Math Grade10

MQ10 VIC ch 01 Page 1 Tuesday, November 20, 2001 10:49 AM

Rational and irrational numbers

1 How does the speed of a car affect its stopping distance in an emergency? Serious car accident scenes are often investigated to identify factors leading up to the crash. One measurement taken is the length of the skid marks which indicate the braking distance. From this and other information, such as the road’s friction coefficient, the speed of a car before braking can be determined. If the formula used is v = 20d , where v is the speed in metres per second and d is the braking distance in metres, what would the speed of a car have been before braking if the skid mark measured 32.50 m in length?

For this scenario, the number you will obtain for the speed is an irrational number. In this chapter, you will find out the difference between rational and irrational numbers and learn to work with both.

MQ10 VIC ch 01 Page 2 Tuesday, November 20, 2001 10:49 AM

2

Maths Quest 10 for Victoria

We use numbers such as integers, fractions and decimal numbers every day. They form part of what is called the Real Number System. (There are numbers which do not fit into the Real Number System, called complex numbers, which you may come across in the future.) Real numbers can be divided into two categories — rational numbers and irrational numbers.

Real numbers Rational numbers Irrational numbers • integers • infinite or non-recurring decimal numbers • fractions • surds • finite (or terminating) decimal numbers • special numbers π and e • recurring decimal numbers This chapter begins with a review of rational numbers such as fractions and recurring decimal numbers. We then move on to consider irrational numbers including surds. As you will see, rational numbers are those numbers which can be expressed as a ratio of a two integers --- where b ≠ 0 (that is, a rational number can be expressed as a fraction). b Why are integers considered to be rational numbers?

Operations with fractions From earlier years, you should be familiar with the main operations of using fractions.

This includes simplifying fractions, converting between mixed numbers and improper fractions and the four arithmetic operations.

Simplifying fractions Fractional answers should always be expressed in simplest form. This is done by dividing both the numerator and the denominator by their highest common factor (HCF).

WORKED Example 1 Write

32 -----44

in simplest form.

THINK 1 Write the fraction and divide both numerator and denominator by the HCF or highest common factor (4).

2 Write the answer.

WRITE 32 8 --------44 11 =

8 -----11

Using the four operations with fractions Addition and subtraction 1. When adding and subtracting fractions write each fraction with the same denominator. This common denominator is the lowest common multiple (LCM) of all denominators in the question.

2. When adding mixed numbers, either change to improper fractions or add the whole numbers and fractions separately.

3. When subtracting mixed numbers, either change to improper fractions or make the second fraction a whole number by adding the same fraction to each part of the question.

MQ10 VIC ch 01 Page 3 Tuesday, November 20, 2001 10:49 AM

Chapter 1 Rational and irrational numbers

WORKED Example 2

Evaluate each of the following. a

3 --5

+

5 --6

b 3 1--2- − 1 4--5-

THINK

WRITE

a

a

1

Write the expression.

2

4

Write equivalent fractions using the lowest common denominator.

Add the fractions by adding the numerators. Keep the denominator the same.

Simplify by writing as a mixed number.

1

Write the expression.

2

Write equivalent fractions using the lowest common denominator.

2 - to both fractions to make the second Add ----10 fraction a whole number.

Subtract the whole number parts.

3

b

3 4

3

3 --5

+

=

18 -----30

=

43 -----30

5 --6

+

25 -----30

-----= 1 13 30

b 3 1--2- − 1 4--55 8 - − 1 -----= 3 ----10 10 7 - − 2 = 3 ----10 7 = 1 ----10

Multiplication and division 1. When multiplying fractions, cancel if appropriate, then multiply numerators and multiply denominators.

2. When dividing fractions, change the division sign to a multiplication sign, tip the second fraction upside down and follow the rules for multiplying fractions (times and tip).

3. Change mixed numbers to improper fractions before multiplying or dividing.

WORKED Example 3

Evaluate each of the following. a

3 --5

×

5 --6

b 2 1--3- ÷

3 --4

THINK

WRITE

a

a

1

Write the expression.

2

Cancel or divide numerators and denominators by the same number where applicable.

Multiply the numerators together and the denominators together and simplify where applicable.

3

b

3 --5

×

13 51 = ------ × ----62 15

=

1 --2

b 2 1--3- ÷

1

Write the expression.

2

Change any mixed numbers to improper fractions.

=

3

Change the division sign to a multiplication sign and tip the second fraction upside down (times and tip).

Multiply the numerators together and multiply the denominators together.

Change the improper fraction to a mixed number.

