Chapter 3

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2345 4 234 6789012

3 5

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48901234

1234

NUMBER

Numbers form many interesting patterns. You already know about odd and even numbers. Pascal’s triangle is a number pattern that looks like a triangle and contains number patterns. Fibonacci numbers are found in many living things: the number of petals on a flower will be a Fibonacci number. You will learn how these different patterns are formed to help you to understand how numbers behave.

03 NCM7 2nd ed SB TXT.fm Page 76 Saturday, June 7, 2008 3:51 PM

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In this chapter you will: Wordbank

• identify special groups of numbers: triangular, square, Fibonacci, Pascal’s triangle and palindromes

• test numbers for divisibility• identify the factors of a number and distinguish

between prime and composite numbers• find the highest common factor of two or more

numbers• express a number as a product of its prime factors• calculate squares and cubes• estimate and calculate square roots and cube

roots• find square roots and cube roots of numbers

expressed as a product of their prime factors.

•

composite number

A number with more thantwo factors.

•

divisibility test

A rule for testing whether a number is divisible by a specific value, for example, divisible by 3.

•

factor

A value that divides evenly into a given number, for example, 3 is a factor of 15.

•

factor tree

A diagram that lists the prime factors of a number.

•

index notation

Using powers to write the repeated multiplication of a number, for example, 3

5

.•

palindrome

A number or word that reads the same forward and backward, for example, ‘2002’, and ‘madam’.

•

prime number

A number with only two factors,1 and the number itself.


78

NEW CENTURY MATHS 7

Start up

1

List the first ten even numbers.

2

Sort these numbers, putting all the even numbers in one group, and the odd numbers in another:

17 2002 371 13460 023 2 748 69190 704 006 1 95 132074 1 000 000 99 999 1256

3

Find all the numbers that divide into 6.

4

Find all the numbers that divide into 24.

5

Find all the even numbers that divide into 36.

6

Find all the odd numbers that divide into 90.

7

How can you tell if a number is even without dividing it?

8

How can you recognise an odd number?

9

Write the next three numbers in each of these patterns:

a

8, 10, 12,

, ,

b

27, 30, 33,

, ,

c

101, 103, 105,

, ,

d

39, 37, 35,

, ,

e

44, 39, 34,

, ,

f

7, 15, 23, , ,

10

What is 8 squared?

11

What is ?

12

Find two numbers that have a product of 48.

3-01 Special number patterns

The numbers 1, 2, 3, 4, 5, … are called the

counting numbers

. There are groups of counting numbers which make special patterns. We will investigate some of them.

1 Triangular numbers

are shown in the diagram below.

Exercise 3-01

Worksheet3-01

Brainstarters 3

Classifying whole numbers

Skillsheet3-01

273

Triangular and square numbers

Worksheet3-02

L 1936

Circus towers:triangular towers

TLF

1 3 6 10


79

CHAPTER 3

EXPLORING NUMBERS

a

Why are they called ‘triangular numbers’?

b

Work out all the triangular numbers less than 100.

c

Complete four more lines of this pattern:

1

=

1

1

+

2

=

3

1

+

2

+

3

=

6

1

+

2

+

3

+

4

=

10

d

Describe how the pattern in part

c

works.

e

Use what you have worked out to help you find the 100th triangular number. (

Hint

: Do you know a quick way to add up all the numbers from 1 to 100?)

2 Square numbers

are shown in the diagram below.

a

Why are these called ‘square numbers’?

b

Work out all the square numbers up to 100.

c

Complete four more lines of this pattern:

1

=

1

1

+

3

=

4

1

+

3

+

5

=

9

1

+

3

+

5

+

7

=

16

d

Describe how the pattern works.

e

Work out another pattern to help you find the square numbers. What is the 50th square number?

f

Complete four more lines of these patterns:

i

1

=

1

2

ii

2

2

=

1

2

+

(1

+

2)1

+

2

+

1

=

2

2

3

2

=

2

2

+

(2

+

3)1

+

2

+

3

+

2

+

1

=

3

2

4

2

=

3

2

+

(3

+

4)

g

Each square number is said to be the sum of two consecutive triangular numbers.

Show that this is true for the square numbers up to 100.

h

Find two numbers that are both triangular numbers and square numbers.

1 4 9 16

Circus towers: square stacks

L 1935TLF

963 + =

+ =


80

NEW CENTURY MATHS 7

3

Leonardo

Fibonacci

was an Italian mathematician who lived in the early 13th century. He discovered this pattern when studying the breeding habits of rabbits:

1, 1, 2, 3, 5, 8, 13, 21, . . .The diagram below illustrates this. The vertical arrows labelled ‘Birth’ indicate the new offspring of a pair of rabbits every two months. The unlabelled arrows indicate the same pair of rabbits. After each month, the number of pairs is a term in Fibonacci’s pattern.

a

How is the Fibonacci pattern formed?

b

Add five more lines to this pattern:111

+

1

=

21

+

2

=

32

+

3

=

53

+

5

=

8

c

Write the first 20 Fibonacci numbers.

i

Write every

third

Fibonacci number, beginning with 2. What number divides evenly into all these numbers?

ii

Write every

fourth


iii

Write every

fifth


d i

Find any triangular numbers in the Fibonacci numbers up to 100.

ii

Find any square numbers in the Fibonacci numbers up to 100.

e

Pairs of Fibonacci numbers are found by counting along the spirals on pine cones. Investigate how and where else Fibonacci numbers occur in nature.

