Chapter01-The History of Numbers

Number

The history of numbers

1

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 HinduArabic 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.

compare the HinduArabic 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 .

numeral A symbol that stands for a number, such as 8 or X. HinduArabic 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 = 4 because 43 = 64.

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?

In this chapter you will:

3

Wordbank

643

Think!

THE HISTORY OF NUMBERS 3

CHAPTER 1

4

NEW CENTURY MATHS 7

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 Write the answers to these:a 10 10 b 4 7 c 900 + 30d 7 + 9 e 10 10 10 f 35 5g 9 9 h 26 8 i 1000 + 200 + 50j 6 5 k 99 11 l 75 16m 18 3 n 7 12 o 128 24p 128 4 q 137 + 45 r 35 12s 452 140 t 280 10 u 3601 59

2 Write each of these numbers in words:a 45 b 120 c 138d 3680 e 5001 f 47 613

3 Write each of the following numbers using numerals:a sixty-eight b seven hundredc two thousand and four d eight hundred and ninety-ninee ten thousand, four hundred and ninety-two

Start up

Worksheet 1-01

Brainstarters 1

Worksheet 1-02

Multiplication facts

Skillsheet 1-01

Reading and writing large

numbers

10

1 2 3 4 5 6 . . . 9

20 . . . 100

(coiled rope) (lotus flower)

200 . . . 1000

10 000 100 000

(bent reed) (fish)

1 000 000 (million)

(man with hands raised in surprise)

THE HISTORY OF NUMBERS 5 CHAPTER 1CHAPTER 1

Example 1

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

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

a b c

Example 2

or or

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

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

3 Write the answer to:

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?

a b c

d e

minusminusminus

plus

plus

minus

a

b

c

d

Exercise 1-01Example 1

Example 2

6 NEW CENTURY MATHS 7

The Australian Aboriginal number systemThe 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 Aborigines1 = Wogin 2 = Booleroo3 = Booleroo Wogin 4 = Booleroo BoolerooKamilaroi Aborigines1 = Mal 2 = Bulan3 = Guliba 4 = Bulan Bulan5 = Bulan Guliba 6 = Guliba Guliba

The Babylonian number systemThe 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 Find the following. (Write your answer using the Aboriginal names for numbers.)a Wogin plus Booleroo Wogin b Guliba times Bulanc Bulan plus Bulan plus Mal d Booleroo times Boolerooe Guliba Guliba minus Guliba f Bulan Bulan minus Mal

2 Explain, in your own words, how the Kamilaroi Aborigines named their numbers.

Exercise 1-02

1 2 3 4 5 . . . 9

10 10 30 . . .. . .20 130120807060

Example 3

Show how a Babylonian would have written each of these numbers:a 15 b 252Solutiona

b For numbers greater than 60, we need to nd how many 60s divide into them. 252 60 = 4 and remainder 12 because 4 60 = 240So 252 = (4 60) + 10 + 2.In Babylonian numerals, 252 is:


The Roman number systemThe 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 5I II III IV V6 7 8 9 10

VI VII VIII IX X50 100 500 1000L C D M

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).

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

Notice that there was no need for a zero.2 Use Babylonian numerals to write each of these amounts.

a 26 b 58 c 107d 300 e 144 f 401

3 Use a dictionary to nd the difference between the words numeral and number.

a b

c d

Exercise 1-03

Example 3

Skillsheet 1-02

Roman numerals

Example 4

Write each of the following in Roman numerals.a 23 b 46 c 101 d 249 Solutiona 23 is XXIII b 46 is XLVI c 101 is CI d 249 is CCXLIX


The modern Chinese number systemChinese people today use the numerals below.

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.

1 Titus, a student in ancient Rome, wrote these numerals. Change them into our numbers:a XXVI b XL c CCLXIV d LIVe MMCLIX f MCMXC g XCVIII h MDVII

2 What would Titus have written for these numbers?a 365 b 36 c 79 d 97e 2600 f 344 g 999 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.5 If Titus had owned a computer, he may have been asked to complete number patterns. Try

the accompanying spreadsheet.

