UNIT 4

ARITHMETIC

INTRODUCTION

Exercise 1. Discuss and answer the questions.

1) Which of these: 0.2p; ; ; 20p does not represent 20 percent of p?

2) Which of these: 2;3;4;5 is the closest square root of 11?
3) What is the solution if you subtract 17 from 3?
4) Round 7.982 to the nearest tenth.
5) What is the smallest factor of a prime number?
6)
7) How do these differ: a factor, a divisor and a multiple?
8) Write 54 .
9) How many types of brackets can you list?
10) Read these notations. Is there only one way to read them?

a) a| ________________________________________________________________
b) (c d) ________________________________________________________________
c) (2 + 7) 1 = 8 ________________________________________________________________
d) 36 ________________________________________________________________
e) ________________________________________________________________
f) ________________________________________________________________
g) a2 ________________________________________________________________
h) {a + x} ________________________________________________________________
i) 103 ________________________________________________________________
j) 6! ________________________________________________________________
k) (6+8)3+1 ________________________________________________________________
l) 7n+1 ________________________________________________________________
m) _________________________________________________________________
n) ________________________________________________________________

Exercise 2. Fill the gaps below with words of suggested meanings.

The ____________________________1 (an area of knowledge that may be considered apart from
related areas) of mathematics ____________________________2 (linked, working, dealing) with

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computations using numbers is called arithmetic. This can____________________________3
(comprise, include) number of specific topics the study of operations on numbers needed to
solve numerical problems; the ____________________________4 (a particular procedure for
accomplishing or approaching something, especially a systematic or established one) needed to
change numbers from one form to another (such as the conversion of fractions to decimals and
vice versa); or the abstract study of the number systems, number theory, and general operations
on sets as defined by group theory and modular arithmetic, for instance.
The word arithmetic comes from the Greek word , constructed from meaning
5
___________________________ (belonging
to the very distant past and no longer in existence) Greece, the term arithmetic referred only to the
____________________________6 (concerned with or involving the theory of a subject or area of
study rather than its practical application) work about numbers, with the word logistics used to
describe the practical everyday computations used in business. Today the term arithmetic is used
in both contexts.

TERMINOLOGY

Exercise 3. Complete the table with formal names of operands and results for
basic arithmetic operations.

SIGN ARITHMETIC OPERATION EXAMPLE NAME OF QUANTITIES

a, b = summands / addends
+ a + b= c
c=
a=
- a b=c b = subtrahend
c=
a =multiplicand
x axb=c b=
a, b = factors
c=
a =dividend
b=
c =quotient

Exercise 4. Math magic just for fun. Follow the instructions.

Take any 3-digit number with 3 distinct digits. Reverse the number. Now subtract the smaller
number from the larger. Reverse the difference. Add these (the difference and the reverse)
together. Add up the digits of the sum and you get ___________. Every time!

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 BINARY OPERATIONS AND THEIR PROPERTIES
Exercise 5. Read the text and complete the rules for properties of some binary
operations below.

An operator is any symbol, term, letter, etc., used to indicate or express a specific
operation or process. For example, + is an operator which is used to add two values.
A binary operation is an operation that requires two inputs. These inputs are known as operands.
The binary operation of addition, multiplication, subtraction, and division takes place on two
operands. Even when we add any three binary numbers, we first add two numbers and then
the third number will be added to the result of the two numbers. Thus, the mathematical
operations which are done with the two numbers are known as binary operations.
The fundamental property of many binary operations is commutativity or commutative
law, and many mathematical proofs depend on it. In mathematics, a binary operation is
commutative if changing the ______________________ 1 of the operands does not change the
______________________ 2. Stated symbolically: a + b = b + a and ab = ba.
The associative property states that when three or more numbers are added (or
multiplied), the sum (or the product) is the same regardless of ______________________________________3
Stated symbolically: (a+b)+c = a+(b+c) and (ab)c = a(bc)
The ________________________________4 property of multiplication and division is one of the
most frequently used properties in math. According to this property, multiplying the sum of two
or more addends by a number will give the same result as multiplying each addend individually
by the number and then adding the products together: a (b+c) = ab+ac and a(b-c) = ab-ac

Exercise 6. Evaluate the reasons for obtaining different results for

the following arithmetic expression. Then summarize the necessary
rules to obtain a correct solution.