Worksheet3-03

Fibonacci numbers

4th generation 5 pairs

3rd generation 3 pairs

2nd generation 2 pairs

1st generation 1 pair

Original pair of rabbits 1 pair

Bir

th

Bir

th

Bir

th

Bir

th


81

CHAPTER 3

EXPLORING NUMBERS

4

Blaise Pascal, a French mathematician who lived in the 17th century, studied a triangle of numbers known to the Chinese as the Yanghui triangle. Each row of the triangle is created using the numbers in the row above it. The triangle is known as

Pascal’s triangle

. The first seven rows are shown at the right.

a

Complete the next four rows of Pascal’s triangle.

b

Describe how the pattern works.

c

Add each row in Pascal’s triangle. What do you notice?

d

The diagonals in Pascal’s triangle produce some interesting patterns. Write the triangular numbers using Pascal’s triangle.

e

We can even find Fibonacci numbers in this pattern. Rewrite the triangle above as a right-angled triangle.

5

A

palindrome

is a word, number or sentence that reads the same forward and backward. The following number, words and sentence are all palindromes:

noon 151

Able was I ere I saw Elba

(Napoleon Bonaparte)

a

Select the palindromes from these numbers.447 373 656 281 37 22 899 191 797 516

b

Find the numbers between 1000 and 2000 that are palindromes.

c

The following steps change any number into a palindrome:• choose any number to start with 64 • reverse the digits and add 46

110• reverse the digits and add 011• repeat until you get a palindrome. 121

Worksheet3-04

Pascal’s triangle

11 1

1 2 11 3 3 1

1 4 6 4 11 5 10 10 5 1

6 15 20 15 6 1

11 11 2 11 3 3 11 4 6 4 11 5 10 10 5 1

Add along the arrows to find the Fibonacci numbers.

Place names can be palindromes.

←


82 NEW CENTURY MATHS 7

Find out how many steps it takes to form a palindrome from each of these numbers.i 26 ii 28 iii 47

iv 75 v 149 vi 273vii 1756 viii 2379 ix 4021

d List some other words and place names that are palindromes.

Using technology

Developing number patternsFollow the instructions below to set up a spreadsheet.• Enter the headings, as shown below, into the given cells in a spreadsheet.

• Enter ‘1’ into cell A2 and ‘2’ into cell A3. Highlight the two cells and Fill Down (click and hold the square in the bottom right-hand corner of cell A3).

• Fill Down to cell A31 (you should see ‘30’ in this cell).

Odd numbers1 Enter the formula =A2 into cell B2.

2 Enter the formula =B2+2 into cell B3.

3 Click on cell B3, and Fill Down to cell B31 to obtain the first 30 odd numbers.

Even numbers1 Enter the formula =A2 into cell C2.

2 Enter the formula =C2+2 into cell C3.

3 Click on cell C3, and Fill Down to cell C31 to obtain the first 30 even numbers.

Square numbers1 Enter the formula =A2^2 into cell D2.

2 Click on cell D3, and Fill Down to cell D31 to obtain the first 30 square numbers.

Triangular numbers1 Enter the formula =A2 into cell E2.

2 Enter the formula =E2+A3 into cell E3.

3 Click on cell E3, and Fill Down to cell E31 to obtain the first 30 triangular numbers.


83CHAPTER 3 EXPLORING NUMBERS

Fibonacci numbers1 Enter the formula =A2 into cell F2.

2 Enter the formula =F2 into cell F3.

3 Enter the formula =F2+F3 into cell F4.

4 Click on cell F4, and Fill Down to cell B31 to obtain the first 30 Fibonacci numbers.

Questions1 Use your spreadsheet to answer the following questions.

a Name the two smallest odd numbers that are also square. Write your answer in cell H1. (Note: separate your answers with a comma.)

b Name all the even numbers less than 30 that are also triangular. Write your answer in cell H2.

c Find all the triangular numbers less than 60 that are also Fibonacci numbers. Write your answer in cell H3.

d Find all the square numbers between 300 and 600. Write your answer in cell H4.e State the 18th Fibonacci number. Write your answer in cell H5.f i In cell H6, write a formula to find the difference between the 24th and 25th

Fibonacci numbers. ii What cell in column F corresponds to your answer in (i)? Write your answer

in cell H7.g i Extend column A to represent the first 50 numbers.

ii Extend the square and triangular numbers to represent the first 50 numbers in each pattern.

h Name the first number over 1000 that is both square and triangular. Write your answer in cell H8.

