a For each of the following, explain which set(s) of numbers they belong to (either natural,
integer, rational, irrational or real). A number may be a member of more than one set.
■ Because this is a whole number (with no decimal places), this is a natural number and also
■ Because it can be written in the form , it is also a rational number.
■ Finally, it is also a real number.
■ The decimal point means that this is definitely not a natural number or an integer.
, so is therefore a rational number.
■ It can be written in the form 25/2, therefore a rational number
■ In addition, it is also a real number.
■ In decimal form, this is 1.4142135… It does not terminate or recur, therefore it is an irrational
■ Just like every other number, it is also a real number.
b Decide whether each of the following statements is true or false.
i All natural numbers are also integers.
This is TRUE. The set of integers includes all natural numbers AND also includes negative
ii A square root is always irrational.
This is FALSE. Most roots (such as 2 or 3 ) are irrational. However, 4 = 2 and 9 = 3.
These are certainly not irrational!
iii A number is either rational or irrational, but never both.
This is TRUE. The definitions of these number sets are opposites: if a number is rational then it
cannot be irrational and vice versa.