Flashcards · 4MA1 · Topic 1

Numbers and the number system flashcards

62 cards on Topic 1, subtopics 1.1 to 1.11. Pick a subtopic or study them all.

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1.1 Integers

Prime number: A number with exactly two factors: 1 and itself. 1 is not prime, and 2 is the only even prime.
Factor and multiple of n: Factor: divides into n with no remainder (6 is a factor of 54). Multiple: in the n times table (54 is a multiple of 6).
Write 600 as a product of powers of its prime factors: Divide by primes: 600 → 300 → 150 → 75 → 25 → 5 → 1 (÷2, ÷2, ÷2, ÷3, ÷5, ÷5). 600 = 2³ × 3 × 5²
Find the HCF and LCM of 90 and 126: 90 = 2 × 3² × 5 and 126 = 2 × 3² × 7. HCF (common primes, lower powers) = 2 × 3² = 18. LCM (every prime, higher powers) = 2 × 3² × 5 × 7 = 630.
Work out 7 + 24 ÷ (5 − 9): Brackets first: 5 − 9 = −4. Then divide: 24 ÷ (−4) = −6. Then add: 7 + (−6) = 1.
What is the difference between (−4)² and −4²?: (−4)² = 16: the negative number is squared. −4² = −(4²) = −16: only the 4 is squared.

1.2 Fractions

Reciprocal: The number you multiply by to get 1. The reciprocal of 6 is 1/6; the reciprocal of 2/5 is 5/2.
Simplest form of a fraction: The numerator and denominator have no common factor except 1. Divide both by their HCF: 42/56 → 3/4 (HCF 14).
Work out 5/6 ÷ 10/9: Keep the first, flip the second, multiply: 5/6 × 9/10 = 45/60 = 3/4
Work out 1 1/2 × 2 2/9: Change to improper fractions first: 3/2 × 20/9 = 60/18 = 10/3 = 3 1/3.
"3/5 of 40" and "24 as a fraction of 40": which operation for each?: 3/5 of 40: multiply, 40 ÷ 5 × 3 = 24. 24 as a fraction of 40: divide, 24/40 = 3/5.

1.3 Decimals

Terminating and recurring decimals: Terminating: stops after a finite number of digits (0.375). Recurring: a digit or block repeats forever (0.3̇ = 0.333…).
Put in order, smallest first: 2.09, 2.9, 2.019, 2.1: Pad to the same number of places and compare from the left: 2.090, 2.900, 2.019, 2.100. Order: 2.019, 2.09, 2.1, 2.9
Write 0.035 as a fraction in its simplest form: 35/1000 = 7/200 (which is 3.5%).
Convert 0.83̇ (0.8333…) to a fractionHigher: 100x = 83.333… and 10x = 8.333… Subtract: 90x = 75, so x = 75/90 = 5/6. One digit comes before the recurring digit, so use 100x − 10x.
Convert 0.6̇3̇ (0.6363…) to a fractionHigher: x = 0.6363…, 100x = 63.6363… 100x − x = 99x = 63, so x = 63/99 = 7/11.

1.4 Powers and roots

Index laws (same base): aᵐ × aⁿ = aᵐ⁺ⁿ aᵐ ÷ aⁿ = aᵐ⁻ⁿ a⁰ = 1 a⁻ⁿ = 1/aⁿ Example: 5³ ÷ 5⁻² = 5⁵.
What is 6⁻²?: 1/6² = 1/36. It is not −36 or −12.
How can prime factors show a square root? Find √1764.: If every power in the prime factorisation is even, the number is a square: halve each power. 1764 = 2² × 3² × 7², so √1764 = 2 × 3 × 7 = 42.
SurdHigher: An irrational root, such as √3 or √10. √36 = 6 is not a surd.
Simplify √12 + √75Higher: √12 = 2√3 and √75 = 5√3 (take out the largest square factor). 2√3 + 5√3 = 7√3. It is not √87.
Rationalise 3/(√7 − 2)Higher: Multiply top and bottom by the conjugate √7 + 2. Denominator: 7 − 4 = 3. 3(√7 + 2)/3 = √7 + 2
Evaluate 64^(−2/3)Higher: Cube root of 64 = 4; 4² = 16; the negative power flips it: 1/16.