Extension: FactorialsFactorials: a definition: 1! = 1

2! = 1 × 2 = 2 3! = 1 × 2 × 3 = 6...n! = 1 × 2 × 3 × … × n

1 Enter the formula =A2 into cell G2.

2 Using the definition given, develop a formula for 2! in cell G3, using cells G2 and A3.

3 Click on cell G4, and Fill Down to cell G31 to obtain the first 30 factorial numbers.

Displaying large numbersa Highlight all the cells that don’t show full numbers (e.g. 4.79E+08). Right click and

choose Format Cells. Change the settings to ‘number’ and ‘0 decimal places’.b ######## in a cell indicates that the column is not wide enough to hold the number

with all digits showing. You may need to widen the column until you can see all numbers down to cell G31.

03 NCM7 2nd ed SB TXT.fm 3 Saturday, June 7, 2008 3:51 PM


3-02 Tests for divisibilityIt is often useful to know if a number is divisible by another number. Here are some simple divisibility tests to help you.

Working mathematically

Figurate numbersNumbers formed from geometric shapes, such as triangular or square numbers, are called figurate numbers. There are many figurate number patterns.

1 Investigate the pentagonal numbers.

2 Investigate the hexagonal numbers.

3 What are the names of the other types of figurate numbers?

More types of numbersInvestigate one or more of the following types of numbers and find out the relationships and patterns in them. You may find the Internet useful. Prepare a short talk for the class on your topic. • Amicable numbers• Perfect numbers• The golden ratio/rectangle• Irrational numbers• Pythagorean triads• Binary, octal and hexadecimal numbers• Factorial numbers, for example the meaning of 5!

Reasoning and communicating

Worksheet3-05

Perfect and amicable numbers

L 1939

Circus towers: square pyramids

TLF

Worksheet3-06

Divisibility tests

669988

55 771010

44332226 040

A number is divisible by 3 if the sum of its digits is divisible by 3.

79 is NOT divisible by 3 since 7 + 9 = 16, and 3 does not go evenly into 16.

A number is divisible by 2 if it ends in 0, 2, 4, 6 or 8.

962

82 7

88

100 0

00 00

0

39 604

5 4 20

679 320 is

A number isdivisible by 4 if its last two digits are divisible by 4.

279 364 805

A number is divisible by 5 if it ends in 0 or 5.

48 ends in 8

4506

ends in 6 4 + 5 + 6 = 15

A number is divisible by 6 if it is divisible by both 2 and 3.

{

There is no simple test for divisibilityby 7.

A number is divisible by 8 if the last three digits are divisible by 8.

748 592

13592

8 + 1 + 2 + 7 + 5 + 4 = 27812 754

A number is divisible by 9 if the sum of its digits is divisible by 9.

300

1 020304050

A number is divisible by 10 if it ends in 0.

84050 0

05

13 592 is divisible by 8.

27 690

4136

679 320

divisible by 4.

4 + 8 = 12

1711 + 7 + 1 = 9



1 Copy this table and work out which of the numbers from 2 to 10 divide exactly into the given numbers (88 has been done for you).

Exercise 3-02

Number 2 3 4 5 6 7 8 9 10

252

600

88 ✓ ✓ ✓

121

6215

3720

747

4753

110 001

40 436

840

75

2 000 646

20 106

7434

601 295

Example 1

Which of the numbers 2 to 10 divide exactly into 112?

Solution2: 112 ends in a 2 so it is divisible by 2.3: 1 + 1 + 2 = 4: 4 is not divisible by 3, so 112 is NOT divisible by 3.4: 12 ÷ 4 = 3 so 112 is divisible by 4.5: 112 does not end in 0 or 5 so is NOT divisible by 5.6: 112 is not divisible by 3, so it is NOT divisible by 6.

7: Check by division: so 112 is divisible by 7.

8: Check by division: so 112 is divisible by 8.

9: 1 + 1 + 2 = 4: 4 is not divisible by 9 so 112 is NOT divisible by 9.10: 112 does not end in 0 so it is NOT divisible by 10.

Answer: 2, 4, 7 and 8 divide exactly into 112.

167 112

148 112

Ex 1



2 Which number is divisible by both 4 and 5? Select A, B, C or D.A 10 B 15 C 20 D 25

3 Write a number less than 100 which is divisible by:a 3 and 5 b 4 and 5 c 6 and 7 d 2 and 6

4 Write a number greater than 100 which is divisible by:a 6 b 5 c 7 d 2 and 3 e 8 and 9

3-03 Factors

Just for the record

Karl GaussKarl Gauss was a German mathematician and astronomer who lived from 1777 to 1855. He invented a new way of finding the positions of heavenly bodies and was one of the first to study electricity.Gauss showed his mathematical ability early in life. When he was in primary school,the class was given the task of adding all the numbers from 1 to 100. The teacher thought this would keep the class busy for some time but Gauss was very quick tofind the answer and even quicker to explain why he was not working on the problem.This is how he did it:

1 + 2 + 3 + 4 + … + 96 + 97 + 98 + 99 + 1001 + 100 = 1012 + 99 = 1013 + 98 = 101, etc.

Find how many pairs of numbers there are and then find the answer to the problem.

Skillsheet3-02

Factors and primes

The factors of a number are those whole numbers that divide exactly into it.!

Example 2

What are the factors of 12?

