Types of number
Section: Number 1 | Syllabus: Cambridge IGCSE Mathematics (0580)
Types of Number
Numbers can be sorted into groups depending on their properties. Recognising which group a number belongs to is the first step in most number problems.
- Natural numbers are the positive counting numbers: 1, 2, 3, 4, ...
- Integers are positive whole numbers, zero and negative whole numbers: ..., -2, -1, 0, 1, 2, ...
- A prime number has exactly two factors: 1 and itself
- 1 is not a prime number, because it has only one factor
- 2 is the smallest prime number, and the only even prime number
- A square number is the result of multiplying a whole number by itself: n × n
- A cube number is the result of multiplying a whole number by itself three times: n × n × n
- Common factors of two or more numbers are the factors they share
- Common multiples of two or more numbers are the multiples they share
Worked Example: Identifying Number Types from a List
- Question: From the list 4, 11, 18, 25, 32, 49, 64, write down (i) the prime numbers, (ii) the square numbers, (iii) a cube number.
- Step 1: Check each number for factors other than 1 and itself - only 11 has no other factors, so 11 is prime
- Step 2: Look for numbers that are n × n: 4 = 2², 25 = 5², 49 = 7², 64 = 8² are all square numbers
- Step 3: Look for a number that is n × n × n: 64 = 4³, so 64 is also a cube number
- Answer: prime: 11; squares: 4, 25, 49, 64; cube: 64
Common Mistakes
MistakeBelieving that 1 is a prime number
Fixa prime number must have exactly two factors; 1 has only one factor (itself), so it is not prime - the smallest prime is 2
MistakeThinking that 0 and negative numbers count as natural numbers
Fixnatural numbers are the positive counting numbers only; integers are the wider group that also includes 0 and negative whole numbers
Prime Factors, HCF and LCM
Every whole number greater than 1 can be broken down into a product of prime numbers. Once a number is written this way, its factors and multiples become much easier to work out.
- Writing a number as a product of its prime factors means expressing it as prime numbers multiplied together, usually using index notation for repeated factors
- The highest common factor (HCF) of two or more numbers is the largest number that divides into all of them exactly
- Found by multiplying the lowest power of each prime factor common to every number
- The lowest common multiple (LCM) of two or more numbers is the smallest number that all of them divide into exactly
- Found by multiplying the highest power of every prime factor that appears in any of the numbers
Branching stops once every number at the end of a branch is prime
Worked Example: Writing a Number as a Product of Prime Factors
- Question: Express 60 as a product of its prime factors, giving your answer in index form.
- Step 1: Divide repeatedly by the smallest prime that fits: 60 ÷ 2 = 30, 30 ÷ 2 = 15, 15 ÷ 3 = 5, 5 ÷ 5 = 1
- Step 2: Collect the primes used: 2, 2, 3, 5
- Answer: 60 = 2^2 × 3 × 5
Worked Example: Finding the HCF and LCM
- Question: Find the HCF and LCM of 60 and 84.
- Step 1: Write each number as a product of primes: 60 = 2^2 × 3 × 5, 84 = 2^2 × 3 × 7
- Step 2: For the HCF, multiply the lowest power of each common prime (2 and 3): HCF = 2^2 × 3 = 12
- Step 3: For the LCM, multiply the highest power of every prime appearing in either number: LCM = 2^2 × 3 × 5 × 7 = 420
- Answer: HCF = 12, LCM = 420
Common Mistakes
MistakeUsing the highest power of each prime for the HCF, mixing up the HCF and LCM rules
Fixthe HCF uses the lowest shared power of only the common primes; the LCM uses the highest power of every prime appearing in either number
MistakeStopping a factor tree before every end branch is prime, e.g. leaving 60 = 2 × 30
Fixkeep branching until every number at the end of a branch is a prime number
Rational Numbers, Irrational Numbers and Reciprocals
Every number is either rational or irrational, depending on whether it can be written exactly as a fraction. Every nonzero number also has a reciprocal, found by dividing 1 by that number.
- A rational number can be written as a fraction (a/b) where a and b are integers and b ≠ 0
- This includes all integers, all terminating decimals, and all recurring decimals
- An irrational number cannot be written as an exact fraction
- Its decimal expansion never terminates and never repeats, e.g. √10, √30, π
- The reciprocal of a number is 1 divided by that number
- For a fraction, the reciprocal is found by swapping the numerator and denominator
- Multiplying a number by its reciprocal always gives 1
- Zero has no reciprocal, because dividing by zero is undefined
Worked Example: Classifying Numbers as Rational or Irrational
- 0.75 is rational: it equals 3/4
- √16 is rational: it equals 4, a whole number, even though it involves a root sign
- √10 is irrational: 10 is not a square number, so its root cannot be written as an exact fraction
- 0.454545... is rational: a recurring decimal can always be written as a fraction
Worked Example: Finding Reciprocals
- The reciprocal of 5 is 1/5
- The reciprocal of 3/8 is 8/3
- The reciprocal of 0.2 is 1 ÷ 0.2 = 5
- Checking: multiplying a number by its reciprocal always gives 1, e.g. 5 × 1/5 = 1
Common Mistakes
MistakeAssuming any number with a root sign is irrational, e.g. saying √16 is irrational
Fixcheck whether the number under the root is a square number first - √16 = 4 is rational; only roots of non-square numbers are irrational
MistakeFinding the reciprocal of a mixed number by only flipping the whole-number part
Fixconvert to an improper fraction first, then flip it: the reciprocal of 2 1/4 is the reciprocal of 9/4, which is 4/9
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