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Chapter

1

Number systems

Contents:

A B C

Different number systems The Hindu-Arabic system Big numbers

10

NUMBER SYSTEMS (Chapter 1)

Archaeologists and anthropologists study ancient civilizations. They have helped us to understand how people long ago counted and recorded numbers. Their findings suggest that the first attempts at counting were to use a tally. For example, in ancient times people used items to represent numbers:

scratches on a cave wall showed the number of new moons since the buffalo herd came through

knots on a rope showed the rows of corn planted

pebbles on the sand showed the number of traps set for fish

notches cut on a branch showed the number of new lambs born

In time, humans learned to write numbers more efficiently. They did this by developing number systems.

OPENING PROBLEM

The number system we use in this course is based on the Hindu-Arabic system which uses the digits 0, 1, 2, 3, 4, 5, 6, 7, 8 and 9.

The number of dots shown here is twenty three. We write this as 23, which means `2 tens and 3 ones'.

How was the number 23 written by: ² ancient Egyptians ² Mayans ² ancient Greeks ² Chinese and Japanese? ² Romans

A

DIFFERENT NUMBER SYSTEMS

The ancient Egyptians used tally strokes to record and count objects.

jjjjjjjjjjjjjjjjjjjjjjj indicated there were 23 objects. In time they replaced every 10 strokes with a different symbol. They chose to represent jjjjjjjjjj . So, 23 was then written as jjj .

We still use tallies to help with counting. Instead of © © jjjjj we now use jjjj . jjjjjjjjjjjjjjjjjjjjjjj objects would be recorded as © © © © jjjj jjjj jjjj jjjj jjj : © © © ©

NUMBER SYSTEMS (Chapter 1)

11

THE EGYPTIAN NUMBER SYSTEM

There is archaeological evidence that as long ago as 3600 BC the Egyptians were using a detailed number system. The symbols used to represent numbers were pictures of everyday things. These symbols are called hieroglyphics which means sacred picture writings. The Egyptians used a tally system based on the number ten. Ten of one symbol could be replaced by one of another symbol. We call this a base ten system. 1 10 100 1000

staff

hock

scroll

lotus flower

10 000

100 000

1 000 000

10 000 000

bent stick

burbay fish

astonished man

religious symbol

The order in which the symbols were written down did not affect the value of the numerals. The value of the numerals could be found by adding the value of the symbols used. So, or would still represent 35.

The Egyptian system did not have place values.

EXERCISE 1A.1

1 a In the Hindu-Arabic number system, 3 symbols are used to write the number 999. How many Egyptian symbols are needed to write the Hindu-Arabic 999? b Write the Egyptian symbols for 728 and 234 124.

2 Convert these symbols to Hindu-Arabic numerals: a b

THE ANCIENT GREEK OR ATTIC SYSTEM

The Ancient Greeks saw the need to include a symbol for 5. This symbol was combined with the symbols for 10, 100, and 1000 to make 50, 500, and 5000. Some examples of Ancient Greek numbers are:

1 20 2 3 30 4 5 50 6 60 7 100 8 400 9 10 500 ¢, H, and X are combined with the symbol d for 5 to make 50, 500 and 5000.

700

1000

5000

This number system depends on addition and multiplication.

12

NUMBER SYSTEMS (Chapter 1)

Example 1

Self Tutor

b 1000 300 20 + 4 1324 6000 700 80 + 1 6781

Change the following Ancient Greek numerals into a Hindu-Arabic number: a

a

b

EXERCISE 1A.2

1 Change the following Ancient Greek numerals into Hindu-Arabic numbers: a d b e c f

2 Write the following Hindu-Arabic numbers as Ancient Greek numerals: a 14 b 31 c 99 d 555 e 4082 f 5601

ROMAN NUMERALS

Like the Greeks, the Romans used a number for five.

The first four numbers could be represented by the fingers on one hand, so the V formed by the thumb and forefinger of an open hand represented 5.

Two Vs joined together

became two lots of 5, so ten was represented by X.