SolutionThe possible ways of multiplying to get 12 are:• 4 × 3 or 3 × 4 • 6 × 2 or 2 × 6 • 1 × 12 or 12 × 1The factors of 12 are: 1, 2, 3, 4, 6 and 12.(Note that 1 will be a factor of every number.)



1 In each of these pairs, is the smaller number a factor of the larger number?a 8, 24 b 3, 39 c 4, 42 d 9, 45 e 8, 54f 7, 91 g 7, 133 h 6, 48 i 5, 57 j 11, 143

2 List all the factors of:a 16 b 21 c 24 d 36 e 35 f 48 g 52 h 80 i 112 j 144k 28 l 100 m 45 n 200 o 363

3 Which of these numbers is not a factor of 45? Select A, B, C or D.A 9 B 5 C 7 D 3

4 Find the common factors for each of these pairs of numbers.a 2, 4 b 9, 6 c 6, 14 d 8, 12e 50, 150 f 46, 69 g 10, 15 h 12, 16i 30, 20 j 18, 24 k 60, 90 l 39, 26m 45, 15 n 36, 39 o 27, 64 p 350, 210

5 Find the common factors for each of these sets of numbers.a 2, 4, 6 b 10, 50, 60 c 22, 33, 121d 24, 36, 144 e 6, 9, 12 f 16, 24, 40, 56g 28, 70, 42, 98 h 30, 90, 75, 135 i 50, 60, 90, 120

6 Find the highest common factor for each of these pairs of numbers.a 12 and 60 b 33 and 22 c 132 and 60d 9 and 21 e 45 and 78 f 64 and 144g 16 and 12 h 8 and 14 i 50 and 150j 18 and 24 k 48 and 72 l 15 and 25m 35 and 21 n 45 and 18 o 75 and 125

Exercise 3-03

Example 3

Find the highest common factor of 24 and 30.

SolutionThe factors of 24 are: 1, 2, 3, 4, 6, 8, 12, 24The factors of 30 are: 1, 2, 3, 5, 6, 10, 15, 30The common factors of 24 and 30 are: 1, 2, 3 and 6.The highest common factor is 6.

The highest common factor (HCF) of two or more numbers is the largest factor that is common to all those numbers. !

Ex 2

Ex 3



7 ‘Every whole number has at least two factors.’ Is this true or false? Why?

8 Using your answers to Question 5, find the highest common factor for each of the given sets of numbers.


Factor path puzzle1 Copy this grid, and try to reach the

100 square at the bottom corner, starting at the 200 square in the top corner and using related factors.You can move from one number to another by going sideways, up or down the page (not diagonally), but only if the numbers have a factor (not 1) in common. So you can move from 80 to 65 (common factor 5), but not from 65 to 72 (no common factor).

2 Find a factor path starting in the bottom corner (105) and finishing top right (195).

3 Choose different starting and finishing positions. Do they all have connectingfactor paths?

200 80 65 91 143 156 195

175 32 96 71 110 77 121

35 28 15 209 87 90 21

39 169 117 95 57 37 81

63 11 29 72 76 75 51

14 98 56 132 48 78 85

105 45 44 187 112 221 100

Applying strategies and reasoning

Just for the record

Korean mathematicsIn Korea, school students find the highest common factor (HCF) using the following method. To find the HCF of 24 and 30:

divide by the first prime number →divide by the next prime number →

Since it is not possible to divide any more, stop.The HCF = 2 × 3 = 6

Use this method to find the HCF of each of the following sets of numbers.a 12 and 15 b 18 and 48c 13, 20 and 28 d 15, 21 and 45e 8, 12, 16 and 48 f 120 and 250g 48 and 120 h 96 and 144i 256, 144 and 48 j 675, 1350 and 825

2 24 303 12 15

4 5



Mental skills 3

Calculating differences and making changeIn every subtraction problem, for example 135 − 47, think of finding the ‘gap’ between the two numbers. That is, find the number in this case that must be added to 47 to get 135.

1 Examine these examples.a 135 − 47

Think: 47 + = 135

Count: ‘47, 50, 100, 135’Add: 3 + 50 + 35 = 88

Answer: 135 − 47 = 88

b 244 − 115Think: 115 + = 244

Count: ‘115, 120, 200, 244’Add: 5 + 80 + 44 = 129

Answer: 244 − 115 = 129

c $60 − $47.65

Count: ‘$47.65, $48, $50, $60’Add: $0.35 + $2.00 + $10.00 = $12.35

Answer: $60 − $47.65 = $12.35

d $100 − $88.45

Count: ‘$88.45, $89, $90, $100Add: $0.55 + $1.00 + $10.00 = $11.55

Answer: $100 − $88.45 = $11.55

2 Now simplify the following.a 176 − 88 b 221 − 54 c 670 − 356d 425 − 340 e 518 − 389 f 199 − 78g $70 − $58.40 h $80 − $73.25 i $45 − $40.30j $100 − $69.95 k $30 − $22.90 l $50 − $17.10

50 100 150

135473 50 35

100 150 200

2441155 80 44

$50 $60 $70

$60$47.65

$2$1035c

$80 $90 $100

$100$88.45

$1$1055c

Maths without calculators



3-04 Prime and composite numbers

Note: 1 is neither prime nor composite. (It has only one factor.)

