43483 ns4.1 part 3 -...
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Mathematics Stage 4
NS4.1 Operations with whole numbers
Part 3 Numbering systems
Part 3 Numbering systems 1
Contents – Part 3
Introduction – Part 3..........................................................3
Indicators ...................................................................................3
Preliminary quiz – Part 3 ...................................................5
Ancient numbering systems ..............................................9
Egyptian numerals ..................................................................10
Babylonian numerals...............................................................11
Other numbering systems ...............................................15
Mayan numerals......................................................................15
Other counting systems ..........................................................17
Roman numerals.............................................................21
Hindu-Arabic numbering�system .....................................25
Why the system works well.....................................................25
Number value, face and place value......................................26
Expanded notation ..................................................................28
Special groups of numbers..............................................31
Figurate numbers ....................................................................32
Palindromic numbers ..............................................................32
More special numbers.....................................................35
Fibonacci numbers..................................................................35
Pascal’s triangle ......................................................................36
Suggested answers – Part 3 ...........................................39
2 NS4.1 Operations with whole numbers
Exercises – Part 3........................................................... 41
Review quiz – Part 3....................................................... 61
Answers to exercises – Part 3 ........................................ 65
Part 3 Numbering systems 3
Introduction – Part 3
While the decimal, or Hindu-Arabic, numbering system in use today is
fairly universal, many civilisations in the past developed their own
systems of numbering to fill their needs at the time. This part describes
how the numbering system in use today, the Hindu-Arabic number
system, compares with systems developed by some of these ancient
societies. It also points out features of our current system that makes it
superior to other systems developed in the past and why it has received
wide acceptance.
This Part also explores some relationships in special groups of numbers
including figurate numbers, palindromic numbers, Fibonacci numbers,
and the numbers in Pascal’s triangle.
IndicatorsBy the end of Part 3, you will have been given the opportunity to work
towards aspects of knowledge and skills including:
• comparing the Hindu-Arabic numbering system with numbering
systems from different societies past and present, especially
Egyptian, Babylonian, Mayan, Aboriginal and Roman
• considering Roman numerals at depth and being able to translate
Hindu-Arabic numbers into Roman numerals and vice versa
• identifying special groups of numbers including figurate numbers,
palindromic numbers, Fibonacci numbers, and the numbers in
Pascal’s triangle
• exploring our current decimal number system showing place value
for the digits.
4 NS4.1 Operations with whole numbers
By the end of Part 3, you will have been given the opportunity to work
mathematically by:
• discussing the strengths and weaknesses of different numbering systems
• describing and recognising the advantages of the Hindu-Arabic
numbering system.
Part 3 Numbering systems 5
Preliminary quiz – Part 3
Before you start this part, use this preliminary quiz to revise some skills
you will need.
Activity – Preliminary quiz
Try these.
1 Write the following as numerals.
a Three hundred and sixty-seven.
___________________________________________________
b Seventeen thousand six hundred and eighty-eight.
___________________________________________________
c Fifty thousand and seventy.
___________________________________________________
2 Sometimes you read a certain number as fifteen hundred. What are
other ways of writing this number? __________________________
3 Write the following numerals in words.
a 302 ________________________________________________
b 5478 _______________________________________________
c 100 240 _____________________________________________
4 Write the numeral that is half a million. What are other ways of
writing this number? _____________________________________
6 NS4.1 Operations with whole numbers
5 Circle the larger number in each pair.
a 729 or 792
b 1002 or 1020
c 20 712 or 20 707
d 14 213 914 or 14 216 902
6 Write these numbers from smallest to largest.
a 243, 234, 432, 423, 342
___________________________________________________
b 4062, 4602, 4206, 4620
___________________________________________________
7 What is the value of each of these Roman numerals?
a VII _______________________________________________
b XV _______________________________________________
c XC _______________________________________________
d CL ________________________________________________
8 Write these numbers as Roman numerals.
a 4 _________________________________________________
b 12 ________________________________________________
c 30 ________________________________________________
9 Write the next two numbers in these patterns.
a 89, 84, 79, 74, __, __
b 1, 2, 4, 8, 16, __, __
c 4, 15, 26, 37, __, __
Part 3 Numbering systems 7
10 How many times bigger is
a 30 than 3? ___________________________________________
b 300 than 3? __________________________________________
11 Write these as simple numbers.
a 5 × 100 + 4 × 10 + 2 × 1 _______________________________
b 3 × 100 + 9 × 1 ______________________________________
12 Determine the missing values in these.
a 348 = ____ × 100 + __ × 10 + __ × 1
b 2609 = ____ × 1000 + __ × 100 + __ × 10 + __ × 1
Check your response by going to the suggested answers section.
8 NS4.1 Operations with whole numbers
Part 3 Numbering systems 9
Ancient numbering systems
There is a story that almost two and a half thousand years ago
King Darius of Persia sent the following command to the Ionians:
The King took a leather thong and tied sixty knots in it. He called
together all of his officers, and said: “each day I want you to untie
one of the knots. If I do not return before all the knots are untied I
want you to leave this place and go to your homes.”
Even before people developed letters and numbers they had a need to
count. Some methods involved putting notches on sticks, or collecting
sticks, stones or shells. For example, a shepherd who had a flock of
sheep would need to keep track of them. If there were only a few, he
might know them all by sight. But as his flock grew he would need a
new way to keep track of them. He might collect pebbles, one for each
sheep, and place them in a bag. The number of pebbles he had would
indicate the number of sheep. He would also have a way of showing
other people how many animals he owned when away from his flock.
The Incas of Central and South America
developed a method of recording numbers which
did not require writing. It involved knots in
strings called a quipu. A number was shown by
knots in the string. If the number 462 was to be
recorded on the string then two touching knots
were placed near the free end of the string, a
space was left, then six touching knots for the
10s, another space, and finally four touching
knots for the 100s.
4 knots
6 knots
2 knots
10 NS4.1 Operations with whole numbers
Egyptian numerals
The Egyptians were only concerned with using numbers in practical
ways. The Egyptians had a decimal system using 7 different symbols.
This system required repeating symbols.
For example, the coil of rope or indicated 100.
So three coils of rope showed 300 is
This table shows the numbers from 1–10 and the remaining symbols
Egyptians used.
1 2 3
4 5 6 7 89
10 100 1 000 10 000 100 000 1 000 000
For example: the number 256 could be written as .
Because there are different symbols for 1, 10, 100 and so on, it doesn’t
matter which way around the numbers are written.
So the coil of rope indicating 100 can be written in either direction: or .