C represented one hundred, and half a One thousand was represented by an

or L became 50. . With a little imagination you should see that an , so D became half a thousand or 500.

split in half and turned on its side became

NUMBER SYSTEMS (Chapter 1)

13

1 I 20 XX

2 II 30 XXX

3 III 40 XL

4 IV 50 L

5 V 60 LX

6 VI 70 LXX

7 VII 80 LXXX

8 VIII 90 XC

9 IX 100 C

10 X 500 D 1000 M

Unlike the Egyptian system, numbers written in the Roman system had to be written in order. For example: IV stands for 1 before 5 or 4 whereas VI stands for 1 after 5 or 6. XC stands for 10 before 100 or 90 whereas CX stands for 10 after 100 or 110. There were rules for the order in which symbols could be used: ² I could only appear before V or X. ² X could only appear before L or C. ² C could only appear before D or M. One less than a thousand was therefore not written as IM but as CMXCIX. Larger numerals were formed by placing a stroke above the symbol. This made the number 1000 times as large. 5000 V 10 000 X 50 000 L 100 000 C 500 000 D 1 000 000 M

EXERCISE 1A.3

1 What numbers are represented by the following symbols? a VIII f LXXXI k D L DCV a 18 3 b XIV g CXXV l DCCXX b 34 c XVI h CCXVI m CDXIX c 279 d 902 d XXXI i LXII n D L V DI e 1046 e CX j MCLVI o MMCCC f 2551

2 Write the following numbers in Roman numerals:

a Which Roman numeral less than one hundred is written using the greatest number of symbols? b What is the highest Roman numeral between M and MM which uses the least number of symbols? Denarii was the c Write the year 1999 using Roman symbols.

unit of currency used by the Romans.

4 Use Roman numerals to answer the following questions. a Each week Octavius sharpens CCCLIV swords for his general. How many will he need to sharpen if the general doubles his order? b What would it cost Claudius to finish his courtyard if he needs to pay for CL pavers at VIII denarii each and labour costs XCIV denarii?

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NUMBER SYSTEMS (Chapter 1)

ACTIVITY 1

What to do: 1 Use a b c d e

IF YOU LIVED IN ROMAN TIMES

Roman numerals to write: your house number and postcode your height in centimetres your phone number the number of students in your class the width of your desk in centimetres.

That's MXXII plus CDL minus CXIX.

2 Use a calendar to help you write in Roman numerals: a your date of birth, for example XXI-XI-MCMXLVI b what the date will be when you are: i XV ii L iii XXI iv C

THE MAYAN SYSTEM

The Mayans originally used pebbles and sticks to represent numbers. They later recorded them as dots and strokes. A stroke represented the number 5.

1

2

3

4

5

6

7

8

9

10

11 12 13 14 15 16 17 18 19 20

Unlike the Egyptians and Romans, the Mayans created a place value by placing one symbol above the other.

The Hindu-Arabic system we use in this course involves base 10. The number 172 is 17 `lots of' 10 plus 2 `lots of' 1. In contrast, the Mayan system used base 20. Consider this upper part represents 8 `lots of' 20 or 160 the lower part represents 12 `lots of' 1 or 12 So, the number represented is 172

The Mayans also recognised the need for a number zero to show the difference between `lots of 1' and `lots of 20'. The symbol our zero. which represented a mussel shell, works like

NUMBER SYSTEMS (Chapter 1)

15

Compare these symbols:

43

40

68

60

149

100

lots of 20 lots of 1

EXERCISE 1A.4

1 Write these numbers using Mayan symbols: a 23 a b 50 b c 99 c d 105 d e 217 e f 303 f 2 Convert these Mayan symbols into Hindu-Arabic numbers:

RESEARCH

OTHER WAYS OF COUNTING

Find out: a how the Ancient Egyptians and Mayans represented numbers larger than 1000 b whether the Egyptians used a symbol for zero c what Braille numbers are and what they feel like d how deaf people `sign' numbers.

1

2

3

4

5

6

7

8

9

0

THE CHINESE - JAPANESE SYSTEM

The Chinese and Japanese use a similar place value system. Their symbols are:

1

2

3

4

5

6

7

8

9

10

100

1000

This is how 4983 would be written:

4 `lots' of 1000 + 9 `lots' of 100 + 8 `lots' of 10 + 3

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NUMBER SYSTEMS (Chapter 1)

EXERCISE 1A.5

1 What numbers are represented by these symbols? a b c

2 Write these numbers using Chinese-Japanese symbols: a 497 3 Copy and complete: Words a b c d thirty seven Hindu-Arabic 37 Roman Egyptian Mayan Chinese-Japanese b 8400 c 1111

CLIX

ACTIVITY 2

MATCHSTICK PUZZLES

Use matchsticks to solve these puzzles. Unless stated otherwise, you are not allowed to remove a matchstick completely.

1 Move just one matchstick to make this correct: 2 Move one matchstick to make this correct: 3 Arrange 4 matchsticks to make a total of 15. 4 Make this correct moving just one matchstick: 5 Remove 3 matchsticks from this sum to make the equation correct.

