Numbers come in families, a bit like animals come in species. Knowing which family a number belongs to tells you how it behaves. Exam questions often give you a list and ask you to pick out the primes, the squares or the irrational numbers, so you need to know each family by sight.
Explained simply
Natural numbers are the counting numbers: 1,2,3,4,… (some books include 0 as well).
Integers are whole numbers, including zero and negatives: …,−2,−1
4The reciprocal of a number is 1 divided by it, so the reciprocal of ba is ab.
,
0
,
1
,
2
,
…
A factor of a number divides into it exactly. The factors of 12 are 1, 2, 3, 4, 6 and 12.
A multiple is a number in its times table. Multiples of 7 are 7, 14, 21, ...
A prime number has exactly two factors: 1 and itself. The first primes are 2, 3, 5, 7, 11, 13. The number 1 has only one factor, so it is not prime.
Square numbers come from multiplying an integer by itself: 1,4,9,16,25,…
Cube numbers come from using it three times: 1,8,27,64,125,…
Triangle numbers are the dots in growing triangles: 1,3,6,10,15,… (add 2, then 3, then 4, ...).
A rational number can be written as one integer over another, like 43, −5=1−5 or 0.4=52.
An irrational number cannot. Its decimal goes on forever with no repeating pattern. Examples are π and square roots of non-square numbers, like 10.
The reciprocal of a number is 1÷ that number. Flip a fraction to find it: the reciprocal of 32 is 23.
You may also need to write numbers in words. Split the digits into groups of three from the right: 3405020 is "three million, four hundred and five thousand and twenty".
Worked example
Here is a list:
−3,0.4,1,32,7,9,15,10,27,π
Integers:−3,1,7,9,15,27. These have no fractional part.
Prime numbers: only 7. Check the others: 9 = 3 × 3, 15 = 3 × 5, 27 = 3 × 9, and 1 is not prime.
Square numbers:1=12 and 9=32.
Cube numbers:1=13 and 27=33.
Triangle number:15, because 1+2+3+4+5=15.
Irrational numbers:10 (10 is not a square number) and π.
Rational numbers: everything else, including 0.4=104=52.
The numbers from the worked example sorted into families: natural numbers sit inside the integers, the integers sit inside the rational numbers, and the irrational numbers √10 and π sit outside the rationals; together they make the real numbers
Reciprocal of 0.4:
0.41=410=2.5
Common mistakes
Calling 1 a prime. It has only one factor. The smallest prime is 2, and it is the only even prime.
Thinking every square root is irrational.16=4 is rational. Only roots of non-square numbers are irrational.
Calling 722 irrational. It is a fraction of integers, so it is rational, even though it is close to π.
Mixing up factors and multiples. Factors are smaller (or equal); multiples are bigger (or equal).
Where you see this in real life
Prime numbers keep your online banking safe: encryption uses the fact that multiplying two huge primes is easy but splitting the answer back into them is incredibly hard. Square and cube numbers appear whenever you work out areas and volumes, and π turns up in every wheel, pipe and pizza.