Exercise 1-04

Example 4

Spreadsheet 1-01

Roman numeral number patterns

Worksheet 1-03

Ancient Chinese rod numerals

1 2 3 4 5

6 7 8 9

10 100 1000 10 000Worksheet

1-04Mayan

numerals

Example 5

Write each of these Chinese numbers using our number system.

Solution

a b

3 100 = 300

7 10 = 70 +

5 = 5375

6 10 = 60 +

4 = 464

a b


The HinduArabic number systemOur 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 rst 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.

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:

1 Use our numerals to rewrite these Chinese numerals Zhang Li wrote.

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

a b c d e


Communicating: Calendar monthMake 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?

Working mathematically

Worksheet1-05

Ancient number systems

Hindu

Hindu

Hindu

Arabic

Spanish

Italian

Caxton(Printer)

200 BC

AD 2

AD 800

AD 900

AD 976

AD 1400

AD 1480

1 2 3 4 5 6 7 8 9 0 10Origin

Numerals

Date

or or or


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

Skillsheet 1-03

Place value

Example 6

Write the place value of each of the digits in 4625.SolutionIn 4625: 5 has a value of 5 or 5 1

2 has a value of 20 or 2 106 has a value of 600 or 6 1004 has a value of 4000 or 4 1000

What is the place value of each of the digits in 501?SolutionIn 501: 1 has a value of 1

0 means there are no tens (zero used to mark a place)5 has a value of 500

Another way to show the meaning of each digit in a number is with a place-value table.

What place value does the digit 5 have in:a 57? b 235? Solutiona 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).

Ten thousands Thousands Hundreds Tens Ones

1 3 8 138

4 6 2 5 46255 0 1 501

8 2 3 5 0 82 350

Example 7

Example 8

Worksheet 1-06

Big numbers

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



Expanded notationOne way to show the place value of each digit in a number is to use expanded notation.

2 Write each of the following using numerals.a eight thousand, seven hundred and ninety-sixb three million and eighty-eightc two thousand, three hundred and eighty-ved six thousand, nine hundred and sevene four hundred and twenty thousand, eight hundred and thirtyf three hundred and nine thousand, two hundred and eleveng one million, two hundred and eighty thousand, four hundred and sixtyh twelve million, nine hundred and one

3 What are the advantages of using a HinduArabic number system?4 Place these numbers in a place-value table, as shown on the facing page.

a 48 b 382 c 2751d 3020 e 15 364 f 44 040

5 What is the place value of the digit 5 in each of these numbers?a 45 b 1057 c 1526d 12 345 e 65 013 f 51 480 260

6 What is the place value of the digit 3 in each of these numbers?a 123 b 2356 c 32 185d 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 6004d 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, 63c 246, 3596, 5369, 432, 16, 6125 d 123, 321, 132, 231, 213e 1045, 450, 145, 82 f 721, 243, 43, 4372, 722g 380 211, 308 022, 300 806, 392 084 h 4 856 231, 4 766 372, 1 429 950, 3 006 853

9 How many times is the rst 3 bigger than the second 3 in each of these numbers?a 1433 b 1343 c 3143 d 2 352 312

Example 7

Example 8

SkillBuilder 1-01

Review of place value

Example 9

Write each of these numbers using expanded notation.a 345 b 3287Solutiona 345 = (3 100) + (4 10) + (5 1)

= 3 102 + 4 10 + 5 1b 3287 = (3 1000) + (2 100) + (8 10) + (7 1)

= 3 103 + 2 102 + 8 10 + 7 1


Note: 102 10 squared means 10 10 = 100103 10 cubed means 10 10 10 = 1000104 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.

1 Write each of these numbers using expanded notation:a 56 b 3562 c 416 d 502 e 1001f 10 253 g 38 002 h 59 644 i 3809 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 1f 3 103 + 0 102 + 5 10 + 7 1g 7 104 + 6 103 + 0 102 + 0 10 + 1 1h 1 104 + 0 103 + 9 102 + 9 10 + 9 1i 3 105 + 4 104 + 4 103 + 2 102 + 2 10 + 0 1j 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?


