Number
Standard form, surds, percentages, ratio and bounds, a quarter of Paper 1 and the arithmetic every other topic stands on.
What "Number" covers, and where it is examined
Number is the widest topic in the syllabus and the one with the most marks quietly hidden inside other questions. A mensuration answer given to the wrong number of significant figures, a trigonometry answer left as 7.2 when the question wanted 3 s.f.. Those marks are lost in Number, not in the topic that was being tested.
The pieces you must be fluent with: types of number (natural, integer, rational, irrational, real), standard form, indices including negative and fractional, surds, percentages including reverse and compound, ratio and proportion, bounds, and exchange rates and money.
Paper 1 is non-calculator. That is not an obstacle to be dreaded. It is a signal. Every Paper 1 number question has been designed so the arithmetic is clean if you take the right route. If you find yourself doing long multiplication of ugly decimals, you have missed the route.
Standard form is a counting exercise
A number in standard form is a × 10ⁿ where 1 ≤ a < 10. That condition is the whole definition: 34.56 × 10⁵ is a correct value written in the wrong form, and it loses the mark.
To convert, move the decimal point until exactly one non-zero digit sits in front of it, and count the hops. Hops to the left mean the original was large: positive power. Hops to the right mean it was small: negative power.
Multiplying and dividing in standard form: deal with the numbers and the powers separately, then fix the form. (2.5 × 10⁴) × (4 × 10⁻²) = 10 × 10², which is not standard form. Rewrite as 1 × 10³.
Adding and subtracting: you cannot add the powers. Convert both to the same power, or to ordinary numbers, then add.
Indices and surds: the rules, then the two that trip people
Indices. aᵐ × aⁿ = aᵐ⁺ⁿ, aᵐ ÷ aⁿ = aᵐ⁻ⁿ, (aᵐ)ⁿ = aᵐⁿ, a⁰ = 1, a⁻ⁿ = 1/aⁿ, and a^(1/n) = ⁿ√a. Fractional powers combine the last two: 27^(2/3) means "cube root, then square", root first, because 3 is a smaller number to work with than 729. Both orders give 9; one of them is pleasant.
Surds. √(ab) = √a × √b, so √50 = √25 × √2 = 5√2. Simplifying means pulling out the largest square factor. Adding: only like surds combine, √50 + √18 = 5√2 + 3√2 = 8√2, but √2 + √3 stays as it is.
Rationalising clears a surd from the bottom of a fraction: 6/√3 = 6√3/3 = 2√3. For (√5 + 2) on the bottom, multiply top and bottom by (√5 − 2): the difference of two squares kills the surd: (√5 + 2)(√5 − 2) = 5 − 4 = 1.
Percentages: forwards, backwards and compound
The single most useful idea is the multiplier. A 20% increase is × 1.20. A 15% decrease is × 0.85. Three years of 15% depreciation is × 0.85³, not × 0.55, the decreases compound because each one acts on a smaller amount.
Reverse percentage. "A shirt costs Rs 1 600 after a 20% discount", the 1 600 is 80% of the original, so the original is 1 600 ÷ 0.8 = Rs 2 000. Students who take 20% of 1 600 and add it back get 1 920, which is wrong for a reason worth understanding: 20% of the discounted price is less than 20% of the original.
A 20% rise then a 20% fall is not zero. × 1.2 × 0.8 = 0.96, a 4% fall. The fall is 20% of a bigger number.
Compound interest is just repeated multipliers: 20 000 × 1.06² = 22 472, so the interest is Rs 2 472. Simple interest would give 2 400; the extra 72 is interest on the first year's interest.
Bounds and ratio, where Paper 2 makes its money
Bounds. A measurement to the nearest unit is an interval half a unit each way. 8.4 cm to the nearest mm is 8.35 ≤ x < 8.45. In a calculation, decide what you want (biggest or smallest result) and pick the bound that pushes each input that way. Largest speed = largest distance ÷ smallest time. Smallest area = both lengths at their lower bound.
Ratio. Sharing 360 in the ratio 2 : 3 : 4 means 9 parts, 40 per part. Combining ratios: A : B = 3 : 4 and B : C = 6 : 5. Make B the same in both (12) to get A : B : C = 9 : 12 : 10. The difference method: two numbers in ratio 5 : 7 with difference 18 means 2 parts = 18, so 1 part = 9 and the larger is 63.
Map scales are ratios: 1 : 25 000 means 1 cm on the map is 25 000 cm = 250 m in the world. Convert the units before you answer, an answer of 160 000 cm scores nothing for a question asking in km.
Where marks go
- Standard form with a ≥ 10 or a < 1. 34.5 × 10³ is not standard form. Neither is 0.5 × 10⁴.
- Adding standard form by adding the powers. 10³ + 10² is not 10⁵.
- Reverse percentage by adding back the same percent. You divide by the multiplier; you do not multiply by it.
- Rounding too early. Keep the full value; round once at the end, to whatever the question asks. If it does not say, 3 s.f. is the convention.
- Upper bound with ≤. It is always <: 8.45 would round to 8.5.
- Thinking 22/7 is π. It is a rational approximation. π is irrational; √7 is irrational; √16 is 4 and rational.
- Forgetting to simplify a surd or a ratio. 8√4 is 16. 18 : 20 is 9 : 10.
What to remember
- Standard form: 1 ≤ a < 10. Count the hops for the power.
- Fractional index: root first, then power. Negative index: reciprocal.
- Surds: pull out the largest square; only like surds add; rationalise by multiplying top and bottom.
- Percentages: think in multipliers. Reverse = divide by the multiplier. Repeated = raise it to a power.
- Bounds: half a unit each way, lower ≤ x < upper; push each input the way you want the answer to go.
- Ratio: find the value of one part. Combine ratios by matching the shared term.
- Answer in the units and the accuracy the question asked for.
Watch one worked through
Here is one done slowly, with the reason for every move. Then one for you.
A price rises by 20% to Rs 7200. Find the original price.
- Spot that 7200 is the new price, not the old one.7200 = 120% of the originalThis is a reverse percentage. Taking 20% off 7200 is the trap and gives the wrong answer.
- Write what one per cent is worth.120% = 7200 1% = 7200 ÷ 120 = 60Going via 1% works for any percentage and needs nothing memorised. It is the safest method under pressure.
- Scale up to 100%.100% = 60 × 100 = Rs 6000The original is always 100%, whatever the change was. Fixing that idea is what makes reverse percentages easy.
- Check forwards.6000 + 20% of 6000 = 6000 + 1200 = 7200 ✓Checking in the forward direction catches the error of subtracting instantly.
Answer: Rs 6000
A price rises by 25% to Rs 500. What was the original in rupees?
Show the answer
400
Practise Number
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Number on paper for O Level, with room to write the working out. Preview the first page free.
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