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Number systems revision notes
Study Number systems with curriculum-aligned Revision Notes resources, practice links, and exam-focused support.
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Number systems
Revision notes
Number Systems: Sets, Positions, Counting and Measurement
Core number sets
- Natural numbers, ℕ: the set of natural numbers includes zero: ℕ = {0, 1, 2, 3, …}. Natural numbers are used for counting.
- Integers, ℤ: the set of integers includes negative whole numbers, zero and positive whole numbers: ℤ = {…, -3, -2, -1, 0, 1, 2, 3, …}.
- Rational numbers, ℚ: a rational number can be written as a fraction, or ratio, of integers. Every integer is rational because it can be written over 1; for example, 7 = 7/1.
- Irrational numbers: an irrational number cannot be written as a fraction. √2 is an example.
- Real numbers, ℝ: the real numbers include the natural numbers, rational numbers and irrational numbers. They represent all possible real-world quantities.
Relationships and distinctions
Natural numbers are a particular group of numbers used for counting. Integers extend this idea to include negative values. Rational numbers include the integers, because each integer can be expressed as a fraction. Irrational numbers are different from rational numbers because they cannot be expressed as a fraction. Real numbers include both rational and irrational numbers, as well as the natural numbers.
Ordinal numbers
Ordinal numbers describe the numerical positions of objects in an ordered collection. If S = {‘a’, ‘b’, ‘c’, ‘d’} is well ordered, then ‘a’ is first, or 1st, ‘b’ is 2nd, ‘c’ is 3rd and ‘d’ is 4th. Ordinal numbers therefore describe position rather than the quantity of objects.
Common errors
- Do not confuse natural and ordinal numbers: natural numbers are used for counting, while ordinal numbers identify positions in an order.
- Do not claim that all numbers are integers. Rational numbers also include values that are written as fractions, and irrational numbers cannot be written as fractions.
- Do not exclude zero from ℕ in this specification: zero is explicitly included.
- Do not describe √2 as rational; it is an irrational number because it cannot be written as a fraction.
- Do not restrict real numbers to integers or rational numbers: ℝ also includes irrational numbers.
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