Every card in Number, Year 11: surds and recurring decimals

A surd

HintIt rhymes with 'word' and has four letters.

Why√2 = 1.41421… goes on for ever without repeating, so the only exact way to write it is √2. A root such as √9 is not one of these, because it equals 3 exactly.

5√3

HintWhich square number divides exactly into the number under the root?

Why75 = 25 × 3 and 25 is a square number, so √75 = √25 × √3 = 5√3. Always look for the largest square factor so that the job is finished in one step.

45

HintTo go back under the root sign, the 3 has to be squared first.

Why3 = √9, so 3√5 = √9 × √5 = √(9 × 5) = √45. This is simplifying run backwards, and it is useful for comparing the sizes of surds.

square

HintThink of 4, 9, 16, 25, 36 …

Why50 = 25 × 2, so √50 = √25 × √2 = 5√2. This works because √(a × b) = √a × √b. Choosing a smaller factor of this kind still works, but the answer then needs simplifying again.

No: √9 + √16 = 7, but √(9 + 16) = √25 = 5

HintEvaluate each side separately as ordinary numbers.

WhyRoots can be split over multiplication and division, √(ab) = √a × √b, but not over addition or subtraction. That is why √2 + √3 cannot be combined into √5 and has to be left as it is.

5√5

HintSimplify each root first; then see whether they are like terms.

Why√20 = 2√5 and √45 = 3√5, so the sum is 2√5 + 3√5 = 5√5. Surds add like algebra terms: only matching roots can be collected.

7

HintMultiply out all four products and watch what happens to the two middle ones.

Why9 − 3√2 + 3√2 − (√2)² = 9 − 2 = 7. The middle terms cancel and √2 × √2 = 2, so the surd disappears. This is the difference of two squares, 3² − (√2)².

2√3

HintMultiply the top and the bottom by the root that is on the bottom.

Why6/√3 × √3/√3 = 6√3/3 = 2√3. Multiplying top and bottom by √3 is multiplying by 1, so the value is unchanged, and √3 × √3 = 3 clears the root from the bottom.

Rationalising the denominator

HintThe name describes turning the bottom number into one that is not a surd.

WhyMultiply the top and the bottom by the surd on the bottom: 1/√2 × √2/√2 = √2/2. The value is the same; it is simply written in the agreed tidy form, which is easier to add to other fractions.

7 + 4√3

HintWrite the bracket out twice and multiply every pair of terms.

Why(2 + √3)(2 + √3) = 4 + 2√3 + 2√3 + 3 = 7 + 4√3. The two middle terms are alike and add; the last term is √3 × √3 = 3.

7/9

HintCall it x, then compare x with ten lots of x.

WhyLet x = 0.777…, so 10x = 7.777…. Subtracting, 9x = 7 and x = 7/9. The subtraction works because both numbers have exactly the same digits after the decimal point.

100; 99; 5/11

HintThe repeating block is two digits long.

Why100x = 45.454545… and x = 0.454545…, so subtracting gives 99x = 45. Then x = 45/99, and dividing top and bottom by 9 gives 5/11.

1/6

HintFind two multiples of x that have identical digits after the decimal point.

WhyLet x = 0.1666…. Then 10x = 1.666… and 100x = 16.666…, so 100x − 10x = 90x = 15. So x = 15/90 = 1/6.

3/10, 0.33, 0.333…

HintWrite all three to four decimal places.

Why3/10 = 0.3000, 0.33 = 0.3300 and the recurring decimal is 0.3333…, which is 1/3. Comparing digit by digit gives the order.

10x and x have exactly the same digits after the decimal point, so those digits cancel

HintWrite the two numbers one above the other and look to the right of the dot.

Why10x = 7.777… and x = 0.777…. Everything to the right of the point matches, so the subtraction leaves a whole number: 9x = 7. Multiplying by the right power of ten is all about making the tails match.