The Egyptians did not have a symbol for zero.
Part 3 Numbering systems 11
Babylonian numerals
The Babylonians used only two symbols for numbers.
There was a symbol sort of like a ‘Y’ for 1, and a symbol looking like
‘<’ for 10. These symbols were made by pushing a pointed stick into wet
clay. They then allowed the clay to dry out to a hard tablet.
The symbols were grouped together to make other numbers, such as:
3 30 45 56
Can you see how the five and the six were made up of symbols stacked
on top of each other.
The last example has 5 ten symbols grouped together to make 50 and
6 one unit symbols grouped together.
The table on the following page shows how the Babylonians would have
written the numbers from 1 to 20.
12 NS4.1 Operations with whole numbers
Grouping ones and tens symbols like this,
they made numbers from 1 to 59.
But what about larger numbers?
They left a space and used another group of
symbols. Instead of using powers of 10 as
you do for the place value positions, they
used powers of 60 (60, 60 × 60 = 3600, and
so on). For example,
indicated 1× 60 + 32 = 92.
Here is another example:
↓ ↓ ↓
4 × 60× 60+ 21× 60 +37
= 14 400+1260+ 37
= 15 697
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
Like the Egyptian numerals, there was no symbol for zero.
You might think 60 a strange number as a base for a number system on
but you still use groups of 60 when measuring time and angles.
You inherited these ideas from the Babylonians.
Most ancient cultures had little need for very large numbers.
The numbering systems they developed served them well for their
purposes. Can you see that as numbers become larger, the number of
symbols needed by the Babylonians and Egyptians became greater?
Writing large numbers was time consuming.
You have been learning about ancient numbering systems as well as
Babylonian and Egyptian numerals. Now check that you can rewrite
these ancient numbers into numbers we use today and vice versa.
Part 3 Numbering systems 13
Go to the exercises section and complete Exercise 3.1 – Ancient
numbering systems.
You might like to investigate ancient number systems further.
Access related sites on the history of numbers by visiting the LMP
webpage below. Select Stage 4 and follow the links to resources for this
unit NS4.1 Operations with whole numbers, Part 3.
<http://www.lmpc.edu.au/mathematics>
14 NS4.1 Operations with whole numbers
Part 3 Numbering systems 15
Other numbering systems
Many early civilisations developed numbers to meet the needs of their
societies. You looked at two of these, the Egyptian and Babylonian
systems in the previous lesson. Remember, the numbering system you
use today had not yet been developed in those times.
Here are some more.
Mayan numerals
The ancient Mayan civilisation emerged some 3000 years ago and
occupied central America. Between 250 and 900 AD it was at its height,
then suddenly disappeared.
The Mayans used dots and bars. The dots are worth 1 and the bars are
worth 5. They also had a symbol for zero and it looked like .
For instance, the number 19 in the Mayan system can
be written like this (3 fives and 4 ones.)
Adding one more dot turns this number into 20. The five dots can be
replaced by another bar. But instead of now drawing four bars, the
Mayans used the symbol for zero to write the number 20.
They wrote 20 as a dot with a below it. The maximum
value of one place is 19 so that four bars, or 20, is too large a
number to fit in one place.
The dot occupies a place in the 20s position, so each dot here is worth 20.
Since five dots equal one bar, each bar is worth 100 in that place.
The Mayan number system was based on the number twenty.
Possibly the reason for base 20 arose from ancient people who counted
16 NS4.1 Operations with whole numbers
on both their fingers and their toes. The Mayan numbering system
requires only three symbols. Numbers were written from bottom to top.
Here are the Mayan numerals up to 29. You do not need to learn them.
0 1 2 3 4 5 6 7 8 9
10 11 12 13 14 15 16 17 18 19
20 21 22 23 24 25 26 27 28 29
The value of a dot in the third place is 400 (20 × 20), and a bar in that
place is worth 5 × 400 = 2000. In each following place, one dot is worth
20 times as much as a dot in the previous place, each bar is worth
20 times as much as a bar in the previous place.
The table summarises the place value of bars and dots in the Mayan
number system.
Each dot in thisposition is worth …
Each bar in thisposition is worth …
8000splace
20 × 20 × 20 8000 5 × 8000 = 40 000
400splace
20 × 20 400 5 × 400 = 2000
20splace
20 20 5 × 20 = 100
1splace
1 1 5 × 1 = 5
Here are some examples.
20s place1s place
2 × 20 + (5 + 1) = 46 (5 × 20) + 8 = 108 4 × 20 + 10 = 90
Part 3 Numbering systems 17
It is very easy to add and subtract using this number system.
8000s
400s
20s
1s
9449 + 10425 = 19874
+ =
This base twenty system is still in use today by such tribes as the Hopi
and the Inuits (Eskimos).
Other counting systems
Numbering and counting systems were developed by a variety of
different cultures. Some of these systems were more advanced than
others. They are presented here so you can get an appreciation of what
was invented to suit people’s needs over the ages. You do not need to
learn them.
Aboriginal
While the Australian Aborigines did not have a written language, they
certainly did have words to express quantity. A numbering system used
by the Torres Strait Islanders is one of the earliest uses of binary.
(Binary is just a numbering system that uses two symbols or conditions
such as 0 or 1, or off and on.)
Here is part of the Torres Strait Islander numbering system:
1 = urapun
2 = okosa
3 = okosa-urapun
4 = okosa-okosa
5 = okosa-okosa-urapun
6 = okosa-okosa-okosa and so on.
18 NS4.1 Operations with whole numbers
Papua New Guinea
Many languages in Papua New Guinea used parts of the body to show
numbers. For example the word ‘hand’ is a common source of the word
for ‘five’ and ‘person’ represented ‘twenty’. Other counting words could
be made up of combinations of the basic words. For example,
‘hand-finger’ could represent 6, while ‘two-hands-one-foot’ could
represent 15.
One extreme example is the Kewa people of Papua New Guinea, who
count from 1 to 68 on different parts of the body. This diagram shows
the body part tally system used by the Fasu people.
12
34
5678
9
1011
12
1314
1516
The Api people
The Api people of the New Hebrides have an interesting way
of counting.
Look carefully to see if you can identify any pattern in the
numbers shown
Here is the pronunciation for the numbers from 1 to 18.
