Number flashcards

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What is the difference between a rational and an irrational number?
A rational number can be written as a fraction \(\frac{a}{b}\) of integers (including terminating and recurring decimals). An irrational number cannot, e.g. \(\pi\) and \(\sqrt{2}\); its decimal never ends or repeats.
List the first few prime numbers and state what makes a number prime.
A prime has exactly two factors, 1 and itself: 2, 3, 5, 7, 11, 13, ... The number 1 is not prime and 2 is the only even prime.
How do you find the HCF and LCM of two numbers?
Write each as a product of prime factors. HCF: multiply the primes common to both, using the smaller powers. LCM: multiply all primes that appear, using the larger powers.
Find the HCF and LCM of 12 and 18.
\(12=2^2\times3\), \(18=2\times3^2\). HCF \(=2\times3=6\). LCM \(=2^2\times3^2=36\).
How do you add or subtract two fractions with different denominators?
Rewrite them over a common denominator (the LCM of the denominators), then add or subtract the numerators and simplify. To divide by a fraction, multiply by its reciprocal.
How do you convert a fraction to a percentage and a decimal to a percentage?
Divide the numerator by the denominator to get the decimal, then multiply by 100 for the percentage. E.g. \(\frac{3}{8}=0.375=37.5\%\).
What is the multiplier for a percentage increase of \(r\%\) and for a decrease of \(r\%\)?
Increase: \(1+\frac{r}{100}\). Decrease: \(1-\frac{r}{100}\). E.g. a 20% increase is \(\times1.2\) and a 15% decrease is \(\times0.85\). Multiply the original amount by the multiplier.
Calculate 15% of 240.
\(0.15\times240=36\).
(Extended) A price after a 20% increase is £60. How do you find the original price, and what is it?
Reverse the multiplier: original \(=\) new \(\div\) multiplier \(=60\div1.2=\)£50. Do not take 20% off £60.
(Extended) State the compound interest formula and what each symbol means.
\(A=P\left(1+\frac{r}{100}\right)^n\): \(P\) is the amount invested, \(r\) the percentage rate per period, \(n\) the number of periods and \(A\) the final amount. Interest earned is \(A-P\).
(Extended) Find the value of £2000 invested at 3% compound interest for 2 years.
\(2000\times1.03^2=2000\times1.0609=\)£2121.80.
What is a ratio, and how do you share an amount in a given ratio?
A ratio compares quantities in the same units. Add the parts to get the total parts, divide the amount by this, then multiply by each part.
Share £150 in the ratio \(2:3\).
Total parts \(=5\), one part \(=150\div5=30\). Shares: \(2\times30=\)£60 and \(3\times30=\)£90.
What is the difference between direct and inverse proportion?
Direct: both quantities increase at the same rate, so the ratio \(y:x\) is constant. Inverse: as one increases the other decreases so that the product \(xy\) is constant.
How do you work out average speed, and how do you rearrange the formula?
Average speed \(=\frac{\text{total distance}}{\text{total time}}\). So distance \(=\) speed \(\times\) time and time \(=\) distance \(\div\) speed. Keep units consistent, e.g. km and hours.
State the rules for multiplying, dividing and raising a power to a power: \(a^m\times a^n\), \(a^m\div a^n\), \((a^m)^n\).
\(a^m\times a^n=a^{m+n}\), \(a^m\div a^n=a^{m-n}\), \((a^m)^n=a^{mn}\). Also \(a^0=1\) and \(a^{-n}=\frac{1}{a^n}\).
(Extended) What do fractional indices \(a^{\frac{1}{n}}\) and \(a^{\frac{m}{n}}\) mean? Evaluate \(27^{\frac{2}{3}}\).
\(a^{\frac{1}{n}}\) is the \(n\)th root of \(a\), and \(a^{\frac{m}{n}}=\left(\sqrt[n]{a}\right)^m\). So \(27^{\frac{2}{3}}=\left(\sqrt[3]{27}\right)^2=3^2=9\).
What are the rules for writing a number in standard form?
Write it as \(a\times10^n\) with \(1\le a<10\) and \(n\) an integer. Large numbers have positive \(n\), small numbers negative \(n\). E.g. \(0.00046=4.6\times10^{-4}\).
Calculate \((4.5\times10^3)\times(2\times10^2)\), giving the answer in standard form.
\(4.5\times2=9\) and \(10^3\times10^2=10^5\), so the answer is \(9\times10^5\).
What is the difference between decimal places and significant figures when rounding?
Decimal places count digits after the point. Significant figures count from the first non-zero digit. Look at the next digit: 5 or more rounds up. E.g. 0.004567 is 0.0046 to 2 s.f.
How do you estimate the answer to a calculation?
Round every number to 1 significant figure, then calculate with the rounded values. E.g. \(\frac{4.8\times19.7}{0.52}\approx\frac{5\times20}{0.5}=200\).
(Extended) What does it mean to write a measurement as \(6.4\) correct to 1 decimal place in terms of bounds?
The true value \(x\) satisfies \(6.35\le x<6.45\). The lower bound is 6.35 and the upper bound is 6.45, half a unit of the rounding place either side.
(Extended) How do you find the upper bound of \(a-b\) and of \(a\div b\)?
Upper bound of \(a-b\): upper bound of \(a\) minus lower bound of \(b\). Upper bound of \(a\div b\): upper bound of \(a\) divided by lower bound of \(b\). Choose bounds to make the result as large as possible.
(Extended) Simplify \(\sqrt{50}\) and rationalise \(\frac{6}{\sqrt{3}}\).
\(\sqrt{50}=\sqrt{25\times2}=5\sqrt{2}\). \(\frac{6}{\sqrt{3}}=\frac{6\sqrt{3}}{3}=2\sqrt{3}\) after multiplying top and bottom by \(\sqrt{3}\).
(Extended) How do you convert the recurring decimal \(0.\dot{3}\dot{6}\) to a fraction?
Let \(x=0.363636...\). Then \(100x=36.3636...\). Subtract: \(99x=36\), so \(x=\frac{36}{99}=\frac{4}{11}\).
What do the set symbols \(\cup\), \(\cap\), \(A'\) and \(\varnothing\) mean?
\(A\cup B\): in A or B or both. \(A\cap B\): in both A and B. \(A'\): not in A (the complement). \(\varnothing\): the empty set. \(n(A)\) is the number of elements in A.
What do \(\mathbb{N}\), \(\mathbb{Z}\), \(\mathbb{Q}\) and \(\mathbb{R}\) stand for, and how do they nest?
\(\mathbb{N}\): natural numbers; \(\mathbb{Z}\): integers; \(\mathbb{Q}\): rationals; \(\mathbb{R}\): all real numbers. Each sits inside the next: \(\mathbb{N}\subset\mathbb{Z}\subset\mathbb{Q}\subset\mathbb{R}\).
How do you calculate a percentage change, and what is the formula?
\(\text{percentage change}=\frac{\text{change}}{\text{original}}\times100\). Always divide by the original value. E.g. £40 to £46 is \(\frac{6}{40}\times100=15\%\) increase.
How do you order and compare positive and negative numbers, fractions and decimals?
Convert everything to decimals (or a common denominator) and compare place by place. On a number line, numbers further left are smaller, so \(-7<-2\).

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