1 Eratosthenes, a mathematician in ancient Greece, found an easy way to work out prime numbers. It is called the Sieve of Eratosthenes and works by deleting multiples of numbers. (Use the link to go to a spreadsheet version of the Sieve.)a Copy the grid below or print out Worksheet 3-07.

b Cross out 1. It is neither prime nor composite.c Except for 2, colour all the multiples of 2 red.

Exercise 3-04

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 30 31 32 33 34 35 36

37 38 39 40 41 42 43 44 45 46 47 48

49 50 51 52 53 54 55 56 57 58 59 60

61 62 63 64 65 66 67 68 69 70 71 72

73 74 75 76 77 78 79 80 81 82 83 84

85 86 87 88 89 90 91 92 93 94 95 96

97 98 99 100 101 102 103 104 105 106 107 108

109 110 111 112 113 114 115 116 117 118 119 120


More factor pathsMake your own 7 × 7 factor path grid similar to the one in the ‘Factor path puzzle’ on page 88. Try making a 4 × 4 grid first, then a 5 × 5 grid and then a 7 × 7 grid.Discuss with your teacher the decisions you need to make as you develop your grid.

Applying strategies

Skillsheet3-02

Factors and primes

A prime number has only two factors: 1 and itself. The prime numbers are 2, 3, 5, 7, 11, 13, 17, 19, 23, . . .A composite number has more than two factors.The composite numbers are 4, 6, 8, 9, 10, 12, 14, 15, 16, . . .

!

Worksheet3-07

Sieve of Eratosthenes



d Except for 3, colour all the multiples of 3 green.e Continue, with different colours, until there are no more multiples. What do you

notice about the numbers that are not coloured?

2 Divide these numbers into two groups (primes and composites).

3 Write any whole numbers which are neither prime nor composite.

4 a List the prime numbers between 36 and 50.b List the composite numbers between 65 and 80. c List the prime numbers less than 20.d List the composite numbers larger than 30 but less than 47.

5 Which number is divisible only by prime numbers, itself and 1? Select A, B, C or D.A 12 B 14 C 16 D 18

6 Look up other meanings for the word ‘composite’. Suggest why this word is used the way it is in mathematics.

3-05 Prime factorsEvery composite number can be written as a product of its prime factors. The prime factors can be found by using a factor tree.

10

9992064 101

17

472

129

27

3341

77

71 62

59967 504

Worksheet3-08

Factor trees

Skillsheet3-03

Prime factors by repeated division

Example 4

Write 24 as a product of its prime factors.

Solution

As a product of prime factors, 24 = 2 × 2 × 2 × 3.

Factor tree

3 and 8 are factors

2 and 4 are factors of 83 is prime

2 is a factor of 42 is prime—stop

24

83

3

3 2 2 2

2 4

×

××

× ××



1 Use factor trees to express each of these numbers as a product of its prime factors.a 8 b 63 c 45 d 36 e 51f 49 g 90 h 27 i 130 j 200k 275 l 342 m 1250 n 1020 o 837

Exercise 3-05

Example 5

Find the highest common factor (HCF) of 1960 and 2000.

Solution

Both numbers contain 2 × 2 × 2 × 5.The HCF is 2 × 2 × 2 × 5 = 40.

1960

19610

2 5 4 49

×

× ××

2000

2 5 2×× 2 7 7× ××

10002

2 100 10

×

× ×

2 10 10×× 10×

2 × 5 2 5×× 2 5 2× ××1960 = 2 × 2 × 2 × 5 × 7 × 7

2000 = 2 × 2 × 2 × 2 × 5 × 5 × 5

Example 6

Write 648 as a product of its prime factors, using index notation (powers).

SolutionSo 648 = 2 × 2 × 2 × 3 × 3 × 3 × 3

= 23 × 34

34 is read ‘3 to the power 4’ and 4 is called the power or index.

648

3242

2 4 81

×

× ×

2 2 9×× 9×

2 × 2 2 3×× 3 3 3× ××

2 ×

Ex 4



2 What are the prime factors of 1260? Select A, B, C or D. A 2 × 3 × 3 × 3 × 5 × 7 B 2 × 2 × 2 × 3 × 5 × 7C 2 × 2 × 3 × 3 × 3 × 7 D 2 × 2 × 3 × 3 × 5 × 7

3 Find the highest common factor of each of these pairs of numbers.a 324 and 486 b 6000 and 1260c 2475 and 3375 d 4900 and 1960e 4950 and 1530 f 1404 and 900

4 Use factor trees to write each number as a product of its prime factors in index notation.a 18 b 20 c 45 d 72 e 98f 196 g 32 h 135 i 200 j 900

Ex 5

Ex 6

Just for the record

Big numbersHere are the names of some very large numbers.

Find the names of some numbers greater than a decillion.

