STUDIUM JĘZYKÓW OBCYCH

POLITECHNIKI ŁÓDZKIEJ

2011/2012

English for Mathematics

a short course for engineering students

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English for Mathematics

a short course for engineering students

NUMBERS!NUMBERS!NUMBERS!NUMBERS!

I.

When do we use the word number and when do we use the word numeral?

Complete the text with the appropriate word.

A n__________ is an abstract entity that represents a count or measurement. In

mathematics, the definition of a number has extended to include fractions, negative,

irrational, transcendental and complex n____________s.

A n___________ is a symbol or group of symbols, or a word in a natural language that

represents a n____________. N____________s differ from n__________s just like words

differ from the things they refer to. The symbols ‘11’, ‘eleven’ and ‘XI’ are different

n__________s, all representing the same n___________. In common usage, n___________s

are often used as labels (e.g. road, telephone and house numbering), as indicators of order

(serial n__________s), and as codes (ISBN)

(Adapted from English for Mathematics)

II.

Read the sentences carefully. Pay close attention to the numbers in brackets.

Use the proper form of a numeral in each sentence according to the context.

1)

Radar was first used in World War (2).

2)

I have a train to catch at (12:48).

3)

Elizabeth (2) comes from the House of Windsor.

4)

I was born on June (3), (1975)

5)

Ben’s telephone number is (205891)

6)

In the last match England beat Poland (2:0).

7)

John McEnroe was leading (30:0) in the (2) game of the (1) set when the match was

broken off due to a thunderstorm.

8)

The dictionary costs ($28.50)

9)

“The match is being watched by (27,498) spectators,” said the voice from the

loudspeakers.

10)

The temperature in Italy rarely falls below (0).

11)

Chris saves (1/2) of his pocket money for summer holidays.

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12)

The area of Canada is (3,851,790) square miles.

13)

Halloween is observed on October (31) and Thanksgiving on the (4) Thursday of

November.

14)

About (3/5) of energy produced in the USA comes from coal and crude oil.

15)

If you want to pass this test, (51%) of your answers must be right.

16)

Pelican Airways are sorry to announce that flight no. (003) to Ouagadougou is

cancelled today because of a dust storm.

17)

A meter is equal to (0.9144) yards.

18)

“Open your books to page (374),” asked the teacher.

19)

The Earth’s volume is about (0.000003) of the Sun’s volume.

20)

This hotel was built in the (1930)’s.

21)

Poland’s foreign debts amount to (40,000,000,000) dollars.

22)

You need a (12) eggs to make this layer cake.

23)

The signature time of a waltz is (3/4).

24)

After the accident, Burt spent (102) days in hospital.

25)

My school is about (2 ½) miles from my house.

26)

Henry (8) reigned in the (1) (1/2) of the (16) century.

27)

3

2

= 9

28)

√9 = 3

29)

6 + 3 = 9

30)

9 - 3 = 6

31)

10 : 2 = 5

32)

5 x 2 = 10

33)

log

7

49 = 2

34)

4! = 24

35)

E = mc

2

36)

Na

2

O + H

2

O → 2NaOH

37)

Janice is (5’4”) tall.

38)

The score is (15:15) and Agassi is on his (2) service.

39)

The USA won the (4x400) relay race in Seoul.

40)

About (2) (20) speakers took part in the parliamentary debate on national defence.

(by Tomasz Kasper)

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III.

Listen and write down the numbers that you hear in the following sentences.
Each sentence is repeated twice.

1.

Current research shows that ___________ Americans stop smoking each year.

2.

Could you give Jack a call at ___________ ?

3.

We're thinking about getting a house. Currently, the average mortgage is about

____________.

4.

____________ new jobs have been created in the high tech sector over the past

____________ years.

5.

Jane is celebrating her __________ birthday next Monday!

6.

___________ of all Americans eat a hamburger at least once a week.

7.

The density of hydrogen is ____________ in that compound.

8.

So, what time shall we get together next week? What do you say if we meet for lunch

at _____________ .

9.

Statistics show that flossing __________ a day can greatly improve general dental

hygiene.

10.

Wall Street closed up _____________ .

(From http://esl.about.com/library/listening/blnumbers1.htm)

IV.