Worksheet 1-07

Base 8 number system

Multiplying by a multiple of 10Place 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:

a 37 10 = 370 b 45 100 = 4500c 16 1000 = 16 000 d 100 1000 = 100 000e 7 90 = 7 9 10 = 630 f 5 400 = 5 4 100 = 2000g 12 300 = 12 3 100 = 3600h 40 800 = 4 10 8 100 = 4 8 10 100 = 32 000

2 Now simplify these:a 18 100 b 26 1000 c 77 10 000 d 10 100e 315 1000 f 1000 1000 g 3 80 h 9 200i 6 50 j 7 30 k 2 6000 l 11 900m 4 400 n 6 700 o 8 500 p 20 70q 400 60 r 50 80 s 3000 40 t 900 2000

Skillbank 1A

SkillTest 1-01

Multiplying by a multiple of 10


The four operationsThere are four basic operations in our number system:

+ addition multiplication subtraction divisionThe old symbols for writing these operations are shown below:

We will now review these operations.

The accompanying spreadsheet might help with problems of this kind.

Googol-plexingThe 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?

Just for the record

Worksheet 1-08

Four operations

Worksheet1-09

Cross numberpuzzle

Example 10

Copy and complete this number grid:

Solution

+ 5 14

8

12

+ 5 14

8

12

+ 5 14

8 13 22

12 17 26

+ 5 14

8 13 22

12 17 26

5 + 8 14 + 8

14 + 125 + 12

Spreadsheet 1-02

Number grids


This exercise is made up of number grids. You may begin with any question, from 1 to 4. Copy the unlled grids into your notebook and complete them.1 Copy and complete these number grids:

a b top row side column c

d e top row side column f top row side column

2 Find the missing numbers:

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

4 Find the missing numbers:

+ 3 4

7

2

54 78

37

26

11 9

8

5

+ 15 41

28

19

36 84

4

3

243 412

128

239

+ 11 9

8

5

+ 10

50

80 100

+ 16

26 28

13

a b c

+ 15

13 28

15

+ 25

45

80 90

+

22 33

6 14

d e f

20 15

8

9

17

9 15

11

7 9 12

11

a b c

5

3 12

28

5

56 40

7

10 6

90

4

a b c



32 64

8

4

8 4

2 24

d e f 72

24 10

5

Number grids1 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:

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

A B C A B C1 Addition table 3 323 + 2 7 + 2 74 3 =A4+B3 =A4+C3 3 5 105 8 =A5+B3 =A5+C3 8 10 156

Using technology

Spreadsheet

Skillsheet 1-04

Spreadsheets

Applying strategies and reasoning: Double-digit dice gameThis is a game for two or more players using one die.

InstructionsStep 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 nds 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.

ScoresheetRoll Tens Units1st2nd3rd4th5th 6th7th

Total



ArithmagonsArithmagons 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 nd the correct numbers to go inside the circles.

The accompanying spreadsheet might help with problems of this kind.

Example 11

Find the numbers missing from the circles in this arithmagon:

SolutionThink 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 + 510 = 9 + 1 = 8 + 2 = 7 + 3 = 6 + 4 = 5 + 5

5 = 4 + 1 = 3 + 2Now, by trial and error, write numbers in the circles to reach a solution:

11 = 10 + 1? 11 = 9 + 2? 11 = 8 + 3?

?

+

?

11 5

?

10

10

+0

11

1

10 9

+1

11 3

2

10 8

+2

11

3

10

wrong1

wrong5

correct

Spreadsheet 1-03

Arithmagons

1 Find the numbers missing from the circles in these arithmagons:

+11

18

+21

28

+30

50

a b c

15 25 40

+42

62

+39

47

+48

29

52 24 25

d e f



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

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

+25

20

+26

10

+44

40

25 16 44

g h i

+144

191

+2890

3784

+10 144

13 460

a b c

173 4114 7604

Changing the orderHave 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 b 38 + 600 + 50 + 12 + 40c 18 + 91 + 9 + 20 d 75 + 33 + 7 + 25e 24 + 16 + 80 + 44 + 10 f 56 + 5 + 20 + 15 + 4g 100 + 36 + 200 + 10 + 90 h 54 + 27 + 9 + 16 + 3

