers, integers, rational numbers and irrational numbers relate to one another, and why each type was needed as mathematics grew.
By the end of this chapter, you should be able to classify any given number into its correct type, represent numbers on the number line, and understand what makes a number rational or irrational. You will also learn how all these number types together form the real number system, which is the foundation for all the algebra and geometry you will study this year and beyond.
Understanding this number system matters because every measurement, every calculation, and every formula you encounter uses these numbers. Whether you are finding the area of a circle or solving an equation, you are working within this world of numbers.
Key ideas
- Natural numbers are the counting numbers 1, 2, 3, 4, … that we use to count objects. They do not include zero or negative numbers.
- Whole numbers are natural numbers together with zero: 0, 1, 2, 3, 4, … They allow us to represent "nothing" as a quantity.
- Integers include all whole numbers and their negatives: …, −3, −2, −1, 0, 1, 2, 3, … They let us represent quantities below zero, like temperatures or debts.
- Rational numbers are numbers that can be written as p/q, where p and q are integers and q is not zero. Examples include 3/4, −2/5, 7 (which is 7/1), and 0.25 (which is 1/4).
- Irrational numbers cannot be expressed as p/q for any integers p and q. Their decimal expansions go on forever without repeating. Examples include √2, √3, and π.
- Real numbers comprise all rational and irrational numbers together. Every point on the number line corresponds to a real number, and every real number corresponds to a point on the number line.
- The decimal expansion of a rational number either terminates (like 0.75) or repeats in a pattern (like 0.333… or 0.142857142857…).
- Every natural number is a whole number, every whole number is an integer, every integer is a rational number, and every rational number is a real number. This forms a nested hierarchy.
Formulas and facts to remember
- A number is rational if and only if it can be written as p/q where p, q are integers and q ≠ 0.
- Terminating decimals have a finite number of digits after the decimal point; they are always rational.
- Repeating decimals have a block of digits that repeats forever; they are also rational.
- Non-terminating, non-repeating decimals are irrational.
- √2, √3, √5, √7 and the square root of any prime number are irrational.
- The sum or difference of a rational and an irrational number is always irrational.
- The product or quotient of a non-zero rational number and an irrational number is irrational.
Worked examples
Example 1: Classifying a number
Classify the number −7/3.
Step 1: Check if it can be written as p/q with integers p and q, and q ≠ 0. Here p = −7 and q = 3, both integers, and 3 ≠ 0.
Step 2: Since it fits the form p/q, it is a rational number.
Step 3: Is it an integer? Dividing −7 by 3 gives −2.333…, which is not a whole number. So −7/3 is rational but not an integer.
Example 2: Showing that √5 is irrational
We prove this by contradiction.
Step 1: Assume √5 is rational, so √5 = p/q where p and q are integers with no common factor and q ≠ 0.
Step 2: Squaring both sides: 5 = p²/q², which gives p² = 5q².
Step 3: This means p² is divisible by 5. Since 5 is prime, p itself must be divisible by 5. Write p = 5m for some integer m.
Step 4: Substituting: (5m)² = 5q², so 25m² = 5q², giving q² = 5m².
Step 5: Now q² is divisible by 5, so q is divisible by 5.
Step 6: But if both p and q are divisible by 5, they share a common factor. This contradicts our assumption.
Therefore, √5 is irrational.
Example 3: Converting a repeating decimal to a fraction
Express 0.272727… as a fraction.
Step 1: Let x = 0.272727…
Step 2: The repeating block has 2 digits, so multiply by 100: 100x = 27.272727…
Step 3: Subtract the original: 100x − x = 27.272727… − 0.272727…
Step 4: This gives 99x = 27.
Step 5: Solving: x = 27/99 = 3/11.
So 0.272727… = 3/11, which is rational.
Common mistakes
- Thinking zero is not a whole number → Zero is included in whole numbers; it represents the absence of quantity.
- Believing all square roots are irrational → Square roots of perfect squares like √16 = 4 or √25 = 5 are rational.
- Writing 22/7 is exactly equal to π → 22/7 is only an approximation; π is irrational and equals 3.14159265…
- Assuming a long decimal must be irrational → Check whether the digits eventually repeat; 0.166666… is rational (1/6).
- Forgetting that integers are also rational → Every integer n can be written as n/1, making it rational.
Quick revision
- Natural ⊂ Whole ⊂ Integers ⊂ Rational ⊂ Real; Irrational numbers fill the gaps.
- Rational means expressible as p/q with integers p, q and q ≠ 0.
- Terminating or repeating decimal → rational; non-terminating non-repeating → irrational.
- √(prime) is always irrational.
- Every point on the number line is a real number.