Listen and write down the numbers that you hear in the following sentences.
Each sentence is repeated twice.

1.

Parsifal was first premiered at Bayreuth in ______________.

2.

Fred's Office Supplies turned an incredible profit of ____________ in this past

quarter.

3.

I'm sure you will find that the ATU ______________ is a remarkable machine.

4.

Athletes from over ____________ countries will be participating in the next meeting

to be held on the __________ of September.

5.

Peter won the bean counting contest with a guess of ____________ beans.

6.

Tiger Woods shot an incredible _____________ under par on the back _________ .

7.

By the time of his death in ____________, Roger Frankline had accumulated over

____________ patents.

8.

It is estimated that the new tax reform will cost the government _______________.

9.

His new computer cooks! He's got ___________ Mb Ram with a _____________ Mhz

processor.

10.

Relax! There are _____________miles left to go.

(From http://esl.about.com/library/listening/blnumbers2.htm)

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BASIC OPERATIONS

ADDITION

6 + 8 = 14

addend addend sum

SYMBOLS

WORDS

7 + 6 = 13

The sum of 7 and 6 is 13

8 + 9 + 6 = 23

The total of 8, 9 and 6 is 23

4 + 7

The number 4 increased by 7

x + 4 = 13

4 more than x is 13

a + b = c

a plus b equals c

a + b = c

a add b equals c

A shortcut for adding is called carrying. It involves three steps:
1.

Write the problem vertically and line up numbers with the same place value.

2.

Add the numbers in each column separately moving from the right to the left.

3.

If the sum of any column is greater than 9, put down the appropriate digit in the
ones place and carry the other digit to the next column to the left.

Example:

199

9 + 8 + 5 = 22

58

Put down 2. Carry 2 to the tens place.

+ 75

2 + 9 + 5 + 7 = 23

Put down 3. Carry 2 to the hundreds place.

The sum is 332.

I.

Fill in missing words in the example of long addition given below.

Step 1

1684

+795

Step 2

4 + 5 = 9

The _____________ of 4 and 5 equals 9.

Step 3

8 + 9 = 17

The number 8 ______________ by 9 is 17. ___________ down 7. __________

1 to the hundreds place.

6 + 7 + 1 = 14
The __________ of 6, 7 and 1 is 14. __________ down 4. ___________ 1 to

the __________ place. The final _________ is 2479.

(Adapted from English for Mathematics)

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BASIC OPERATIONS

SUBTRACTION

9 – 3 = 6

minuend subtrahend difference

SYMBOLS

WORDS

9 – 3 = 6

The difference between 9 and 3 is 6

13 – 4

13 decreased by 4

17 – 9 = 8

9 from 17 is 8

x – 5 = 9

5 less than x is 9

b – a

Subtract a from b

A shortcut for subtracting is called borrowing. It involves three steps:
1.

Write the problem vertically and line up numbers with the same place value.

2.

Subtract the numbers in each column separately moving from the right to the left

3.

If the digit in the minuend is less than the digit that has the same place value in
the subtrahend, rewrite the minuend by borrowing 1 from the digit immediately
to the left of the smaller digit and adding 10 to the smaller digit.

Example:

62

Since in the ones column 2 < 7, we must borrow.

- 37

Since 1 ten = 10 ones, borrow 1 from 6 in the tens
column to get 5 tens, and add 10 to the 2 in the ones
column to get 5 ones.

Subtract 12 – 7 = 5

Subtract 5 – 3 = 2

The result is 25.

I.

Fill in the missing words in the example of long subtraction given below.

Step 1

1365

- 978

0

2 5

Step 2 and 3

1

1

3

1

6

1

5

- 9 7 8

3 8 7

Since in the ones column 5 < 8, ________ 1 from the 6 in the __________ column to get
5 tens. __________ 10 to the 5 in the ones column. _________ 8 from 15. 15 – 8 = 7.

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We are now left with 5 in the tens column. Since in the tens column 5 < 7, __________ 1
from the ________ column to get 2 hundreds. ________ 10 to the 5 in the tens column.
_________ 7 from 15.

Since in the hundreds column 2 <9, _________ 1 from the _________ column. Since 1
thousand = 10 hundreds, _________ 10 to the 2 in the hundreds column. The final result
is 387.