Skillbank 1B SkillTest 1-02Changing the order


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

i 70 + 50 + 30 + 25 + 25 j 32 + 120 + 40 + 80 + 40k 8 4 5 l 50 7 2m 3 5 6 n 5 11 40o 12 2 3 p 2 4 25 8q 3 20 7 5 r 6 8 5 2s 2 3 2 11 t 4 4 9 25

Worksheet 1-08

Four operations

Example 12

Divide $312 among 12 people.SolutionMethod 1: Long division Method 2: Preferred multiples

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

Simplify 296 21.SolutionMethod 1: Long division Method 2: Preferred multiples

Answer = 14 , so 296 = 14 21 + 2

2612 312 12 into 31 is 2

24 72 12 into 72 is 6

720

Example 13

14 remainder 221 296

210 10 times86

84 4 times2 14 times

14 remainder 221 296 21 into 29 is 1

2186 21 into 86 is 4

842

221------

2612 312

120 10 times192

120 10 times72

72 6 times0 26 times


Order of operations

1 Find the answers for the following:a 180 15 b 462 22 c 731 17d 666 18 e 992 31 f 78 13 g 900 25 h 667 23 i 85 17

2 Carry out the following divisions and write your answers in the form: = + .

a 304 12 b 505 14 c 99 26d 917 19 e 958 34 f 869 28 g 594 27 h 79 13 i 815 40


Example 13

The order of operations rulesFirst: 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.Solution

(5 + 13) 2 work out brackets= 18 2 division

= 9 answer

1 Find the value of 15 5 8.Solution

15 5 8 division

= 3 8 multiplication

= 24 answer

{ {

Example 15

{ {


2 Find the value of 5 + 6 2 7.Solution

5 + 6 2 7 multiplication= 5 + 12 7 addition= 17 7 subtraction= 10 answer

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 rst= 25 [14 + 4] grouping symbols= 25 18 subtraction= 7 answer

{{ {Example 16

{{ {{

1 Evaluate (nd the value of):a 12 (3 + 5) b (16 3) 2 c (60 + 12) 6d (3 2) 5 e (2 + 5) 6 f (12 4) 4 g 7 (25 12) h 36 (14 10) i (5 7) 16j 120 (34 24) k 5 + 6 (50 10) l (77 11) 7

2 Evaluate:a 3 + 5 2 b 20 2 5 c 5 + 3 2 7d 19 4 4 1 e 24 5 5 + 7 f 17 + 8 3 2 g 2 10 9 + 28 h 42 7 5 i 9 + 28 12j 4 8 3 3 k 109 + 36 4 l 60 8 4 + 20

3 Find the answer to:a (24 4) 5 + 7 b 2 (10 9) + 28c (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) + 7i (5 + 8) 2 (25 5) j 4 [(5 + 11) 2] (15 2)k 100 [12 + (3 5) 3] l 120 {16 + [(2 5) + 4]}m {15 [3 (12 9) + 1]} [(44 2) + 12] 50n [(16 4) 10] [(45 3) + 25]o 86 + [(15 3) + (65 5)] 2p [20 (5 4) 2] {[(4 + 5) 3] [15 (30 5)]}


Example 15

Example 16

CAS 1-01

BODMAS

THE HISTORY OF NUMBERS

21

CHAPTER 1CHAPTER 1

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.

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.

Symbol Meaning Symbol Meaning

+

plus, add, sum square root (

=

5)

minus, subtract, difference cube root (

=

2)

multiply, times, product

therefore

divided by, quotient

or

approximately equal to

=

equal to 3

2

squared (3

3)

not equal to 5

3

cubed (5

5

5)

less than ( ) parentheses or brackets

less than or equal to [ ] square brackets

greater than { } braces

greater than or equal to

4 Put grouping symbols where necessary to make each of the following statements true. The rst one has been done for you.a 5 2 4 = 12 becomes (5 2) 4 = 12b 3 + 8 7 = 4 c 15 3 5 = 60d 15 3 5 = 0 e 8 + 4 3 2 = 10f 8 + 4 3 2 = 6 g 8 + 4 3 2 = 18h 6 + 4 0 = 6 i 6 + 4 0 = 0j 100 10 + 10 = 5 k 100 10 + 10 = 20