Name Numeral Power of 10

one 1 100

ten 10 101

hundred 100 102

thousand 1000 103

million 1 000 000 106

billion 1 000 000 000 109

trillion 1 000 000 000 000 1012

quadrillion 1 000 000 000 000 000 1015

quintillion 1 000 000 000 000 000 000 1018

sextillion 1 000 000 000 000 000 000 000 1021

septillion 1 000 000 000 000 000 000 000 000 1024

octillion 1 000 000 000 000 000 000 000 000 000 1027

nonillion 1 000 000 000 000 000 000 000 000 000 000 1030

decillion 1 000 000 000 000 000 000 000 000 000 000 000 1033



3-06 Squares, cubes and rootsFinding square roots and cubesRaising a number to the power 2 gives its square.Raising a number to the power 3 gives its cube.

The , and keys on a calculator can be used to find the square, cube and other powers of a number.

Finding square roots and cube rootsThe process that undoes squaring is finding the square root (symbol ). The square root of a given number is the positive value which, when multiplied by itself, produces the given number.The process that undoes cubing is finding the cube root (symbol ).


Goldbach’s conjectureIn 1742, Christian Goldbach said: Every even number greater than 2 can be written as the sum of two prime numbers.

Show that his theory is true for all the even numbers between 1 and 100. Primes may be repeated, for example 10 = 5 + 5, and a number can have more than one pair of prime numbers.

Applying strategies and reasoning

Skillsheet3-04

Square roots andcube roots

Worksheet3-09

Powers and roots

Example 7

Find a the square of 11 b the cube of 5

Solutiona 112 = 11 × 11 = 121 b 53 = 5 × 5 × 5 = 125

The square of 11 is 121. The cube of 5 is 125.

x2 x3 ^

3

Square root sign first used in 1220 Cube root sign created in 1525



The and keys on a calculator can be used to find the square root and cube

root of a number. To calculate , press 40 . The result is 6.324555…,a more accurate answer than our estimate above.

Finding square roots and cube roots using a factor tree

Example 8

What is:a the square root of 64? b the cube root of 27?

Solution

a The square root of 64 = = 8 (because 8 × 8 = 64).

b The cube root of 27 = = 3 (because 3 × 3 × 3 = 27).

64

273

Example 9

Estimate the value of

SolutionThere is no exact answer for the square root of 40, because there isn’t a number which, if squared, equals 40 exactly. Instead, we find a number whose square is close to 40.

Looking at the square numbers 52 = 25, 62 = 36, 72 = 49, we can tell must be between 6 and 7. Because 40 is closer to 36 than to 49, the square root must be closer to 6.

As an estimate, ≈ 6.3.

40.

40

40

3

40 =

Example 10

Use a factor tree to find the value of .

SolutionSo 196 = 2 × 2 × 7 × 7

= = 2 × 7= 14

(Note: = 2)

196

196

494

2 2 7 7

×

× ××

196 2 2 7 7×××

2 2×



1 Copy and complete the following table.

2 Which number(s) from Question 1 are both square and cube numbers?

3 Use your calculator to find the square of each of these numbers.a 84 b 123 c 24 d 42

4 Use your calculator to find:a 112 b 153 c 1002

d 673 e 0.12 f 3.53

5 Find the square root of:a 9 b 16 c 81 d 121e 25 f 4 g 36 h 100

6 Find the cube root of each of these numbers, using the table from Question 1.a 8 b 125 c 343d 1000 e 729 f 27 000

7 Between which two numbers does lie? Select A, B, C or D (without using a calculator).

A 4 and 5 B 5 and 6 C 26 and 28 D 756 and 784

8 Between which two numbers does lie? Select A, B, C or D.A 14 and 16 B 4 and 5 C 196 and 225 D 3 and 4

Exercise 3-06

Number 1 2 3 4 5 6 7 8 9 10 11 12

Number squared 16

Number cubed 512

Example 11

Use a factor tree to find the value of .

SolutionSo 216 = 2 × 2 × 2 × 3 × 3 × 3

= = 2 × 3= 6

(Note: = 2)

2163

216

544

2 2 27

×

× ×

2 2 3×× 9×

2 2 ×× ×

2 ×

2×

2 3 3 ×× 3

2163 2 2 2 3 3 3×××××3

2 2 2××3

Ex 7

Ex 8

Ex 8

Ex 9 27

15



9 Between which two consecutive whole numbers does lie?

10 Give estimates for the following, then use a calculator to check.

a b c

d e f

11 Find the following square roots, using factor trees.

a b c

d e f

g h i

12 Find the following, using a factor tree.

a b c

d e f

80

50 142 1000

663 9993 1233

Ex 10

484 1764 625

900 784 256

196 400 3136

Ex 11

10 6483 27443 33753

64 0003 49133 92613

Using technology

Absolute cell referencingUsing a spreadsheet to calculate powers 1 Set up your spreadsheet by entering the information in the cells as shown below.

2 We need to use absolute cell referencing to complete this task easily. We use this technique to maintain a particular value in a cell without changing it when writing a formula. For example: to write 21 in cell B2, enter =$B$1^A2. This formula will not change the 2, but will change the power to each consecutive number as we Fill Down (i.e. 21, 22, 23, etc.)

3 Click on cell B2 and Fill Down to cell B13. Your spreadsheet will now show the first 12 powers of 2.

4 By modifying the formula given in point 2, repeat this process, using the appropriate cells, absolute cell referencing and Fill Down for columns C, D and E to show the first 12 powers of 3, 5 and 7.



