Build the whole family of number sets one step at a time: natural, whole, integers, rational, irrational and real numbers. Learn how to tell them apart, why √2 is irrational, and how prime, composite and co-prime numbers fit in.
“Number systems” is not one single thing. It is a family of sets of numbers, each one built by relaxing a rule of the set before it. Counting numbers cannot answer 3 − 5, so we add negatives. Whole numbers cannot answer 3 ÷ 4, so we add fractions. Fractions cannot measure the diagonal of a square, so we add irrational numbers. This guide builds the family one set at a time, shows how to tell the sets apart (including how to prove that √2 is irrational), and then covers the other ways we sort numbers: by sign, by even and odd, and as prime, composite or co-prime.
A set is a collection of things. A number system is a set of numbers with a shared property, and it has a name. Some sets sit completely inside bigger ones, like Russian dolls. Mathematicians use capital letters for the main sets: ℕ for natural numbers, ℤ for integers, ℚ for rational numbers and ℝ for real numbers. The symbol ∈ means “belongs to”, so 5 ∈ ℕ is read “5 belongs to the natural numbers”, and the symbol ⊂ means “is inside”.
Natural numbers are the counting numbers: 1, 2, 3, 4 and so on, without end. They are the numbers we use to count objects, and you would not say “I have zero apples” while counting a pile.
A note on conventions. This guide, like most textbooks in Pakistan and India, starts the natural numbers at 1. Some books and some computer languages start them at 0 instead. If an exam question depends on this, follow the definition in your own course book.
1, 2, 3, 50 and 1,000,000 are natural numbers. Zero, −4, 1/2 and 3.7 are not.
Whole numbers are the natural numbers together with zero: 0, 1, 2, 3 and so on. Every natural number is a whole number, but zero is a new member.
This depends on which convention your course follows.
The fix: in the convention used here, natural numbers start at 1, and zero is the extra member that turns them into whole numbers. So 0 is a whole number but not a natural number. Always check which definition your syllabus uses, and state it if the question allows.
Integers are the whole numbers and their negatives: … −3, −2, −1, 0, 1, 2, 3 … An integer never has a fraction or decimal part. We needed integers to answer questions like 3 − 5, which has no answer among the whole numbers.
−100, −7, 0, 4 and 1,000 are integers. The numbers −2.5 and 3/4 are not integers.
A rational number is any number that can be written as a fraction p/q, where p and q are integers and q is not zero. Every integer is rational, because n = n/1. Rational numbers include all fractions, all terminating decimals and all recurring decimals.
When you divide the top of a fraction by the bottom, the decimal either terminates (stops) or recurs (a block of digits repeats for ever). Both are rational. For example 3/8 = 0.375 stops, but 1/3 = 0.333… and 5/12 = 0.41666… recur.
You can tell in advance. Write the fraction in lowest terms. If the denominator has no prime factors except 2 and 5, the decimal terminates. Otherwise it recurs.
| Fraction | Denominator in primes | Decimal |
|---|---|---|
| 3/8 | 2 × 2 × 2 | 0.375 (terminates) |
| 7/20 | 2 × 2 × 5 | 0.35 (terminates) |
| 1/3 | 3 | 0.333… (recurs) |
| 5/12 | 2 × 2 × 3 | 0.41666… (recurs) |
| 1/7 | 7 | 0.142857142857… (recurs) |
(a) Write 0.2727… as a fraction. Let x = 0.2727… Two digits repeat, so multiply by 100: 100x = 27.2727… Subtract: 100x − x = 27, so 99x = 27 and x = 27/99 = 3/11.
(b) Write 0.1666… as a fraction. Let x = 0.1666… Then 10x = 1.666… and 100x = 16.666… Subtract the second from the third: 100x − 10x = 15, so 90x = 15 and x = 1/6.
The idea is to multiply by 10, 100 or 1000 so that the repeating parts line up, then subtract to remove them.
An irrational number cannot be written as a fraction of two integers. Its decimal goes on for ever without any repeating block. Famous examples are π = 3.14159265… and √2 = 1.41421356… Roots of numbers that are not perfect squares, such as √3, √5 and √20, are irrational too.
