USING NUMBERS
USING NUMBERS
GLOSSARY
base The number used for counting in a particular system, for example the decimal system has base 10.
binary A numerical system based on two digits: 1 and 0.
denominator In a simple fraction (one number over another) the denominator is the bottom number and cannot be zero.
e A mathematical constant, also called Euler’s number, which is approximately equal to 2.71828. It is the number whose natural logarithm is 1.
greatest common divisor (GCD) The largest positive integer that divides into two other integers.
irrational number A number that cannot be represented as a simple fraction.
machine learning A statistical and computational algorithm that learns features from data through trial and error of known data.
n! A factorial of n. Where all positive whole numbers up to n are multiplied together.
natural logarithm A logarithm to base e.
numerator In a simple fraction, the numerator is the top number.
place-value A system of writing and giving value to numbers based on the relative position of their digits.
whole number A number with no decimal point or fraction, also called an integer.
NUMERICAL BASES
the 30-second calculation
A numerical base corresponds to the number of different digits, including 0, in a positional number system. Our usual base ten system runs from 0 to 9 and then repeats, without confusion, by shifting a digit into a new position. After counting ten of something, we can count up to ten lots of ten, and then move to ten lots of those, and so on. We owe this system to having ten fingers. The number 123 usually means 1 hundred plus 2 tens plus 3 – this is implied by the positions of the digits. Writing numbers in this way – called ‘place-value’ – is efficient, since it avoids requiring new symbols for each number; each position represents a power of ten. Other bases work in the same way, by representing numbers in units and then increasing powers of the base, as they shift to the right. Had we evolved with eight fingers we might count in base eight, and write the number 123 to represent 1 sixty-four (sixty-four is eight eights) plus 2 eights plus 3. To write a number unambiguously in a base other than the usual ten the base may be used as a subscript, for example 1238. Any whole number greater than one can be the base for a positional number system. The ancient Babylonians used base 60, which has echoes in our counting of seconds and minutes. Computers represent numbers internally using base two, known as binary. The digits in binary are just 0 and 1.
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A numerical base is the number used for counting in a particular system. We usually count in tens (base ten), but base two (in computing) and others are used.
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At the most fundamental level, computers use binary – base two. Base 16, using digits 0–9 along with A–F, is used in numerical internet identifiers. Other bases such as 12 (inches and hours) and 60 (seconds and minutes) persist in our culture and language.
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3-SECOND BIOGRAPHY
GOTTFRIED WILHELM LEIBNIZ
1646–1716
German polymath who invented the binary number system now used in computing
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John McDermott
We count using base ten but computers count in base two.
LOGARITHMS
the 30-second calculation
A logarithm is a quantity representing the power to which a fixed positive number, the base, must be raised to produce a given target number. That is, it tells us how many of the base we must multiply together to produce the target. For example, the logarithm in base 2 of 8 is 3, since 2 × 2 × 2 = 23 = 8. Logarithms are written using the abbreviation ‘log’ and the base as a subscript, so that ‘the logarithm in base 2 of 8 is 3’ is shortened to ‘log2 (8) = 3’. The value of a logarithm need not always be a whole number; for example, log2 (10) is approximately 3.313. Logarithms in base 10 are widely used to reduce the size of numbers involved in calculations with and display of data, since log10 (100) = 2, log10 (1000) = 3, and so on. Tables of logarithms facilitated complex calculations in the days before the widespread availability of calculators and computers. Such calculations were vital in navigation, engineering, medicine, science and technology. The (so-called) irrational number e, which is approximately 2.718, is known as the base of the natural logarithm. The natural logarithm has its own notation so that, for example, loge (8) and loge (10) are usually written ln (8) and ln (10). This special treatment is earned by the vital role e plays in calculus, probability and numerous other branches of mathematics.
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The logarithm of a positive number relative to another number, the base, is the power to which the base must be raised to give that positive number.
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Logarithms are used to describe values that grow or shrink too fast to be easily compared or displayed otherwise. The Richter scale for measuring the strength of earthquakes is logarithmic: an earthquake scoring a 6 on this scale is approximately ten times as strong as one scoring a 5.
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3-SECOND BIOGRAPHY
JOHN NAPIER
1550–1617
Scottish mathematician who invented logarithms. He also popularized the use of the decimal point in representing numbers
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John McDermott
Logarithms aided calculations before computers but are still useful in understanding computing and big data.