Exercises:

Write the following problems vertically and give step-by-step instructions for:

a)

936 + 685

b)

36 + 87 + 12

and fill in the missing words in the instructions for subtraction:

c)

1004 – 237

Since 4 < 7, __________ 1 from the tens column. Yet, the tens column is zero, so we
move to the hundreds column and finally to the thousands column. We have to borrow 1
_________= 10 hundreds = 10 x 10 tens. Now, we are able to borrow from the
_________ column.

14 – 7 = 7
14 __________ by 7 equals 7.
In the tens column, we are now left with 9.

9 – 3 = 6
3 _________ than 9 is 6.
In the hundreds column, we are now left with 9 units, too.

9 – 2 = 7
2 _________ 9 is 7.
In the thousands column, we are now left with 0.
The final ___________ is 767.

Check the result by ____________ the _____________ to the subtrahend.

Now, follow the example above and do the same for:

d)

352 – 228

e)

743 – 184

(Adapted from English for Mathematics)

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BASIC OPERATIONS

MULTIPLICATION

7 x 8 = 56

multiplier multiplicand product

factors

If the multiplication problem is written vertically, by convention, the larger number is
considered the multiplicand and written on top.

SYMBOLS

WORDS

8 x 7

8 multiplied by 7

8 times 7

The product of 8 and 7

To multiply whole numbers:
1.

Write the problem vertically and place the number with the longer number digit on
top and the smaller below it.

2.

Multiply each digit of the top number (multiplicand) by the ones digit in the bottom
number (multiplier), moving from right to left.

3.

For a product that exceeds 9, carry the rightmost digit to the next column on the left
and write it above the multiplicand. Circulate the next product and be sure to add to
that product the digit that was carried.

4.

Multiply each digit in the multiplicand by the next digit to the left in the multiplier.
Place each product under the previously calculated one, but displaced one column to
the left.

5.

Repeat step 4 for all remaining digits in the multiplier.

6.

Add the products to get the final result.

I.

Solve the multiplication problem and complete the missing words in the
instructions.

325 x 68 = ?

Multiply 325 by 8.

8 x 5 = 40

Put ________ 0, carry 4.

8 x 2 = 16

16 + 4 = 20

___________ 0, carry 2.

8 x 3 = 24

24 + 2 = 26

___________ 26.

___________ 325 _______6.

6 x 5 = 30

___________ 0, ________ 3.

6 x 2 = 12

12 + 3 = 15

___________ 5, ________ 1.

6 x 3 = 18

18 + 1 = 19

___________ 19.

Now, _________ the products.
The final result is ___________.

(Adapted from English for Mathematics)

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BASIC OPERATIONS

DIVISION

a : b = c

dividend divisor quotient

SYMBOLS

WORDS

a : b = c

a divided by b equals c

a/b

a over b

a/b

The quotient of a and b

If you want to divide 31 by 4, write the problem as shown.

________

quotient

31 : 4

Think what biggest integer multiplied by 4 will give a product less than or equal to 31.
It is 7. Write 7 in the space for the quotient.
Multiply 7 x 4 = 28.
Subtract 28 from 31. 31 – 28 = 3, the remainder.
The quotient is 7, the remainder is 3.

I.

Complete the instruction for solving ________

(quotient) using the words

369 : 7

from the box. There are more words than necessary.

PLACED ○ QUOTIENT ○ REMAINDER ○ ABOVE ○ DIVISOR ○ QUOTIENT ○ NUMBER

BROUGHT ○ SUBTRACTED ○ RIGHT ○ DIVISOR ○ DIVIDEND ○ RESULT ○ SUBTRACTED

Division starts from the left of the _____________, and the ____________ is written on
the line above. Start from the left, the divisor is divided into the first digit or set of digits
it divides into. In this case, 7 is divided into 36, the __________ is 5, which is placed
above 6. It is then multiplied by the ___________ and the product is ___________ from
the set of digits in the dividend first selected. 5 x 7 equals 35, 35 subtracted from 36
equals 1. The next digit to the ___________ in the dividend is them brought down and
the divisor is divided into this number. Here, 9 is brought down and the divisor is divided
into 19, the result is 2, which is placed __________ the 7. The result is multiplied by the
____________ and the product is ____________ from the last number used in division.
7 x 2 = 14; 14 subtracted from 19 equals 5. This process is repeated until all digits in the
dividend have been ___________ down. The result of the last subtraction is the
__________. The number placed above the dividend is the ___________.