(a) √16 = 4, which is a whole number, so it is rational.
(b) √20: 20 is not a perfect square (4² = 16 and 5² = 25), so √20 is irrational.
(c) 22/7 is a fraction, so it is rational. It is only an approximation to π: 22/7 = 3.142857… while π = 3.141593…
(d) 3.14 is a terminating decimal, so it is rational. It is a rounded value of π, not π itself.
Some square roots simplify.
The fix: √9 = 3 exactly, and 3 is rational. A root is irrational only when the number under it is not a perfect square (not 1, 4, 9, 16, 25, 36 …). Always try to simplify first. Similarly, “π is 3.14” and “π is 22/7” are both wrong: they are rational approximations of an irrational number.
The ancient Greeks asked whether the diagonal of a unit square could be written as a fraction. Here is the classic argument, a proof by contradiction.
The starting assumption must be false, so √2 cannot be written as a fraction. It is irrational.
The real numbers are all the rational numbers and all the irrational numbers together. Every real number has exactly one place on the number line, and every point on the line is a real number. The word “real” is a technical name. It does not mean that negative numbers or irrational numbers are somehow “not real”.
Irrational numbers are not just decimals you cannot finish writing. You can place some exactly using geometry. A right-angled triangle with two sides of length 1 has a long side of length √2, by Pythagoras: 1² + 1² = 2. Swing that length onto the number line with a compass and you have marked √2 exactly.
Does “real” mean the same as in everyday English?
The fix: a real number is any number with a position on the number line. −7 is just as real as 7, because it has a place to the left of zero. (Numbers that are not real exist in higher mathematics, but you will not meet them at this level.)
Each new set was invented because the old one could not answer a question. We can see this by asking which operations never leave the set. A set is closed under an operation if the answer is always inside the set.
| Set | Add | Subtract | Multiply | Divide (not by 0) |
|---|---|---|---|---|
| Natural | Yes | No (3 − 5) | Yes | No (3 ÷ 4) |
| Integers | Yes | Yes | Yes | No (3 ÷ 4) |
| Rational | Yes | Yes | Yes | Yes |
| Irrational | No | No | No | No |
The irrational row shows “No” because, for example, √2 + (−√2) = 0 and √2 × √2 = 2 are not irrational. The whole numbers behave like the natural numbers. The real numbers are closed under all four operations like the rationals, and they also fill every gap on the number line.
A number is positive if it is greater than 0 and negative if it is less than 0. Zero is neither: it is the dividing point of the number line.
An integer is even if it is a multiple of 2 and odd if it is not. So 0, 2, −6 and 100 are even, and 1, 3, −5 and 99 are odd. Zero is even, because 0 ÷ 2 = 0 exactly. Even and odd apply only to integers: 2.5 is neither.
A prime number has exactly two factors: 1 and itself. A composite number has more than two factors. The number 1 has only one factor, so it is neither prime nor composite. The number 2 is the only even prime.
| Number | Factors | Type |
|---|---|---|
| 1 | 1 | Neither |
| 2 | 1, 2 | Prime (the only even prime) |
| 9 | 1, 3, 9 | Composite |
| 17 | 1, 17 | Prime |
| 91 | 1, 7, 13, 91 | Composite (7 × 13) |
Every composite number can be written as a product of primes in exactly one way (apart from the order). This is called its prime factorisation. For example, 84 = 2 × 2 × 3 × 7 = 2² × 3 × 7. To find it, divide by the smallest prime that fits, again and again, until you reach 1.
Count its factors.
The fix: 1 has just one factor (itself), but a prime needs exactly two. So 1 is neither prime nor composite. One reason this matters: if 1 were prime, the prime factorisation of 84 could be written as 2² × 3 × 7 or 1 × 2² × 3 × 7 or 1 × 1 × 2² × 3 × 7, so it would no longer be unique.
Co-prime is a property of a pair of numbers: their only common factor is 1, so their HCF is 1. For example, 8 and 15 are co-prime (8 = 2³ and 15 = 3 × 5 share nothing), although neither is prime. The numbers 6 and 9 are not co-prime, since both are multiples of 3. For co-prime numbers the LCM is just the product: the LCM of 8 and 15 is 120.