PERCENTAGES
the 30-second calculation
Percentages represent parts of a whole, expressed in hundredths, and provide an alternative to the use of decimals or fractions. Twenty-five per cent (25%) is the same as 0.25 or 25/100, which is the same as a quarter, as there are four 25s in 100. To find 25% of another number, multiply by 0.25, or divide by four: 25% of 60 is one quarter of 60, i.e. 15. To find a more awkward percentage, convert to an equivalent decimal and multiply: 17% of 60 is 0.17 × 60 = 10.2. Underlying these basic calculations is a fact that sometimes causes confusion: percentages are relative. If a shop raises its prices by 10%, then an item previously costing £100 will now cost £110 (£100 plus 10% of £100). If the shop later sells the item at ‘10% off’, the price will be £99 (£110 reduced by 10% of £110, i.e. by £11). The increase and reduction do not cancel exactly. Savings and borrowings are commonly subject to interest payments, measured in percentage rates. Interest may be applied simply, so that a rate of 5% per year on a loan of £100 for two years would see the borrower repay £100 + £10 (£5 for each year) at the end. However, interest is usually compounded, meaning that in the second year, the interest to be paid will be calculated on the total amount owed so far, i.e. it will not be £5 but 5% of (£100 + £5) = £5.25.
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A percentage describes a proportion, in parts per hundred, of a whole.
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To calculate percentages without a calculator, a useful trick is to first find 10% (divide by 10), and then smaller amounts such as 5% (divide in half again) and 1% (divide the 10% figure by 10 again). Then 17% = 10% + 5% + 1% + 1%, so 17% of 60 is 6 + 3 + 0.6 + 0.6 = 10.2.
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3-SECOND BIOGRAPHY
JACOB BERNOULLI
1655–1705
A member of a famous Swiss mathematical family, he discovered the hugely important constant, e – the base of the natural logarithm – while investigating the subject of compound interest
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John McDermott
Percentages are widely used to describe price changes, especially to trumpet retail discounts.
FRACTIONS
the 30-second calculation
Dividing one whole number by another gives a fraction. The first (top) number is called the numerator while the second (bottom) is the denominator. A fraction may represent a proportion, some parts of a whole, so 3/4 represents three out of four equal parts. Three slices of a cake divided into four pieces is exactly as good as six slices of the same cake divided into eight; so 3/4 = 6/8. This works in general: multiplying or dividing top and bottom by the same number gives an equal fraction. A fraction in which the numerator is the larger number represents some whole unit(s) plus some fractional part. So 4/3 may represent one whole cake and one third of another and may be written as 11/3. Fractions can be added, subtracted, multiplied and divided. To add two fractions, first write them with a common denominator, then add the numerators. For example, to add 3/4 and 5/8, first re-write 3/4 as 6/8, so the sum becomes 6/8 + 5/8 = 11/8, or 13/8. To multiply fractions, multiply the two numerators together and the two denominators together. So, 3/4 × 5/8 = 3×5/4×8. To divide one fraction by another, invert the second (switch its numerator and denominator) and then multiply this by the first. For example, 3/4 ÷ 5/8 = 3/4 × 5/5 = 24/20, which can be simplified to 6/5 = 11/5.
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A fraction is a number formed from one whole number divided by another, written with the first above the second and a bar between; for example, 3/4.
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Fractions are in everyday use in great variety – describing quantities of food in recipes or for sale, as well as price discounts and special offers. Fractions are depicted on fuel gauges and describe positions on sports fields, and of course we use them to tell the time.
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FIBONACCI
c. 1170–c. 1250
The way in which we write fractions – the ‘bar’ notation with one digit above the other – was popularized by the Italian mathematician Fibonacci (real name Leonardo of Pisa)
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John McDermott
Fractions are parts of a whole.
ALGORITHMS
the 30-second calculation
Algorithms specify how to solve classes of problems in mathematics and computer science. Examples include the Euclidean algorithm for finding the greatest common divisor (GCD) of two numbers, a whole class of algorithms for sorting data, and the process by which the shortest route from A to B is found by a satnav system. An algorithm is a procedure with precise step-by-step instructions to be followed in order. This procedure may have one or more inputs, such as the start and finish points of a journey, and it will produce the solution, such as the shortest route, as an output. Due to this precise and mechanical nature, algorithms are suited to automation by computer in the form of programs. They underpin the functionality of most modern technology, so that each time we use a touchscreen, make a call, start an application or press a button in a car, we rely on an algorithm to work. Two algorithms devised for the same class of problems may have different qualities, in terms of efficiency or the nature of the solutions produced. Deciding, in an automatic way, which to select to tackle a problem is a task for artificial intelligence. This in turn includes the area of machine learning, which uses algorithms to look for patterns in data and then adapts the behaviour in its programs accordingly.
