Chapter 1

August 25, 2017 | Author: Mena Gorgy | Category: Spreadsheet, Bracket, Numbers, Google, Multiplication
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01 NCM7 2nd ed SB TXT.fm Page 2 Saturday, June 7, 2008 7:07 PM

NUMBER

Mathematics began with people counting, and many civilisations came up with symbols to represent numbers. As people around the world started to cross paths, a common number system was needed. Eventually the Hindu–Arabic system was adopted all over the world. It is important to understand how our number system works and the rules it follows.

1

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01 NCM7 2nd ed SB TXT.fm Page 3 Saturday, June 7, 2008 2:30 PM

0123456789

48901234 0123456789

0123456789 567890123

4567890123 8901234 0123456789

3 8901234

In this chapter you will: 456789 012345 0123456789 678901234

56789012345678 5678

567890123

4567890123 8901234 Wordbank 0123456789 4567890123456789012345 1234567890123456789

56789012345678 5678

• compare the Hindu–Arabic number system with • cube root The value which, if cubed, will give the 234567890123 45678901234 0123456789 0123456789 56789012 456789012345678901234567890123456789012345678 56789012 number systems from7890123456789012 different societies, past and required number, for example 3 64 = 4 because 8 3456789012345678 present 901234567890123456789012345678901234567890123456789012345678 789012345678 3 4 = 64. 567890123456789012345678901234567890123456789012345 • recognise, read 56789012345678901234567890 and convert Roman numerals 1234567890123456789 0123456789 • evaluate To find the value of a numerical 78901234 5678 234567890123 45678901234 • state the place value of any digit in large numbers 0123456789 expression. 456789013456789012345678901234567890123456 7890123456789012 34567890 • order numbers of any size, in ascending and 456789012 9012345 6789012345678901234567890123456789012345678901234567890123456789 • expanded notation A way of writing a number 6789012345678901234567890123456789012345678901234567890123 descending order shows the place value of every digit. 89012345678 345678 9012345 67890123456789012345678 that 90123456789012345678901234567890123456 7890123456789012 345678901234567890123 901234567890123 123456789012345678901234 5678 012345678901234 5678901 • record large numbers using01234567890 expanded notation • Hindu–Arabic number system The number 0123 4567890 1234567890123456789 012345678901234567 890123456789012345678901234567890123456789012345678901234567890123456789012345678 45678901234567890123456 789012345678901234567 8901234 678901234567890 234567 • revise the four basic operations on whole numbers system we use, with the numerals 0, 1, 2, 3, 4, 5, 2345678901 756789012345678901234 45678901234 6789012345678901 34567890123456789012345678901234567 0123456789012345 901234567890123456789 5678901234567890123456789012345678901234567890123456789012345678901234567890123456789012345678901234567890123456789012345678901234567 6, 7, 8 and 9. • apply order of operations to simplify expressions 4567890123456789012340123456789012345678901234567890123456789012345678901234567890123456789012345678901234567890123456789012345678901234567890123456789012 • numeral A symbol that stands for a number, such 890123456789012345678 567890123456789012345678901234567890123456789012345678901234567890123456789012345678901234567890123456789012345678901234567890 123456 • divide two-digit and three-digit numbers by a two345678901234567890123901234567890123456789012345678901234567890123456789012345678901234567890123456789012345678901234567890123456789012345678901234567890 as 8 or X. digit number 4567890123456789012345678901234567890123456789012345678901234567890123456789012345678901234567890123456789012345678901234567890123456 890123456789012345678 234567890123456789012 9012345678901234567890123456789012345678901234567890123456789012345678901234567890123456789012345678901234567890 12345678901234567890 • order of operations The rules for calculating an • use3456789012345678901234567890123456789012345678901234567890123456789012345678901234567890123456789012345678901234567890123456789012345 the symbols of mathematics, including 789012345678901234567 3 234567890123456789012 8901234567890123456789012345678901234567890123456789012345678901234567890123456789012345678901234567890123456789012345678901234567890 expression containing mixed operations, such as and . 678901234567890123456 34567890123456789012345678901234567890123456789012345678901234567890123456789012345678901234567890 1234567890123456789012345678901234 14 − 2 × 4 + 1. 1234567890123456789017890123456789012345678901234567890123456789012345678901234567890123456789012345678901234567890123456789012345678901234567890123456789 6789012345678901234562345678901234567890123456789012345678901234567890123456789012345678901234567890123456789012345678901234567890123456789012345678901234 that the position of a digit • place value The way123456789012345678901234567890123456789012345678 012345678901234567890 789012345678901234567890123456789012345678901234567890123456789012345678901234567890 5678901234567890123451234567890123456789012345678901234567890123456789012345678901234567890123456789012345678901234567890123456789012345678901234567890123 in a number tells us its value. 0123456789012345678906789012345678901234567890123456789012345678901234567890123456789012345678901234567890123456789012345678901234567890123456789012345678 456789012345678901234 1234567890123456789012345678901234567890123456789012345678901234567890 12345678901234567890123456789012345678901234567890123456789012 9012345678901234567895678901234567890123456789012345678901234567890123456789012345678901234567890123456789012345678901234567890123456789012345678901234567 4567890123456789012340123456789012345678901234567890123456789012345678901234567890123456789012345678901234567890123456789012345678901234567890123456789012 890123456789012345678 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01 NCM7 2nd ed SB TXT.fm Page 4 Saturday, June 7, 2008 2:30 PM

Start up Worksheet 1-01 Brainstarters 1 Worksheet 1-02 Multiplication facts

Skillsheet 1-01

1 Write the answers to the following. a 10 × 10 b 4×7 d 7+9 e 10 × 10 × 10 g 9×9 h 26 − 8 j 6×5 k 99 ÷ 11 m 18 × 3 n 7 × 12 p 128 ÷ 4 q 137 + 45 s 452 − 140 t 280 × 10

c f i l o r u

2 Write each of these numbers in words. a 45 b 120 d 3680 e 5001

c 138 f 47 613

900 + 30 35 ÷ 5 1000 + 200 + 50 75 − 16 128 − 24 35 × 12 3601 − 59

3 Write each of the following numbers using numerals. a sixty-eight b seven hundred c two thousand and four d eight hundred and ninety-nine e ten thousand, four hundred and ninety-two

Reading and writing large numbers

1-01 The ancient Egyptian number system The ancient Egyptians used one of the earliest number systems about 5000 years ago. Pictures called hieroglyphs represented words or sounds. They were written on papyrus (a type of paper made from reeds) or painted on walls. The hieroglyphic symbols used by the Egyptians were: 1

2

10

3

20

...