(Adapted from English for Mathematics)

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HIERARCHY OF MATHEMATICAL OPERATIONS

Most mathematical operations: addition, subtraction, multiplication and division are
normally performed in a particular order or sequence. Multiplication and division are
done prior to addition and subtraction. Usually, mathematical operations are performed
from left to right. The use of parentheses is common to set apart operations to be
performed in a certain order.


I.

Complete the instructions for solving equations with the words from the box.
There are more words than you need.

(4 x 2) + (3 + 2) +

(

)

= ?

OUTSIDE ○ BEFORE ○ ALL ○ INSIDE ○ PRIOR ○ OPERATIONS ○ MOVE

1.

___________ from left to right within the equation and within the set of parentheses

2.

First, perform all ___________ within the parentheses.

4 x 2 = 8
3 + 2 = 5

(

)

= = 4

Addition of 5 and 3 was performed __________ to division.

3.

Perform ________ operations _________ the parentheses. Move from left to right.
8 + 5 + 4 = 17

II.

Solve the equation [3 x (2 + 4) – 5 + 2] x 3.
Match the operations with their descriptions.

1.

2 + 4 = 6

2.

(3 x 6 – 5 + 2) x 3

3.

(18 – 5 + 2) x 3 = (18 – 3) x 3 = 15 x 3

4.

15 x 3 = 45

a)

Perform multiplication outside the brackets. _____

b)

Rewrite the equation. _____

c)

Perform operations in the innermost set of parentheses. _____

d)

Perform multiplication prior to addition and subtraction within the brackets. _____

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BRACKETS

{} – braces, curly brackets
[] – square brackets, brackets
() – parentheses (sing parenthesis), round brackets
<> - angle brackets

The word bracket is commonly used to mean any bracket if there is only
one set of brackets involved.

III.

Reconstruct the rules. There is always one word you do not need.

1.

Expanding brackets, or removing brackets, is writing an ___________ such as 3(x + 2)
in an ___________ form, in this case 3x + 6, _________ any brackets.

EQUIVALENT, WITHOUT, SIMILAR, EXPRESSION

2.

To multiply out a ___________ of brackets, for example (x + 5)(x + 10), each
__________ in the second bracket is multiplied __________ the first bracket.

TERM, PAIR, AGAINST, OVER

3.

In the expression 4(2 + 3), we say that 4 ____________ both the bracketed numbers
or 4 ____________ itself ___________ 2 and 3.

MULTIPLIES, OVER, DISTRIBUTES, MULTIPLIED

4.

We can ___________ expressions nested in various sets of brackets. In order to do
that we have to __________ from the _________ out.

WORK, INSIDE, ACT, SIMPLIFY

5.

To keep our notation easy to understand, we follow the __________ that working
from the inside out, we write the ___________ in parentheses, then in brackets, and
then in ___________.

BRACES, ROUND BRACKETS, EXPRESSIONS, CONVENTION

6.

To factorize 7(3 + x), the common ___________ must be written __________ the
bracketed ________, in other words, it has to be taken out of the brackets.

QUOTIENT, OUTSIDE, FACTOR, TERM

IV.

Solve the equation and reconstruct the rules.

[(5 – 3) + (4 x 3) – (8 : 4)] : 2 = ?

1.

Perform math operations ____ each set of parentheses.

2.

Perform addition and subtraction ____ left ____ right.

3.

Perform division _____ the brackets.

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V.

Complete the crossword.

1

2

3

4

5

6

7

8

9

10

11

12

13

14

15

16

17

18

19

1.

[…]

2.

a plus b

3.

<

4.

… system

5.

a x b

6.

a times b

7.

a + b + 2d = c

8.

a decreased by b

9.

(…)

10.

The number that divides

11.

The result of

6

12.

a – b

13.

nought

14.

the number remaining after the

procedure of

17

is completed

15.

the result of

17

16.

the result of

8

17.

a divided by b

18.

the number divided into another

number

19.

the result of

2

What is the phrase in the vertical column?