5 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 = 64c 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 b 1, 2, 3, 5 c 3, 4, 6, 8d 2, 6, 8, 11 e 2, 4, 6, 8 f 2, 5, 8, 10 g 3, 5, 7, 9 h 4, 5, 7, 9 i 2, 5, 7, 10

12 18 4121 10 442 8 60

SkillBuilder 1-05

Order of operations

253 83

22

NEW CENTURY MATHS 7

Example 17

Find the answer for each of the following:a 62 b cSolutiona 62 = 6 squared = 6 6 b = the square root of 9

= 36 = 3 since 32 = 3 3 = 9c = the cube root of 125

= 5 since 53 = 5 5 5 =125

Write the meaning of each of the following:a 3 7 b 5 5Solutiona 3 7 3 is less than or equal to 7.b 5 5 5 is greater than or equal to 5.

9 1253

9

1253

Example 18

1 Here is a list of words that relates to the four basic operations +, , and .plus minus times multiply and dividesubtract share decrease product difference lessincrease 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 12c 12 is greater than 5 d 5 is not equal to 3 plus 6e the product of 7 and 8 f the square root of 16g 36 divided by 4 h 5 squaredi 8 more than 12 j 6 less than 13k increase 3 by 13 l the quotient of 39 and 3m the difference between 25 and 8 n the cube root of 125o 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 b the sum of 26 and 14c the total of 6, 8 and 22 d 9 times 8e 7 squared f the quotient of 36 and 4g the number that is 14 more than 8 h decrease 33 by 11i incease 83 by 27 j 7 lots of 13k the cube root of 64 l the difference between 135 and 29

+

Exercise 1-12

THE HISTORY OF NUMBERS

23

CHAPTER 1CHAPTER 1

4 Write whether each of the following is true (T) or false (F):a 16 2 b 42 = 8c 300 5 100 d 3602 = 3600e = 5 f 8 201 8 200g 2 h product of 2 and 15 = 17i 63 3 60 5 j 33 = 27k 52 3 = 7 l 72 73m 16 0 7 0 n (30 6) 5 12 10o = 6 p = 1q 53 = 15 r 4

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

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.a 13 b 5 c 8 d 42

e 3 = 8 f 11 g = 2 h 5 + 8

25273

36 13

24

3

Example 18

Example 17

Applying strategies and reecting: The four 4s puzzleForm 10 groups (Group A, Group B, Group C, etc.). Use only four 4s and any ofthe 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 reecting: Brain benderVarious forms of brain benders are common in daily newspapers and magazines. Here is one for you. Copy the grid and ll 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.

5 + =

3 =

+ 4 =

45


Spreadsheet 1-04

Brain bender


Cryptic arithmeticSimple 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 gure out which letter replaces which number.The addition: 99 could become: KK

+ 22 + DD121 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 N I NE Hint: Try R = 0 and N = 5 FOUR

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

3 FORT Y Hint: T = 8 and Y = 6TE N

+ TE NS 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

+ FOURS 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 nd?

5 On a holiday, Carlos ran short of money. He sent a telegram to his parents:S END

+ MOREMONE 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, nd out how much money he asked for.

6 a R E AD b RE AD+ TH I S TH I S

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

Power plus


7 Now for a cheery message:A

MER RY No hints this time!+ XMAS

TURK EY

8 Try to create a cryptic arithmetic question of your own. (It is not as easy as it seems!)Magic squaresMagic 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 rst 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?

3 Make these squares magic by nding the missing numbers:

42 14 34

22 30 38

26 46 18

21 0 15

12 6 18

3 30 5

a b c 38 8 28

16 24 32

20 30 12

29 19 33

21 35

44

39 49

34

a b c 21 6

12 45

48 27

33 3 42


4 Another famous magic square appears in a woodcut by the German artist Albrecht Drer, 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 identies a year that occurred during Drers lifetime.

b What is the magic sum for this 4 4 square?

c Find ve 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, nd four numbers each from a different row and column that add to the magic sum. There are more than two solutions.