=

7 --3 7 --3

=

28 -----9

4 5

5 --6

÷ ×

= 3 1--9-

3 --4 3 --4 4 --3

MQ10 VIC ch 01 Page 4 Tuesday, November 20, 2001 10:49 AM

4

Maths Quest 10 for Victoria

Graphics Calculator tip! Fraction calculations As with any calculation involving fractions, if you wish to have an answer expressed as a fraction then each calculation needs to end by pressing MATH , selecting 1:Frac and pressing ENTER .

The calculation for worked example 2(a) would be entered as 3 รท 5 + 5 รท 6 then you would press MATH , select 1:Frac and press ENTER .

When entering mixed numbers, it is necessary to use brackets. This allows the correct order of operations to occur.

The calculations for worked example 3(b) can be viewed in the screen shown. Note that the answers are given as improper fractions.

remember remember 1. To write fractions in simplest form, divide numerator and denominator by the HCF of both.

2. To add or subtract fractions, write each fraction with the same denominator first.

3. To add mixed numbers: either (i) change them to improper fractions first and then add or (ii) add the whole numbers first and then the fraction parts.

4. To subtract mixed numbers: either (i) change them to improper fractions first and then subtract or (ii) write the fraction parts with the same denominator and make the second fraction a whole number by adding the same fraction to both parts of the question.

5. To multiply fractions, cancel if possible, then multiply the numerators together and the denominators together. Simplify if appropriate.

6. To divide fractions, change the division sign to multiplication, tip the second fraction upside down then multiply and simplify if appropriate (times and tip).

MQ10 VIC ch 01 Page 5 Tuesday, November 20, 2001 10:49 AM

Chapter 1 Rational and irrational numbers

1A

WORKED

Example

2

WORKED

Example

i

15 -----27

f

25 -----45

j

c

16 -----20

d

16 -----25

16 -----30

g

9 -----54

h

10 -----40

56 -----63

k

55 --------132

l

36 -----60

1 --2

d

2 --5

HEET

HEET

2 Evaluate each of the following: a

1 --4

+

1 --3

b

1 --6

+

2 --3

c

e

1 --2

−

2 --9

f

5 --6

−

7 -----12

g 1 1--4- +

i

1 3--4- −

j

1 1--6- −

8 --9

+

3 --4 4 --5

k 2 1--7- − 1 2--5-

5 -----12

+

7 -----10

h 1 5--8- + l

2 --3

e

5 -----12

i

5 --8

×

3 --4

×

b

2 --7

3 --4

f

7 -----15

× 2 3--4-

j

2 1--2- × 3 5--6-

×

HEET

8 --9

×

HEET

c

3 --5

×

5 --6

g 1 2--5- ×

5 --8

d 4 --9

3 -----10

×

6 -----11

7 - × h 1 ----10

6 -----17

3b

1 2--7- × 3 1--9-

d

11 -----12

4 Evaluate each of the following: 1 --2

e

7 -----10

i

5 --6

÷

3 --5

÷

b

4 --7

2 --3

f

15 -----16

÷

5 --8

j

2 7--8-

÷

1 4--5-

c

5 --8

÷

3 --4

÷

1 --3

L Spread XCE

÷ ÷

2 --5

1 1--3-

g 1 1--4- ÷ k

-----2 11 12

2 --3

÷

7 --9

3 - ÷ h 1 ----10

l

3÷

A b

5 --7

A c

3 --4

is equal to: 15 -----25

B

60 -----72

12 -----18

32 -----40

C

9 -----10

C 1 3--5-

1 D 2 ----21

15 -----32

-----D 1 17 32

D

E

30 -----48

sheet

5 --8

Multiplying fractions L Spread XCE

5 multiple choice a

1 4--5-

7 -----10

sheet

a

Adding and subtracting fractions E

Example

l

1.4

L Spread XCE

E

WORKED

k 1 1--3- × 2 5--8-

1.3

sheet

2 --3

1.2

3 2--5- − 1 3--4-

3 Evaluate each of the following: a

1.1

SkillS

e

b

6 -----15

E

3a

a

8 -----12

SkillS

1

1 Write each of the following fractions in simplest form.

SkillS

Example

Operations with fractions

SkillS

WORKED

5

Dividing fractions

+ 1 1--3- is equal to: 9 -----21

B

E 1 2--3-

÷ 1 3--5- is equal to: B 1 1--5-

C

E 1 1--4d If 1--3- of a glass is filled with lemonade and 1--2with water, what fraction of the glass has no liquid?

