Number systems study guide

A reliable classification method

When given a number, begin with the definition rather than its appearance. Ask whether it belongs to ℕ, ℤ, ℚ, or the irrational numbers, and then consider its relationship with ℝ.

1. Check for natural-number membership. A value in ℕ is one of 0, 1, 2, 3, …. These numbers are appropriate when counting.
2. Check for integer membership. An integer may be negative, zero or positive, as shown by ℤ = {…, -3, -2, -1, 0, 1, 2, 3, …}.
3. Check for rational membership. If the value can be written as a fraction of integers, it is rational. For example, 7 is rational because 7 = 7/1. This is why the set of rational numbers includes the integers.
4. Check for irrationality. If the value cannot be written as a fraction, it is irrational. The specification gives √2 as an example.
5. Use the real-number category correctly. Real numbers include natural, rational and irrational numbers, so ℝ is the broad set for possible real-world quantities.

Worked reasoning

Consider the statement: “Seven is not rational because it is written as a whole number.” This is incorrect. The written form does not prevent a number from being rational: 7 can be rewritten as 7/1, which is a fraction made from integers. Therefore, 7 is rational. It is also a natural number and an integer, and consequently belongs to the real numbers.

Consider the statement: “An ordered list has four objects, so the objects should be labelled with counting numbers only.” This confuses counting with position. Natural numbers can describe how many objects there are, whereas ordinal numbers describe where each object occurs. In S = {‘a’, ‘b’, ‘c’, ‘d’}, ‘a’ is 1st and ‘b’ is 2nd. The labels express positions in the order.

Counting versus measurement

Use natural numbers for counting discrete objects or items. Use real numbers for measurement, because real numbers include the possible real-world quantities identified by ℝ. In an exam response, explain the purpose of the number: counting asks “how many?”, while measurement describes a quantity. Keep this distinction separate from ordinal use, which asks “which position?”.

Self-check questions

• Can you write ℕ exactly as specified, including zero?
• Can you explain why every integer is rational using the example 7 = 7/1?
• Can you state the defining difference between a rational and an irrational number?
• Can you identify the role of ordinal numbers in the ordered set {‘a’, ‘b’, ‘c’, ‘d’}?
• Can you justify why natural numbers suit counting and real numbers suit measurement without confusing either use with ordinal position?

For each answer, use the relevant definition and give a short justification rather than relying only on a label.