Types Of Numbers Quizzes
Visual overview of Types Of Numbers.
Introduction
Knowing how numbers are classified (natural, whole, integers, prime, composite, rational, irrational, and parity) underpins GCSE Maths. It guides factorisation, HCF/LCM, fractions/decimals/percentages, and algebra.
Example: recognising that 7 is prime while 12 is composite immediately suggests different factor strategies.
Core Concepts
Natural Numbers
Counting numbers starting at 1: \(1,2,3,4,5,\ldots\)
Used for counting and ordering.
Whole Numbers
Natural numbers including zero: \(0,1,2,3,4,5,\ldots\)
Integers
All whole numbers and their negatives: \(\ldots,-3,-2,-1,0,1,2,3,\ldots\)
Model gains/losses, temperature, elevation, and direction.
Prime Numbers
Integers \(>1\) with exactly two factors (1 and itself).
- Examples: \(2,3,5,7,11,13,17,19,\ldots\)
- Note: 2 is the only even prime. 1 is not prime.
Composite Numbers
Integers \(>1\) with more than two factors.
- Examples: \(4,6,8,9,10,12,14,\ldots\)
Rational Numbers
Numbers that can be written as a fraction \(\tfrac{a}{b}\) with integers \(a,b\) and \(b\neq0\).
- Examples: \(\tfrac{1}{2},\;-\tfrac{3}{4},\;5,\;0.75\)
- All integers are rational: \(k=\tfrac{k}{1}\).
- Decimals that terminate or recur are rational.
Irrational Numbers
Cannot be expressed as \(\tfrac{a}{b}\) with integers. Decimal expansion is non-terminating and non-recurring.
- Examples: \(\pi \approx 3.14159\ldots\), \(\sqrt{2}\approx1.41421\ldots\)
Even and Odd Numbers
- Even: divisible by 2 (e.g. \( -8,0,2,4,6 \))
- Odd: not divisible by 2 (e.g. \( -7,1,3,5 \))
Square Numbers
Numbers of the form \(n^2\) for integer \(n\).
- Examples: \(1^2=1,\;2^2=4,\;3^2=9,\;4^2=16,\ldots\)
Cube Numbers
Numbers of the form \(n^3\) for integer \(n\).
- Examples: \(1^3=1,\;2^3=8,\;3^3=27,\;4^3=64,\ldots\)
Factors and Multiples
- Factor: divides exactly. Factors of \(12:\;1,2,3,4,6,12\).
- Multiple: obtained by multiplying by an integer. Multiples of \(4:\;4,8,12,16,\ldots\)
Worked Examples
Example 1 (Foundation): Classify
Classify \(0,\,7,\,-5,\,12,\,\tfrac12,\,\sqrt{2}\).
- \(0\) → whole, integer, rational, even
- \(7\) → natural, whole, integer, prime, rational, odd
- \(-5\) → integer, rational, odd
- \(12\) → whole, integer, composite, rational, even
- \(\tfrac12\) → rational
- \(\sqrt{2}\) → irrational
Example 2 (Foundation): Squares and Cubes
Decide whether \(9,16,27,64,81\) are square or cube numbers.
- \(9=3^2\) → square
- \(16=4^2\) → square
- \(27=3^3\) → cube
- \(64=8^2=4^3\) → square and cube
- \(81=9^2\) → square
Example 3 (Higher): Prime or Composite
Classify \(2,15,19,21,23,25\).
- \(2\) prime
- \(15=3\times5\) composite
- \(19\) prime
- \(21=3\times7\) composite
- \(23\) prime
- \(25=5\times5\) composite
Example 4 (Higher): Rational or Irrational
Classify \(4,\,-7,\,0.\overline{3},\,\pi,\,\sqrt5\).
- \(4\) rational
- \(-7\) rational
- \(0.\overline{3}=\tfrac13\) rational
- \(\pi\) irrational
- \(\sqrt5\) irrational
Example 5: Even or Odd
Classify \(-8,\,0,\,7,\,15,\,22\).
- \(-8\) even
- \(0\) even
- \(7\) odd
- \(15\) odd
- \(22\) even
Common Mistakes
- Calling \(1\) prime (it is neither prime nor composite).
- Confusing integers with natural numbers (negatives are not natural).
- Forgetting that terminating and recurring decimals are rational.
- Mixing up squares and cubes.
- Forgetting \(0\) is even.
Applications
- Fractions: simplifying using prime factors and HCF.
- Algebra: restricting solutions to integers or naturals.
- Geometry: recognising square/cube numbers in area/volume.
- Number theory: multiples, factors, divisibility tests.
Strategies & Tips
- Memorise primes to \(50\) and squares to \(20^2\); cubes to \(10^3\).
- Use factor trees to classify composites quickly.
- Check decimals: terminating or recurring → rational; non-terminating and non-recurring → irrational.
- Be systematic: apply the checklist to each number.
Summary / Call-to-Action
Number types are the grammar of maths. Mastering classifications makes factorisation, equations, and problem solving faster and more accurate.
- Practise rapid classification on mixed lists.
- Drill primes, squares, and cubes for speed.
- Apply the checklist in HCF/LCM and algebra questions.