Magic squaresIn 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.

Challenge1 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.

A B C D E1 Magic squares 3 32 =A5+B4+C33 =A3+B3+C34 =A4+B4+C45 =A5+B5+C56 =A3+A4+A5 =B3+B4+B5 =C3+C4+C5 =A3+B4+C5789 Magic squares 4 4

10 =A14+B13+C12+D1111 =A11+B11+C11+D1112 =A12+B12+C12+D1213 =A13+B13+C13+D1314 =A14+B14+C14+D1415 =A11+A12+A13+A14 =B11+B12+B13+B14 =C11+C12+C13+C14 =D11+D12+D13+D14 =A11+B12+C13+D14

Using technology

Spreadsheet


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.

Language of mathsbraces cube root differencedigit evaluate expanded notationgrouping symbols HinduArabic long divisionmillion number system numeralorder of operations parentheses place valuepreferred multiples product quotientsquare brackets square root 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?

a b 3

Worksheet 1-10

Number find-a-word

Order of operations

Four operations

HinduArabic numerals0, 1, 2, 3, 4, 5, 6, 7, 8, 9

Place value

Symbols +, , , ,

3

Early number systems Egyptian Aboriginal

HISTORY

OF

NUMBERS


Chapter 1 Review

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

2 Write each of the following using numerals:a six hundred and twelveb nine hundred and forty-threec ve thousand, four hundred and ninety-nined six thousand and twoe nine million, seven hundred and fty thousand and seventy-six

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

4 What is the place value of the digit 4 in:a 47? b 3024?c 8412? d 146 235?

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

6 Find the answers to these:a 36 + 58 b 127 + 81c 39 17 d 78 39e 2501 + 58 f 26 9g 123 5 h 36 11i 36 4 j 252 7k 750 6 l 3500 10

7 Find the answers to these. Write your answer in the form: = + .

a 384 16 b 912 19c 784 17 d 877 23

8 Find the value of each of these:a 16 (5 3) b 6 + 5 3c 30 10 2 d (16 2) + (18 11)e (320 120) 12 f 35 (19 17) 20g (36 14) 2 4 h 36 (28 13) + (20 3 5)i (256 120) 17 j [394 + (30 5)] (440 11)k 36 (4 3) (35 23) l 2 000 000 [(300 100) + 1]

Ex 1-04

Ex 1-06

Ex 1-06

Ex 1-06

Ex 1-07

Ex 1-08

Ex 1-10

Ex 1-11

Topic testChapter 1

THE HISTORY OF NUMBERS 29 CHAPTER 1

9 Use order of operations to calculate:a 12 + 7 2 3 b 15 2 4 + 6 (8 5)c 24 + 16 4 16 4 + 9 d 15 + (64 + 2) 3 16e 18 + 6 3 3 + 2 5 f 166 + 12 3 48 4

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

11 Write whether each of these is true (T) or false (F):a 5 8 b 7 2 + 4c 52 10 d 6 7 43

e 23 5 + 1 f = 6

Ex 1-11

Ex 1-11

Ex 1-12

36

Student textImprint pageTable of contentsPrefaceHow to use this bookHow to use the CD-ROMAcknowledgementsSyllabus reference grid1 The history of numbersDifferent number systemsThe HinduArabic number systemPlace valueExpanded notationThe four operationsArithmagonsDividing by a two-digit numberOrder of operationsThe symbols of mathematicsTopic overviewChapter review

2 AnglesNaming anglesComparing angle sizeThe protractorDrawing anglesAngle geometryNaming linesAngles and parallel linesFinding parallel linesTopic overviewChapter review

3 Exploring numbersSpecial number patternsTests for divisibilityFactorsPrime and composite numbersPrime factorsIndex notationSquares, cubes and rootsTopic overviewChapter review