1 tai 10 lualuna
2 lua 11 lualuna tai
3 tolu 12 lualuna lua
4 vari 13 lualuna tolu
5 luna 14 lualuna vari
6 otai 15 toluluna
7 olua 16 toluluna tai
Part 3 Numbering systems 19
8 otolu 17 toluluna lua
9 ovari 18 toluluna tolu
Ancient Greeks
The ancient Greeks gave number values to their letters and used them as
numerals. This alphabetical system is still used by the Greeks, just like
you still use Roman numerals.
This numbering system needed 27 letters. Since the Greek alphabet has
only 24 letters three disused letters were added, one in each group: 6 is
(digamma or stigma), 90 is (koppa), 900 is (sampi).
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
30
40
50
60
70
80
90
100
200
300
400
500
600
700
800
900
For example 847 can be written asωµζ ' .
You have been learning about other numbering systems used in different
civilisations. Now check that you understand what you have learned, and
that you can rewrite these numbers in any form.
Go to the exercises section and complete Exercise 3.2 – Other ancient
numbering systems.
You might like to investigate the history of numbers further.
Access related sites on the history of numbers by visiting the LMP
webpage below. Select Stage 4 and follow the links to resources for this
unit NS4.1 Operations with whole numbers, Part 3.
<http://www.lmpc.edu.au/mathematics>
20 NS4.1 Operations with whole numbers
Part 3 Numbering systems 21
Roman numerals
The Romans were active in trade and commerce, and from the time they
developed a method of writing they needed a way to indicate numbers.
The system they invented lasted in Europe for many hundreds of years.
Roman numerals were developed around 500 BC
partly from early Greek alphabet symbols.
Today you still see them on things such as clocks,
and the front pages of many books.
You have probably already learned how to read
and write Roman numerals. Here is a summary of
that information.
XIIXI
XIX
VIII
VII VI V
IVIII
II
I
Roman numerals are based on seven capital letters, each of which has a
specific value:
I = 1 V = 5 X = 10 L = 50 C = 100 D = 500 M = 1000
All numbers can be written by using a combination of these letters.
In this, like many other systems, there was no symbol for zero.
In Roman numerals, letters come together according to these basic rules.
• Smaller values to the right of a larger number are added:
VI = 6 (5 + 1); XV = 15 (10 + 5); CX = 110 (100 + 10).
• Smaller values to the left of a larger number are subtracted:
IV = 4 (5 – 1); XL = 40 (50 – 10); XC = 90 (100 – 10).
• When subtracting, only I can come before V or X; only X can come
before L or C; and only C can come before D or M:
45 is XLV (50 – 10 + 5), not VL.
22 NS4.1 Operations with whole numbers
• No more than three identical values can follow a larger one:
I = 1 II = 2 III = 3 ...but IV = 4
VI = 6 VII = 7 VIII = 8 ...but IX = 9
• Only one value can be subtracted from each larger value.
For example:
• 19 is XIX (10 + 10 – 1) but 18 is XVIII, not XIIX
• 94 is XCIV (100 – 10 + 5 – 1), not VIC.
When converting large numbers, it is easier to split them into thousands,
hundreds, tens and units. For example:
• 1974 = 1000 + 900 + 70 + 4, and
1000 = M 900 = CM 70= LXX 4 = IV,
giving MCMLXXIV.
• 649 = 600 + 40 + 9, and
600 = DC 40 = XL; 9 = IX,
giving DCXLIX.
Work the opposite way when changing Roman numerals into our
numbering system. For example:
• MCDLXXIV means
M = 1000 CD = 500 – 100 = 400
LXX = 50 + 10 + 10 = 70 IV = 4.
So the final sum is 1000 + 400 + 70 + 4 = 1474.
• CMXCIX means
CM = 1000 – 100 = 900 XC = 100 – 10 = 90
IX = 10 – 1 = 9.
So the final sum is 900 + 90 + 9 = 999.
To deal with very large numbers, Romans placed a bar over the letter.
This multiplied the value of the letter by 1000.
V = 5000 X = 10 000 M = 1 000 000
But such large numbers were rarely used.
Part 3 Numbering systems 23
Use the activity below to practise reading and writing Roman numerals.
Activity – Roman numerals
Try these.
1 Write these Roman numerals using our numbering system.
a LXXVIII
___________________________________________________
b MXCV
___________________________________________________
2 Write these numbers using Roman numerals.
a 47
___________________________________________________
b 382
___________________________________________________
Check your response by going to the suggested answers section.
Romans used their numeral system especially for daily life purposes such
as stating prices of goods at the market, or distances on milestones, or to
indicate seat numbers in circuses and theatres. Roman numerals were not
practical at all for higher calculations.
You have been practicing reading and writing Roman numerals.
Now check that you can do this by yourself.
Go to the exercises section and complete Exercise 3.3 – Roman numerals.
24 NS4.1 Operations with whole numbers
You might like to investigate Roman numerals further.
Access related sites on Roman numerals by visiting the LMP webpage
below. Select Stage 4 and follow the links to resources for this unit
NS4.1 Operations with whole numbers, Part 3.
<http://www.lmpc.edu.au/mathematics>
Part 3 Numbering systems 25
Hindu-Arabic numbering�system
You use the Hindu-Arabic numbering system. It was developed by the
Indians and Arabs and became popular in Europe about 1000 years ago
and should probably be called the Indo-Arabic numbering system.
This system only needs 10 symbols: 0, 1, 2, 3, 4, 5, 6, 7, 8, and 9.
It counts in bundles of 10. You call this a decimal system.
The value of a particular symbol depends not only on what it is, but also
where it is. When you see the number 64, you assume that each unit in
the right place is worth 1 and each unit in the left place is worth 10.
Say the number aloud: sixty-four. It means six tens and four: 6 × 10 + 4.
When you write 582, you mean 5 8 2
↓ ↓ ↓
5 ×100 + 8 ×10 + 2 ×1
In fact you read it this way: five hundred and eighty-two.
Why the system works well
Some quantities do not fill all the places they use, which is why all place
notation systems, like this one, need a zero. Zeros represent places
which are there but do not have anything in them.
For example, 4070 means
4 0 7 0
↓ ↓ ↓ ↓
4 ×1000 + 0 ×100 + 7 ×10 + 0 ×1
In this example there are no hundreds and no units (ones). Putting a zero
in these places avoids confusion such as those you have already
described with other numbering systems.
26 NS4.1 Operations with whole numbers
Number value, face andplace�value
A digit has a different value depending on its position. To find the actual
value of a digit you multiply it by its place value.
Follow through the steps in this example. Do your own working in the
margin if you wish.
What is the value of ‘5’ in each of the following?
