Numbers
Types of numbers
Numbers in a group together may be called a series or set of numbers. If the order in which
they occur is significant then they may be called a sequence of numbers. 1, 4, 9, 16, 25 is a
sequence of numbers, for example - it represents the numbers 1 to 5 squared.
1,3,5, 7 ... = odd numbers; 2, 4, 6, 8 ... = even numbers; 2, 3, 5, 7, 11 ... = prime numbers.
The highest number in a group is the maximum and the lowest is the minimum. The room
holds a maximum of 50 and we u/on't run the class without a minimum of 12 students.
An approximate number is one which is roughly correct but is not the precise or exact
number. Look at the figures and work out in your head what the approximate answer is
tikely to be. Then use a calculator to find the exact number.
An aggregate is a number reached by totalling a set of numbers = the total. The average
mark achieved in the exam is calculated by taking the aggregate of alt the marks and
dividing by the number of exam entries.
A discrete number or unit is something which is separate and cannot be divided into smaller
numbers or units of the same thing. The opposite of discrete is continuous. A bag of apples,
for example, could be considered as consisting of discrete items whereas apple sauce could be
considered - by mathematicians, at least - as continuous.
A constant number or quantity is one that does not change. In the experiment u/e varied
[changed] the amount of water in the beaker but kept the amount of salt added constant.
A random number is one chosen by chance, i.e. it isnot predictable.
Working with numbers
The word figure is often used to refer to the symbol used for a number. Write the total
number in words and figures.
Verbs that are frequently used with the word number include calculate [work out] a number,
estimate ' a number, round a number up/down/, total [add up] a set of numbers. Numbers
can also tally", My figures don't seem to tally with yours. You can also deduct [take away,
subtract] one number from another number.
I make a rough guess at 2 make a fraction, e.g. *or 0.78 into the nearest whole number
3 match, agree
Values and variables are also useful terms when working with numbers. Values are individual
numbers in a set of data. The graph shows the temperature values for different months of the
year. Variables are characteristics that can take on different values for different members of a
group or set being studied. In investigating living standards you must take key variables such
as social provision and cost of tiving into account.
The incidence of something refers to how frequently it occurs. The incidence of twins in the
population is growing. When talking about numbers, magnitude simply refers to the size of
something, whereas in other contexts it indicates large size or importance. Write down the
numbers in order of magnitude, beginning with the smaltest.
When making calculations in, say, an exam, it is often a good idea to make an estimate" first
of what the answer is likely to be. Then you will see if your final answer is in the right area5
or not. Exam candidates are also often advised ro show their workings" so that the marker
can see how they arrived at their answer and they may get credit for their method even if the
final answer is incorrect.
4 rough guess 5 approximately the same 6 leave all their calculations on the page
58 Academic Vocabulary in Use
25.
25.2
Exercises
25.1 Answer these questions.
1 What is five squared?
2 What is the next prime number after 19?
3 How is this sequence of numbers created? 3, 9, 27, 81
4 What is the aggregate of this set of test marks? 6, 8,9, 5, 6, 7
5 If you round up 6.66, what number do you have?
6 ~ and 4 - which is a whole number and which is a fraction?
7 ln your country is tax automatically deducted from employees' earnings?
8 Is an accountant pleased or displeased if figures that he/she is checking tally?
25.2 Dr Syal is advising one of his dissertation students who is interested in pollution in road
tunnels. Complete the conversation. You are sometimes given the first letter to help you.
Dr Syal: You could Cp the total number of private cars that use the tunnel each
week, based on the day-to-day figures, and get an a.p.p.p.P.P.P.P.P.P..P.P.P.figure for how
much carbon they're all emitting.
Melissa: How p..P.P P.P.PPPP.P.would that figure have to be?
Dr Syal: Oh, it doesn't ha ve to be exact, you just need to e... .P....Pmore or less what
the tatal pollution will be. Then you can check to see if those figures t.
with the figures that have already been published for similar tunnels. And the
figure won't be c of course; iťll go up and down depending on lots of
factors such as weather conditions, average speed, etc.
Melissa: But can we say if the figur es will be true for the future too?
Dr Syal: Well, we do know that the traffic growth has been cP.P....P.P.P.P.P.P..P.P....over the past ten
years; it hasn't ever gone down, so I think you can ma ke some useful predictions.
Melissa: Should I present each daily total as a dpp ..p.P.P.P.P.item or can I just put them all
together into one figure for each week?
