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© State of New South Wales, Department of Education and Training (CLI) 2004

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


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.


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.


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? _____________________________________


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, __, __


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.


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


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.


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.


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.


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>


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


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


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.


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


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.



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.


• 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.


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

___________________________________________________


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.


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.



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.


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 _____________________________________________



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?


Try this.

2 Write these numbers from smallest to largest:

7623, 2367, 3627, 6327, 2673.

_______________________________________________________


The value of each digit is its place value. The next section will look at

the total value and meaning of the number itself.


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.


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.


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

_______________________________________________________



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.


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, ___, ___, ___


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.


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.


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.



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.


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? ______________________________________

_______________________________________________________


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


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


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


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.


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.


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.


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


Activity – Roman numerals

1 a 78 b 1095

2 a XLVII b CCCLXXXII


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


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


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?

___________________________________________________

___________________________________________________


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? _______________________


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

___________________________________________________


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.)

__________________


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?

___________________________________________________

___________________________________________________


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? _______________________________________________


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

___________________________________________________


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________________________________________________

___________________________________________________


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 ______________________________________________


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?

___________________________________________________

___________________________________________________

___________________________________________________


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

___________________________________________________


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.

___________________________________________________


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?

___________________________________________________


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

___________________________________________________


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?

_______________________________________________________


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?

___________________________________________________


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? ____________


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


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.


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


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

___________________________________________________


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 __________________________


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?


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


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’


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


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


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


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 (\)

43483 ns4.1 part 3 -...

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