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An algorithm is a procedure for solving a mathematical or computational problem in a finite number of steps.
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Algorithms are used in weather forecasting, to handle share transactions and to provide suggestions based on user likes for Amazon, Netflix, Spotify and others. Google’s PageRank Algorithm determines the order in which search results are returned to the user and has been called the ‘50 trillion dollar algorithm’.
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3-SECOND BIOGRAPHY
MUHAMMAD IBN MUSA AL-KHWARIZMI
c. 780–c. 850
Persian mathematician, astronomer and geographer, from whose name the word ‘algorithm’ derives
30-SECOND TEXT
John McDermott
An algorithm is the recipe for a task suited to the working of a computer.
FALSE POSITIVES & FALSE NEGATIVES
the 30-second calculation
In a test with two possible outcomes, such as a blood test for the presence of a disease, there are two types of error that can occur: a false positive, or type I error, which incorrectly indicates the presence of the disease, or a false negative, or type II error, which fails to detect that the disease is present. The relative importance of the types of error is highly dependent on context. Failure to detect a disease – a false negative – can be fatal; the cost of a false positive in an email spam filter – loss of a genuine email – probably won’t be. Test reliability depends, in a non-intuitive way, on actual incidence of the condition. Suppose that a test is always correct for infected people, but only 99% of the time for people who are not infected. This test seems good, having a 0% false negative rate and 1% false positive rate. In a population of 1,000, if 20% are infected there will be 200 people with the disease, all correctly diagnosed – and 800 people without the disease, eight of whom will be incorrectly diagnosed as infected. The likelihood that a person diagnosed as having the disease actually has it is 200 in 208, or about 96%. Here the test works well. If the infection rate is a much lower 0.1%, then just one person in 1,000 has the disease. Of the 999 uninfected, approximately ten will be wrongly diagnosed. In this setting only 1 in 11, or 9%, of positive diagnoses is right.
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A false positive occurs when a test wrongly indicates the presence of a condition. A false negative occurs when lack of the condition is wrongly indicated.
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False positives and negatives arise in settings including medicine, business and computing, but also in the legal arena. The counterintuitive dependence of rates of these errors on incidence of a condition in the population (the base rate), may lead to poor decision-making or judgements.
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DANIEL KAHNEMAN
1934–
Nobel prize-winning psychologist who helped to develop the cognitive psychology of human error due to bias
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John McDermott
The consequences of a false positive or negative may be alarming, or even deadly.
PERMUTATIONS & COMBINATIONS
the 30-second calculation
A permutation of a set is an arrangement of some or all members of the set. Counting the number of ways in which this can be done for a given set and selection size is fundamental to understanding statistics and probability. Given four tasks to be completed, in how many ways can the order be chosen? Any one of the four tasks could be chosen to be first, leaving any one of the remaining three to be chosen second. Then two tasks remain to be chosen as third and, finally, there is just one option – the remaining unchosen task – to be completed last. So the number of ways of ordering the tasks is 24 = 4 × 3 × 2 × 1. The product 4 × 3 × 2 × 1 has its own mathematical shorthand: 4! (pronounced ‘four factorial’). To arrange three blue balls, one green ball and one red ball, two choices which differ only by a rearrangement of the blues (which can be done in 3! ways) are indistinguishable, so the number of different arrangements is 5!/3! = 20. A combination of items is chosen from a set without regard to order. There are 8!/(3! × (8 - 3)!) = 8!/(3! × 5!) ways to choose three items from eight, since the number of rearrangements of the three chosen, or the remaining five not chosen, should be discounted. The general formula for choosing a combination of r items from n objects is n!/(r! × (n - r)!).
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A permutation is a selection of members from a set in a specific order. A combination is a selection from a set where the order does not matter.
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Permutations and combinations arise in biology, computing, optimization and probability, on which lotteries are based. Take a lottery draw that selects six numbers, disregarding order, from a set of 59. There are 59!/(6! × 53!) ways of choosing these, giving 1 chance in more than 45 million of choosing the matching set.
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3-SECOND BIOGRAPHY
ÉVARISTE GALOIS
1811–32
French mathematician who studied the permutations of roots of polynomials and founded the theory named for him, which underpins the branch of algebra known as group theory
30-SECOND TEXT
John McDermott
Counting permutations or combinations allows us to understand probabilities.