8 : 2 * (2+2) = 1
8 : 2 * (2+2) = 16
Rule 1:
Rule 2:
Rule 3:

GREATEST COMMON DIVISOR AND

LEAST COMMON MULTIPLE

Exercise 7. Recall the criteria of divisibility and then complete the gaps in the
following sentences.

A natural number is divisible by number 2 if the natural number is _______________________.

A natural number is divisible by 9 if the sum __________________________________________.
A natural number is divisible by 3 if ________________________________________________.
A natural number is divisible by 5 if ________________________________________________.

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Exercise 8. Read the text and fill in the missing verbs.

Natural numbers _____________1 a lot of interesting properties. One of the important concepts is that
of the greatest common divisor (also called the greatest common factor (GCF)).
The greatest common divisor (GCD) of two natural numbers is the greatest (largest) positive
integer that _____________2 the numbers evenly, i.e. without a remainder.
Consider the two natural numbers 45 and 54. Find the GCD of 45 and 54
Step 1: _____________3 the divisors of given numbers: The divisors of 45 are: 1, 3, 5, 9, 15, 45
The divisors of 54 are: 1, 2, 3, 6, 9, 18, 27, 54
Step 2: Find the greatest number that these two lists _____________4 in common. In this example the
GCD is 9.
The least common multiple (LCM) of two natural numbers is the smallest (least) natural number
that is a multiple of each of the two given numbers. The least common multiple can also be
_____________5 of as the smallest (least) natural number that is _____________6 by both of the given
numbers. _____________7 the two natural numbers 6 and 8. Find the LCM of 6 and 8. The multiples
of 6 are: 6, 12, 18, 24, 30, . . . The multiples of 8 are: 8, 16, 24, 32, 40, . . . So, the Lowest Common
Multiple is 24. Any composite number can be _____________8 as a product of prime factors.
statement is the fundamental theorem of arithmetic.
The process of finding the prime factors of the given number, which when _____________9 together
give the original, is called prime factorization.

GRAMMAR

TENSES 3: EXPRESSING THE FUTURE

Exercise 9. Match the rules with suitable examples (A J).

Will-future is used a) to make predictions _____

b) to express spontaneous suggestions, decisions and
offers made at the time of speaking _____
c) to talk about future events that we see as facts _____
Going to is used a) to express planned events or intentions. These events
or intentions are decided on before the moment of
speaking _____
b) for future predictions based on physical evidence _____
Present continuous for planned or personally scheduled events _____
Present simple to express future events that are based on a timetable or
calendar _____
Future continuous is used to express an activity that will be in progress at a specific
time in the future _____
Future perfect is used to speak about what will have been finished by a time in
the future _____
Future perfect continuous is used to speak about how long something will have been
happening up to a point of time in the future _____

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 A going to ask my boss for a pay rise next week.
B
C
D
E Their share price will probably rise when the market recovers.
F
G Next year, the company will be 20 years old.
H I'll help you with the project if you want.
I By the end of this week, we will have been working on the project for a month.
J

Exercise 10. Complete the sentences by putting the verbs in brackets into the
correct future tense.

1) By the time all the papers are ready, the deadline _______________________________ (pass).
2) The flight ________________________ (leave) at 2 pm and ________________________ (arrive) at 4.20.
3) By the end of the term, she ______________________________________ (study) for 3 years.
4) We are late. By the time we get there the meeting _______________________________ (finish).
5) I _______________________________ (go) to Vienna on Monday.
6) Our costs were too high last year. This year, we _________________________ (reduce) our costs.
7) Would you mind waiting for a moment? It _______________________________ (not be) long.
8) What _______________________________ (you / learn) by the end of your course?
9) If you want me to, I_______________________________ (explain) how the equipment works.
10) Angela _______________________________ (find) a new job by the end of the year.

Exercise 11. Complete the text using the correct form of the verbs in brackets.

