The Number Family

Numbers belong to families, and one number can belong to several families at the same time. This guide from Quantum Leap Star Limited introduces the main number types in clear language, with examples for learners from 11+ through GCSE and into A-level study.

The main number sets

Curly brackets { } enclose a set. The symbol ∈ means “belongs to”, ∉ means “does not belong to”, and … means the pattern continues.

1. Natural numbers: ℕ

The counting numbers, starting at one.

ℕ = {1, 2, 3, 4, 5, …}

Some books include zero in ℕ. Here, we use the convention starting at 1.

2. Whole numbers

Zero and all the counting numbers.

Whole numbers = {0, 1, 2, 3, 4, …}

Whole numbers have no universally agreed symbol.

3. Integers: ℤ

Whole numbers and their negatives.

ℤ = {…, −3, −2, −1, 0, 1, 2, 3, …}

4. Rational numbers: ℚ

Numbers that can be written as one integer divided by another non-zero integer.

ℚ = {p/q : p, q ∈ ℤ, q ≠ 0}

Examples include −3, 0, 1/4, 7/13 and 0.75. Rational decimals end or eventually repeat a fixed block of digits. All integers are rational: for example, 4 = 4/1.

5. Irrational numbers: ℝ ∖ ℚ

Real numbers that cannot be written as a ratio of two integers. Their decimals never end and never eventually repeat.

ℝ ∖ ℚ = {x ∈ ℝ : x ∉ ℚ}

Examples include √2, √3, ∛2, ∛3, π and e. These are examples, not a complete list of irrational numbers.

6. Real numbers: ℝ

All numbers on the number line, including both rational and irrational numbers.

ℝ = {x : x is rational or irrational}

Examples include −5, 0, 1/2, 7/13, √2 and π.

Rationals and irrationals are both subsets of the reals. Neither contains the other, and they have no members in common.

Decimals: ending, repeating and non-repeating

A decimal that ends

1/4 = 0.25

The decimal ends, so the number is rational. Another example is 3/4 = 0.75.

A decimal with a repeating block

7/13 = 0.538461 538461 538461 …

The spaces highlight the six-digit block 538461. This block repeats forever, so 7/13 is rational.

Decimals that never eventually repeat

√2 = 1.414213562373095…
√3 = 1.732050807568877…
π = 3.141592653589793…

All three decimals continue forever without settling into a repeating block. All three numbers are irrational. Individual digits can still appear many times.

A short calculator display alone cannot prove that a number is irrational. The non-repeating properties of √2, √3 and π are mathematical facts, not guesses from the digits shown.

Both types are real: on the number line, 1/2 lies halfway between 0 and 1; √2 lies between 1 and 2; and π lies between 3 and 4. Decimal repetition distinguishes rational from irrational numbers, not real from complex numbers.

Special number families

7. Even numbers

Integers that divide exactly by two.

Even numbers = {…, −4, −2, 0, 2, 4, …}

Zero is even because 0 ÷ 2 = 0.

8. Odd numbers

Integers that do not divide exactly by two.

Odd numbers = {…, −3, −1, 1, 3, 5, …}

9. Prime numbers

Positive integers greater than 1 that divide exactly only by 1 and themselves. They have exactly two distinct positive factors.

Prime numbers = {2, 3, 5, 7, 11, 13, 17, …}

1 is NOT prime. It has only one positive factor, 1, not two. Saying “1 and itself” still names the same factor twice. The number 2 is the only even prime.

10. Composite numbers

Positive integers greater than one with more than two positive factors.

Composite numbers = {4, 6, 8, 9, 10, 12, 14, 15, …}

For example, 6 has positive factors 1, 2, 3 and 6. 1 is neither prime nor composite.

11. Square numbers

Numbers made by multiplying an integer by itself: n × n = n².

Square numbers = {0, 1, 4, 9, 16, 25, 36, …}

For example, 5² = 25.

12. Cube numbers

Numbers made by multiplying three copies of an integer: n × n × n = n³.

Cube numbers = {…, −27, −8, −1, 0, 1, 8, 27, …}

For example, 3³ = 27 and (−2)³ = −8.

13. Triangular numbers

Dot counts in a triangle with rows of 1, 2, 3 and so on.

Triangular numbers = {1, 3, 6, 10, 15, 21, …}

The nth triangular number is n(n + 1)/2, for n = 1, 2, 3, … . For example, the fourth is 4 × 5 ÷ 2 = 10.

14. Fibonacci numbers

Starting with 0 and 1, each next term is the sum of the previous two.

Sequence: 0, 1, 1, 2, 3, 5, 8, 13, …

Sometimes the sequence starts at 1, 1 instead. We show it as a sequence to preserve its order and the repeated 1.

15. Star numbers: enrichment

Dot counts in a growing, regular six-pointed star pattern.

Star numbers = {1, 13, 37, 73, 121, 181, …}

Enrichment means extra exploration: here, star numbers extend our study of number patterns rather than form a core list to memorise.

16. Complex numbers: a brief introduction

Complex numbers have the form a + bi, where a and b are real and i² = −1.

ℂ = {a + bi : a, b ∈ ℝ, i² = −1}

Examples include 2 + 3i, −i and 4 = 4 + 0i. Real numbers are included when b = 0.

So 4 belongs to all these groups at once: natural, whole, integer, even, composite, square, rational, real and complex. It does not belong to every group: for example, it is neither odd nor prime nor irrational.

How the sets fit together

ℕ ⊂ ℤ ⊂ ℚ ⊂ ℝ ⊂ ℂ

The symbol ⊂ means “is a proper subset of”. Every set shown is contained in the next. Whole numbers sit between the natural numbers and the integers under our convention.

Check your understanding

Try these eight questions before checking the answers below.

1. Classify a number. List every group in this lesson that contains 4.
2. Is 1 prime? A student says: “1 is prime because it divides by 1 and itself.” Explain why this is incorrect.
3. Prime or composite? Classify 1, 2, 9, 13 and 21 as prime, composite or neither.
4. Rational or irrational? Classify 0.25, 7/13, √2, √9, π and −3. Which are real?
5. Decimal patterns. Write the repeating block in 7/13 = 0.538461538461… . Explain how its decimal differs from √3.
6. Continue the sequences. Write the next two terms:
   (a) 1, 4, 9, 16, …
   (b) 1, 8, 27, 64, …
   (c) 1, 3, 6, 10, …
   (d) 0, 1, 1, 2, 3, 5, …
7. Triangular numbers. Use n(n + 1)/2 to find the 8th triangular number. Show your calculation.
8. True or false? Correct any false statement.
   (a) Zero is even.
   (b) Every real number is rational.
   (c) A number can be both rational and irrational.
   (d) 4 is complex because 4 = 4 + 0i.

Original explanations and practice questions prepared for Quantum Leap Star Limited.