4

100

(coiled rope)

4

5

200

...

6

...

1000

(lotus flower)

10 000

100 000

1 000 000 (million)

(bent reed)

(fish)

(man with hands raised in surprise)

NEW CENTURY MATHS 7

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01 NCM7 2nd ed SB TXT.fm Page 5 Saturday, June 7, 2008 2:30 PM

Example 1 Show how an ancient Egyptian would have written each of these numbers. a 25 b 126 c 3468 Solution a

b

c

Example 2 If ancient Egyptian numerals could be written in any order, how could 125 be written? Solution or

or

Exercise 1-01 1 If you were an ancient Egyptian student, how would you write these numerals? a 7 b 37 c 165 d 268 301 e 3 251163 f 1253

Ex 1

2 Use our numerals to write the numbers represented by these Egyptian numerals. a b c d

e

3 Write the answer to these in Egyptian numerals. a minus

b plus

c plus

d minus

CHAPTER 1 THE HISTORY OF NUMBERS

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01 NCM7 2nd ed SB TXT.fm Page 6 Saturday, June 7, 2008 2:30 PM

4 State one advantage and one disadvantage of working with ancient Egyptian numerals. 5 Why do you think a picture of a surprised man was used by the ancient Egyptians to represent a million?

1-02 Australian Aboriginal number systems The Australian Aboriginal way of life had no need for a complicated number system. Their society relied on story-telling, using the spoken language rather than writing, and Aboriginal people did not have symbols for numbers. Different regions had their own names for numbers. The Belyando River people of central Queensland used only two words to name their numbers: 1 = wogin 2 = booleroo 4 = booleroo booleroo 3 = booleroo wogin The Kamilaroi people lived in northern New South Wales, including the regions surrounding Moree and Tamworth. They used three words to name their numbers. 1 = mal 2 = bularr 4 = bularr bularr 3 = guliba 6 = guliba guliba 5 = bularr guliba

Exercise 1-02 1 How did the Belyando River people form words for the numbers 3 and 4? 2 How did the Kamilaroi people form words for 4, 5 and 6? 3 Answer the following, using the correct Aboriginal words: a wogin + booleroo wogin b guliba × bularr c bularr + bularr + mal d booleroo × booleroo e guliba guliba − guliba f bularr bularr − mal 4 State one advantage and one disadvantage of working with Aboriginal numbers.

1-03 The Babylonian number system The ancient kingdom of Babylon existed from about 3000 to 200 BC where Iraq is today. Babylonian writing used wedge shapes called cuneiform. The wedges were stamped into clay tablets which were then baked. Babylonian numerals also used cuneiform. While our number system is based on 10 and 100, the Babylonian number system was based on 10 and 60. This wedge stood for 1: A sideways wedge stood for 10: A larger wedge stood for 60:

6

1

2

10

20

3

4

30

NEW CENTURY MATHS 7

...

5

60

70

...

80

9

...

120

130

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Example 3 Show how a Babylonian would have written each of these numbers. a 15 b 252 Solution a b For numbers greater than 60, we need to find how many 60s divide into them. 252 ÷ 60 = 4 and remainder 12 because 4 × 60 = 240 So 252 = (4 × 60) + 10 + 2. In Babylonian numerals, 252 is:

Exercise 1-03 1 How would you write each of these numerals using our numerals? a b

c

d

Notice that there was no need for a zero. 2 Use Babylonian numerals to write each of these amounts. a 26 b 58 c 107 d 300 e 144 f 401

Ex 3

3 State one advantage and one disadvantage of working with the Babylonian number system.

1-04 The Roman number system The Roman empire was one of the greatest empires. Roman numerals were invented about 2000 years ago. They were used until the end of the 16th century. Today they are used mainly in clocks and for some page numbers in books. The Romans used the following numerals: 1 2 3 4 5 I II III IV V 6 7 8 9 10 VI VII VIII IX X 50 L

100 C

500 D

Skillsheet 1-02 Roman numerals Worksheet 1-03 Roman numerals

1000 M

CHAPTER 1 THE HISTORY OF NUMBERS

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The Romans had an unusual method of writing certain numbers: • Instead of writing 4 as IIII, they wrote IV meaning V − I (that is 5 − 1 = 4). • Instead of writing 9 as VIIII they wrote IX meaning X − I (that is 10 − 1 = 9). • For 90, they wrote XC (that is 100 − 10 = 90).

Example 4 Write each of the following in Roman numerals. a 23 b 46 c 101

d 249

Solution a 23 is XXIII

d 249 is CCXLIX

b 46 is XLVI

c 101 is CI

Exercise 1-04 1 Titus, a student in ancient Rome, wrote these numerals. Change them into our numbers. a XXVI b XL c CCLXIV d LIV e MMCLIX f MCMXC g XCVIII h MDVII 2 What would Titus have written for these numbers? a 365 b 36 c 79 e 2600 f 344 g 999

Ex 4

d 97 h 3473

3 Why do you think Roman numerals are no longer widely used? 4 The Roman word for hundred was ‘centum’ which is why C stands for 100. List some words beginning with ‘cent’ that mean one hundred of something.

1-05 The modern Chinese number system Chinese people today use the numerals below.

Worksheet 1-04

1

2

3

4

6

7

8

9

10

100

5

Ancient Chinese rod numerals

1000

10000

Worksheet 1-05

• The Chinese write from top to bottom. • The symbols in a number are grouped in pairs and the numbers in each pair are multiplied together. • The products are added to give the number.

Mayan numerals

8

NEW CENTURY MATHS 7

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Example 5 Write each of these Chinese numbers using our number system. a b

Solution a 6 × 10 = 60

b

3 × 100 = 300

+

4= 4 64

7 × 10 = 70 5=

+

5 375

Exercise 1-05 1 Use our numerals to rewrite these Chinese numerals Zhang Li wrote. a

b

c

d

Ex 5

e

2 If you were writing to Zhang Li, how would you write each of these numbers using Chinese numerals? a 13 b 46 c 175 d 999 3 What are the difficulties in working with modern Chinese numerals?

Working mathematically

Communicating

Calendar month Make a calendar for the month of your birthday using a different type of number system. Are some number systems easier to use than others? Why?

