WJEC Mathematics for AS Level - Pure

1 Proof 10 1.1 Real and imaginary numbers A feature of real numbers is that when they are squared, positive answers are obtained. For example, 2 2 = 4, (3.002) 2 = 9.012004, (−50) 2 = 2500. Numbers involving square roots of negative real numbers are called imaginary numbers. For example, √ −5, √ −9, √ −53 are all imaginary numbers. The main feature of an imaginary number is that when it is squared, a negative (real) number is obtained. For example, ( √ −4) 2 = −4, ( √ −11) 2 = −11, ( √ −67) 2 = −67. The important point may be summarised as follows: If a number m is such that m 2 < 0 then m cannot be a real number but is instead an imaginary number. This result will be used later. 1.2 Rational and irrational numbers Rational numbers A rational number is a number that can be expressed as a fraction (i.e. a b  , where b ≠ 0) with the numerator (i.e. the top number) and the denominator (i.e. the bottom number) both being whole numbers. All whole numbers are rational. For example, 2 is rational as it can be written as the fraction, 2 1  . Fractions are always rational no matter how complicated they are, so 13209 121342 is rational. Decimals with repeating units such as 0.166666666… can be expressed as fractions (i.e. 1 6 in this case) and are therefore rational. Another way of deϐining a rational number is to say it can be accurately placed on a number line. For example, the number 1 3 (or 0.333333...) can be accurately positioned on the number line as one third of the way between the number 0 and 1. Irrational numbers An irrational number is a number that cannot be expressed as a fraction. When expressed as decimals, irrational numbers have endless non-repeating numbers to the right of the decimal point. Examples of irrational numbers are π (i.e. 3.141592…) and √ 2 (i.e. 1.414213…). Irrational numbers cannot be positioned on the number line. 1.3 Proof by exhaustion Proof by exhaustion involves using all allowable values to prove the mathematical statement is either true or false. This only works when there are only a small number of possible values to try. Imaginary numbers are being mentioned here as a way of illustrating when a number is not real. Imaginary numbers will not be assessed in GCE Maths AS Unit 1. There are lots of specialist terms included in this topic. In exam questions these terms will be used so make sure you understand each term and can give an example. BOOST G ra d e ⇪⇪⇪⇪