Solution
2415 � 5 (five units, 5 × 1)
2451 � 50 (five tens, 5 × 10)
2541 � 500 (5 × 100)
5214 � 5000 (5 × 1000)
The digit 5 has a face value of five in each case but the place value
changes depending on its position: 1, 10, 100, 1000.
Now check if you understand place value in the following.
Activity – Hindu-Arabic numbering system
Try these.
1 What is the value of ‘9’ in each of the following?
a 952 _______________________________________________
b 19 867 _____________________________________________
Check your response by going to the suggested answers section.
Part 3 Numbering systems 27
Consider these numbers:
265 256 652 526 562 625
Each number has the same three digits. Their values are different
because the positions of the digits are different. The three positions have
place values of hundreds, tens and units.
To find the largest number look for the largest digit in the
hundreds position.
Then look for the next largest digit in the next position.
The remaining digit goes in the units place.
So 652 is the largest number in this group.
6 _ _
6 5 _
6 5 2
Can you find the smallest number? Did you see it is 256?
Activity – Hindu-Arabic numbering system
Try this.
2 Write these numbers from smallest to largest:
7623, 2367, 3627, 6327, 2673.
_______________________________________________________
Check your response by going to the suggested answers section.
The value of each digit is its place value. The next section will look at
the total value and meaning of the number itself.
28 NS4.1 Operations with whole numbers
Expanded notation
Meanings of 102, 103, 104, and so on (the powers of 10).
102 = 10 ×10 =100
103 = 10 ×10 ×10 =1000
104 = 10 ×10 ×10 ×10 =10 000
105 = 10 ×10 ×10 ×10 ×10 =100 000
106 = =1 000 000 one million( )
109 = = 1 000 000 000 one billion( )
Notice how every group of three digits to the numbers on the right is
separated by a gap. With only four digits in the number, you have a
choice of writing 1000 or 1 000. Don’t use commas any more to separate
digits because in some countries a decimal point is shown by
a comma.
The number 2374 is read as:
“two thousand three hundred and seventy-four”.
The 2 is worth 2000
3 is worth 300
7 is worth 70
4 is worth 4
which is 2374
The number 2374 can be written as;
2 lots of a thousand + 3 lots of a hundred + 7 lots of ten + 4 lots of one
= 2 ×1000 + 3 ×100 + 7 ×10 + 4 ×1
= 2 ×103 + 3×102 + 7 ×10 + 4 ×1
This is called writing the number in expanded form or
expanded notation.
Part 3 Numbering systems 29
Follow through the steps in this example. Do your own working in the
margin if you wish.
a Write 25 386 in expanded notation.
b Write the numeral for this expanded notation:
4 × 103 + 2 × 102 + 9 × 1
Solution
a 25 386 is
2 lots of ten thousand + 5 lots of a thousand +
3 lots of a hundred + 8 lots of ten + 6 lots of 1= 2 ×10 000 + 5 ×1000 + 3 ×100 + 8 ×10 + 6 ×1
= 2 ×104 + 5 ×103 + 3×102 + 8 ×10 + 6 ×1
b 4 ×1000 +2 ×100 +9
= 4000 +200 +9
= 4209
All numbers can be rewritten using powers of 10
The next activity will give you practice at rewriting numerals in expanded
notation and vice versa.
Activity – Hindu-Arabic numbering system
Try these.
3 Write these numbers in expanded notation.
a 3406 ______________________________________________
b 12 598 _____________________________________________
4 Write the numeral for this expanded notation:
3 × 105 + 2 × 103 + 5 × 102 + 7 × 10
_______________________________________________________
Check your response by going to the suggested answers section.
30 NS4.1 Operations with whole numbers
Writing numerals in expanded notation gives you a better understanding
of the meaning of numbers.
You can now review all the work on the Hindu-Arabic numbering
system.
Go to the exercises section and complete Exercise 3.4 – Hindu-Arabic
numbering system.
Part 3 Numbering systems 31
Special groups of numbers
There are certain groups of numbers that have interesting properties.
For example: 1, 4, 9, 16, 25, 36, 49, … .
Did you recognise these as the perfect square numbers?
12 = 1 22 = 4 32 = 9 42 = 16 52 = 25 62 = 36
Remember, square numbers or perfect squares are the answers you get
when you multiply a number by itself.
Activity – Special groups of numbers
Try these.
Write down the next three numbers in this pattern:
1, 4, 9, 16, 25, 36, 49, ___, ___, ___
Check your response by going to the suggested answers section.
Mathematicians often study numbers because of the patterns or
relationships they form, just like some people study butterflies because of
the patterns on their wings. Sometimes what mathematicians discover by
doing this can later be applied to solve real problems.
32 NS4.1 Operations with whole numbers
Here are some other special groups of numbers.
Figurate numbers
Figurate numbers are numbers that can be shown by a regular
geometrical arrangement of equally spaced points. Here are some shapes
that make figurate numbers.
triangularnumbers
squarenumbers
pentagonalnumbers
hexagonalnumbers
For example, the triangular numbers can be shown like this.
1 3 6 10 15
The triangular numbers are shown by the number of dots in each
diagram. Did you know that the ten pins in tenpin bowling, looking from
the top, are arranged in a triangular number pattern?
Palindromic numbers
Have you seen the following?
A MAN, A PLAN, A CANAL, PANAMA!
Try reading it backwards. (Ignore spaces and punctuation marks.)
Did you find it reads the same thing? Palindromes are sentences, words
or numbers that read the same forwards and backwards.
Part 3 Numbering systems 33
There are many words that read the same in both directions.
Examples are mum, level, and madam. But here you are only interested
in palindromic numbers.
Here is a palindromic number: 34743.
Here is a palindromic date: 20/02/2002 (20th February, 2002).
There are many special groups of numbers. In this section you have been
shown three of them. Review your understanding in the following
exercise.
Go to the exercises section and complete Exercise 3.5 – Special groups of
numbers.
You might like to investigate Figurate numbers and Palindromic numbers
further.
Access related sites on Figurate numbers and Palindromic numbers by
visiting the LMP webpage below. Select Stage 4 and follow the links to
resources for this unit NS4.1 Operations with whole numbers, Part 3.
<http://www.lmpc.edu.au/mathematics>
34 NS4.1 Operations with whole numbers
Part 3 Numbering systems 35
More special numbers
In this section you will explore more special numbers.
Fibonacci numbers
Leonardo Fibonacci (1170–1250) is remembered for helping introduce
the Hindu-Arabic numbering system in use today into Europe. He also
came up with the sequence of numbers named after him.