NUMBER SYSTEMS (Chapter 1)

17

B

THE HINDU-ARABIC SYSTEM

The number system we will use throughout this course was developed in India 2000 years ago. It was introduced to European nations by Arab traders about 1000 years ago. The system was thus called the Hindu-Arabic system. The marks we use to represent numbers are called numerals. They are made up using the symbols 1, 2, 3, 4, 5, 6, 7, 8, 9 and 0, which are known as digits.

ordinal number Hindu-Arabic numeral

modern numeral

one

two

three

four

five

six

seven

eight

nine

1

2

3

4

5

6

7

8

9

The digits 3 and 8 are used to form the numeral 38 for the number `thirty eight' and the numeral 83 for the number `eighty three'. The numbers we use for counting are called natural numbers or sometimes just counting numbers. The possible combination of natural numbers is endless. There is no largest natural number, so we say the set of all natural numbers is infinite. If we include the number zero or 0, then our set now has a new name, which is the set of whole numbers. The Hindu-Arabic system is more useful and more efficient than the systems used by the Egyptians, Romans, and Mayans. ² It uses only 10 digits to construct all the natural numbers. ² It uses the digit 0 or zero to show an empty place value. ² It has a place value system where digits represent different numbers when placed in different place value columns. Each digit in a number has a place value. For example: in 567 942

hundred thousands

ten thousands

thousands

hundreds

5

6

7

9

4

units

tens

2

18

NUMBER SYSTEMS (Chapter 1)

Example 2

Self Tutor

b 5709 c 127 624?

What number is represented by the digit 7 in: a 374

a In 374, the 7 represents `7 lots of 10' or 70: b In 5709, the 7 represents `7 lots of 100' or 700: c In 127 624, the 7 represents `7 lots of 1000' or 7000:

EXERCISE 1B

1 What number is represented by the digit 8 in the following? a 38 e 1981 i 60 008 b 81 f 8247 j 84 019 c 458 g 2861 k 78 794 d 847 h 28 902 l 189 964

2 Write down the place value of the 3, the 5 and the 8 in each of the following: a 53 486 3 b 3580 c 50 083 d 805 340

a Use the digits 6, 4 and 8 once only to make the largest number you can. b Write the largest number you can using the digits 4, 1, 0, 7, 2 and 9 once only. c What is the largest 6 digit numeral you can write using each of the digits 2, 7 and 9 twice? d How many different numbers can you write using the digits 3, 4 and 5 once only? the following numbers in ascending order: 57, 8, 75, 16, 54, 19 660, 60, 600, 6, 606 1080, 1808, 1800, 1008, 1880 45 061, 46 510, 40 561, 46 051, 46 501 236 705, 227 635, 207 653, 265 703 554 922, 594 522, 545 922, 595 242

4 Put a b c d e f

5 Write the following numbers in descending order: a 361, 136, 163, 613, 316, 631 b 7789, 7987, 9787, 8779, 8977, 7897, 9877 c 498 231, 428 931, 492 813, 428 391, 498 321 d 563 074, 576 304, 675 034, 607 543, 673 540

Ascending means from smallest to largest. Descending means from largest to smallest.

NUMBER SYSTEMS (Chapter 1)

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Example 3

Self Tutor

a Express 3 £ 10 000 + 4 £ 1000 + 8 £ 10 + 5 £ 1 in simplest form. b Write 9602 in expanded form. a 3 £ 10 000 + 4 £ 1000 + 8 £ 10 + 5 £ 1 = 34 085 b 9602 = 9 £ 1000 + 6 £ 100 + 2 £ 1 6 Express the following in simplest form: a 8 £ 10 + 6 £ 1 b 6 £ 100 + 7 £ 10 + 4 £ 1 c 9 £ 1000 + 6 £ 100 + 3 £ 10 + 8 £ 1 d 5 £ 10 000 + 2 £ 100 + 4 £ 10 e 2 £ 10 000 + 7 £ 1000 + 3 £ 1 f 2 £ 100 + 7 £ 10 000 + 3 £ 1000 + 9 £ 10 + 8 £ 1 g 3 £ 100 + 5 £ 100 000 + 7 £ 10 + 5 £ 1 h 8 £ 100 000 + 9 £ 1000 + 3 £ 100 + 2 £ 1 7 Write in expanded form: a 975 b 680 e 56 742 f 75 007 8 Write the following in numeral form: a twenty seven c six hundred and eight e eight thousand two hundred b eighty d one thousand and sixteen c 3874 g 600 829 d 9083 h 354 718

