Chapter01-The History of Numbers

1

Number

The history of numbers

Mathematics began with the invention of numbers. Early tribes used notches on sticks, pebbles, and knots in ropes to represent numbers. We use the numerals 0, 1, 2, 3, 4, 5, 6, 7, 8, 9. This is called the Hindu–Arabic system, but there have been many other number systems before it. It is important to be able to read and write numbers in our own number system and to understand the rules our numbers follow.

In this chapter you will: ■ ■ ■ ■ ■ ■ ■ ■ ■

compare the Hindu–Arabic number system with number systems from different societies, past and present recognise, read and convert Roman numerals state the place value of any digit in large numbers order numbers of any size, in ascending and descending order record large numbers using expanded notation revise the four basic operations on whole numbers apply order of operations to simplify expressions divide two-digit and three-digit numbers by a two-digit number use the symbols of mathematics, including and 3 .

Wordbank ■ ■ ■ ■ ■ ■ ■

numeral A symbol that stands for a number, such as 8 or X. Hindu–Arabic number system The number system we use, with the numerals 0, 1, 2, 3, 4, 5, 6, 7, 8 and 9. place value The way that the position of a digit in a number tells us its value. expanded notation A way of writing a number that shows the place value of every digit. order of operations The rules for calculating an expression containing mixed operations, such as 14 − 2 × 4 + 1. evaluate To find the value of a numerical expression. cube root The value which, if cubed, will give the required number, for example 3 64 = 4 because 4 3 = 64.

Think! Suppose we could only write numbers using eight digits (0, 1, 2, 3, 4, 5, 6, 7) instead of 10 digits. That is, suppose that the digits 8 and 9 were never invented. ■ How would we write eight or nine (without inventing new symbols)? ■ Would 14, for example, still mean ‘fourteen’ under this system? ■ How was it decided that we use 10 digits anyway? Why 10 digits?

THE HISTORY OF NUMBERS

3

CHAPTER 1

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

1 Write the answers to these: a 10 × 10 d 7+9 g 9×9 j 6×5 m 18 × 3 p 128 ÷ 4 s 452 − 140

b e h k n q t

4×7 10 × 10 × 10 26 − 8 99 ÷ 11 7 × 12 137 + 45 280 × 10

c f i l o r u

2 Write each of these numbers in words: a 45 b 120 d 3680 e 5001 Skillsheet 1-01 Reading and writing large numbers

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

c 138 f 47 613

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

Different number systems 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)

5

200

...

6

...

1000

(lotus flower)

10 000

100 000

1 000 000 (million)

(bent reed)

(fish)

(man with hands raised in surprise)

4

NEW CENTURY MATHS 7

9

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

Example 1

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

d

Example 2

e

3 Write the answer to: a minus

b plus

c plus

d minus

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? THE HISTORY OF NUMBERS

5

CHAPTER 1

The Australian Aboriginal number system The Australian Aboriginal way of life had no need for a complicated number system. Their society relied on story-telling and did not have symbols for numbers. Different tribes had their own names for numbers. Here are two examples: Belgando River Aborigines 1 = Wogin 3 = Booleroo Wogin

2 = Booleroo 4 = Booleroo Booleroo

Kamilaroi Aborigines 1 = Mal 3 = Guliba 5 = Bulan Guliba

2 = Bulan 4 = Bulan Bulan 6 = Guliba Guliba

Exercise 1-02 1 Find the following. (Write your answer using the Aboriginal names for numbers.) a Wogin plus Booleroo Wogin b Guliba times Bulan c Bulan plus Bulan plus Mal d Booleroo times Booleroo e Guliba Guliba minus Guliba f Bulan Bulan minus Mal 2 Explain, in your own words, how the Kamilaroi Aborigines named their numbers.

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:

1

2

3

4

10

10

20

30

...

5

...

9

60

70

80

...

120

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. because 4 × 60 = 240 252 ÷ 60 = 4 and remainder 12 So 252 = (4 × 60) + 10 + 2. In Babylonian numerals, 252 is:

6

NEW CENTURY MATHS 7

130

Exercise 1-03 1 How would you write each of these numerals using our number system? 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

Example 3

3 Use a dictionary to find the difference between the words ‘numeral’ and ‘number’. Skillsheet 1-02 Roman numerals

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 VI

7 VII

8 VIII

9 IX

50 L

100 C

500 D

1000 M

10 X

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

b 46 is XLVI

c 101 is CI

d 249 is CCXLIX THE HISTORY OF NUMBERS

7

CHAPTER 1

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 Example 4

2 What would Titus have written for these numbers? a 365 b 36 c 79 e 2600 f 344 g 999

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. Spreadsheet 1-01 Roman numeral number patterns

5 If Titus had owned a computer, he may have been asked to complete number patterns. Try the accompanying spreadsheet.

The modern Chinese number system Worksheet 1-03 Ancient Chinese rod numerals

Worksheet 1-04 Mayan numerals

Chinese people today use the numerals below. 1

2

3

4

6

7

8

9

10

100

1000

5

10 000

• 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.