Dr Syal: A weekly total is fine, and you can PP..P.ppit up or to the
nearest 100.
Melissa: Right, OK. Thanks so much for your help.
25.3 Rewrite these spoken sentences so that they are more appropriate for writing, using the
word in italics in an appropriate formo
1 There were fewer car accidents last year. incidence
2 We made a rough guess at what the final figure might be. estimate
3 The graph shows the results from the lowest to the highest. magnitude
4 A computer program helped us work out the significance of the different variables. calculate
5 Taking x away from y will help you arrive at the correct answer. subtract
6 The results from the first experiment were not the same as those we got from the repeat
experiment. tally
25.4 Fill in the gaps in this advice a maths lecturer is giving her students.
ln the exam, don't forget to show all your (1) .P as we want to see how you
(2) ..P.P.PP...P.P.P.PP.....P.Pat your results. Make your (3) p ...very carefully - you'd be amazed
at how many people submit answers that are hardly even in the right (4)ppppppp . And
please write legibly - we must be able to distinguish all your (5) ..... .p.....pp ! When doing
graphs, plot your (6) .. ......carefully and if asked to describe an experiment don't
forget to take all significant (7) into account. Good luck!
I
·Find some examples of the use of numbers in your own subject area. Note down some
interesting phrases or sentences.
Academic Vocabu/ary in Use 59
~. . r
Statistics
Basic statistical terms
Notice the key vocabulary in these three short texts about statistics.
~~------~ I
.. Six children are 7, 8, 8, 8, 11 and 12 years old.. f
( Their average age is 9 years old (the sum of their ~
~ ages divided by six)..Th~ mode (tne most fre.quent t
\ value) is 8. The med lan 158 (the halfway point \.
I between the two extremes of the range). ~
l .,...,.--·--......-...-,.,-.~-"""--_.-...-
t
;
A normal dístríbutton ot data means tnat !I
most of the examples in a set of data are close í
\
to the average, while relatively few examples \
tend to one extreme ar the other. Normally (
\ distributed data shown on a chart will typically J
show a bell (urve. It will often be necessary 1
l
to work aut the extent to which individuals ~
deviate' from the norm? and to calculate the }..
figure that represents standard deviatiorr'. )
--~~~~....-...".,..-..---."...-~..--~~.--'
(
1;
Statisticians are often concerned with working aut
corretatlcns" - the extent to which, say, left-handedness
correlates with intelligence. They must ensure that any
data they collect is valid, i.e. that it is measuring what it
claims to measure - all the subjects in the sample ' must be
appropriately and accurately assessed as left- ar right-handed,
for example. The figures must also be reliable, i.e. they would
be consistent" if the measurements were repeated. Usually,
statisticians hope that their calculations will show/indicate
a tendency, e.g. that left-handed people will be shown to be
significantly7 more intelligent than right-handed people.
~ . .~---1
1 differ 2 the average 3 average difference from the norm 4 connections, often as cause and
effect 5 the subjects of the experiment or group representing the total population measured
6 the same 7 noticeably
A probability' problém
Notice the vocabulary in this problem from a statistics textbook.
Sue picks a card at random/ from an ardinary pack of 52 cards. If the card is a king, she
stops. If not, she continues to pick cards at random, without replacing them, until either
a king is picked ar six cards have been picked. The random variable",C, is the total
number of cards picked. Construct a diagram to illustrate the possible ourcomes"of the
experiment, and use it to calculate the probability distribution5 of C.
1 likelihood of something happening 2 by chance 3 number or element of a situation that can
change 4 results 5 assessment of probabilities for each possible value of C
Other useful nouns for talking about statistics
In a class of 8 women and 4 men, what proportion ' are male? Answer: one third
In the same class what is the female to male ratio/? Answer: 2:1
The figures show a trend ' towards healthier eating habits.
The study investigates the increase in the volume" of traffic on the roads.
1 number compared with another number 2 relationship between two numbers showing how
much bigger one is 3 change in a particul<l;.rdirection 4 amount, quantity
----
\)~. ~
,{-niC
We say 10 per cent (NOT the 10 per ceRt or 10 perceRtage) of students got an A for
their exam but the percentage of students achieving an A has increased.
60 Academic Vocabu/ary in Use
can
Exercřses
26.1 Complete the sentences.
1 The six subjects who took the test scored 24, 22, 16, 16, 16, and 14 points out of 30. The
......................................was 16. The . score was 16 and the . . score was 18.