(Adapted from English for Mathematics)

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FRACTIONS, ROOTS AND POWERS

A manufacturer is thinking about giving both metric measurements (for example,

millimetres) and imperial measurements (for example, inches) in its product

specifications. One of the company’s engineers is giving his opinion on the idea in a

meeting.

‘One problem is, when you convert from metric to imperial you no longer have whole

numbers – you get long decimal numbers. For example, one millimetre is nought point

nought three nine three seven inches as a decimal. So to be manageable, decimals have to

be rounded up or down. You’d probably round up that number to two decimal places, to give

you zero point zero four. Now, you might say the difference is negligible – it’s so small it’s

not going to affect anything. But even if it’s just a tiny fraction of a unit – one hundredth of

an inch (1/100), or one thousandth of an inch (1/1000) – and those numbers are then used in

calculations, the rounding error can very quickly add up to give bigger inaccuracies.’

1 mm = 0.03937 inches ≈ 0.04 inches

I. Write the numbers in words.

1.

…………………………………………………………………………………………………..

2.

3.0452

………………………………………………………………………………………………….

3.

………………………………………………………………………………………………….

4.

………………………………………………………………………………………………….

5.

………………………………………………………………………………………………….

6.

0.25

…………………………………………………………………………………………………

7.

…………………………………………………………………………………………………

8.

0.16

…………………………………………………………………………………………………

II.

Complete the descriptions of the numbers using words from the text above.

1.

0.25 = ¼

The first number is a decimal, and the second is a …………………. .

2.

0.6368 ≈ 0.637

The second number is …………………… ………………….. to three

…………………. …………………. .

3.

7.5278 ≈ 7.5

The second number is ………………….. …………………… to one

………………… …………………. .

4.

8, 26, 154

The numbers aren’t fractions or decimals.

They’re ……………………. numbers.

5.

Error: 0.00001%

The error is so small that it’s ………………………... .

6.

0.586 kg x 9,000 = 5,274 kg

0.59 kg x 9,000 = 5,310 kg

This difference is the result of a ……………… ……………….. .

(Adapted from Professional English in Use)

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III.

How are these values spoken?

1.

x²

2.

x³

3.

xⁿ

4.

x

1

−

n

5.

x

n

−

6.

x

7.

3

x

8.

n

x

IV.

Practise reading these expressions:

1.

x

p

−

=

p

x

1

2.

x

q

p /

=

q

p

x

3.

x² - a² = (x + a) (x - a)

4.

y = ae

kx

5.

x =

n

m

mx

nx

+

+

2

1

6.

y - y

1

=

)

(

1

2

1

2

x

x

y

y

−

−

( x - x

1

)

7.

1

2

2

2

2

2

2

=

+

+

c

z

b

y

a

x

8.

d =

[

]

2

2

1

2

2

1

2

2

1

)

(

)

(

)

(

z

z

y

y

x

x

−

+

−

+

−

9.

b

2

= a

2

( 1 – e

2

)

10.

x

2

+ y

2

+ 2gx + 2fy + c = 0

(Adapted from Basic English for Science)

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READING MATHEMATICAL EXPRESSIONS

I.

Read out these equations:

1. x =

c

b

a

+

2.

b

a

A

y

x

−

=

+

3. I = a + (n - 1) d

4. V= IR

5.

f

v

u

1

1

1

=

+

6. v = u + at

7. Ft = mv – mu

8.

EI

M

R

−

=

1

9.

q

dz

dQ

−

=

10. E = T + P – c + e

II.

Here is the Greek alphabet. Make sure you know how this is read.

α Α

β Β

γ Γ

δ Δ

ε Ε

ζ Ζ

η Η

θ Θ

ι Ι

κ Κ

λ Λ

μ Μ

ν Ν

ξ Ξ

ο Ο

π Π

ρ Ρ

σ Σ

τ Τ

υ Υ

φ Φ

χ Χ

ψ Ψ

ω Ω

Listen and repeat.

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III. Practise reading out the expressions:

1.

f =

LC

π

2

1

2.

E =

4

T

δ

3.

W

S

=

P

f

π

2

4.

