Worked examples 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. i. 35 ■ Because this is a whole number (with no decimal places), this is a natural number and also an integer. ■ Because it can be written in the form , it is also a rational number. ■ Finally, it is also a real number. ii. 12.5 ■ 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. iii. √3 ■ In decimal form, this is 1.4142135… It does not terminate or recur, therefore it is an irrational number. ■ 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 numbers. 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.
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