CHAPTER 1 THE HISTORY OF NUMBERS

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1-06 The Hindu–Arabic number system Worksheet 1-06 Ancient number systems

Our number system goes back to the Hindus (who lived in India) and came to Europe through the Middle East/Arabia. Our system needs only ten symbols called digits: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9. It is easier to use because it has a zero and the position of each numeral determines its value. This is called place value. The numerals first appeared in Europe in the 10th century, but were different to the ten numerals we use today. The following table shows how our numerals have changed over time. Numerals Date

Origin

200 BC

Hindu

1

2

Hindu

AD

800

Hindu

AD

900

Arabic

AD

976

Spanish

AD

1400

Italian

AD

1480

Caxton (Printer)

AD

2

3

4

5

6

7

8

9

0

10

The Hindus called the zero ‘sunya’ meaning a void. Other names used were ‘cipher’, ‘nought’ and the Arabic ‘sifr’. Even today, different cultures use different symbols: or

or

or

Place value Skillsheet 1-03

We can write any number using only ten symbols or digits. When we write numbers, each column has a special value called the place value.

Place value

Example 6 Worksheet 1-07 Big numbers

Write the value of each of the digits in 4625. Solution In 4625:

10

5 has a value of 5 2 has a value of 20 6 has a value of 600 4 has a value of 4000

NEW CENTURY MATHS 7

or or or or

5×1 2 × 10 6 × 100 4 × 1000

01 NCM7 2nd ed SB TXT.fm Page 11 Saturday, June 7, 2008 2:30 PM

Example 7 What is the value of each of the digits in 501? Solution In 501:

1 has a value of 1 (zero used to mark a place) 0 means there are no tens 5 has a value of 500 Another way to show the meaning of each digit in a number is with a place-value table. Ten thousands

Thousands

4

8

2

Hundreds

Tens

Ones

1

3

8

138

6

2

5

4625

5

0

1

501

3

5

0

82 350

Example 8 What value does the digit 5 have in: a 57?

b 235?

Solution a In 57, the 5 has a value of 50 (or 5 tens). b In 235, the 5 has a value of 5 (or 5 units).

Exercise 1-06 1 Write the value of each digit in the following numbers, then write each number in words. a 609 b 1039 c 70 104 d 504 860 e 9 134 671 f 5 837 000 g 4001 h 205 689 i 34 000 036

Ex 6

2 Write each of the following using numerals. a eight thousand, seven hundred and ninety-six b three million and eighty-eight c two thousand, three hundred and eighty-five d six thousand, nine hundred and seven e four hundred and twenty thousand, eight hundred and thirty f three hundred and nine thousand, two hundred and eleven g one million, two hundred and eighty thousand, four hundred and sixty h twelve million, nine hundred and one CHAPTER 1 THE HISTORY OF NUMBERS

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3 What are the advantages of using a Hindu–Arabic number system? 4 What is the value of 7 in 237 601? Select A, B, C or D. A 7 hundred B 7 thousand C 70 thousand D 7 hundred thousand 5 In 2 982 645, which digit is in the ten thousands place? Select A, B, C or D. A2 B9 C8 D6 Ex 7

6 Place these numbers in a place-value table, as shown on the previous page. a 48 b 382 c 2751 d 3020 e 15 364 f 44 040

Ex 8

7 What is the value of the digit 5 in each of these numbers? a 45 b 1057 c 1526 d 12 345 e 65 013 f 51 480 260 8 What is the value of the digit 3 in each of these numbers? a 123 b 2356 c 32 185 d 85 532 e 1 385 264 f 3 485 260 9 What is the value of the digit 4 in each of these numbers? a 4281 b 124 386 c 6004 d 4 316 725 e 362 154 f 1 426 813 10 Arrange the numbers in each of these sets in order, from smallest to largest. a 321, 17, 8000 b 17, 707, 27, 63 c 246, 3596, 5369, 432, 16, 6125 d 123, 321, 132, 231, 213 e 1045, 450, 145, 82 f 721, 243, 43, 4372, 722 g 380 211, 308 022, 300 806, 392 084 h 4 856 231, 4 766 372, 1 429 950, 3 006 853 11 How many times is the first 3 bigger than the second 3 in each of these numbers? a 1433 b 1343 c 3143 d 2 352 312

Worksheet 1-08 Base 8 number system

1-07 Expanded notation One way to show the place value of each digit in a number is to use expanded notation.

Example 9 Write each of these numbers using expanded notation. a 345 b 3287 Solution a 345 = (3 × 100) + (4 × 10) + (5 × 1) = 3 × 102 + 4 × 10 + 5 × 1 b 3287= (3 × 1000) + (2 × 100) + (8 × 10) + (7 × 1) = 3 × 103 + 2 × 102 + 8 × 10 + 7 × 1

12

NEW CENTURY MATHS 7

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102 10 squared means 10 × 10 = 100 103 10 cubed means 10 × 10 × 10 = 1000 104 10 to the power of 4 means 10 × 10 × 10 × 10 = 10 000 The power of 10 shows how many zeros follow the 1 in the number.

!

Exercise 1-07 1 Write each of these numbers using expanded notation. a 56 b 3562 c 416 d 502 f 10 253 g 38 002 h 59 644 i 3809

Ex 9

e 1001 j 120 435

2 Write each of these as a single number. a (5 × 100) + (2 × 10) + (4 × 1) b (6 × 1000) + (5 × 100) + (3 × 10) + (7 × 1) c (4 × 102) + (2 × 10) + (9 × 1) d (6 × 103) + (4 × 102) + (7 × 10) + (3 × 1) e 8 × 104 + 2 × 103 + 3 × 102 + 4 × 10 + 3 × 1 f 3 × 103 + 0 × 102 + 5 × 10 + 7 × 1 g 7 × 104 + 6 × 103 + 0 × 102 + 0 × 10 + 1 × 1 h 1 × 104 + 0 × 103 + 9 × 102 + 9 × 10 + 9 × 1 i 3 × 105 + 4 × 104 + 4 × 103 + 2 × 102 + 2 × 10 + 0 × 1 j 9 × 105 + 0 × 104 + 0 × 103 + 9 × 102 + 9 × 10 + 9 × 1 3 What is 9047 in expanded notation? Select A, B, C or D. A 9 × 1000 + 4 × 100 + 7 × 10 B 9 × 1000 + 4 × 10 + 7 × 1 C 9 × 1000 + 4 × 100 + 7 × 1 D 9 × 100 + 4 × 10 + 7 × 1 4 Find out what ‘to expand’ means. Is the dictionary meaning the same as the one in mathematics?

