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Flashcards · 8300 · Topic 1

Number flashcards

29 cards on Topic 1, subtopics 3.1.1 to 3.1.3. Pick a subtopic or study them all.

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3.1.1 Structure and calculation

Prime number
A number with exactly two factors: 1 and itself. 1 is not prime; 2 is the only even prime.
How do you find the HCF and LCM from prime factors?
1. Write each number as a product of primes in index form. 2. HCF: the primes they share, each to the lowest power. 3. LCM: every prime that appears, each to the highest power.
Find the HCF and LCM of 48 and 180
48 = 2⁴ × 3 and 180 = 2² × 3² × 5. HCF = 2² × 3 = 12. LCM = 2⁴ × 3² × 5 = 720.
Priority of operations
Brackets → powers and roots → × and ÷ (left to right) → + and − (left to right). Example: 4 − 2 × (−3)² = 4 − 18 = −14.
−3² compared with (−3)²
−3² = −9: the power acts before the minus sign. (−3)² = 9: the brackets make the whole −3 squared.
Index laws for the same base a
aᵐ × aⁿ = aᵐ⁺ⁿ aᵐ ÷ aⁿ = aᵐ⁻ⁿ (aᵐ)ⁿ = aᵐⁿ a⁰ = 1 a⁻ⁿ = 1/aⁿ, so 2⁻³ = 1/8 (a positive number).
Standard form
A × 10ⁿ with 1 ≤ A < 10 and n an integer. Example: (8 × 10⁻³) × (7 × 10⁶) = 56 × 10³ = 5.6 × 10⁴.
Fractional indices: what do a^(1/n) and a^(m/n) mean?Higher
a^(1/n) is the nth root of a. a^(m/n) = (nth root of a)ᵐ: take the root first, then the power. A negative index also means take the reciprocal. Example: 4^(5/2) = 2⁵ = 32.
How do you simplify a surd such as √180?Higher
1. Find the largest square number that divides it (36). 2. Split: √180 = √36 × √5. 3. Square-root the square: 6√5.
Product rule for countingHigher
If there are m ways to make one choice and n ways to make the next, there are m × n ways altogether. Example: 6 shirts, 4 trousers, 3 pairs of shoes → 6 × 4 × 3 = 72 outfits.

3.1.2 Fractions, decimals and percentages

Terminating and recurring decimals
Terminating: stops after a finite number of digits (0.125). Recurring: a digit or block repeats forever (0.1666…).
Which fractions give terminating decimals?
In its simplest form, a fraction terminates only when its denominator has no prime factors other than 2 and 5. So 3/16 = 0.1875 terminates; 3/14 recurs (14 has a factor 7).
Dot notation: 0.1̇6̇ compared with 0.16̇
0.1̇6̇ (dots on both digits) = 0.161616… 0.16̇ (dot on the 6 only) = 0.1666…
How do you order fractions, decimals and percentages?
1. Convert every value to a decimal. 2. Pad with zeros to the same number of decimal places. 3. Compare column by column from the left. 4. Write the answer using the original values.
Multiplier for an increase or decrease of r%
Increase by r%: × (1 + r/100). Decrease by r%: × (1 − r/100). Example: decrease by 4.5% → × 0.955.
How do you combine two successive percentage or fraction changes?
Multiply the multipliers; never add the percentages. Example: increase by 2/3 then decrease by 2/5: 5/3 × 3/5 = 1, so no overall change.
30% of 50 compared with increasing 50 by 30%
30% of 50 = 0.3 × 50 = 15. Increase 50 by 30% = 1.3 × 50 = 65.
In the ratio a : b, what fraction is the first quantity of the total, and of the second quantity?
Of the total: a/(a + b). Of the second quantity: a/b. Example: in 3 : 5 the first is 3/8 of the total but 3/5 of the second.
How do you convert a recurring decimal to a fraction?Higher
1. Let x = the decimal. 2. Multiply by 10ⁿ, where n is the number of digits in the repeating block. 3. Subtract so the recurring tails cancel. 4. Solve for x and simplify.
Write 0.7̇2̇ as a fraction in its simplest formHigher
x = 0.7272…, 100x = 72.7272… 100x − x = 99x = 72 x = 72/99 = 8/11

3.1.3 Measures and accuracy

Rounding compared with truncation
Rounding goes half a unit either way; truncation goes a whole unit upwards only.
Decimal places compared with significant figures
Decimal places are counted after the point; significant figures start at the first non-zero digit. 0.0503 is 4 d.p. but 3 s.f.
Conversion factors for length, area and volume (cm to m)
Length: 100 for cm to m. Area: 10 000 for cm² to m². Volume: 1 000 000 for cm³ to m³.
Decimal time compared with clock time
1.3 hours is 1 h 18 min, not 1 h 30 min (0.3 × 60 = 18).
How do you estimate a calculation?
1. Round every number to 1 significant figure. 2. Write the rounded calculation down. 3. Work it out. Example: 397 × 0.512 ÷ 19.6 ≈ 400 × 0.5 ÷ 20 = 10.
Error interval for a value rounded: p = 14.7 to 1 d.p.
Halve the unit of rounding (0.1 → 0.05), then go that far either way: 14.65 ≤ p < 14.75
Error interval for a truncated value: r = 9.2 truncated to 1 d.p.
The lower limit is the value itself; the upper limit is one whole unit more: 9.2 ≤ r < 9.3
Upper and lower bounds of a − b and a ÷ bHigher
Maximum of a − b = UB of a − LB of b. Minimum of a − b = LB of a − UB of b. Maximum of a ÷ b = UB of a ÷ LB of b. Minimum of a ÷ b = LB of a ÷ UB of b.
How do you give an answer to an appropriate degree of accuracy using bounds?Higher
1. Find the upper and lower bounds of the answer. 2. Round both to 1 s.f., 2 s.f., 3 s.f. … 3. Stop at the most figures where they still agree, and give that as the answer with the reason.

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