1.5 Set language and notation

Universal set ℰ: Everything under consideration in the question. Every other set is drawn from it.
Empty set ∅: A set with no elements. It is not {0}, which has one element.
Complement A′: The elements of ℰ that are not in A.
A ∩ B and A ∪ B: A ∩ B (intersection): elements in both A and B. A ∪ B (union): elements in A or B or both.
(A ∪ B)′ and (A ∩ B)′: which regions?: (A ∪ B)′: one region, outside both circles (in neither). (A ∩ B)′: everything except the overlap.
Formula for n(A ∪ B), and what n(A) meansHigher: n(A) is the number of elements in A. n(A ∪ B) = n(A) + n(B) − n(A ∩ B)
Of 30 people, 18 have a cat, 15 have a dog and 6 have neither. How many have both?Higher: At least one pet: 30 − 6 = 24. 18 + 15 − both = 24, so both = 9.
Meaning of ⊂ and of {x : …}Higher: A ⊂ B: A is a subset of B (every element of A is in B). {x : …}: the set of all x such that … Example: {x : x is an integer, 0 ≤ x < 4} = {0, 1, 2, 3}.

1.6 Percentages

Multiplier for a percentage increase or decrease of r%: Increase: × (1 + r/100). Decrease: × (1 − r/100). Example: decrease by 18% → × 0.82. A 40% reduction is × 0.6, not × 0.4.
Percentage change formula: (change ÷ original) × 100 Example: £125 to £140 is a change of 15, so 15/125 × 100 = 12%. Divide by the original, not the new value.
A price is 172.80 after an 8% rise. Find the original price.: Divide by the multiplier: 172.80 ÷ 1.08 = 160. Check: 160 × 1.08 = 172.80.
Compound interest and depreciation formulae: Growth: P × (1 + r/100)ⁿ Depreciation: P × (1 − r/100)ⁿ P = starting amount, r = rate per year (%), n = years. Subtract P for the interest. Example: £3200 at 2.5% for 2 years = £3362.
A value rises by 40% then falls by 25%. Find the overall percentage change.Higher: Multiply the multipliers: 1.4 × 0.75 = 1.05, an overall increase of 5%. Do not add the percentages.
A machine falls from £50 000 to £40 500 in 2 years at a fixed rate. Find the rate.Higher: m² = 40 500 ÷ 50 000 = 0.81, so m = 0.9. Rate = (1 − 0.9) × 100 = 10% per year.

1.7 Ratio and proportion

Share 96 in the ratio 5 : 3: Total parts = 8; one part = 96 ÷ 8 = 12. Shares: 5 × 12 = 60 and 3 × 12 = 36.
Write 12 : 30 in the form 1 : n: Divide both parts by the first part: 30 ÷ 12 = 2.5, so 1 : 2.5.
a : b = 2 : 7. What fraction of the total is a, and what fraction of b is a?: a is 2/9 of the total (2 out of 9 parts). a is 2/7 of b.
A map has scale 1 : 20 000. A path is 4.5 cm on the map. Find its real length in km.: 4.5 × 20 000 = 90 000 cm = 900 m = 0.9 km. Multiply by n first, then convert units.
y is directly proportional to x, and y = 21 when x = 6. Find y when x = 10.: k = y ÷ x = 21 ÷ 6 = 3.5, so y = 3.5x. y = 3.5 × 10 = 35.

