Numbers and the number system flashcards

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What is the difference between the HCF and the LCM of two numbers?
The HCF is the largest number that divides into both. The LCM is the smallest number that both divide into (the smallest common multiple).
How do you find the HCF and LCM using prime factors? Do it for 72 and 60.
Write \(72=2^3\times3^2\), \(60=2^2\times3\times5\). HCF: lowest powers of common primes, \(2^2\times3=12\). LCM: highest power of every prime, \(2^3\times3^2\times5=360\).
What is a prime number? List the primes below 20.
A prime has exactly two factors, 1 and itself. So 1 is not prime. Primes below 20: 2, 3, 5, 7, 11, 13, 17, 19. Note 2 is the only even prime.
State the three main laws of indices.
\(a^m\times a^n=a^{m+n}\), \(a^m\div a^n=a^{m-n}\), \((a^m)^n=a^{mn}\). The bases must be the same for the first two.
Why is \(a^0=1\), and what do negative and fractional powers mean?
\(a^n\div a^n=a^0=1\). \(a^{-n}=\dfrac1{a^n}\). \(a^{1/n}=\sqrt[n]{a}\) and \(a^{m/n}=(\sqrt[n]{a})^m\).
Evaluate \(8^{-2/3}\).
\(8^{1/3}=2\), squared is 4, and the negative power gives the reciprocal: \(8^{-2/3}=\dfrac14\).
What makes a number correctly written in standard form?
It is \(a\times10^n\) with \(1\le a<10\) and \(n\) an integer. For example \(0.00047=4.7\times10^{-4}\) and \(52\,000=5.2\times10^{4}\).
Work out \((3\times10^4)\times(2\times10^5)\) in standard form.
Multiply the numbers and add the powers: \(6\times10^{9}\). For division, divide the numbers and subtract the powers; re-adjust if the front number is not between 1 and 10.
How do you round to significant figures, and what is 0.004068 to 2 s.f.?
Start counting at the first non-zero digit, then look at the next digit to round. 0.004068: the 2 s.f. are 4 and 0, the next digit 6 rounds up, giving 0.0041.
How do you estimate the value of a calculation?
Round each number to 1 significant figure, then calculate. For \(\dfrac{4.87\times19.6}{0.52}\) use \(\dfrac{5\times20}{0.5}=200\).
How do you divide one fraction by another? Work out 2/3 ÷ 4/5.
Keep the first, flip the second, multiply: \(\dfrac23\times\dfrac54=\dfrac{10}{12}=\dfrac56\).
How do you add mixed numbers? Work out \(1\tfrac12+2\tfrac34\).
Convert to improper fractions or add wholes and fractions separately: \(3+\tfrac{2}{4}+\tfrac34=3+\tfrac54=4\tfrac14\).
How do you work out a percentage change?
\(\dfrac{\text{new}-\text{old}}{\text{old}}\times100\). Price rises from £80 to £92: \(\dfrac{12}{80}\times100=15\%\) increase.
What is a reverse percentage problem, and how do you solve it?
You are given the value after a change and need the original. After a 20% increase the multiplier is 1.2, so original \(=\) new \(\div\,1.2\). New price £60 gives \(60\div1.2=£50\).
Give the compound interest formula and use it for £2000 at 3% a year for 4 years.
\(A=P\left(1+\dfrac r{100}\right)^n\). So \(A=2000\times1.03^4=£2251.02\) (to the nearest penny). Interest earned is £251.02.
What multiplier represents a 15% decrease, and how is it used for repeated change?
\(1-0.15=0.85\). For a repeated change raise it to a power: after \(n\) years the value is the original \(\times\,0.85^n\).
How is simple interest calculated? Find it for £500 at 4% for 3 years.
\(I=\dfrac{PRT}{100}\), interest the same each year. \(\dfrac{500\times4\times3}{100}=£60\).
How do you share a quantity in a given ratio? Share £84 in the ratio 2 : 5.
Add the parts (2 + 5 = 7), find one part (84 ÷ 7 = 12), then multiply. The shares are £24 and £60, which add back to £84.
What is the difference between direct and inverse proportion?
Direct: both quantities rise together at a constant rate, \(y=kx\) (a straight line through the origin). Inverse: one rises as the other falls, \(y=\dfrac kx\), so \(xy\) stays constant.
Give the formulae linking speed, distance, time and density, and convert 20 m/s to km/h.
speed \(=\dfrac{\text{distance}}{\text{time}}\), density \(=\dfrac{\text{mass}}{\text{volume}}\). \(20\text{ m/s}\times3.6=72\text{ km/h}\).
(Higher) How do you convert the recurring decimal 0.363636... to a fraction?
Let \(x=0.\dot3\dot6\). Then \(100x=36.\dot3\dot6\). Subtract: \(99x=36\), so \(x=\dfrac{36}{99}=\dfrac4{11}\).
(Higher) What are the rules for surds? Simplify \(\sqrt{72}\).
\(\sqrt a\times\sqrt b=\sqrt{ab}\) and \(\sqrt{\dfrac ab}=\dfrac{\sqrt a}{\sqrt b}\). Take out the largest square factor: \(\sqrt{72}=\sqrt{36}\times\sqrt2=6\sqrt2\).
(Higher) How do you rationalise a denominator? Do \(\dfrac6{\sqrt3}\) and \(\dfrac1{2+\sqrt3}\).
Multiply top and bottom by the surd, or by the conjugate. \(\dfrac6{\sqrt3}\times\dfrac{\sqrt3}{\sqrt3}=2\sqrt3\). \(\dfrac1{2+\sqrt3}\times\dfrac{2-\sqrt3}{2-\sqrt3}=\dfrac{2-\sqrt3}{4-3}=2-\sqrt3\).
What is the difference between rational and irrational numbers?
A rational number can be written as a fraction of two integers (its decimal ends or recurs). An irrational number cannot, for example \(\pi\) and \(\sqrt2\). Surds and \(\pi\) are left exact in answers.
(Higher) A length is 8.4 cm to 1 d.p. State its lower and upper bounds.
Half a unit of the last digit either side: \(8.35\le L<8.45\). The upper bound is not reached, but it is used in calculations.
(Higher) Sides are 8.4 cm and 5.2 cm (1 d.p.). Find the upper bound of the rectangle's area.
Use both upper bounds: \(8.45\times5.25=44.3625\text{ cm}^2\). For a lower bound of a speed, use lower distance \(\div\) upper time.
What do the symbols ∈, ∪, ∩, A′, ∅ and ξ mean in set notation?
∈ is 'is an element of'; ∪ is union (in either); ∩ is intersection (in both); A′ is the complement (not in A); ∅ is the empty set; ξ is the universal set.
In a universal set of 40 people, n(A) = 18, n(B) = 15 and n(A ∩ B) = 6. Find n(A ∪ B) and n(A ∪ B)′.
\(n(A\cup B)=18+15-6=27\). Outside both sets: \(40-27=13\).
What is the correct order of operations?
Brackets, then indices (powers and roots), then division and multiplication, then addition and subtraction (BIDMAS). Work left to right within the same level.

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