Number, Year 11: surds and recurring decimals

A surd

HintIt rhymes with 'word' and has four letters.

The 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?

The 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.

The 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 …

The 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.

The 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.

The 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.

The 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.

The 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.

The 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.

The 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.

The 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.

The 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.

The 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.

The 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.

The 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.