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

Solution a

b 6 × 10 = 60 4= 4 64

8

NEW CENTURY MATHS 7

+

3 × 100 = 300 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

Example 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

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?

The Hindu–Arabic number system

Worksheet 1-05 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 10 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

AD 2

Hindu

AD 800

Hindu

AD 900

Arabic

AD 976

Spanish

AD 1400

Italian

AD 1480

Caxton (Printer)

1

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 THE HISTORY OF NUMBERS

9

CHAPTER 1

Skillsheet 1-03 Place value Worksheet 1-06 Big numbers

Place value 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.

Example 6 Write the place value of each of the digits in 4625.

Solution In 4625:

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

5×1 2 × 10 6 × 100 4 × 1000

or or or or

Example 7 What is the place value of each of the digits in 501?

Solution In 501:

1 has a value of 1 0 means there are no tens 5 has a value of 500

(zero used to mark a place)

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 place value does the digit 5 have in: a 57?

b 235?

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

Exercise 1-06 Example 6

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

10

NEW CENTURY MATHS 7

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 3 What are the advantages of using a Hindu–Arabic number system? 4 Place these numbers in a place-value table, as shown on the facing page. a 48 b 382 c 2751 d 3020 e 15 364 f 44 040

Example 7

5 What is the place 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

Example 8

6 What is the place 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 7 What is the place 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 8 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

SkillBuilder 1-01 Review of place value

9 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

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 THE HISTORY OF NUMBERS

11

CHAPTER 1

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

Exercise 1-07 Example 9

Worksheet 1-07 Base 8 number system

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

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 Find out what ‘to expand’ means. Is the dictionary meaning the same as the one in mathematics?

Skillbank 1A SkillTest 1-01 Multiplying by a multiple of 10

Multiplying by a multiple of 10 Place value allows us simply to 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: b 45 × 100 = 4500 a 37 × 10 = 370 c 16 × 1000 = 16 000 d 100 × 1000 = 100 000 e 7 × 90 = 7 × 9 × 10 = 630 f 5 × 400 = 5 × 4 × 100 = 2000 g 12 × 300 = 12 × 3 × 100 = 3600 h 40 × 800 = 4 × 10 × 8 × 100 = 4 × 8 × 10 × 100 = 32 000 2 Now simplify these: a 18 × 100 e 315 × 1000 i 6 × 50 m 4 × 400 q 400 × 60

12

NEW CENTURY MATHS 7

b f j n r

26 × 1000 1000 × 1000 7 × 30 6 × 700 50 × 80

c g k o s

77 × 10 000 3 × 80 2 × 6000 8 × 500 3000 × 40

d h l p t

10 × 100 9 × 200 11 × 900 20 × 70 900 × 2000

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 two billion web pages in less than one second.

How many googols are there in a googolplex?

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

Worksheet 1-08 Four operations

÷ division

Worksheet 1-09 Cross number puzzle

We will now review these operations.

Example 10 Copy and complete this number grid:

+

5

14

8 12

Solution +

5

14

+

8

8

12

12

5

14

5+8

14 + 8

13

22

5 + 12

14 + 12

17

26

+

5

14

8

13

22

12

17

26 Spreadsheet 1-02

The accompanying spreadsheet might help with problems of this kind. THE HISTORY OF NUMBERS

Number grids

13

CHAPTER 1

Exercise 1-08 Example 10

This exercise is made up of number grids. You may begin with any question, from 1 to 4. Copy the unfilled grids into your notebook and complete them. 1 Copy and complete these number grids: a b top row − side column +