Just for the record

Googol-plexing The number 10100, the googol, is 1 followed by one hundred zeros. The name ‘googol’ was created by the 9-year-old nephew of American mathematician Dr Edward Kasner. The number 10googol, that is 1 followed by a googol zeros, is called the googolplex. The googol is a very big number but it is rarely used for practical purposes. Even the number of particles in the observable universe, estimated at being between 1072 and 1087, is less than a googol! The Internet search engine Google was named after the googol, to reflect the huge size of the world wide web. It was invented in 1996 by two Stanford University students, Larry Page and Sergey Brin. Google is a powerful search engine because it can find information from at least 25 billion web pages in less than 1 second. How many googols are there in a googolplex?

CHAPTER 1 THE HISTORY OF NUMBERS

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Mental skills 1A

Maths without calculators

Multiplying by a multiple of 10 Place value allows us to simply add zeros to the end of a number whenever we multiply by a power of 10. The zeros at the end shift all the other digits one or more places to the left which results in them having higher place values. 1 Examine these examples. a 37 × 10 = 370 b 45 × 100 = 4500 c 16 × 1000 = 16 000 d 100 × 1000 = 100 000 e 7 × 90 = 7 × 9 × 10 = 63 × 10 = 630 f 5 × 400 = 5 × 4 × 100 = 20 × 100 = 2000 g 12 × 300 = 12 × 3 × 100 = 36 × 100 = 3600 h 40 × 800 = 4 × 10 × 8 × 100 = 4 × 8 × 10 × 100 = 32 × 100 = 32 000 2 Now simplify these. a 18 × 100 e 315 × 1000 i 3 × 80 m 2 × 6000 q 5 × 80

b f j n r

26 × 1000 1000 × 1000 8 × 200 11 × 900 25 × 20

c g k o s

77 × 10 000 294 × 10 6 × 50 4 × 400 300 × 60

d h l p t

10 × 100 475 × 100 7 × 30 5 × 700 900 × 4000

1-08 The four operations Worksheet 1-09 Four operations

There are four basic operations in our number system: + addition × multiplication − subtraction The old symbols for writing these operations are:

÷ division

We will now review these operations.

Example 10 Copy and complete this number grid.

+

5

14

8 12 Solution +

5

14

8

8

12

14

+

12

NEW CENTURY MATHS 7

5

14

5+8

14 + 8

13

22

5 + 12 14 + 12

17

26

+

5

14

8

13

22

12

17

26

01 NCM7 2nd ed SB TXT.fm Page 15 Saturday, June 7, 2008 2:30 PM

Example 11 Copy and complete this number grid.

+

12 30

20

65

Solution +

+

12

30 − 12

30 20

12

18

65

20

30 20 + 12

32

65 − 20

+

12

45

63

18

30

63

65

20

32

65

45

18 + 45

Exercise 1-08 Use the link to Worksheet 1–10 to print the number grids in this exercise. 1 Copy and complete these number grids. a

+

3

4

b

+

15

41

c

+

7

28

8

2

19

5

11

Ex 10

9

Worksheet 1-10 Number grids Worksheet 1-11 Arithmagons

2 Copy and complete these number grids. a top row − side column b top row − side column −

19



25

54

TLF

c top row − side column −

78

7

37

128

12

26

239

The take-away bar: go figure

243 412

TLF

×

11

9

b

×

L 90

The multiplier: go figure

TLF

L 2006

The divider: with or without remainders

3 Copy and complete these number grids. a

L 102

2

5

c

×

8

15

10

5

23

17

12

20

CHAPTER 1 THE HISTORY OF NUMBERS

TLF

L 82

The multiplier: make your own hard multiplications

15

01 NCM7 2nd ed SB TXT.fm Page 16 Saturday, June 7, 2008 2:30 PM

4 Copy and complete these number grids. a top row ÷ side column b top row ÷ side column ÷

Ex 11

36

÷

48

32

c top row ÷ side column ÷

64

4

8

4

3

4

5

60

100

5 Copy and complete these number grids. a

+

b

10

+

50 80

16 26

100

c

+ 22

28

6

13

33 14

6 Find the missing numbers (top row − side column). a



20

15

b

8



c

17

9

7

15

9

− 9

12 11

11

7 Find the missing numbers. a

×

b

5

3

×

5

12

56

28

7

c

×

10

6

90

40 4

8 Find the missing numbers (top row ÷ side column). a

÷

6

16

b

24 3

8

4

2

NEW CENTURY MATHS 7

c

÷ 4 24

÷

72 24

10 5

01 NCM7 2nd ed SB TXT.fm Page 17 Saturday, June 7, 2008 2:30 PM

Using technology Skillsheet 1-04

What is a spreadsheet?

Spreadsheets

A spreadsheet is like a calculator. We can enter data and solve many problems more easily, using an Excel spreadsheet. Spreadsheets are made up of many cells. As we go across the page, we change the column (A, B, C, D, etc.). As we go down the page, the row changes (1, 2, 3, 4, etc.). Using formulas To write a formula in a cell, always start with an equal sign ‘=’. A spreadsheet uses special symbols to do calculations. Consider these basic operations: a =A1+A2+A3 or =sum(A1:A3) means add the values in cells A1, A2 and A3 b =A5-A4 means subtract the value in cell A4 from the value in cell A5 c =A1*A3 means multiply the value in cell A1 by the value in cell A3 d =A5/4 means divide the value in cell A5 by 4 (/ is used instead of ÷) e =A2^2 means square the value in cell A2 (instead of (A2)2) f =average(A1:A5) means find the average of all values in cells A1 to A5 1 Enter the following numbers into cells as shown below, where m represents the value in cell B1, n is the value in cell B2, p is the value in cell B3, and so on.

2 Enter the following formulas into the given cells. a C1, q − 7 (means enter the formula into cell C1 as shown above) b C2, n − m c C3, 2 × r − 7 d C4, 3 × ( p + q) e C5, p × q × r f C6, p2

3×m g C7, -------------2

r–p h C8, ----------3 j C10, average of m, n, p, q and r

i C9, m + n + p + q + r q r k C11, ---- − ---m p

3 Choose different values and enter them into cells B1 to B5. Consider the new answers obtained in column C, for the formulas entered from question 2.