0, 1, 1, 2, 3, 5, 8, 13, …
(The three dots … just indicate that the sequence continues on.)
Start with 0 and 1. The next number is found by just adding the last two
previous numbers together.
0 + 1 = 1
1 + 1 = 2
1 + 2 = 3
2 + 3 = 5 and so on.
The simplest way of finding a Fibonacci number is by knowing the
previous two numbers and adding them together.
36 NS4.1 Operations with whole numbers
Activity – More special numbers
Try these.
1 Here are some Fibonacci numbers: 0, 1, 1, 2, 3, 5, 8, 13.
Write down the next three numbers in this sequence.
_______ _______ _______
2 Is 87 a Fibonacci number? _________________________________
How do you know? ______________________________________
_______________________________________________________
Check your response by going to the suggested answers section.
Note: finding large Fibonacci numbers can be rather time consuming
when using the method where you must add the two previous numbers
together to get the next number.
Pascal’s triangle
In 1653, the French mathematician Blaise Pascal described an
arrangement of numbers in a triangle.
1
1 1
1 6 15 20 15 6 1
1 5 10 10 5 1
1 4 6 4 1
1 3 3 1
1 2 1
Part 3 Numbering systems 37
This arrangement looks like a Christmas tree with 1s down
the sides. Can you see how the other numbers are obtained?
Just add the two numbers above and either side of it.
10
4 6
4+6=10
Activity – More special numbers
Try this.
3 Here are two rows from Pascal’s triangle.
Write in the missing values.
1 9 36 84 126 84 36 9 1
1 __ 45 __ 210 210 __ __ __ 1
Check your response by going to the suggested answers section.
Some interesting patterns can be obtained using Pascal’s triangle.
For example, suppose all the numbers in the triangle divisible by 5 were
coloured in. This is what the pattern would look like for the first
127 rows.
You have been learning about Pascal’s triangle and Fibonacci numbers.
The following exercise will help you review these special numbers.
Go to the exercises section and complete Exercise 3.6 – More special
numbers.
38 NS4.1 Operations with whole numbers
You might like to investigate Fibonacci numbers and Pascal triangles
further.
Access related sites on Fibonacci numbers and Pascal triangles by
visiting the LMP webpage below. Select Stage 4 and follow the links to
resources for this unit NS4.1 Operations with whole numbers, Part 3.
<http://www.lmpc.edu.au/mathematics>
You have now finished the learning for this part.
Complete the review quiz for this part and return it to your teacher.
It contains problems based on work from throughout this entire part.
Part 3 Numbering systems 39
Suggested answers – Part 3
Check your responses to the preliminary quiz and activities against these
suggested answers. Your answers should be similar. If your answers are
very different or if you do not understand an answer, contact your teacher.
Activity – Preliminary quiz
1 a 367 b 17 688 c 50 070
2 One thousand five hundred, 1500
3 a Three hundred and two
b Five thousand four hundred and seventy-eight
c One hundred thousand two hundred and forty
4 500 000, five hundred thousand
5 a 792 b 1020 c 20 712
d 14 216 902
6 a 234, 243, 342, 423, 432
b 4062, 4206, 4602, 4620
7 a 7 b 15 c 90
d 150
8 a IV b XII c XXX
9 a 69, 64 b 32, 64 c 48, 59
10 a 10 b 100
11 a 542 b 309
12 a 3, 4, 8 b 2, 6, 0, 9
40 NS4.1 Operations with whole numbers
Activity – Roman numerals
1 a 78 b 1095
2 a XLVII b CCCLXXXII
Activity – Hindu-Arabic numbering system
1 a 9 hundred b 9 thousand
2 2367, 2673, 3627, 6327, 7623
3 a 3 × 103 + 4 × 102 + 6
b 1 × 104 + 2 × 103 + 5 × 102 + 9 × 10 + 8
4 302 570
Activity – Special groups of numbers
1 64, 81, 100
Activity – More special numbers
1 21, 34, 55
2 No. If you continue the pattern from the previous question the next
number is 89.
3 The complete row is 1, 10, 45, 120, 210, 210, 120, 45, 10, 1
Part 3 Numbering systems 41
Exercises – Part 3
Exercises 3.1 to 3.6 Name ___________________________
Teacher ___________________________
Exercise 3.1 – Ancient numbering systems
1 An ancient cave dweller comes across a herd of 15 to 20 animals.
Suggest how he could record how many animals there were so he
could inform his tribe at home.
_______________________________________________________
_______________________________________________________
_______________________________________________________
2 a What did each knot in Darius’ thong indicate?
___________________________________________________
___________________________________________________
b What is another way that King Darius could have recorded the
number 60?
___________________________________________________
___________________________________________________
c King Darius could easily have made 60 marks in the sand.
Would this have been a suitable recording method?
Why or why not?
___________________________________________________
___________________________________________________
42 NS4.1 Operations with whole numbers
d King Darius did not have a quipu. But even if he did, why
would this not be suitable for his purpose?
___________________________________________________
___________________________________________________
3 What numbers are shown on the following quipus?
a b
5 knots
8 knots
6 knots
9 knots
1 knot
5 knots
4 What problem might there be by trying to show the number like 301
using a quipu?
_______________________________________________________
5 In the Egyptian numbering system, what number is shown by:
a a frog? _____________________________________________
b a bent finger? ________________________________________
c a lotus plant? ________________________________________
d a figure of a god with arms raised? _______________________
Part 3 Numbering systems 43
6 What numbers do these Egyptian numbers show?
a_______________________
b_______________________
7 Write the following numbers using Egyptian numerals.
a 1043
b 335 490
(It is not important that you learn to write Egyptian numerals in this
course. By drawing the Egyptian numbers in this question, did you
see that as the number became larger it took more of your time to
draw it?) What is one advantage of using our numbering system to
that of using the Egyptian numbering system?
_______________________________________________________
8 What numbers do these Babylonian numbers show?
a
___________________________________________________
___________________________________________________
b
___________________________________________________
44 NS4.1 Operations with whole numbers
9 Explain why indicates 2, but shows 61.
_______________________________________________________
_______________________________________________________
10 What values do these show
a
___________________________________________________
b
___________________________________________________
c
___________________________________________________
11 Here are the two numbers, 3604 and 64 written using
Babylonian symbols.
↓ ↓ ↓ ↓
1× 60× 60 +0× 60+ 4 = 3604 1× 60+ 4 = 64
Explain why these two numbers might be confused.
_______________________________________________________
_______________________________________________________
12 (Harder) What number does this Babylonian number represent?