1.8 Degree of accuracy

Significant figures: Digits counted from the first non-zero digit. Leading zeros in a decimal do not count; zeros between digits do.
Round 0.080 96 to 3 significant figures, and 61 540 to 2 significant figures: 0.080 96 → 0.0810 (keep the final 0: it is the third significant figure). 61 540 → 62 000 (not 62).
A length is 40 cm to the nearest 5 cm. Write its bounds.: Half of 5 is 2.5. 37.5 ≤ x < 42.5
How do you estimate a calculation? Estimate (0.587 × 3.94) ÷ 0.213: Round every number to 1 significant figure and write the rounded calculation: (0.6 × 4) ÷ 0.2 = 2.4 ÷ 0.2 = 12.
Bounds: which values give the upper bound of a − b and of a ÷ b?Higher: Upper bound of a with the lower bound of b, for both. For the lower bound: lower a with upper b. Sums and products: upper with upper, lower with lower.
How do you give an answer to a suitable degree of accuracy using bounds?Higher: Work out the upper and lower bounds of the answer. Round both to 1, 2, 3… significant figures. The answer is the most accurate value on which both bounds agree.

1.9 Standard form

Standard form: a × 10ⁿ, with 1 ≤ a < 10 and n an integer. Examples: 92 000 = 9.2 × 10⁴; 0.000 605 = 6.05 × 10⁻⁴.
Multiplying and dividing in standard form: Multiply: multiply the a parts and add the powers. Divide: divide the a parts and subtract the powers. Then adjust: 25 × 10⁻⁶ = 2.5 × 10⁻⁵.
Work out 2.4 × 10⁵ + 6 × 10⁴ in standard form: Match the powers or use ordinary numbers: 240 000 + 60 000 = 300 000 = 3 × 10⁵. It is not 8.4 × 10⁵.
How many times bigger is 6 × 10⁸ than 1.5 × 10⁵?Higher: Divide: (6 ÷ 1.5) × 10⁸⁻⁵ = 4 × 10³ times (4000).
Work out √(3.6 × 10⁷)Higher: Make the power even first: 3.6 × 10⁷ = 36 × 10⁶. √(36 × 10⁶) = 6 × 10³ = 6000.

1.10 Applying number

Converting between decimal hours and minutes: Decimal hours to minutes: × 60 (0.85 h = 51 min). Minutes to hours: ÷ 60 (35 min = 35/60 h, not 0.35 h). 1.5 h is 1 h 30 min, not 1 h 50 min.
How do you decide which pack is the best buy?: Find the cost of one common amount (per item, per 100 g, per litre) for each pack, show both values, then state which is better value. Example: 6 for £2.94 is 49p each; 10 for £4.75 is 47.5p each, so the 10-pack.
The rate is £1 = €1.15. Change €230 into pounds.: From the other currency into pounds, divide by the rate: 230 ÷ 1.15 = £200.
Area and volume conversions from m to cm: 1 m² = 10 000 cm² (square the length factor). 1 m³ = 1 000 000 cm³ (cube it). 1 cm³ = 1 ml; 1000 cm³ = 1 litre; 1 m³ = 1000 litres.
Everyday problems: when do you round up and when down?: Round up when you must have enough (tins of paint, buses, packs). Round down when only whole items fit or can be afforded (boxes in a lift, tickets on a budget).

1.11 Electronic calculators

Using a calculator: how do you enter a fraction with a sum on top and bottom?: Put brackets round the whole numerator and the whole denominator (or use the fraction template). Example: (3.8 + 4.6) ÷ (2.1 × 0.5) = 8.4 ÷ 1.05 = 8.
Why type brackets when squaring a negative number on a calculator?: Without brackets the calculator works out −3² = −9. With brackets, (−3)² = 9.
Interpret the display 2.25 as a time, and 14.7 as money in pounds: 2.25 hours = 2 h 15 min (0.25 × 60 = 15). 14.7 → £14.70 (money to 2 decimal places).
A calculator shows 4.6E+05. How do you write it?: 4.6 × 10⁵ (not 4.6⁵). Always write proper standard form notation.