CHAPTER 1 THE HISTORY OF NUMBERS

17

01 NCM7 2nd ed SB TXT.fm Page 18 Saturday, June 7, 2008 2:30 PM

Working mathematically

Applying strategies and reasoning

Double-digit dice game This is a game for two or more players using one die. Instructions Step 1: Copy the scoresheet shown on the right. Step 2: Each player rolls the die seven times and, for each roll, can choose to write the number in either the tens column or the units column of his or her scoresheet. Step 3: Each player finds the total of his or her seven numbers. The winner is the person with a total closest to 99. Step 4: Play the game again and work out a strategy to improve your score.

Scoresheet Roll

Tens

Units

1st 2nd 3rd 4th 5th 6th 7th Total

Mental skills 1B

Maths without calculators

Dividing by a multiple of 10 Place value allows us to remove zeros from the end of a number when we divide by a power of 10. The deleted zeros shift all the other digits one or more places to the right which results in them having lower place values. 1 Examine these examples. a 2000 ÷ 10 = 2000 ÷ 10 = 200 b 1800 ÷ 100 = 1800 ÷ 100 = 18 c 37 000 ÷ 100 = 37 000 ÷ 100 = 370 d 6 000 000 ÷ 1000 = 6 000 000 ÷ 1000 = 6000 e 6000 ÷ 200 = 6000 ÷ 100 ÷ 2 = 60 ÷ 2 = 30 f 350 ÷ 70 = 350 ÷ 10 ÷ 7 = 35 ÷ 7 = 5 g 2800 ÷ 40 = 2800 ÷ 10 ÷ 4 = 280 ÷ 4 = 70 h 40 000 ÷ 5000 = 40 000 ÷ 1000 ÷ 5 = 40 ÷ 5 = 8 2 Now simplify these. a 200 ÷ 10 e 1900 ÷ 10 i 180 ÷ 30 m 4200 ÷ 60 q 24 000 ÷ 600

18

b f j n r

NEW CENTURY MATHS 7

6000 ÷ 100 2600 ÷ 100 300 ÷ 50 21 000 ÷ 700 24 000 ÷ 3000

c g k o s

45 000 ÷ 100 530 ÷ 10 1600 ÷ 400 44 000 ÷ 2000 64 000 ÷ 80

d h l p t

30 000 ÷ 1000 720 000 ÷ 1000 45 000 ÷ 5000 1600 ÷ 200 5400 ÷ 900

01 NCM7 2nd ed SB TXT.fm Page 19 Saturday, June 7, 2008 2:30 PM

1-09 Dividing by a two-digit number In primary school, you studied division by a single-digit number. We will now divide numbers by a two-digit number using two different methods.

Worksheet 1-09 Four operations

Example 12 Divide $312 among 12 people. Solution Method 1: Long division 26 12 312 12 into 31 is 2 −24 ↓ 72 12 into 72 is 6 −72 0

Method 2: Preferred multiples 26 12 312 −120 192 −120 72 −72 0

Each person receives $26.

10 times 10 times 6 times 26 times

Example 13 Simplify 296 ÷ 21. Then complete: 296 = Solution Method 1: Long division 14 remainder 2 21 296 21 into 29 is 1 −21↓ 86 21 into 86 is 4 −84 2

×

+

.

Method 2: Preferred multiples 14 21 296 −210 86 −84

remainder 2 10 times

2

14 times

4 times

2 - , so 296 = 14 × 21 + 2 Answer = 14 ----21

Exercise 1-09 1 Find the answers for the following. a 180 ÷ 15 b 462 ÷ 22 d 666 ÷ 18 e 992 ÷ 31 g 900 ÷ 25 h 667 ÷ 23

Ex 12

c 731 ÷ 17 f 78 ÷ 13 i 85 ÷ 17

CHAPTER 1 THE HISTORY OF NUMBERS

19

01 NCM7 2nd ed SB TXT.fm Page 20 Saturday, June 7, 2008 2:30 PM

Ex 13

2 Carry out these divisions and write your answers in the form: = a 304 ÷ 12 b 505 ÷ 14 c 99 ÷ 26 d 917 ÷ 19 e 958 ÷ 34 f 869 ÷ 28 g 594 ÷ 27 h 79 ÷ 13 i 815 ÷ 40

×

+

.

3 At a party 275 lollies are shared equally among 25 children. How many lollies does each child get? 4 A piece of wood 390 cm in length is to be cut into 15 equal pieces. How long is each piece?

Working mathematically Worksheet 1-12

Reasoning

Magic squares Magic squares have every row, column and diagonal adding to the same magic sum. The Lo-Shu magic square dates back to about 2200 BC. It appeared on an ancient Chinese tablet and was first drawn on a tortoise shell given to the Emperor Yu.

Magic squares

1 a Draw a 3 × 3 magic square frame. Write the Lo-Shu magic square into your frame using the numbers 1 to 9. (Hint: Count the dots. Top left-hand corner is a 4.) b What is the magic sum for the Lo-Shu square? 2 Which of these squares are not magic? a

42

14

34

22

30

26

46

b

21

0

15

38

12

6

18

3

30

c

38

8

28

18

16

24

32

5

20

30

12

3 Make these squares magic by finding the missing numbers. a

29

19

33

b

44 39

21

35

c

21

49

12

34

48 33

20

NEW CENTURY MATHS 7

6

3

45 27 42

01 NCM7 2nd ed SB TXT.fm Page 21 Saturday, June 7, 2008 2:30 PM

4 Another famous magic square appears in a woodcut by the German artist Albrecht Dürer, who lived from 1471 to 1528. It is called the magic square of Jupiter. a Find the 4-digit numeral contained within the square that identifies a year that occurred during Dürer’s lifetime. b What is the magic sum for this 4 × 4 square? c Find five 2 × 2 squares within the magic square for which the numbers have the same total as the magic sum. d Apart from the two diagonals, find four numbers each from a different row and column that add to the magic sum. There are more than two solutions.

Using technology

Sorting data Sort the set of numbers {60, 107, 85, 6, 28, 45, 265} using a spreadsheet, by following the instructions shown below. 1 a Enter the numbers, in the given order, into column A.

b Highlight cells A1 to A8 and choose Data and Sort.