(This one is hard. If you do attempt it you may need a calculator.)
__________________
Part 3 Numbering systems 45
Exercise 3.2 – More ancient numbering systems
1 In the Mayan system of numbers, why can there not be more than
four dots in a single place?
_______________________________________________________
2 Write the value for these Mayan numbers.
a
___________________________________________________
b
___________________________________________________
3 Write these numbers using Mayan numerals.
a 34
b 57
4 Write the answer to these additions using Mayan symbols:
a + _____________________________________
b+
____________________________________
5 a How would you pronounce 9 in the Torres Strait Islander
numbering system? ___________________________________
b Why would such a numbering system be awkward to use?
___________________________________________________
___________________________________________________
46 NS4.1 Operations with whole numbers
6 How many different Greek letters were used for the:
a units? ______________________________________________
b hundreds ___________________________________________
7 Using the Greek numbering system, how would you write
a 84? ________________________________________________
b 359? _______________________________________________
Part 3 Numbering systems 47
Exercise 3.3 Roman numerals
1 Give four examples of where you might see Roman numerals
used today.
_______________________________________________________
_______________________________________________________
2 C and M come from the Latin words centum, meaning 100, and
mille, meaning 1000. For example, century means 100 years and
millimetre is one thousandth part of a metre. Give three more words
in English that include these Latin words meaning 100 and 1000.
_______________________________________________________
_______________________________________________________
_______________________________________________________
3 Write the following Romans numerals in our numbering system.
a XLIX_______________________________________________
b CCLXXV ___________________________________________
c XMXX_____________________________________________
4 The trailer in a movie indicated it was made in MCMXLVIII.
Which year was this?
_______________________________________________________
5 Write the following using Roman numerals.
a 88
___________________________________________________
b 826
___________________________________________________
48 NS4.1 Operations with whole numbers
6 Write this year in Roman numerals.
_______________________________________________________
7 Patrice wrote 1999 as MIM. Demeter thought this was wrong and
wrote it as MCMXCIX. Who is correct? Explain.
_______________________________________________________
_______________________________________________________
_______________________________________________________
8 Each of these Roman numerals has been written incorrectly.
Explain briefly why it is incorrect, then write the number correctly.
a VIIII _______________________________________________
___________________________________________________
b VC ________________________________________________
___________________________________________________
c CVV _______________________________________________
___________________________________________________
d DIC________________________________________________
___________________________________________________
Part 3 Numbering systems 49
Exercise 3.4 Hindu-Arabic numbering system
1 Look up the meaning of the word decimal.
a What does decimal mean?
___________________________________________________
___________________________________________________
b Why can our numbering system be called a decimal system?
___________________________________________________
___________________________________________________
2 Explain why is it important in our numbering system to have a
zero (0)?
_______________________________________________________
_______________________________________________________
3 What is the value of 4 in each of these numbers?
a 743
___________________________________________________
b 4219
___________________________________________________
4 The digits 2, 8, and 5 are written on three pieces of cardboard.
By moving the pieces about, what is the largest number you
can make?
_______________________________________________________
5 Write the following numerals in words
a 6059 _______________________________________________
___________________________________________________
b 34 607 ______________________________________________
50 NS4.1 Operations with whole numbers
6 Write numerals for the following numbers:
a three million, four hundred and seventeen thousand two hundred
___________________________________________________
b a quarter of a million _________________________________
7 Put the digits of each number in their correct place value column.
(Don’t forget to put in any zero placeholders needed.te
n m
illio
ns
mill
ion
s
hu
nd
red
th
ou
san
ds
ten
th
ou
san
ds
tho
usa
nd
s
hu
nd
red
s
ten
s
on
es
a forty thousandand ninety-two
b fourteen millionand fifteenthousand
8 The odometer of a car indicates the number of kilometres it has
travelled. The odometer readings at the beginning and end of a trip
are shown.
a How far has the car travelled
between odometer readings?
0 0 5 0 7 2
0 1 2 7 8 0
___________________________________________________
___________________________________________________
b Why do odometers start with zeros?
___________________________________________________
___________________________________________________
___________________________________________________
Part 3 Numbering systems 51
9 Expand the following then write as one number:
a 3 × 102
___________________________________________________
b 7 × 105
___________________________________________________
10 Write the following numbers in expanded notation.
a 76 000
___________________________________________________
b 627
___________________________________________________
11 What numbers are represented by the following expanded forms?
a 8 ×106
___________________________________________________
b 9 ×104 + 5×102 + 7 ×10 + 6 ×1
___________________________________________________
52 NS4.1 Operations with whole numbers
Exercise 3.5 Special groups of numbers
1 Look at these diagrams showing the first 5 triangular numbers.
1 3 6 10 15
a Draw the shape to produce the next triangular number.
b What is this number? _________________________________
c What pattern do you notice showing how to
find the next triangular number?1 3 6 10
+2 +3 +4
___________________________________________________
___________________________________________________
d Write down the first 10 triangular numbers.
___________________________________________________
Part 3 Numbering systems 53
2 These diagrams show the first 5 square numbers.
a Write down the first 5 square numbers. ____________________
b Draw the shape to produce the next square number above.
c What is this number? __________________________________
d What pattern do you notice for the square numbers?
___________________________________________________
54 NS4.1 Operations with whole numbers
3 These diagrams show the first 5 pentagonal numbers.
You can use matchsticks or toothpick to build them.
a Why are they called pentagonal numbers?
___________________________________________________
b The first few are 1, 5, 12, 22, 35, 51, 70, ... .
Can you describe a pattern to show how to find the next
pentagonal number?
___________________________________________________
c Write down the next two pentagonal numbers. _____________
4 These diagrams show the first 5 hexagonal numbers.
a Why are they called hexagonal numbers?
___________________________________________________
b The first six hexagonal numbers are: 1, 6, 15, ____, 45, 66.
The fourth number is missing. What is it?
c Is 91 a hexagonal number? _____________________________
5 Write a number that is a palindrome with
a 4 digits
___________________________________________________
b 7 digits
___________________________________________________
Part 3 Numbering systems 55
6 a Write down all the palindromic numbers with 2 digits.
___________________________________________________
b How many are there? _________________________________
7 There are nine palindromic numbers with only one digit: 1, 2, 3, 4, 5,
6, 7, 8, 9, and there are nine with two digits: 11, 22, 33, 44, 55, 66,
77, 88, 99. The table shows how many palindromes there are for
numbers containing differing numbers of digits.