Displaying large numbersIf the numbers cannot be seen properly follow these steps:

a Highlight all the cells that don’t show full numbers (e.g. 2·44E + 08). Right click and choose Format cells. Change the settings to ‘number’ and ‘0 decimal places’.

b ######## in a cell indicates that the column is not wide enough to hold the number with all digits showing. You may need to widen the column until you can see all numbers.

Using a spreadsheet to calculate the lowest common multiple5 This activity finds the lowest common multiple (LCM) of sets of numbers. The

LCM is the smallest number that all numbers in a particular set divide into; e.g. for the pair of numbers 6 and 20, the LCM = 60. This is the smallest number both 6 and 20 divide into.Find the LCM of 8, 12 and 16. Open a new sheet and enter the information shown.

a In cell B2, enter the formula =$B$1*A2. Use Fill Down to find the first 15 multiples of 8.

b In cells C2 and D2, enter similar formulas and Fill Down to find the first 15 multiples of 12 and 16. [Hint: Only change the absolute cell reference.]

c Now, compare the columns and identify the LCM of 8, 12 and 16.

6 Modify your spreadsheet from part a above to find the LCM of the following sets of numbers. Note: you may need to extend beyond the first 15 multiples.

a 6 and 15 b 12 and 18 c 3, 7 and 15 d 48, 60 and 75 Try other combinations of numbers and calculate each LCM.

Power plus

1 Evaluate each of the following.a 72 − 33 b 8 × 42 c 53 × 22

d 33 + 62 − 42 e 34 + 33 + 32 + 3 f 12 + 22 + 32 + 42



2 Arrange each of these sets of index terms in order, from the smallest value to the largest.a 23, 32, 35, 53, 25, 52 b 44, 73, 35, 82, 52, 63 c 1002, 114, 27, 34, 54

3 a Copy and complete:12 =112 =1112 =11112 =

b Based on the patterns in your part a answers, write the squares of these numbers:i 11 111 ii 1 111 111 iii 111 111 111 iv 1 111.1111

4 a Copy and complete this number pattern. 1 = 1

2 + 3 + 4 = 1 + = (1)3 + (2)3

5 + 6 + 7 + 8 + 9 = 8 + 27 = ( )3 + ( )3

10 + 11 + 12 + 13 + 14 + 15 + 16 = + = ( )3 + ( )3

b Write the next two lines of the pattern in part a.c Find the sum without adding each time. Show how you did it.

i 50 + 51 + … + 63 + 64 ii 82 + 83 + … + 99 + 100iii 290 + 291 + … + 323 + 324 iv 577 + 578 + … + 624 + 625

5 Try finding the square root of each number. (They’re not as hard as they look!)a 2500 b 8100 c 10 000d 1 000 000 e 1 210 000 f 100 000 000g 640 000 h 176 400 i 10 000 000 000

6 Find the cube root of each of these numbers. (What you discovered in Question 5 should help.)a 8000 b 343 000 c 1 000 000d 64 000 000 e 1 000 000 000 f 27 000

7 Find the square root of:a 3 × 3 × 2 × 2 b 5 × 5 × 4 × 4 × 3 × 3c 36 × 49 d 16 × 25 × 4

8 Find the cube root of:a 2 × 2 × 2 b 4 × 4 × 4 × 5 × 5 × 5c 7 × 7 × 7 × 6 × 6 × 6 d 8 × 27e 125 × 64 × 1000 f 343 × 729

9 Find the value of:

a b c

d e f

10 Two prime numbers that differ by 2 are called twin primes. For example, 11 and 13 are twin primes, but 23 and 29 are not. Find the sets of twin primes between1 and 100.

92 34 54 26×

293 33 56×3 164



Chapter 3 review

Language of mathscomposite number cube cube root divisibility testestimate factor factor tree Fibonacci numberhighest common factor index notation palindrome Pascal’s trianglepower prime factors prime number productsquare square root triangular number

1 Describe in your own words how the Fibonacci numbers are formed.

2 Find the non-mathematical meaning of:a ‘factor’ b ‘index’

3 The date 30/11/03 was a palindromic date. When will be the next palindromic date?

4 Describe what a ‘factor tree’ does.

5 Find as many meanings for these words as you can.a product b prime

Topic overview• Write in your own words what you have learnt about number patterns and the way

numbers behave.• What was your favourite part of this topic?• What parts of this topic did you not understand? Talk to your teacher or a friend about

them.• Give examples of where some of the number patterns in this chapter occur or are used.• This diagram provides a summary of this chapter of work. Copy it into your workbook

and complete it. Use bright colours, add your own pictures, and change it, if necessary, to be sure you understand it.