CHAPTER 1 THE HISTORY OF NUMBERS

21

01 NCM7 2nd ed SB TXT.fm Page 22 Saturday, June 7, 2008 2:30 PM

c Choose Sort by Column A and Ascending as shown below.

d The data should now be sorted from smallest number (cell A1) to the largest number (cell A7). e A set of numbers can also be sorted in descending order. Highlight the cells and choose Sort by Column A and Descending. 2 Now sort these sets of numbers in the columns given, by repeating this method. a Enter {55, 89, 36, 21, 19, 4, 95} in column B b Enter {263, 141, 940, 508, 836, 392, 1063} in column C c Enter {4987, 4200, 8740, 9005, 2601, 2514, 4810} in column D d Enter {16 101, 12 167, 10 010, 11 412, 10 107, 10 761, 11 214} in column E 3 Sort the data from question 2 in descending order, for each of columns B to E.

1-10 Order of operations

Example 14 Find the value of (5 + 13) ÷ 2. Solution (5 + 13) ÷ 2 =

{ {

!

The order of operations rules First: Work out the value within any grouping symbols, starting with the innermost grouping symbols: parentheses or round brackets ( ) square brackets [ ] braces { }. Second: Work out multiplication or division as you come to it, going from left to right. Third: Work out addition or subtraction as you come to it, going from left to right.

=

22

÷2

18

9

NEW CENTURY MATHS 7

work out grouping symbols division answer

01 NCM7 2nd ed SB TXT.fm Page 23 Saturday, June 7, 2008 2:30 PM

Example 15 1 Find the value of 15 ÷ 5 × 8.

=

×8

3

24

Solution 5+6×2−7

division

{ {{

=

{ {

Solution 15 ÷ 5 × 8

2 Find the value of 5 + 6 × 2 − 7. multiplication

multiplication

= 5 + 12 − 7

addition

answer

=

subtraction

=

−7

17

10

answer

Example 16 Find the value of 25 − [7 × (5 − 3) + 4]. Solution 25 − [7 × (5 − 3) + 4]

{ { { {

innermost grouping symbols

= 25 − [7 × 2 + 4]

grouping symbols: inside multiplication first

= 25 − [14 + 4]

grouping symbols

= 25 −

subtraction

=

18

7

answer

Exercise 1-10 1 Evaluate (find the value of) each of the following. a 12 × (3 + 5) b (16 − 3) × 2 d (3 − 2) × 5 e (2 + 5) × 6 g 7 × (25 − 12) h 36 ÷ (14 − 10) j 120 ÷ (34 − 24) k 5 + 6 × (50 − 10)

c f i l

(60 + 12) ÷ 6 (12 − 4) ÷ 4 (5 × 7) − 16 (77 ÷ 11) − 7

2 Evaluate the following. a 3+5×2 d 19 − 4 × 4 − 1 g 2 × 10 − 9 + 28 j 4×8−3×3

c f i l

5+3×2−7 17 + 8 − 3 × 2 9 + 28 − 12 60 − 8 × 4 + 20

b e h k

20 − 2 × 5 24 − 5 ÷ 5 + 7 42 ÷ 7 − 5 109 + 36 ÷ 4

3 12 ÷ 4 + 8 × 5 = ? Select A, B, C or D. A5 B 16

C 43

Ex 14

Ex 15

D 55

4 Find the answer to each of the following. a (24 − 4) ÷ 5 + 7 b 2 × (10 − 9) + 28 c (8 + 2) × (17 − 7) d 7 + 7 + (11 − 8) e (16 − 5 + 8) × 9 f (8 + 8 − 5) × (7 + 4) g 9 + 3 × (15 − 4) − 5 × 6 h 16 × 3 − 4 × (15 − 6 × 2) + 7 i (5 + 8) × 2 − (25 ÷ 5) j 4 × [(5 + 11) ÷ 2] − (15 × 2) CHAPTER 1 THE HISTORY OF NUMBERS

Ex 16

23

01 NCM7 2nd ed SB TXT.fm Page 24 Saturday, June 7, 2008 2:30 PM

k 100 − [12 + (3 × 5) ÷ 3] l 120 ÷ {16 + [(2 × 5) + 4]} m {15 − [3 × (12 − 9) + 1]} − [(44 × 2) + 12] ÷ 50 n [(16 − 4) × 10] ÷ [(45 ÷ 3) + 25] o 86 + [(15 ÷ 3) + (65 ÷ 5)] × 2 p [20 ÷ (5 − 4) × 2] − {[(4 + 5) × 3] ÷ [15 − (30 ÷ 5)]} 5 Put grouping symbols where necessary to make each of the following statements true. The first one has been done for you. a 5 − 2 × 4 = 12 becomes (5 − 2) × 4 = 12 b 3+8−7=4 c 15 − 3 × 5 = 60 d 15 − 3 × 5 = 0 e 8 + 4 − 3 × 2 = 10 f 8+4−3×2=6 g 8 + 4 − 3 × 2 = 18 h 6+4×0=6 i 6+4×0=0 j 100 ÷ 10 + 10 = 5 k 100 ÷ 10 + 10 = 20 6 Put grouping symbols where necessary to make each of the answers correct. a 84 ÷ 3 + 9 × 15 − 11 = 152 b 84 ÷ 3 + 9 × 15 − 11 = 64 c 84 ÷ 3 + 9 × 15 − 11 = 94 7 Use the four numbers in each set only once (in any order), with the operations +, −, ×, ÷ or grouping symbols, to make an equation that equals the number in the red box. a 2, 7, 8, 9

12

b 1, 2, 3, 5

18

c 3, 4, 6, 8

41

d 2, 6, 8, 1

21

e 2, 4, 6, 8

10

f 2, 5, 8, 10

44

g 3, 5, 7, 9

2

h 4, 5, 7, 9

8

i 2, 5, 7, 10

60

1-11 The symbols of mathematics Worksheet 1-13

Mathematics does not only involve numbers. It has a language of its own and uses symbols recognisable throughout the world. This table shows some of the most common symbols.

Cross number puzzle

Symbol

Meaning

+

plus, add, sum



minus, subtract, difference

×

multiply, times, product

÷

divided by, quotient

Symbol

Meaning square root ( 25 = 5)

3

∴  or 

cube root ( 3 8 = 2) therefore approximately equal to

=

equal to

32



not equal to

53

cubed (5 × 5 × 5)



less than

( )

parentheses or brackets



less than or equal to

[ ]

square brackets



greater than

{ }

braces



greater than or equal to

squared (3 × 3)

The square root of a given number is the positive value which if squared will give that number. The cube root of a number is the value which if cubed will give the number.