DEMO

DEMO

f nineteen thousand five hundred and thirty eight g seventy five thousand four hundred and three h six hundred and two thousand eight hundred and eighteen. 9 What number is: a one less than eight c e g h four more than seventeen seven greater than four thousand four more than four hundred thousand 26 more than two hundred and nine thousand? b two more than eleven d one less than three hundred f 3 less than 10 000

10 The number 372 474 contains two 7s and two 4s.

first 7 second 7

a How many times larger is the first 7 compared with the second 7? b How many times smaller is the second 7 compared with the first 7? c Which of the 4s represents a larger number? By how much is it larger than the other one?

20

NUMBER SYSTEMS (Chapter 1)

C

For example:

BIG NUMBERS

2; 954 two thousand, nine hundred and fifty four 4; 234; 685 four million, two hundred and thirty four thousand, six hundred and eighty five

Commas or , are sometimes used to make it easier to read numbers greater than 3 digits.

When typed, we usually use a space instead of the comma. Can you suggest some reasons for this? Millions hundreds tens 5 Thousands hundreds tens units 4 7 9 Units hundreds tens 6 8

units 3

units 2

The number displayed in the place value chart is 53 million, 479 thousand, 682. To make the number easier to read the digits are arranged into the units, the thousands, and the millions. With spaces now used to separate the groups, the number on the place value chart is written 53 479 682.

A MILLION

One million is written 1 000 000. Just how large is one million? Consider the following: is a diagram of a cube with sides 1 mm.

is a diagram of a cube with sides 1 cm. Each 1 cm = 10 mm. This cube contains 1000 cubes with sides 1 mm.

DEMO

A cube which has sides 10 cm is made up of 10¡£¡10¡£¡10¡=¡1000 cubes with sides 1 cm, and each cube with sides 1 cm is made up of 1000 cubes with sides 1 mm. So, it is made up of 1000¡£¡1000 or 1¡000¡000 cubes with sides 1 mm.

A BILLION AND A TRILLION

A billion is 1000 million or 1 000 000 000. We saw previously that a 10 cm £ 10 cm £ 10 cm cube contains 1 000 000 cubic millimetres. A billion cubic millimetres are contained in a cube which is 1 m £ 1 m £ 1 m. A trillion is 1000 billion or 1 000 000 000 000.

1m 1m

1m

NUMBER SYSTEMS (Chapter 1)

21

Trillions H T U 6 3

Billions H T U 5 8 4

Millions H T U 2 0 1

Thousands H T U 5 7 1

H 9

Units T U 2 6

The number displayed in the place value chart is 63 trillion, 584 billion, 201 million, 571 thousand 9 hundred and 26.

EXERCISE 1C

1 In the number 53 479 682, the digit 9 has the value 9000 and the digit 3 has the value 3 000 000. Give the value of the: a 8 b 5 c 6 d 4 e 7 f 2 2 Write the value of each digit in the following numbers: a 3 648 597 b 34 865 271

3 Read the following stories about large numbers. Write each large number using numerals. a A heart beating at a rate of 70 beats per minute would beat about thirty seven million times in a year. b Austria's largest hamburger chain bought two hundred million bread buns and used seventeen million kilograms of beef in one year. c The Jurassic era was about one hundred and fifty million years ago. d One hundred and eleven million, two hundred and forty thousand, four hundred and sixty three dollars and ten cents was won by two people in a Powerball Lottery in Wisconsin USA in 1993. e A total of twenty one million, two hundred and forty thousand, six hundred and fifty seven Volkswagen `Beetles' had been built to the end of 1995. f In a lifetime the average person will blink four hundred and fifteen million times. g One Megabyte of data on a computer is one million, forty eight thousand, five hundred and seventy six bytes. 4 Arrange these planets in order of their distance from the Sun starting with the closest. Venus 108 200 000 kms Saturn 1 427 000 000 kms Earth 149 600 000 kms Uranus 2 870 000 000 kms Mercury 57 900 000 kms Jupiter 778 300 000 kms Pluto 5 900 000 000 kms Neptune 4 497 000 000 kms Mars 227 900 000 kms

22 5

NUMBER SYSTEMS (Chapter 1)

a Use the table to answer the following: i Which continent has the greatest area? ii Name the continents with an area greater than 20 million square kilometres. b Which continents are completely in the Southern Hemisphere?

