Download Number systems - Haese Mathematics

Chapter
1
Number systems
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Contents:
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Different number systems
The Hindu-Arabic system
Big numbers
IB MYP_1
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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?
A
² Romans
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
©
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jjjjjjjjjjjjjjjjjjjjjjj objects would be recorded as
© jjjj
© jjjj
© jjjj
© jjj :
©
©
©
©
jjjj
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IB MYP_1
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,
would still represent 35.
or
The Egyptian system did not have place values.
EXERCISE 1A.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.
1
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
2
3
4
20
30
700
1000
5
50
6
7
8
60
100
9
10
400
¢, H, and X are
combined with the
symbol d for 5 to make
50, 500 and 5000.
500
5000
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This number system depends on addition and multiplication.
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IB MYP_1
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NUMBER SYSTEMS (Chapter 1)
Self Tutor
Example 1
Change the following Ancient Greek numerals into a Hindu-Arabic number:
a
b
1000
a
6000
b
300
700
20
80
+ 4
+ 1
1324
6781
EXERCISE 1A.2
1 Change the following Ancient Greek numerals into Hindu-Arabic numbers:
a
b
c
d
e
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.
became two lots of 5, so ten was represented by X.
or L became 50.
C represented one hundred, and half a
. With a little imagination you should see that an
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100
, so D became half a thousand or 500.
split in half and turned on its side became
50
One thousand was represented by an
75
Two Vs joined together
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IB MYP_1
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NUMBER SYSTEMS (Chapter 1)
1
I
2
II
3
III
4
IV
5
V
6
VI
7
VII
8
VIII
9
IX
10
X
20
XX
30
XXX
40
XL
50
L
60
LX
70
LXX
80
LXXX
90
XC
100
C
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.
10 000
X
5000
V
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
b XIV
g CXXV
k D L DCV
l DCCXX
c XVI
h CCXVI
d XXXI
i LXII
e CX
j MCLVI
m CDXIX
n D L V DI
o MMCCC
2 Write the following numbers in Roman numerals:
a 18
3
b 34
c 279
d 902
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f 2551
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.
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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e 1046
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unit of currency
used by the
Romans.
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NUMBER SYSTEMS (Chapter 1)
ACTIVITY 1
IF YOU LIVED IN ROMAN TIMES
What to do:
1 Use
a
b
c
d
e
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
which represented a mussel shell, works like
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our zero.
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NUMBER SYSTEMS (Chapter 1)
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
b 50
c 99
d 105
e 217
f 303
e
f
2 Convert these Mayan symbols into Hindu-Arabic numbers:
a
b
c
d
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.
1
2
3
4
5
6
7
8
9
10
100
1000
Their symbols are:
This is how 4983 would be written:
4 ‘lots’ of 1000
+
9 ‘lots’ of 100
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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
b 8400
c 1111
3 Copy and complete:
a
Words
Hindu-Arabic
thirty seven
37
Roman
Egyptian
Mayan
Chinese-Japanese
b
c
CLIX
d
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:
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5 Remove 3 matchsticks from this sum to make
the equation correct.
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IB MYP_1
NUMBER SYSTEMS (Chapter 1)
B
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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
one
two
three
four
five
six
seven
eight
nine
1
2
3
4
5
6
7
8
9
Hindu-Arabic
numeral
modern numeral
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.
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For example: in 567 942
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NUMBER SYSTEMS (Chapter 1)
Self Tutor
Example 2
What number is represented by the digit 7 in:
a 374
b 5709
c 127 624?
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
b 3580
c 50 083
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?
3
4 Put
a
b
c
d
e
f
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
Ascending means from
smallest to largest.
Descending means from
largest to smallest.
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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
5
d 805 340
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NUMBER SYSTEMS (Chapter 1)
19
Self Tutor
Example 3
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
c 3874
g 600 829
DEMO
d 9083
h 354 718
DEMO
8 Write the following in numeral form:
a twenty seven
b eighty
c six hundred and eight
e eight thousand two hundred
d one thousand and sixteen
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
b two more than eleven
four more than seventeen
seven greater than four thousand
four more than four hundred thousand
26 more than two hundred and nine thousand?
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
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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?
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NUMBER SYSTEMS (Chapter 1)
C
BIG NUMBERS
Commas or , are sometimes used to make it easier to read numbers greater than 3 digits.
For example:
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
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
3
Units
hundreds tens
6
8
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?
DEMO
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.
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
1m
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.
1m
A billion cubic millimetres are contained in a cube
which is 1 m £ 1 m £ 1 m.
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A trillion is 1000 billion or 1 000 000 000 000.
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NUMBER SYSTEMS (Chapter 1)
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
21
Units
H T U
9 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.
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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
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NUMBER SYSTEMS (Chapter 1)
5
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.
Continent
Africa
Antarctica
Asia
Australia
Europe
North America
South America
b Which continents are completely in the
Southern Hemisphere?
ACTIVITY 3
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
NUMBER SEARCH PROBLEMS
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.
PRINTABLE
WORKSHEET
Draw or click on the icon to print these grids then insert the given numbers.
Search 1:
2 digits
89, 92, 56
6 digits
949 875
3 digits
183
7 digits
8 097 116
3 291 748
6 709 493
7 264 331
4 387 096
3 872 095
4 digits
6680
5 digits
69 235
8 digits
62 658 397
79 408 632
10 343 879
91 863 432
81 947 368
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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
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IB MYP_1
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
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
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
b 569
3 Give the numbers represented by the Roman numerals:
a XVIII
b LXXIX
4 Write the year 2012 using Roman numerals.
5 Write the following numbers using the Mayan system:
a 46
b 273
6 Give the numbers represented by the Chinese-Japanese symbols:
a
b
7 Give the number represented by the digit 4 in:
a 3409
b 41 076
8 What is the place value of the 8 in the following numbers?
a 3894
b 856 042
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.
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100
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75
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5
95
100
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75
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5
95
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5
95
100
50
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5
12 Write seventeen thousand three hundred and four in numeral form.
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IB MYP_1
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
b 245
2 Give the numbers represented by the Egyptian symbols:
a
b
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
b 2113
5 What number is represented by the digit 7 in the following?
a 3174
b 207 409
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
b 306 428
9 What number is:
a five more than eighteen
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.
1st
2nd
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5
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5
95
100
50
75
25
0
5
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?
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IB MYP_1