Worksheet3-10

Exploring numbers crossword

Compo

site Prim

e

Factor trees

Factors

Squares, cubes, roots

Divisibility

tests

Exploringnumbers

Fibonacci

Triangular

Num

ber patterns



Chapter revision1 Write the next three numbers in each of these patterns.

a 1, 3, 5, 7, … b 2, 4, 6, 8, … c 1, 3, 6, 10, …d 1, 4, 9, 16, … e 1, 1, 2, 3, 5, 8, … f 60, 55, 50, 45, …

2 In Question 1, which set of numbers are the:a square numbers? b triangular numbers? c Fibonacci numbers?

3 Write the following.a a triangular number between 10 and 20 b the highest Fibonacci number below 40c the next palindrome after 2002 d the next prime number after 29e the first five composite numbers f the square number between 40 and 50

4 Write the next three lines of Pascal’s triangle as shown on the right.

5 Which of the numbers from 2 to 10 divide exactly into:a 81? b 327? c 228? d 170? e 4326?

6 a Write all the factors of 60.b Write all the factors of 42.

7 a Find the common factors of 42 and 60.b Find the highest common factor of 20 and 48.c Find the highest common factor of 36 and 84.

8 Find the prime numbers from:27 93 6 29 19 39 96 31 57 2 51100 65 37 17 13 1 67 73 83 89 27

9 Draw factor trees to find the prime factors of these numbers.a 24 b 60 c 27 d 200e 36 f 45 g 72 h 144

10 Write your answers from Question 9 using index notation.

11 Find the square of each of the following.a 4 b 9 c 6 d 11

12 Find the square root of each of the following.a 64 b 25 c 49 d 144

13 Find the cube of each of the following.a 2 b 6 c 9 d 8

14 Find the cube root of each of the following.a 27 b 64 c 125 d 1000

15 Use factor trees to find:a b c d

16 Between which two consecutive whole numbers does lie?

Exercise 3-01

Exercise 3-01

Exercise 3-01

11 1

1 2 11 3 3 1

1 4 6 4 1

Exercise 3-01

Exercise 3-02

Exercise 3-03

Exercise 3-03

Exercise 3-04

Exercise 3-05

Exercise 3-05

Exercise 3-06

Exercise 3-06

Exercise 3-06

Exercise 3-06

Exercise 3-06

225 256 1764 58323

55 Exercise 3-06

Topic test 3



Mixed revision 11 Use our Hindu–Arabic numerals to write the Babylonian number on the right.

2 Write the Roman numeral XXIX using Hindu–Arabic numerals.

3 Use Hindu–Arabic numerals to write the Chinese number shown on the right.

4 What is the value of the digit 6 in each of the following?a 261 b 1006 c 63 110 d 210 632

5 Write each of the following in expanded notation.a 38 b 201 c 3987

6 Complete these number grids.a b top row − side column

c d top row ÷ side column

7 Find the answers to each of these.a 390 ÷ 15 b 294 ÷ 21 c 259 ÷ 14

8 Evaluate each of the following.a 15 − 3 × 5 b 7 × 3 + 2 × 5 c 26 ÷ 2 − 14 ÷ 7d 7 + 5 × (12 − 3) e 414 ÷ 18 f [(3 + 5) × 2 − (20 ÷ 5)] × 5

9 True or false?a 12 � 20 b 100 � 25 × 5 c = 10

d 62 − 6 = 6 e 12 × 5 � 4 × 15 f 3 �

10 Measure these angles.

Exercise 1-03

Exercise 1-04

Exercise 1-05

Exercise 1-06

Exercise 1-07

Exercise 1-08

+ 12

3 9

27

− 17

9 15

11

÷ 36

4 12

6

× 9

7 35 28

27

40

Exercise 1-09

Exercise 1-10

Exercise 1-11

100

83

Exercise 2-02

a b c


103MIXED REVISION 1

11 Draw an example of:a an obtuse angle b an acute angle c a reflex angle

12 Name this angle using three letters and write the name of its parts.

13 Find the complement of:a 66° b 85°c 12° d 89°

14 Find the supplement of:a 32° b 90° c 105° d 153°

15 Draw a diagram and mark in vertically opposite angles.

16 a Draw a pair of parallel lines and cut them with a transversal.b Mark a pair of alternate angles with (•).c Mark a pair of corresponding angles with (*).d Mark a pair of co-interior angles with (+).

17 Find the size of each angle marked by a letter.

18 Write the next three numbers in each of these patterns.a 1, 4, 7, … b 1, 3, 6, … c 1, 1, 2, 3, 5, … d 11, 9, 7, …

19 Which of the numbers from 2 to 10 divide exactly into:a 68? b 294? c 6152?

20 Find the factors of:a 18 b 45 c 360

21 Using factor trees, write each of the numbers in Question 20 as a product of its prime factors. Write your answers using index notation.

22 Simplify each of the following.a 62 b 33 c d e f

Exercise 2-04

Exercise 2-01

a

b

b

P

AS

Exercise 2-05

Exercise 2-05

Exercise 2-06

Exercise 2-11

i kj

m°

50°

110°

b°

142°p°

l

98°

a°

36°p° 88°

b°

a°

38°c°

m°

m°

70°

56°

37°

p°

m°

a c d

e gf h

x°

x°45°

75°

58°

122°y°

b

64°

Exercise 2-12

Exercise 3-01

Exercise 3-02

Exercise 3-03

Exercise 3-05

Exercise 3-06

25 121 1253 643


Chapter 3

Documents

Transcript of Chapter 3