Digits in number Palindromes possible
1
2
3
4
5
6
9
9
90
90
900
#
Look at the pattern in the table. How many palindromes are possible
for numbers with:
a 6 digits?
___________________________________________________
b 7 digits?
___________________________________________________
c 8 digits?
___________________________________________________
8 The date 20th February, 2002 (20-02-2002) is palindromic.
What are the next two palindromic dates?
_______________________________________________________
56 NS4.1 Operations with whole numbers
Exercise 3.6 More special numbers
1 Here are the first eight Fibonacci numbers: 0, 1, 1, 2, 3, 5, 8, 13.
a Write down the next four Fibonacci numbers. ______________
b Is 144 a Fibonacci number? ____________________________
c Choose the pattern which follows the Fibonacci numbers?
i even-even-odd-even-even-odd
ii even-odd-odd-even-odd-odd
iii even-odd-even-odd.
2 There is a river with a number of stepping
stones across it. Now you can step from one
stone to another, or jump over a stone.
You can’t jump over two stones.
With one stone in the river you
can step-step or jump across.
With two stones you can step-step-
step, jump-step, or step-jump.
step step
jump
step step
jump
step
step
jumpstep
a How many different ways can you get across the river with three
stepping stones? List them.
___________________________________________________
b List the different ways you can get across the river with four
stepping stones?
___________________________________________________
c What pattern do you notice?
___________________________________________________
Part 3 Numbering systems 57
3 Suppose you have a brick that has a length twice
as long as its height. You can use it to build a
brick wall two units tall. You can make our wall
in a number of patterns, depending on how long
you want it:
length
heig
ht
One unit wide:
only one wall pattern made by
putting the brick on its end.
Two units wide:
two patterns: two bricks long-
ways up put next to each other or
two side-ways bricks laid on top
of each other and.
1
2
3
a How many patterns are there for a wall three units wide?
___________________________________________________
b Describe the pattern arrangements for bricks in a wall three
units wide.
___________________________________________________
c Draw the different patterns for a wall 4 units wide.
d How many patterns are there for a wall of length:
i 4 units? _____________________________________________
ii 5 units? _____________________________________________
e Are the number of patterns Fibonacci numbers? ____________
58 NS4.1 Operations with whole numbers
4 The first seven rows of Pascal’s triangle are shown.
Underneath, write down the next two rows.
1
1 1
1 6 15 20 15 6 1
1 5 10 10 5 1
1 4 6 4 1
1 3 3 1
1 2 1
5 Colour in the numbers divisible by 2 to form a pattern in
Pascal’s triangle.
1 13 78 186 715 1287171617161287 715 186 78 13 1
1 12 66 220 495 792 924 792 495 220 66 12 1
1 11 55 165 330 462 462 330 165 55 11 1
1 10 45 120 210 252 210 120 45 10 1
1 9 36 84 126 126 84 36 9 1
1 8 28 56 70 56 28 8 1
1 7 21 35 35 21 7 1
1 6 15 20 15 6 1
1 5 10 10 5 1
1 4 6 4 1
1 3 3 1
1 2 1
1 1
1
Part 3 Numbering systems 59
6 Here is a copy of Pascal’s triangle.
(Copies are in the Additional resources.)
1 13 78 186 715 1287171617161287 715 186 78 13 1
1 12 66 220 495 792 924 792 495 220 66 12 1
1 11 55 165 330 462 462 330 165 55 11 1
1 10 45 120 210 252 210 120 45 10 1
1 9 36 84 126 126 84 36 9 1
1 8 28 56 70 56 28 8 1
1 7 21 35 35 21 7 1
1 6 15 20 15 6 1
1 5 10 10 5 1
1 4 6 4 1
1 3 3 1
1 2 1
1 1
1
a How many rows are shown? ____________________________
b Add up the numbers in each row. Do this for the first five rows.
What do you notice about the number pattern?
___________________________________________________
___________________________________________________
c What do you think the sum of the sixth row should be?
___________________________________________________
Check your guess by finding the sum of the numbers in the sixth
row. ________________________________________________
d Look down a diagonal (either / or \). Can you find the diagonals
that give
i the counting numbers 1, 2, 3, 4, … .
How many diagonals do this? _______________________
ii the triangular numbers 1, 3, 6, 10, … .
How many diagonals do this? _______________________
You have now completed the exercises and tasks for this part.
Complete the review quiz section and return it to your teacher.
60 NS4.1 Operations with whole numbers
Part 3 Numbering systems 61
Review quiz – Part 3
Name ___________________________
Teacher _________________________
1 What number is shown on this quipu? ________________________
5 knots
8 knots
6 knots
2 Use Egyptian symbols to write the number 478
3 Write 23 using Babylonian symbols?
4 Which of the following societies invented a symbol for zero?
a Egyptian b Babylonian
c Mayan d Greek
62 NS4.1 Operations with whole numbers
5 What was the greatest value of the symbols placed in any one place
in the Mayan numbering system?
_______________________________________________________
6 In the Torres Strait Islander numbering system:
1 = urapun
2 = okosa
3 = okosa-urapun
4 = okosa-okosa
5 = okosa-okosa-urapun
6 = okosa-okosa-okosa
How would you say these numbers using this system?
a 7
___________________________________________________
b 8
___________________________________________________
7 Use the ancient Greek numbering system to write:
a 29
___________________________________________________
b 355
___________________________________________________
8 Use our numerals to write the number for these Roman numerals
a XVII
___________________________________________________
b LIX
___________________________________________________
c MMCCCXIX
___________________________________________________
Part 3 Numbering systems 63
9 Write these numbers in Roman numerals.
a 16
___________________________________________________
b 45
___________________________________________________
c 1760
___________________________________________________
10 What is the value of the 8 in each of these numbers?
a 856
___________________________________________________
b 1489
___________________________________________________
11 Write these as ordinary numbers:
a 103
___________________________________________________
b 2 × 101
___________________________________________________
12 Write these in expanded form
a 237
___________________________________________________
b 4509
___________________________________________________
13 What numbers are shown by the following expanded forms?
a 1×102 + 3 ×10 + 5 ×1 __________________________________
b 2 ×104 + 6 ×102 + 9 ×10 + 8 ×1 __________________________
64 NS4.1 Operations with whole numbers
14 a What is special about this number: 247 742?
___________________________________________________
b Is it divisible by 11? If so, is the answer a palindrome?
___________________________________________________
15 Write the next two numbers in these patterns:
a 1, 3, 6, 10, 15, 21, ___, ___
b 1, 4, 9, 16, 25, 36, ___, ___
c 0, 1, 1, 2, 3, 5, 8, 13, ___, ___
16 Replace each triangle with the correct number in this
Pascal’s triangle.