2 The .... ..of all donations to the charity in 2003 was $3,938. The smallest donation
was $10 and the largest was $130. Most were around the point of $60.
3 Each questionnaire item asked respondents to choose one of a . . ..of six options,
with the two being 'very dissatisfied indeed' and 'completely satisfied'.
26.2 Use the correct form of the words in the box to complete this text.
distribute trend varysignificant probable random correlation outcome
Life insurance companies base their calculations on the laws of . , that is
they assess the likely. . , given the different.... such as age, sex,
lifestyle and medical history of their clients. The premiums are therefore not chosen at
......................................but are carefully calculated. The , of ages at which death occurs
and causes of death are studied to see if they . with other factors to be
taken into account in setting the premiums. Naturally, the companies also monitor social
..........................and react to any changes which might affect mortality rates.
26.3 Answer the questions.
1 There are 12 male students and 6 female students in the class. What is the ratio of males
to females? And what proportion of the class is male?
2 If I am collecting data on course choices among second-year undergraduates and my
sample is too small, what exactly do I need to do?
3 If my data show that students ha ve a tendency to choose the type of clothing their
friends choose, does it mean that they always, often or rarely choose similar clothes?
4 If I repeat the same experiment three times and the results are not consistent, is my
method reliable?
5 If 20 out of 200 students fail an exam, what proportion, in percentage terms, failed?
6 If the average score in a test is 56, and Barbara scores 38, by how many points has she
deviated from the norm?
7 If the volume of court cases increases, what changes: the type of case, the size of each
case or the total number of cases?
8 What does standard deviation tell us? (a) What the standard of something is, (b) what
the norm is, or (c) what the average difference from the norm is?
9 If a general survey of teenage eating habits asks questions about what teenagers eat for
breakfast and lunch, is the survey likely to be valid?
10 Here is a graph showing how many 25,---------------
students got scores within each 10-mark
band in a biology test. Are the scores
normally distributed? What is the shape of
the graph called?
.1' What kinds of statistical data are likely to be
discussed in your discipline? Find a relevant
• chart, graph or table and write about it using
some terms from this unit.
30-39 40-49 50-59 60-69 70-79 80-89 90-100
20-l------f---~------
E
'"~15+-----r----~~----
'O
~ 10+-----~------~----
E
"c:
o+---~--~--_r--_r--~--~--~
range of scores
Academic Vocabulary in Use 6 I
27
cross-section
Graphs and diagrams
table
Types of dlagrams
pí, chart Loobar chart ~ histogram
Diagrams are visual ways of presenting data concisely. They are often also called
figures. In an academic article they are usually labelIed Fig. (Figure) 1, Fig. 2, etc.
A pie chart is a circle divided into segments from the rniddle (like slices of a cake) to
show how the total is divided up. A key or legend shows what each segment represents.
A bar chart is a diagram in which different amounts are represented by thin vertical or
horizontal bars which ha ve the same width but vary in height or length. A histogram is
a kind of bar chart but the bar width also varies to indicate different values.
A table is a grid with columns and rows of numbers.
A cross-section is something, or a model of something, cut across the middle so that you
can see the inside. A cross-section of the earth's crust, for example, shows the different
layers that ma ke it up. A label gives the name of each part of the cross-section. Crosssection
can also be used to mean a small group that is representative of all the different
types within the total group (e.g. the survey laaked at a cross-section af saciety).
A flowchart is a diagram which indicates the stages of a process.
flowchart
D-D-D
1:1Weekly
packet
maney
received bor
teenagers
in the UK
A graph '"30
'"
The graph presents data relating to teenagers and pocket ~ 25
" 20
money. A random sample of 1,000 teenagers were .§ 15
surveyed and the average pocket money received at ~ 10
each age has been plotted on the graph. The x axis or ~ 5
horizontal axis indicates age and the y axis or vertical '" 0-+1'3-1"4---ri5-1T"'6---'l'7-'T"ia-1;'g
axis shows the amount of money received per week. The age
graph shows that 1S-year-olds receive twice as much pocket money as 13-year-olds. From
the graph we can see that the amount received reaches a peak at the age of 18 and then
starts to decline. This decline can perhaps be explained by the fact that many teenagers start
earning and stop receiving pocket money at the age of 18.