3

−

4

54

c ×

78

7

37

8

2

26

5

e top row ÷ side column

d +

15

÷

41

36

f

9

top row − side column −

84

11

28

4

128

19

3

239

243

412

2 Find the missing numbers: a

+

11

9

b

+

8

+

50

5 d

c

10

80

+

15 13

e

28

+

26 100 f

25

+ 22

80

28

13

45

15

16

90

6

33 14

3 Find the missing numbers (top row − side column): a

−

20

15

b

8

− 9

15

9

14

− 7

9

11

4 Find the missing numbers: a × 5 3

c

17

b

×

5

12

56

28

7

NEW CENTURY MATHS 7

12 11

c

×

10 90

40 4

6

d

÷

32

e

64

f

÷

8

8

4

2

÷

4

72 24

10

24

5

Using technology Number grids 1 Use a spreadsheet program such as Excel, to automatically complete a number grid. For instance, for an addition table with 3 rows and 3 columns, enter the following formulas: A

1

B

C

A

B

Skillsheet 1-04 Spreadsheets

C

Addition table 3 × 3

2 3

+

2

7

+

2

7

4

3

=A4+B3

=A4+C3

3

5

10

5

8

=A5+B3

=A5+C3

8

10

15

Spreadsheet

6

2 Change the numbers in A4, A5, B3 and C3. 3 Use a spreadsheet to design your own number grid.

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.

Scoresheet Roll

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.

Tens

Units

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

THE HISTORY OF NUMBERS

15

CHAPTER 1

Arithmagons Arithmagons are number puzzles made from triangles. The circled numbers are added together to give the number on the line joining the circles. The challenge is to find the correct numbers to go inside the circles.

Example 11 Find the numbers missing from the circles in this arithmagon:

? 11

+

?

5 ?

10

Solution Think about the pairs of numbers that add together to give the numbers on the lines. 11 = 10 + 1 = 9 + 2 = 8 + 3 = 7 + 4 = 6 + 5 10 = 9 + 1 = 8 + 2 = 7 + 3 = 6 + 4 = 5 + 5 5 =4+1=3+2 Now, by trial and error, write numbers in the circles to reach a solution: 11 = 9 + 2? 11 = 8 + 3? 11 = 10 + 1? 2

1 11 10

Spreadsheet 1-03 Arithmagons Example 11

+

wrong 1

11 9

0

10

+ 10

3 wrong 3

11

1

8

correct 5

+ 10

2

The accompanying spreadsheet might help with problems of this kind.

Exercise 1-09 1 Find the numbers missing from the circles in these arithmagons: a

b 11

+

15

c 21

18

25

+

52

62 NEW CENTURY MATHS 7

+

40

50

e 42

30

28

d

16

+

f 39

+ 47

24

48

+ 29

25

g

h 25

+

25

i 26

20

+

16

44

10

+

44

40

2 These arithmagons use larger numbers, but are solved the same way as the others. (You may use a calculator.) a b c 144

+

173

191

2890

+

4114

10 144

+

7604

13 460

3784

3 a What does the word ‘arithmetic’ mean? b What is the difference between ‘arithmetic’ and ‘mathematics’?

SkillTest 1-02 Changing the order

Skillbank 1B Changing the order Have you noticed that 4 + 7 = 7 + 4? Have you also noticed that 3 × 5 = 5 × 3? Numbers can be added or multiplied in any order. We can use this property to make our calculations simpler. 1 Examine these examples: a 19 + 5 + 5 + 1 = (19 + 1) + (5 + 5) = 20 + 10 = 30 b 13 + 8 + 20 + 27 + 80 = (13 + 27) + (20 + 80) + 8 = 40 + 100 + 8 = 148 c 2 × 36 × 5 = (2 × 5) × 36 = 10 × 36 = 360 d 25 × 11 × 4 × 7 = (25 × 4) × (11 × 7) = 100 × 77 = 7700 2 Now simplify these: a 45 + 16 + 45 + 4 + 7 c 18 + 91 + 9 + 20 e 24 + 16 + 80 + 44 + 10 g 100 + 36 + 200 + 10 + 90

b d f h

38 + 600 + 50 + 12 + 40 75 + 33 + 7 + 25 56 + 5 + 20 + 15 + 4 54 + 27 + 9 + 16 + 3 THE HISTORY OF NUMBERS

17

CHAPTER 1

i k m o q s

70 + 50 + 30 + 25 + 25 8×4×5 3×5×6 12 × 2 × 3 3 × 20 × 7 × 5 2 × 3 × 2 × 11

j l n p r t

32 + 120 + 40 + 80 + 40 50 × 7 × 2 5 × 11 × 40 2 × 4 × 25 × 8 6×8×5×2 4 × 4 × 9 × 25

Dividing by a two-digit number Worksheet 1-08 Four operations

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.

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 10 times 192 −120 10 times 72 −72 6 times 0 26 times

Answer = 26, so 312 = 26 × 12 Each person receives $26.

Example 13 Simplify 296 ÷ 21.