24

NEW CENTURY MATHS 7

01 NCM7 2nd ed SB TXT.fm Page 25 Saturday, June 7, 2008 2:30 PM

Example 17 Find the answer for each of the following. a 62 b 9 Solution a 6 2 = 6 squared = 6 × 6 = 36

c 3 125 b

9 = the square root of 9 since 32 = 3 × 3 = 9 =3

c 3 125 = the cube root of 125 = 5 since 53 = 5 × 5 × 5 = 125

Example 18 Write the meaning of each of the following. a 37 b 55 Solution a 3 is less than or equal to 7.

b 5 is greater than or equal to 5.

Exercise 1-11 1 Here is a list of words that relates to the four basic operations +, −, × and ÷. plus minus times multiply and divide subtract share decrease product difference less increase total lots of quotient take away more than Draw a table with column headings as shown below in your notebook, and write each of the given words in the appropriate column. +



×

÷

2 Rewrite these questions using mathematical symbols. a 15 minus 6 b 48 plus 12 c 12 is greater than 5 d 5 is not equal to 3 plus 6 e the product of 7 and 8 f the square root of 16 g 36 divided by 4 h 5 squared i 8 more than 12 j 6 less than 13 k increase 3 by 13 l the quotient of 39 and 3 m the difference between 25 and 8 n the cube root of 125 o 13 is not equal to 3 p 999 is approximately equal to 1000 3 Write the answer to each of the following. a the number 6 less than 18 b the sum of 26 and 14 d 9 times 8 e 7 squared g the number 14 more than 8 h decrease 33 by 11 j 7 lots of 13 k the cube root of 64 l the difference between 135 and 29

c the total of 6, 8 and 22 f the quotient of 36 and 4 i increase 83 by 27

CHAPTER 1 THE HISTORY OF NUMBERS

25

01 NCM7 2nd ed SB TXT.fm Page 26 Saturday, June 7, 2008 2:30 PM

4 Which of these statements is true? Select from A, B, C or D. 36 = 18

A Ex 17

D 72  12

C 6 × 4  15

5 Write whether each of the following is true (T) or false (F). a 16  2 b 42 = 8 c 300  5 × 100 d 3602 = 3600 g 2  3 27 j 33 = 27

e 25 = 5 h product of 2 and 15 = 17 k 52 − 3 = 7

f 8 × 201  8 × 200 i 63 ÷ 3  60 ÷ 5 l 72  73

m 16 × 0  7 × 0

n (30 − 6) × 5  12 × 10

o

36 = 6

r

24  4

p Ex 18

B 18 ÷ 2 ≠ 9

3

1 =1

q

53

= 15

6 Complete the blank with  or  to make each statement true. a 7130 860 b 2001 2010 c 352 140 4 082 716 d 2651 2561 e 3602 3206 f 13 253 1353 g 8079 8097 h 1432 1483 7 For each of the following statements, select all the numbers from this list of seven numbers that make the statement true: 2, 3, 7, 8, 11, 36, 41. a  13 b 5 c 8 d  42 e

3

=8

f

g 3

 11

Working mathematically

h 5+

=2

8

Applying strategies and reflecting

The four 4s puzzle Form 10 groups (Group A, Group B, Group C, etc.). Use only four 4s and any of the mathematical symbols =, −, ×, ÷, brackets, a decimal point (.), factorial (!) or square root ( ) to make expressions for all the numbers from 1 to 100. Group A does the numbers 1 to 10, Group B does 11 to 20, … Group J does 91 to 100. Here are some suggestions: • 4 + 4 × 4 + 4 = 4 + 16 + 4 = 24 • 4 × 4 − 4 ÷ 4 = 16 − 1 = 15 • 4! + 4 × 4 ÷ 4 = 24 + 4 = 28 • 4 × 4 + 4 × 4 = 16 + 16 = 32 (Hint: 4! = 4 × 3 × 2 × 1)

Brain bender Various forms of ‘brain benders’ are common in daily newspapers and magazines. Here is one for you. Copy the grids and fill in the six gaps to complete each of the lines, using the remaining digits from 1 to 9 only once. Be sure to use the ‘order of operations’ rules. The aim is to make the sum of the answers for the three lines total 45.

26

NEW CENTURY MATHS 7

5

+ × −

3

×

=



=

+

4

= 45

01 NCM7 2nd ed SB TXT.fm Page 27 Saturday, June 7, 2008 2:30 PM

Using technology

Fruit picking

Skillsheet 1-04 Spreadsheets

An orchardist employed people to pick fruit in his orchard over the summer. The table below shows the types of fruit grown and numbers of bins of fruit picked each day in a particular week. 1 Copy the table, as shown, into a spreadsheet.

2 To find the total number of bins of fruit picked on Monday, type the formula =sum(B2:B5) in cell B6.

3 To copy this formula into cells C6 to F6, click on cell B6 and Fill Right by grabbing the bottom right-hand corner of the cell and dragging across to cell F6. Let go of the mouse and you will see the totals for each day.

4 Use the sum formula in cell G2 to find the number of bins of apples picked in this particular week. Use Fill Down to copy the formula into cells G3 to G6. Centre the totals calculated in the ‘G’ column. 5 Answer the following questions in the given cell. In cell: a A8, type the number of bins of fruit pieces picked on Wednesday b A9, write a formula to find how many more bins of oranges than apples were picked in this week. c A10, write a formula to find how much more fruit was picked on Wednesday compared to Monday in this week. d A11, write a formula to find how many bins of lemons and mandarins in total were picked in this week. e A12, write the day of the week on which the most fruit was picked. f A13, write the day of the week on which the least fruit was picked. g A14, type the total number of bins of fruit picked in this particular week.

CHAPTER 1 THE HISTORY OF NUMBERS

27

01 NCM7 2nd ed SB TXT.fm Page 28 Saturday, June 7, 2008 2:30 PM

Power plus Cryptic arithmetic Simple codes can be made by replacing letters with other letters, symbols or numbers. Number codes are studied in a branch of mathematics called cryptic arithmetic. Your challenge is to figure out which letter replaces which number. The addition: 99 could become: KK + 22 + DD 121 RDR where K = 9, D = 2 and R = 1. Note that K + D gives an answer bigger than 10 so carrying will be involved. To solve cryptic arithmetic problems, you need to know about carrying digits when adding. Choose any of the following problems from 1 to 7. 1 ON + ON + ON + ON = GO

Hint: Set it out as a column sum.