The name is misleading.
The fix: only the relationship matters. 8 and 15 are co-prime even though neither is prime, while 3 and 3 (both prime) are not co-prime, since they share the factor 3. To test a pair, find their HCF: if it is 1, they are co-prime.
| Number | Natural | Whole | Integer | Rational | Irrational |
|---|---|---|---|---|---|
| 7 | Yes | Yes | Yes | Yes | No |
| 0 | No | Yes | Yes | Yes | No |
| −4 | No | No | Yes | Yes | No |
| 3/4 | No | No | No | Yes | No |
| 0.333… | No | No | No | Yes | No |
| √16 | Yes | Yes | Yes | Yes | No |
| √20 | No | No | No | No | Yes |
| π | No | No | No | No | Yes |
All eight numbers in the table are real. To classify any number, work through this order: simplify it first (for example √16 = 4), then ask whether it is a counting number, a whole number, an integer, a fraction, or neither.
Type a number, or tap one below. You can use a whole number, a fraction such as 3/4, a decimal such as 0.75, a root such as √20 (or sqrt(20)), or pi. The explorer shows every set the number belongs to and why.
Tap a number above to get started.
Enter a top and a bottom number. The checker divides, shows whether the decimal stops or repeats, and finds the repeating block.
Tap the correct answer for each question. Your best score is saved on this device.
1. It terminates, so it is rational. 0.45 = 45/100 = 9/20.
2. 84 = 2² × 3 × 7 and 90 = 2 × 3² × 5. They share the factors 2 and 3 (HCF = 6), so they are not co-prime.
3. √2 + √2 = 2√2 is irrational. √2 × √2 = 2, which is rational. This shows that operations on irrational numbers can give either kind.
1. −7: integer and rational. 0: whole, integer and rational. 2/3: rational only. √16 = 4: natural, whole, integer and rational. √20: irrational only (20 is not a perfect square).
2. Let x = 0.4545… Then 100x = 45.4545…, so 99x = 45 and x = 45/99 = 5/11.
3. 9/25: 25 = 5², so it terminates, and 9/25 = 0.36. 5/12: 12 = 2² × 3 contains a 3, so it does not terminate. 7/40: 40 = 2³ × 5, so it terminates, and 7/40 = 0.175.
4. 360 ÷ 2 = 180, ÷ 2 = 90, ÷ 2 = 45, ÷ 3 = 15, ÷ 3 = 5, ÷ 5 = 1. So 360 = 2³ × 3² × 5. Check: 8 × 9 × 5 = 360.
5. (a) 18 = 2 × 3² and 35 = 5 × 7 share no prime factor, so their HCF is 1 and they are co-prime. (b) 18 = 2 × 3² and 24 = 2³ × 3, so the HCF is 2 × 3 = 6.
6. (a) √2 × √8 = √16 = 4, which is rational. (b) A rational number plus an irrational number is irrational, so 3 + √2 is irrational.
7. (a) 23 and 29. (b) For example √2 + (−√2) = 0, or (3 + √2) + (3 − √2) = 6, which is rational.
Learners usually memorise the names of the sets but not the reason for them. Start from the questions each set could not answer: 3 − 5, 3 ÷ 4, and the diagonal of a square. This gives the nesting a purpose. Ask learners to place a few numbers on a large number line on the floor, and to explain why each one belongs. The Number Explorer on this page lets them test a number and see every set it belongs to, along with the reason. Be explicit that definitions of the natural numbers vary between books. The reasoning for the irrationality of √2 is a good first taste of proof, and it is worth walking through slowly. For decimals, the Fraction Decimal Checker shows the recurring block, which makes the idea of a repeating pattern concrete.
For historical background, the MacTutor History of Mathematics archive at the University of St Andrews is a trusted free source. Check the number topics in your own examination syllabus and past papers, because definitions (for example whether 0 is natural) can differ between boards. This article is original EDUSAMBAM writing.
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