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 remainder 2 21 296 −210 10 times 86 −84 4 times 2 14 times

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

18

NEW CENTURY MATHS 7

Exercise 1-10 1 Find the answers for the following: b 462 ÷ 22 a 180 ÷ 15 e 992 ÷ 31 d 666 ÷ 18 h 667 ÷ 23 g 900 ÷ 25

Example 12

c 731 ÷ 17 f 78 ÷ 13 i 85 ÷ 17 Example 13

2 Carry out the following 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 h 79 ÷ 13 i 815 ÷ 40 g 594 ÷ 27

Order of operations 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.

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

=

(5 + 13) ÷ 2

work out brackets

{ {

Solution

division

÷2

18

=

9

answer

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

= =

15 ÷ 5 × 8

division

{ {

Solution

multiplication

3

×8

24

answer

THE HISTORY OF NUMBERS

19

CHAPTER 1

2 Find the value of 5 + 6 × 2 − 7.

Solution

5+6×2−7

{ { {

multiplication

= 5 + 12 − 7

addition

=

subtraction

−7

17

=

10

answer

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

Solution

25 − [7 × (5 − 3) + 4]

Example 14

Example 15

Example 16

{ { { {

= 25 − [7 × 2 + 4]

grouping symbols: inside multiplication first

= 25 − [14 + 4]

grouping symbols

= 25 −

subtraction

= CAS 1-01 BODMAS

innermost grouping symbols

18

7

answer

Exercise 1-11 1 Evaluate (find the value of): a 12 × (3 + 5) d (3 − 2) × 5 g 7 × (25 − 12) j 120 ÷ (34 − 24)

b e h k

(16 − 3) × 2 (2 + 5) × 6 36 ÷ (14 − 10) 5 + 6 × (50 − 10)

c f i l

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

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

b e h k

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

c f i l

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

3 Find the answer to: b 2 × (10 − 9) + 28 a (24 − 4) ÷ 5 + 7 d 7 + 7 + (11 − 8) c (8 + 2) × (17 − 7) f (8 + 8 − 5) × (7 + 4) e (16 − 5 + 8) × 9 g 9 + 3 × (15 − 4) − 5 × 6 h 16 × 3 − 4 × (15 − 6 × 2) + 7 j 4 × [(5 + 11) ÷ 2] − (15 × 2) i (5 + 8) × 2 − (25 ÷ 5) l 120 ÷ {16 + [(2 × 5) + 4]} k 100 − [12 + (3 × 5) ÷ 3] 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)]}

20

NEW CENTURY MATHS 7

4 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 c 15 − 3 × 5 = 60 b 3+8−7=4 d 15 − 3 × 5 = 0 e 8 + 4 − 3 × 2 = 10 g 8 + 4 − 3 × 2 = 18 f 8+4−3×2=6 h 6+4×0=6 i 6+4×0=0 k 100 ÷ 10 + 10 = 20 j 100 ÷ 10 + 10 = 5 5 Put grouping symbols where necessary to make each of the answers correct: b 84 ÷ 3 + 9 × 15 − 11 = 64 a 84 ÷ 3 + 9 × 15 − 11 = 152 c 84 ÷ 3 + 9 × 15 − 11 = 94 6 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, 11

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

SkillBuilder 1-05 Order of operations

The symbols of mathematics As you have probably already discovered, mathematics does not simply deal with numbers. Mathematics has a language of its own and uses symbols recognisable throughout the world. The table below shows some of the most common symbols. 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

squared (3 × 3)



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

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

THE HISTORY OF NUMBERS

21

CHAPTER 1

Example 17 Find the answer for each of the following: b 9 a 62

c

3

125

Solution a 6 2 = 6 squared = 6 × 6 = 36 c

3

b

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

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

Example 18 Write the meaning of each of the following: a 3
7

b 55

Solution a 3
7 b 55

3 is less than or equal to 7. 5 is greater than or equal to 5.