17 In the space provided write out the next two rows in this Pascal’s
triangle.
1
1 1
1 5 10 5 1
1 4 6 4 1
1 3 1
1 2 1
__________________________________________________________
__________________________________________________________
18 In the 20th row, the first number is 1. What is the second number?
Part 3 Numbering systems 65
Answers to exercises – Part 3
This section provides answers to questions found in the exercises section.
Your answers should be similar to these. If your answers are very
different or if you do not understand an answer, contact your teacher.
Exercise 3.1 – Ancient numbering systems
1 Notches on a piece of wood, collect leaves, count on fingers and
toes. There are many others you can think of.
2 a One day.
b Examples include notches on a tent pole, pebbles in a bag.
c Not permanent enough. Winds could cover over the markings, or rain
could wash them away.
d Each knot on the thong was untied for each day. On a quipu a
knot in a higher position indicates 10. Untying one of these
indicated 10 days, not one. The remaining 9 days would need to
have knots retied lower down.
This complicates things.
3 a 586 b 915
4 There is a gap between the groups of knots for the units, the tens, the
hundreds, and so on. With no tens in 301, how big must the gap be
so it is not confused with numbers such as 31 and 3001?
5 a 100 000 b 10 000 c 1000 d 1000 000
6 a 3224 b 21 237
66 NS4.1 Operations with whole numbers
7 a
b
One advantage of our numbering system is the symbols for the digits
are easy to write. It takes much less time to write a number like
335 490 using our symbols than in drawing Egyptian numerals.
8 a 43 b 55
9 Two ones joined together is 1 + 1 = 2. But the gap between the two
ones indicates that the first 1 is multiplied by 60: 1 × 60 + 1 = 61.
10 a 3 b 121 c 3661
11 They use the same symbols. The only difference is the size of the
gap between them. Only when the two different numbers are placed
side by side can we compare the relative gap sizes. The gap size is
confusing as to whether the first symbol indicates 60 × 60 or just 60.
There was no zero in Babylonian mathematics.
12 424 000
Exercise 3.2 – More ancient numbering systems
1 Five dots is replaced by a bar indicating 5.
2 a 21 b 128
3 a b
4 a b
5 a okosa-okosa-okosa-okosa-urapun
b The name becomes longer as the number increases.
6 a 9 b 9
7 a πδ’ b τνθ’
Part 3 Numbering systems 67
Exercise 3.3 – Roman numerals
1 Clock faces, tombstones, front pages of books, trailers of movies,
numbering kings and queens of England (eg George VI,
Elizabeth II), dates on older buildings. See also numbering parts of
questions, such as in these answers.
2 centum: centurion, cents, centipede, centigrade, centenary,
centimetre
milli: millennium, millipede, milligram, millisecond
3 a 49 b 275 c 11 022
4 1948
5 a LXXXVIII b DCCCXXVI
6 Answers will vary: 2004 is MMIV; 2005 is MMV; 2006 is MMVI
7 Demeter is correct. Only C can be written before M, so 1999 has to
be written as 1000 + (1000 – 100) + (100 – 10) + (10 – 1)
8 a Only 4 of the same symbol can be written together; IX
b Only X can be placed before C; so 95 is XCV
c VV just means X (10); CX
d The number 599 should be written as DXCIX
Exercise 3.4 – Hindu-Arabic numbering system
1 a Relating to 10 parts
b There are 10 digits, and the place value for each digit is a power
of 10 (1, 10, 100, 1000 and so on).
2 As a place holder for where there are no other digits eg 507 there are
no tens.
3 a 40 b 4000
4 852
5 a six thousand and fifty-nine
b thirty four thousand, six hundred and seven
68 NS4.1 Operations with whole numbers
6 a 3 417 200 b 250 000
7 The digits in each column are:
a 0 0 0 4 0 0 9 2
b 1 4 0 1 5 0 0 0
8 a 7708 km
b To show there are no values in that power of 10. (There are no
blanks in odometers.) For example, 5072 is written as 005072
to show there are no values in the 10 000 or 100 000 columns.
9 a 300 b 700 000
10 a 7 × 10 000 + 6 × 1000
b 6 × 100 + 2 × 10 + 7 × 1
11 a 8 000 000 b 90 576
Exercise 3.5 – Special groups of numbers
1 a
b 21
c Add one more than you added before to get the next number
d 1, 3, 6, 10, 15, 21, 28, 36, 45, 55
2 a 1, 4, 9, 16, 25
b
c 36
Part 3 Numbering systems 69
d squares of the counting numbers: 12, 22, 32, 42, and so on
3 a The dots form the shape of a pentagon
b 1 + 4 = 5; 5 + 7 = 12; 12 + 10 = 22; 22 + 13 = 35; 35 + 16 = 51
Start by adding 4, then keep adding 3 more each time to get the
next number
c 92, 117
4 a The dots form the shape of a hexagon
b 28 c yes
5 a 3443 b 2349432
(there are many different examples you could write)
6 a 11, 22, 33, 44, 55, 66, 77, 88, 99
b 9
7 a 900 b 9000 c 9000
8 February 1, 2010 (01-02-2010); February 11, 2011 (11-02-2011)
Exercise 3.6 – More special numbers
1 a 21, 34, 55, 89 b yes c ii
2 a 5 ways. step-step-step-step; jump-step-step; step-jump-step;
step-step-jump; jump-jump
b step-step-step-step-step; jump-step-step-step; step-jump-step-
step; step-step-jump-step; step-step-step-jump; jump-jump-step;
jump-step-jump; step-jump-jump
c Fibonacci numbers
3 a 3
b Three bricks long-way up, or one brick long-way up then two
side-ways bricks laid on top of each other, or two side-ways
bricks laid on top of each other then one brick long-way up
c
d i 5 ii 8
70 NS4.1 Operations with whole numbers
e yes
4 1, 7, 21, 35, 35, 21, 7, 1 and 1, 8, 28, 56, 70, 56, 28, 8, 1
5 If you have done it correctly you should obtain an interesting pattern.
6 a 14
b 1, 2, 4, 8, 16. The sum is double the sum of the previous row.
Also the numbers are powers of 2: 20 = 1 (use a calculator to
show this), 21 = 2, 22 = 4, 23 = 8, 24 = 16.
c 32
d i Two diagonals: second from the left (/) and second from the
right (\)
ii Two diagonals: third from the left (/) and third from the
right (\)