2

Hint: Try R = 0 and N = 5

N I NE − F OUR

F I VE There are 71 other possible solutions. In many of these (but not all) R = 0 and N = 5. Can you find two other solutions? How many different solutions can the class find? 3

FORT Y T EN + T EN

Hint: T = 8 and Y = 6

S I XTY The key to this problem is to decide what value is N + N and what value is E + E. 4

THRE E + FOUR S EVEN For this puzzle there are 38 possible solutions. Hint: Try E = 6 and V = 0 for one solution. Try E = 5 and V = 1 for another solution. Try H = 9 and R = 4 for another. How many different solutions can the class find?

5 On a holiday, Carlos ran short of money. He sent an email to his parents: S E ND +MOR E MON E Y The value of ‘MONEY’ is the amount Carlos asked for. If Carlos asked for more than $10 000 and less than $20 000, find out how much money he asked for. 6 a

RE AD + TH I S

b

READ − TH I S

P AG E P AG E These are two different problems, so R and the other letters have a different value in each problem. 7 Try to create a cryptic arithmetic question of your own. (It is not as easy as it seems!)

28

NEW CENTURY MATHS 7

01 NCM7 2nd ed SB TXT.fm Page 29 Saturday, June 7, 2008 2:30 PM

Chapter 1 review Language of maths braces evaluate long division order of operations product sum

Worksheet 1-14

cube root expanded notation million parentheses quotient

difference grouping symbols number system place value square brackets

digit Hindu–Arabic numeral preferred multiples square root

1 What is ‘expanded notation’? Explain in your own words. 2 What is a thousand thousands? 3 What is the Roman numeral for 500? 4 Write and name the three types of grouping symbols. 5 With which arithmetic operation would you associate the word: a quotient? b difference? 6 What is the meaning of each of these symbols? a  b 3

Topic overview • In your own words, write what you have learnt about the history of numbers. • Is there anything you did not understand? Ask a friend or your teacher for help. • Copy this overview into your workbook and complete it using what you have learnt in this chapter. Ask your teacher to check your overview. Order of operations • • •

Four operations • • • •

Place value • • •

Hindu–Arabic numerals 0, 1, 2, 3, 4, 5, 6, 7, 8, 9

H I S T O R Y

Symbols • +, −, ×, ÷ • • •

OF

NUMBERS

,

3

Early number systems • Egyptian • Aboriginal • • •

CHAPTER 1 THE HISTORY OF NUMBERS

29

Number find-a-word

01 NCM7 2nd ed SB TXT.fm Page 30 Saturday, June 7, 2008 2:30 PM

Topic test 1

Exercise 1-01

Chapter revision 1 Write these using Egyptian numerals. a 13

b 2402

Exercise 1-02

2 Write these using the words of the Kamilaroi Aboriginal people. a 3 b 5

Exercise 1-03

3 Write these using Babylonian numerals. a 32

b 110

Exercise 1-04

4 Write each of the following in Roman numerals. a 12 b 40 c 179 d 2004

Exercise 1-05

5 Write these using modern Chinese numerals. a 17 b 82

Exercise 1-06

6 Write each of the following using numerals. a six hundred and twelve b nine hundred and forty-three c five thousand, four hundred and ninety-nine d six thousand and two e nine million, seven hundred and fifty thousand and seventy-six

Exercise 1-06

7 Arrange the numbers in each of these sets in order, from largest to smallest. a 16, 21, 38, 19, 14 b 89, 36, 101, 98, 88 c 2356, 2534, 2635, 2300, 2533 d 12 391, 12 913, 11 990, 11 391, 12 300

Exercise 1-06

8 What is the place value of the digit 4 in: a 47? c 8412?

Exercise 1-07

Exercise 1-08

b 3024? d 146 235?

9 Write each of these using expanded notation. a 19 b 283 c 665 d 42 891 10 Find the answers to these. a 36 + 58 c 39 − 17 e 2501 + 58 g 123 × 5 i 36 ÷ 4 k 750 ÷ 6

30

NEW CENTURY MATHS 7

b d f h j l

127 + 81 78 − 39 26 × 9 36 × 11 252 ÷ 7 3500 ÷ 10

01 NCM7 2nd ed SB TXT.fm Page 31 Saturday, June 7, 2008 2:30 PM

11 Find the answers to these. Write your answer in the form: = × + . a 384 ÷ 16 b 912 ÷ 19 c 784 ÷ 17 d 877 ÷ 23 12 Find the value of each of these. a 16 − (5 × 3) c 30 − 10 ÷ 2 e (320 − 120) × 12 g (36 − 14) × 2 ÷ 4 i (256 − 120) ÷ 17 k 36 − (4 × 3) ÷ (35 − 23) 13 Use ‘order of operations’ to calculate: a 12 + 7 − 2 × 3 c 24 + 16 ÷ 4 × 16 − 4 + 9 e 18 + 6 ÷ 3 − 3 + 2 × 5

Exercise 1-09

Exercise 1-10

b d f h j l

6+5×3 (16 ÷ 2) + (18 − 11) 35 × (19 − 17) × 20 36 − (28 − 13) + (20 − 3 × 5) [394 + (30 ÷ 5)] ÷ (440 ÷ 11) 2 000 000 − [(300 × 100) + 1] Exercise 1-10

b 15 − 2 × 4 + 6 ÷ (8 − 5) d 15 + (64 + 2) ÷ 3 − 16 f 166 + 12 × 3 − 48 ÷ 4

14 Use grouping symbols and operations signs (+, −, ×, ÷) to make each of these true. a 7?3?1=9 b 10 ? 5 ? 5 = 10 c 8?3?6?2=8 d 28 ? 4 ? 7 = 49 e 6 ? 4 ? 3 ? 5 = 40 f 19 ? 1 ? 5 ? 3 ? 1 = 0

Exercise 1-10

15 Write whether each of these is true (T) or false (F). a 58 b 72+4 2 c 5 10 d 6 × 7  43 e 23  5 + 1

f

Exercise 1-11

36 = 6

CHAPTER 1 THE HISTORY OF NUMBERS

31

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