Exercise 1-12 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 that is 6 less than 18 c the total of 6, 8 and 22 e 7 squared g the number that is 14 more than 8 i incease 83 by 27 k the cube root of 64

22

NEW CENTURY MATHS 7

b d f h j l

the sum of 26 and 14 9 times 8 the quotient of 36 and 4 decrease 33 by 11 7 lots of 13 the difference between 135 and 29

4 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 25 = 5 f 8 × 201  8 × 200 e g i k m o

2  3 27 63 ÷ 3  60 ÷ 5 52 − 3 = 7 16 × 0  7 × 0 36 = 6

q 53 = 15

Example 17

h j l n

product of 2 and 15 = 17 33 = 27 72  73 (30 − 6) × 5  12 × 10

p

3

1 =1 24  4

r

5 Complete the blank with  or  to make each statement true. a 7130 860 b 2001 2010 4 082 716 d 2651 2561 c 352 140 3206 f 13 253 1353 e 3602 8097 h 1432 1483 g 8079 Example 18

6 For each of the following statements, select all the numbers from this list of seven numbers to make the statement true: 2, 3, 7, 8, 11, 36, 41.  13 b
5 c 8 d
42 a e

3

=8

f

 11

g

3

h 5+

=2

8

Working mathematically 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 operations =, −, ×, ÷, 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

Applying strategies and reflecting: Brain bender Various forms of ‘brain benders’ are common 5 + in daily newspapers and magazines. Here is one for you. Copy the grid 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. Use the accompanying Excel spreadsheet to help you.

3

×

=

−

=

+

4

= 45

THE HISTORY OF NUMBERS

Spreadsheet 1-04 Brain bender

23

CHAPTER 1

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 8. 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 − FOUR

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

FORT Y TEN + TEN

Hint: T = 8 and Y = 6

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

THRE E + FOUR S EVE N 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 a telegram to his parents: S END + MOR E MONE Y The result of the additions 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 R E AD + TH I S

b RE AD − TH I S

P AG E PA GE These are two different problems, so R and the other letters have a different value in each problem.

24

NEW CENTURY MATHS 7

7 Now for a cheery message: A MER RY + XMAS

No hints this time!

TURK E Y 8 Try to create a cryptic arithmetic question of your own. (It is not as easy as it seems!)

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.

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

6 12

34

48 33

45 27

3

THE HISTORY OF NUMBERS

42

25

CHAPTER 1

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 Spreadsheet

Magic squares In a magic square, every row, column and diagonal adds to the same number. Use a spreadsheet, such as Excel, to create the following tables. The formulas have been designed to add every row, column and diagonal automatically. The spreadsheet can be used to check whether a table is a magic square or not. A

B

C

D

E

1 Magic squares 3 × 3 2

=A5+B4+C3

3

=A3+B3+C3

4

=A4+B4+C4

5

=A5+B5+C5

6 =A3+A4+A5

=B3+B4+B5

=C3+C4+C5

=A3+B4+C5

7 8 9 Magic squares 4 × 4 10

=A14+B13+C12+D11

11

=A11+B11+C11+D11

12

=A12+B12+C12+D12

13

=A13+B13+C13+D13

14

=A14+B14+C14+D14

15 =A11+A12+A13+A14 =B11+B12+B13+B14 =C11+C12+C13+C14 =D11+D12+D13+D14 =A11+B12+C13+D14

Challenge 1 Create your own magic square with the numbers 1, 2, 3, 4, 5, 6, 7, 8 and 9. 2 Create your own 4 × 4 magic square.

26

NEW CENTURY MATHS 7

Worksheet 1-10 Number find-a-word

Language of maths braces digit grouping symbols million order of operations preferred multiples square brackets

cube root evaluate Hindu–Arabic number system parentheses product square root

difference expanded notation long division numeral place value quotient sum

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? b a


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 workbooks 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 OF

NUMBERS

Symbols • +, −, ×, ÷ • ,3 • •

Early number systems • Egyptian • Aboriginal • • •

THE HISTORY OF NUMBERS

27

CHAPTER 1

Chapter 1

Review

Topic test Chapter 1

Ex 1-04

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

Ex 1-06

2 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

Ex 1-06

3 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

Ex 1-06

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

b 3024? d 146 235?

Ex 1-07

5 Write each of these using expanded notation: a 19 b 283 c 665 d 42 891

Ex 1-08

6 Find the answers to these: a 36 + 58 c 39 − 17 e 2501 + 58 g 123 × 5 i 36 ÷ 4 k 750 ÷ 6

Ex 1-10

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

7 Find the answers to these. Write your answer in the form: = × + . a 384 ÷ 16 c 784 ÷ 17

Ex 1-11

b d f h j l

8 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)

28

NEW CENTURY MATHS 7

b 912 ÷ 19 d 877 ÷ 23 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]

9 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

Ex 1-11

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

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

Ex 1-11

11 Write whether each of these is true (T) or false (F): b 72+4 a 58 c 52 10

d 6 × 7
43

e 23  5 + 1

f

Ex 1-12

36 = 6

THE HISTORY OF NUMBERS

29

CHAPTER 1