Mixed revision 14 SolidsNaming solidsConvex and non-convex solidsPolyhedraPrisms and pyramidsCylinders, cones and spheresClassifying solidsEulers ruleEdges of a solidThe Platonic solidsDrawing and building solidsDifferent views of solidsTopic overviewChapter review

5 IntegersNumber linesNumbers above and below zeroDirected numbersOrdering directed numbersAdding and subtracting integersMultiplying integersDividing integersThe four operations with integersReading a map gridThe number planeThe number plane with negative numbersTopic overviewChapter review

6 Patterns and rulesNumber rules from geometric patternsUsing pattern rulesThe language of algebraTables of valuesFinding the ruleFinding harder rulesFinding rules for geometric patternsAlgebraic abbreviationsSubstitutionSubstitution with negative numbersTopic overviewChapter review

Mixed revision 27 DecimalsPlace valueUnderstanding the pointOrdering decimalsDecimals are special fractionsAdding and subtracting decimalsMultiplying and dividing by powers of 10Multiplying decimalsCalculating changeDividing decimalsDecimals at workConverting common fractions to decimalsRecurring decimalsRounding decimalsMore decimals at workTopic overviewChapter review

8 Length and areaThe history of measurementThe metric systemConverting units of lengthReading measurement scalesThe accuracy of measuring instrumentsEstimating and measuring lengthPerimeterAreaConverting units of areaArea of squares, rectangles and trianglesAreas of composite shapesMeasuring large areasTopic overviewChapter review

9 Geometric figuresPolygonsClassifying trianglesNaming geometric figuresConstructing trianglesClassifying quadrilateralsConstructing perpendicular and parallel linesConstructing quadrilateralsTopic overviewChapter review

Mixed revision 310 FractionsHighest common factor and lowest common multipleNaming fractionsEquivalent fractionsOrdering fractionsAdding and subtracting fractionsAdding and subtracting mixed numeralsFractions of quantitiesMultiplying fractionsDividing fractionsTopic overviewChapter review

11 Volume, mass and timeVolumeVolume of a rectangular prismCapacity and liquid measureMassTimelinesConverting units of timeTime calculationsWorld standard timesTimetablesTopic overviewChapter review

12 AlgebraAlgebraic expressionsAlgebraic abbreviationsFrom words to algebraic expressionsLike termsMultiplying algebraic termsExpanding an expressionExpanding and simplifyingAlgebraic substitutionTopic overviewChapter review

13 Interpreting graphs and tablesPicture graphsColumn graphs and divided bar graphsSector graphsLine graphsTravel graphs and conversion graphsStep graphsReading tablesTopic overviewChapter review

Mixed revision 4General revisionAnswersIndex

GlossaryABCDEFG HI JK LMNOPQRSTU VW X Y Z

All files menuChapter 1Chapter 2Chapter 3Chapter 4Chapter 5Chapter 6Chapter 7Chapter 8Chapter 9Chapter 10Chapter 11Chapter 12Chapter 13

Explanation menuChapter 1Chapter 2Chapter 3Chapter 4Chapter 5Chapter 7Chapter 8Chapter 9Chapter 10Chapter 11Chapter 12Chapter 13

Learning technology menuChapter 1Chapter 2Chapter 3Chapter 4Chapter 5Chapter 6Chapter 7Chapter 8Chapter 9Chapter 11Chapter 12

Practice menuChapter 1Chapter 2Chapter 3Chapter 4Chapter 5Chapter 6Chapter 7Chapter 8Chapter 9Chapter 10Chapter 11Chapter 12Chapter 13

Revision menuChapter 1Chapter 2Chapter 3Chapter 4Chapter 5Chapter 6Chapter 7Chapter 8Chapter 9Chapter 10Chapter 11Chapter 12Chapter 13

Using technology menuChapter 1Chapter 2Chapter 3Chapter 4Chapter 5Chapter 6Chapter 7Chapter 8Chapter 9Chapter 10Chapter 11Chapter 13

HelpLicence agreement

sb0101: sb0105: menu:

glossary:

Date post:	19-Oct-2015
Category:	Documents
Upload:	len16328
View:	53 times
Download:	0 times

Chapter01-The History of Numbers

Documents