Graphs are drawn by plotting points on them and then drawing a line to join adjacent
points. If there are two lines on a graph - separate lines, for example, to indicate boys' and
girls' pocket money - then the lines would probably cross or intersect at various points. Lines
that run paralIel to one another never intersect.
Graphs show how numbers increase or decrease. The nouns increase and decrease have the
stress on the first syllable, but the verbs have the stress on the second syllable. Numbers can
also be said to rise or grow and fall, drop or decline. The nouns rise, growth, fall, drop and
decline, like increase and decrease are followed by in (to explain what is rising) or of (to
explain the size of the change), e.g. a rise af 10% in the number af cars. Other verbs used
about growth include double', soar', multiply', appreciate" and exceeď.
I grow to twice the size; opposite = halve 2 (dramatic word) rapid movement upwards; opposite
= plummet 3 grow rapidly to a very large number 4 used about the value of something, e.g.
a painting or car; opposite = depreciate 5 go over, expresses a number in relation to another
number; opposite = fall below
Note that graph is a noun and graphlc [relating to drawing: vivid, especially when
describing something unpleasant] is usually an adjective. The economics textbook contains
a lot offascínating graphs. My nephew studied graphic design. The book contains some very
.graphic descriptions of the massacre. Graphics can be used as a plural noun to refer to
pictorial material, e.g. The graphics in that computer game are bríllíant
V,,,
~
62 Academic Vocabulary in Use
Weekly
packet
maney
received by
teenagers
in the UK
start
Exerc;ses
27.1 Look at the chart. Complete the commentary with words from the opposite page.
cars entering downtown West City
270,000
,
O number
- af cars
per day
-
-
~
260,000
~
ě 250,000
'O
1: 240,000
E
::>
c: 230,000
220,000
210,000
2 4 5 6 7
year
8
The chart.H...H ...H... the number of cars entering the downtown area
of West City each day over an eight-year period (years 1-8). The totals
are listed on the .......HH H axis (gíve two answers), while the years
are listed on the ..H.. HH axis (gíve two answers). To the right of
the graph we see the .. ..HH. The number of cars ..
over the period. The total rose in the first few years and "HHHH'HH'HH'Ha
HH'HHHHH'HH'H'Hin year 5, after which the numbers started tO..H .... HHHHH
This can beHHH'HH'H by the .. 'HHH'H' that a new mass transit
railway was opened in year 6, which is a "H'H' illustration of how
good public transport can dramatically affect car use.
27.2 Answer the questions.
1 Draw examples of a pie chart and a bar chart.
2 What would be the best type of diagram to present the different layers of rock in the
Grand Canyon?
3 ln a table, what is the difference between columns and rows?
4 What would be the best type of diagram to present the different stages in a research
project you did?
5 How many segments are there in the pie chart opposite?
6 If you look at two adjacent columns in a table, are they next to each other ar separated?
7 What is another name for a legend in a diagram?
8 What type of data collection are you doing if you survey the first 50 people you come across?
9 What do two lines on a graph do if (a) they intersect and (b) they run parallel to each other?
27.3 Make the rather informal words in bold sound more precise and academic.
1 The different bits of the pie chart show the numbers of people in each age group.
2 She kept a record by marking the midday temperature on a graph for a month.
3 People's salaries usually reach their highest point when they are in their late 40s.
3 This flowchart shows the different bit s of our project over the next five years.
5 The two line s on the graph cross each other at point A.
6 Draw a line connecting the points that are next to each other.
7 The governmenťs popularity in the opinion polls is beginning to fall,
8 If you look along the top line of the table you can see the figures for the 1950s.
27.4 Change the sentences using words with the same meanings as the words in bold.
1 Populations of some bird species in South Asia have crashed by 97% in recent years.
The number of cases of death by poisoning has increased sharply.
2 ln 2007 the child mortality rate was lower than 60 deaths per 1,000.
3 The average family car in the UK goes down in value by 20% per year. This means its
value has fallen by more than half after just three years.
4 A typical piece of land on the edge of the city will go up in value by 15% per year, and
house prices have gone up rapidly in the last six months.
5 Business courses have increased great1y in number while science programmes have gone down.
6 The temperature was higher than 45°C in some parts of the country during the heatwave.
7 Between 1983 and 2006, the number of this species of condor: went up from 22 pairs to
273. Other bird populations have gone up by two times in the same period.
8 The numbers of old soldiers attending regimental reunions are becoming smaller each year.
".large birds from South America
Academic Vocabulary in Use 63