F

R

W

π

γ

4

0

=

5.

μ

0

= 4 π × 10

7

−

Hm

1

−

6.

C =

2

2

2

L

R

L

ω

+

IV. Now listen and write down the formulae you hear.

References:

Donovan P., Basic English for Science, Oxford, OUP 1997.

Ibbotson M., Professional English in Use, Cambridge, Cambridge University Press 2009.

Krukiewicz-Gacek A., Trzaska A., English for Mathematics, Kraków, AGH University of

Science and Technology Press 2010.

Websites:

www.mathwords.can

www.about.com

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English for Mathematics

English for Mathematics

English for Mathematics

English for Mathematics

Glossary

acute angle – kąt ostry

add – dodawać

addend – składnik sumy

addition – dodawanie

adjacent – przyległy

angle – kąt

base – podstawa

base-ten system – system dziesiątkowy

bisector – symetralna odcinka, dwusieczna kąta

bottom – dolny

bracket – nawias

broken line – linia przerywana

circle – okrąg, koło

circumcircle – okrąg opisany

circumference – obwód koła

circumscribe about – opisać na

common fraction – ułamek zwykły

common logarithm – logarytm zwykły, dziesiętny

congruent – przystający

curve – krzywa

decimal fraction – ułamek dziesiętny

denominator – mianownik

derivative – pochodna

diagonal – przekątna

diameter – średnica

difference – różnica

digit – cyfra

displace – przenosić, przesuwać

divide – dzielić

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dividend – dzielna

division – dzielenie

divisor – dzielnik

dotted line – linia kropkowana

equation – równanie

equilateral triangle – trójkąt równoboczny

even numer – liczba parzysta

expanded notation – zapis w formie rozszerzonej

extract a root – wyciągać pierwiastek

factor – czynnik

factorial – silnia

factorize – rozkładać na czynniki

formula – wzór

fraction – ułamek

greatest common factor/divisor – największy wspólny dzielnik

height – wysokość

horizontal – poziomy

hypotenuse – przeciwprostokątna

inequality – nierówność

inscribe in – wpisać w

integer – liczba całkowita

isosceles triangle – trójkąt równoramienny

LCD (the least common denominator) – najmniejszy wspólny mianownik

leg – przyprostokątna

minuend – odjemna

multiplicand – mnożna

multiplication – mnożenie

multiplier – mnożnik

multiply – mnożyć

natural number – liczba naturalna

naught/nought – zero

negative number – liczba ujemna

number – liczba

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numeral – cyfra (np. arabska lub rzymska)

numerator – licznik

obtuse angle – kąt rozwarty

octagon – ośmiokąt

odd number – liczba nieparzysta

operation – działanie

ordinal number – liczba porządkowa

parallel – równoległy

parallelogram – równoległobok

pentagon - pięciokąt

perimeter – obwód

perpendicular (to) – prostopadły, wysokość (np. trójkąta)

positive number – liczba dodatnia

power - potęga

prime number – liczba pierwsza

product – iloczyn

proper fraction – ułamek właściwy

quotient – iloraz

raise a number to a power – podnosić liczbę do potęgi

rational number – liczba wymierna

real number – liczba rzeczywista

reciprocal – wielkość odwrotna

rectangle – prostokąt

recurring decimal – ułamek dziesiętny okresowy

reduce to lowest terms – skrócić/uprościć ułamek

remainder – reszta

repeating decimal – ułamek dziesiętny okresowy

rhomboid – równoległobok

rhombus – romb

right angle – kąt prosty

root – pierwiastek

round – zaokrąglić (np. liczbę)

satisfy an equation – spełnić równanie

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semi-circle - półkole

side – bok

sketch a graph – narysować wykres

solution – rozwiązanie

solve an equation – rozwiązać równanie

square – kwadrat

square root – pierwiastek kwadratowy

subscript – indeks dolny

subtract – odejmować

subtraction – odejmowanie

subtrahend – odjemnik

sum – suma

superscript – indeks górny

take a root – wyciągnąć pierwiastek

tangent (to) – styczna (z)

top - górny

trapezium/trapezoid – trapez

vertex - wierzchołek

wavy line - linia falująca

zigzag – linia łamana