Mathematics (Core)Exam code: 0580
Types of Numbers
Number
Rational numbers
- Any value expressible as one integer over another, with a non-zero denominator, is a rational number
- Every integer is rational, because it can be written over a denominator of 1
- Every terminating decimal is rational
- 0.35 = 7/20
- Every recurring decimal is rational
- 0.333… = 1/3
Irrational numbers
- An irrational number admits no such integer-over-integer form
- Its decimal form never terminates and never settles into a repeating block
- The square root of a positive integer that is not a square number is irrational
- √7 is irrational, but √36 is not, because it equals 6
- π is irrational
- A non-zero rational number multiplied by an irrational number is still irrational
- 5√2 and π/4 are both irrational
Worked example
Naming an irrational number in a range
Write down an irrational number between 3 and 4. Explain how you know it is irrational.
Solution:
- Square both ends of the range: 3² = 9 and 4² = 16
- Any square root of a number between 9 and 16 therefore lies between 3 and 4
- Choose a value in that range that is not a square number, such as 10
- √10 is one possible answer, and √11, √12, √13, √14 and √15 all work equally well
- It is irrational because 10 is not a square number, so its square root neither terminates nor recurs
- √16 would not do, because 16 is a square number and √16 = 4, which is rational
Worked example
Picking numbers from a list
Here is a list of numbers: 21, 29, 36, 57, 64, 75, √30, √144. From the list, write down a prime number, a square number, a cube number, a factor of 150 and an irrational number.
Solution:
- Test each number in the list for the property asked, because some numbers that look prime are not
- 21 = 3 × 7, and the digits of 57 add to 12, a multiple of 3, so 57 = 3 × 19
- 36 and 64 are even, 75 ends in 5, and √144 = 12, so none of them is prime
- 29 is not divisible by 2, 3 or 5, and 6 × 6 = 36 is already more than 29, so no larger factor needs testing: the prime number is 29
- 36 = 6 × 6 and 64 = 8 × 8, so the square number is 36 or 64: either one answers the question
- 64 = 4 × 4 × 4, so the cube number is 64; 36 is not a cube, since 3³ = 27 and 4³ = 64
- 150 = 2 × 75, so the factor of 150 is 75
- 30 is not a square number, since 5² = 25 and 6² = 36, so the irrational number is √30
- √144 is rational, because it equals 12