Continent Africa Antarctica Asia Australia Europe North America South America

Area in square km 30 271 000 13 209 000 44 026 000 7 682 000 10 404 000 24 258 000 17 823 000

ACTIVITY 3

NUMBER SEARCH PROBLEMS

PRINTABLE WORKSHEET

Number searches are like crossword puzzles with numbers going across and down. The aim is to fit all of the numbers into the grid using each number once. There is only one way in which all of the numbers will fit.

Draw or click on the icon to print these grids then insert the given numbers.

Search 1:

2 digits 89, 92, 56 3 digits 183 4 digits 6680 5 digits 69 235

6 digits 949 875 7 digits 8 097 116 3 291 748 6 709 493 7 264 331 4 387 096 3 872 095

8 digits 62 658 397 79 408 632 10 343 879 91 863 432 81 947 368

Search 2: ² seven hundred and nine ² five hundred and eighty six ² sixty thousand, two hundred and eighty four ² seven hundred and ninety three thousand and forty two ² four hundred and forty nine thousand, seven hundred and sixty eight ² three million eight hundred and two thousand, seven hundred and forty eight 0 ² two million six hundred and eighty three thousand, one hundred and forty eight ² seventy million, two hundred and eighty three thousand, six hundred and forty two ² nineteen million, three hundred and eighty four thousand, and three ² five hundred and eighty three million, seventy nine thousand, six hundred and forty six ² three hundred and forty five million, six hundred and ninety seven thousand and fifty one

NUMBER SYSTEMS (Chapter 1)

23

Did you know? The milk from 1 000 000 litre cartons would fill a 50 metre long by 20 metre wide pool to a depth of 1 metre.

KEY WORDS USED IN THIS CHAPTER

² ² ² ² ² ² Ancient Greek system counting number Hindu-Arabic system million numeral tally ² ² ² ² ² ² billion digit infinite natural number place value trillion ² ² ² ² ² ² Chinese-Japanese system Egyptian system Mayan system number system Roman numeral whole number

REVIEW SET 1A

1 Give the numbers represented by the Ancient Greek symbols: a b

2 Write the following numbers using Egyptian symbols: a 27 a XVIII b 569 b LXXIX 3 Give the numbers represented by the Roman numerals: 4 Write the year 2012 using Roman numerals. 5 Write the following numbers using the Mayan system: a 46 a b 273 b 6 Give the numbers represented by the Chinese-Japanese symbols:

7 Give the number represented by the digit 4 in: a 3894 b 856 042

a 3409

b 41 076

8 What is the place value of the 8 in the following numbers? 9 Use the digits 8, 0, 4, 1, 7 to make the largest number you can. 10 Write these numbers in ascending order (smallest first): 569 207, 96 572, 652 097, 795 602, 79 562 11 Express 2 £ 1000 + 4 £ 100 + 9 £ 10 + 7 £ 1 in simplest form. 12 Write seventeen thousand three hundred and four in numeral form.

24

NUMBER SYSTEMS (Chapter 1)

13 Write the value of each digit in the number 4 532 681. 14 The total area of Canada is approximately nine million, nine hundred and eighty four thousand, seven hundred square kilometres. Write this number using numerals.

REVIEW SET 1B

1 Write the following numbers using Ancient Greek symbols: a 78 a b 245 b 2 Give the numbers represented by the Egyptian symbols:

3 Which Roman numeral between 100 and 200 uses the greatest number of symbols? 4 Write these numbers using the Chinese-Japanese system: a 386 a 3174 b 2113 b 207 409 5 What number is represented by the digit 7 in the following? 6 What is the largest 6 digit number you can write using each of the digits 0, 5 and 8 twice? 7 Write in descending order (largest first): 680 969, 608 699, 6 080 699, 698 096, 968 099 8 Write the following numbers in expanded form: a 2159 9 What number is: a five more than eighteen b 306 428 b nine less than one thousand?

10 Write the value of each digit in the number 37 405 922. 11 The average person will travel five million, eight hundred and ninety thousand kilometres in a lifetime. Write this number using numerals. 12 Consider the number 2 000 000 000. a Write this number in words. b Copy and complete: 2 000 000 000 is ...... lots of one million.

1st 2nd

13 The number 2 552 667 contains two 2s, two 5s and two 6s. a How many times larger is the first 2 compared with the second 2? b How many times smaller is the second 5 compared with the first 5? c Which of the 6s represents a larger number? By how much is it larger than the other 6?

1st 2nd

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