Real Numbers

Mathematics — Second Preparatory

Worked examples

Comparison aspect	Main idea	Keywords
Example 1	Is 0.25 a rational number?	rationalnumber
Solution	0.25 = 25/100 = 1/4 which is a fraction with integer numerator and denominator. So 0.25 is rational	fractionintegernumeratordenominator
Example 2	Prove that 0.333… is rational then write it as a fraction	Proverationalthenwrite
Solution	Let x = 0.333… so 10x = 3.333…
Subtract	10x − x = 3.333… − 0.333… so 9x = 3 and x = 3/9 = 1/3. The number is rational	numberrational
Example 3	Is √18 a rational number?	rationalnumber
Solution	18 is not a perfect square. We write √18 = √9 × √2 = 3√2. If 3√2 were rational then √2 would be rational which is false. So √18 is irrational	perfectsquarewritewere
Example 4	Arrange these numbers from smallest to largest: 2.5 2.6	Arrangethesenumberssmallest
Solution	√5 ≈ 2.236 and √7 ≈ 2.646. So the order is: √5 < 2.5 < 2.6 < √7	order

Real Numbers · 5 / 7