As the manager of our team, I have several exciting projects lined up for the future. Next month,
I ________________________1 (travel) to our headquarters in New York for a series of meetings with
senior executives. During my visit, I ________________________2 (discuss) our upcoming product
launches and ________________________3 (present) our marketing strategy for the next quarter.
By the end of the year, I ________________________4 (complete) a comprehensive review of our team's
performance and identified areas for improvement. We ________________________5 (implement) new
training programs to enhance our skills and productivity. By fostering a culture of creativity and
teamwork, I am confident that we ________________________6 (achieve) our goals and surpass
expectations in the years to come.
While I am away on business trips, my assistant ________________________7 (handle) day-to-day
operations and ________________________8 (ensure) that everything runs smoothly in my absence.
By the time I return, she ________________________9 (organize) all necessary documentation and
________________________10 (schedule) follow-up meetings with key stakeholders.

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READING
THE HISTORY OF ELECTRONIC CALCULATORS

Exercise 12. Read the paragraphs A-D and find words whose meanings are given
below.

advanced - ___________________________________ limitation - ____________________________________

affordable - __________________________________ machine - _____________________________________
to appear - ______________________________________ perform - ____________________________________
average - ______________________________________ reasonable - _________________________________
capacity - ______________________________________ results - ______________________________________
to create - _______________________________________ short-term - ___________________________________
crucial - ______________________________________ widespread - ________________________________
entirely - _____________________________________

A.
The temporary memory has a limited size, which can be readily illustrated by the simple expedient
of trying to remember a list of random items and seeing when errors begin to creep in. According
to research by psychologist George Miller the number of objects a typical human can hold in
working memory (known a
That's why people have been using aids to help them calculate since
ancient times. Indeed, the word calculator comes from the Latin calculare, which means to count
up using stones.

B.
The very first calculator was the Pascaline adding and subtracting device created by Blaise Pascal
in 1642. The golden age of calculators began in the 1800's. Technological and mechanical
challenges were faced by all those who devised early calculators. These restrictions often caused
early calculators to not function correctly, if at all. However, in the 1800's technological and
mechanical capabilities had improved enough for reliable mechanical calculators to be built and
operated.

C.
Mechanical calculators (ones made from gears and levers) were in extensive use from the late-
19th to the late-20th century. That's when the first cheap, pocket, electronic calculators started to
turn up, thanks to the development of silicon microchips in the late 1960s and early 1970s.
Calculators have much in common with computers: they share much of the same history and work
in a similar way, but there's one vital difference: a calculator is a completely human-operated
machine for processing math, whereas a computer can be programmed to operate itself and do a
whole range of more general-purpose jobs. Simple calculators carry out only the basic four
functions of addition, subtraction, multiplication, and division. More sophisticated calculators can
perform trigonometric, statistical, logarithmic, and other advanced calculations.

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D.
Although the calculator can shorten the time it takes to perform computations, we should keep in
mind that the calculator provides results that supplement, but do not replace, our knowledge of
mathematics. We must use our mathematical knowledge to determine whether
the calculator's outcomes are sensible and how the results can be used to answer a question.

Exercise 13. Complete the text with any suitable word (nouns, adjectives,
-).

1 steadily

declined since 2003. The latest Program for International Student Assessment (PISA) in 2018
showed only 10% of Australian teenagers ______________ 2 in the top two levels, compared
______________ 3 44% in China and 37% in Singapore.
Despite attempts to reform how we teach maths, it is ______________ 4
improve if they are not engaging ______________ 5 their lessons.
______________ 6 teachers, parents, and policymakers may not be aware of is research shows students
-
There are four main non-thinking behaviours:
- slacking: where there is no attempt at a task. The student may talk or do nothing.
- stalling: where there is not real attempt at a task. This may involve legitimate off-task
behaviours, such as sharpening a ______________ 7.
- faking: where a student pretends to do a task but achieves nothing. This may involve
legitimate on-task behaviours such as drawing pictures or writing numbers.
- mimicking: this includes ______________ 8 attempts to complete a task and can often involve
completing it. It involves referring to others or previous examples.
One ______________ 9 for students slacking and stalling is teachers are doing most of the talking and
directing, and not providing enough ______________ 10 for students to think.
11 reduce the prevalence of non-

thinking behaviours?
Here are two research-based ideas.
Form random groups Our studies found random groupings improved ______________ 12
willingness to collaborate, reduced social stress ______________ 13 caused by self-selecting groups,
and increased enthusiasm for mathematics learning.
Get kids to stand up We found groups of about three students standing together and working
on a whiteboard can ______________ 14 thinking behaviours. Just the physical act of standing can
eliminate slacking, stalling, and faking behaviours. The additional ______________ 15 of only allowing

beneficial.
Abridged.Muir, T., Liljedahl, P. (2023, May 30). From whiteboard work to random groups, these
simple fixes could get students thinking more isimple fixes could get students thinking more in
maths lessons. The conversation. Retrieved Sept 10, 2024 from
https://theconversation.com/from-whiteboard-work-to-random-groups-these-simple-fixes-
could-get-students-thinking-more-in-maths-lessons-203059 )

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 REVISION
Exercise 1. Fill the gaps with suitable expressions.

Arithmetic is the oldest and most _e____________________1 (the most basic) branch of mathematics.
The term arithmetic also refers to the properties of integers related to prime numbers, divisibility,
and the solution of equations in integers. P_____________________2 (a series of actions/operation pl.)
that take one or more numbers as input and produce a number as output are called numerical
operations.
______________________3 operations take a single input number and produce a single output number.
______________________4 operations take two input numbers and produce a single output number.
______________________5 is the basic operation of arithmetic. In its simplest form, it combines two
numbers, the _____________________6 or terms, into a single number, the ____________________7 of
the numbers. It is commutative and associative so the order the terms are added does not matter.
The opposite operation is _____________________8. It finds the difference between two numbers, the
_______________________9 minus the _________________________10. This operation is neither commutative
nor associative. Multiplication is the second basic operation of arithmetic. It combines two
numbers into a single number, the ____________________11. The two original numbers are called the
__________________12 and the ___________________13 (or they are both simply called _________________14).
Multiplication is commutative and associative; further it is distributive over addition.
____________________15 is essentially the opposite of multiplication. It finds the __________________16 of
two numbers, the ___________________17 divided by the __________________18. Natural numbers have a lot
of interesting properties. The _____________________19 of two natural numbers is the largest positive
integer that divides the numbers without a remainder. Another concept that goes hand in hand
with the previous one is the ___________________20 - the smallest (least) natural number that is a
multiple of each of the two given numbers.

Exercise 2. Complete the sentences by putting the verbs in brackets into the
correct future tense.

1) Have you heard the news? AMC _______________________________ (buy) C-tech.

2) They _______________________________ (study) for five hours by six o'clock.
3) I think I _______________________________ (change) my Internet Service Provider.
4) In the future videoconferences _______________________ (probably replace) many international
meetings.
5) We _______________________________ (open) a factory in Hungary next year.
6) How long _______________________________ (work) here when you retire?
7) Dr. Jones _________________________ (give) the same talk in room 193 at 10.00 next Thursday.
8) You _______________________________ (learn) enough German to communicate with the delegates by
the day of the conference.

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Exercise 3. Complete each gap with only one word. Choose from the words
below. Change the word form if necessary. Each word should be used
only once unless indicated otherwise.

multiplication 3x number zero add 3x yield

principal undefined undo one definition 2x

Subtraction may be ___________________1 as the inverse of ___________________2, and division may be

___________________3 as the inverse of ___________________4. This is one reason for regarding addition and
multiplication as the ___________________5 operations in arithmetic and algebra.

The meaning of inverse in mathematics is similar to its meaning in common everyday experiences.
We can think of it as an ___________________6 operation.

The major laws for addition and multiplication, such as the commutative, associative, and
distributive laws, have been discussed. We shall now observe some other properties of the
operations that are useful in algebra. These additional properties are not required for a number
system but are necessary in constructing other kinds of mathematical systems.

Zero is called the identity element for _______________7 because zero added to any
________________8 a gives the same number.
We call a the inverse element for addition (or the ________________9 inverse of a) because, for
each replacement of a from the set of real numbers, -a is a number which on addition to a
gives ___________________10.
One is called the identity element for ________________11 because, when one is multiplied by any
number a, it ________________ 12 that same number.
We call the inverse element for multiplication (or the ________________ 13 inverse of a)
because, for each replacement of a, a 0, from the set of real numbers, is a number which
on multiplication by a gives ________________ 14.
Note that zero has no multiplicative inverse, since the symbol 1/0 is an ________________ 15
symbol.

(by A.B. Evenson, Modern mathematics, introductory concepts and their implications, p.50-52,
abridged )

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