A

1 --2

B

5 --6

D

2 --3

E

1 --6

C

3 --5

Math

cad

2 A 2 ----15

Operations with fractions

MQ10 VIC ch 01 Page 6 Tuesday, November 20, 2001 10:49 AM

6

Maths Quest 10 for Victoria

6 Five hundred students attended the school athletics carnival. Three-fifths of them wore sunscreen without a hat and 1--4- of them wore a hat but no sunscreen. If 10 students wore both a hat and sunscreen, how many students wore neither?

7 Phillip earns $56 a week doing odd jobs. If he spends 5--8- of his earnings on himself and saves 1--5- , how much does he have left to spend on other people?

GAM

E

8 A pizza had been divided into four equal pieces. i Bill came home with a friend and the two boys shared one piece. How much of the pizza was left? ii Then Milly came in and ate 1--3- of one of the remaining pieces. How much of the pizza did she eat and how much was left? iii Later, Dad came home and ate 1 1--3- of the larger pieces which remained. How much did he eat and how much of the pizza was left?

time

QUEST

GE

S

EN

M AT H

Rational and irrational numbers — 001

CH

A

LL

1 Without using a calculator, find the value of: 1 − 2 + 3 − 4 + 5 − 6 + 7 − 8 + . . . − 98 + 99 − 100.

2 Find a fraction that is greater than

5----11

but less than

6----13 .

3 If this calculation continued forever, what would you expect the answer to be?

1 1 1 1 1 --- + --- + ------ + ------ + --------- + º 3 9 27 81 243

4 If this calculation continued forever, what would you expect the answer to be?

1

1 1 1 1 1 1 – --- + --- – --- + ------ – ------ + ------ – º 2 4 8 16 32 64

MQ10 VIC ch 01 Page 7 Tuesday, November 20, 2001 10:49 AM

Chapter 1 Rational and irrational numbers

7

Finite and recurring decimal numbers

WORKED Example 4 Express each of the following fractions as a recurring decimal number. a

7 -----12

b

3 --7

THINK

WRITE

a

a

b

1

Write the fraction.

2

Divide the numerator by the denominator until a recurring pattern emerges.

3

Write the answer.

1

Write the fraction.

2

Divide the numerator by the denominator until a recurring pattern emerges.

3

Write the answer.

7 -----12

0.583 33 12 ) 7.000 00

7 -----12

b

= 0.583Ë™

3 --7

0.428 571 428 571 4 7 ) 3.000 000 000 000 0

3 --7

= 0.428 571

If asked to convert a fraction to a decimal number without specifying the number of decimal places or significant figures required, work until a pattern emerges or a finite answer is found. Some recurring patterns will quickly become obvious.

To convert recurring decimal numbers to fractions requires some algebraic skills. We call the recurring decimal number x, then multiply this by 10, 100, 1000, etc. We aim to form a decimal number, such that when we subtract the equations the decimal part will disappear.

1.5

SkillS

HEET

HEET

SkillS

The four basic operations when applied to decimal numbers are very straightforward using a calculator. It is important that you are able to convert between the fractional and decimal forms of a rational number.

All fractions can be written as finite or recurring decimal numbers. Finite (or terminating) decimal numbers are exact and have not been rounded. Recurring decimal numbers repeat the last decimal places over and over again. They are represented by a bar or dots placed over the repeating digits. Many calculators round the last digit on their screens, so recurring decimal patterns are sometimes difficult to recognise.

Converting between fractions and terminating decimal numbers was covered in earlier years and can be revised by clicking on the skillsheet icons here or in exercise 1B. To convert a fraction to a recurring decimal number requires you to recognise the recurring pattern when it appears.

1.7

MQ10 VIC ch 01 Page 8 Tuesday, November 20, 2001 10:49 AM

8

Maths Quest 10 for Victoria

WORKED Example 5 Convert each of the following to a fraction in simplest form. a 0.63 b 0.63˙ THINK a 1 Write the recurring decimal and its expanded form.

2 Let x equal the expanded form and call it equation [1].

Multiply both sides of equation [1] by 3 100 because there are two repeating digits and call the new equation [2].

4 Subtract [1] from [2] in order to eliminate the recurring part of the decimal number.

WRITE a 0.63 = 0.636 363 . . .

Solve the equation and write the answer in simplest form.

x=

5

b

1 2 3

4

Write the recurring decimal and its expanded form.

Let x equal the expanded form and call it [1].

Multiply both sides of equation [1] by 10 because there is one repeating digit and call the new equation [2].

Subtract [1] from [2] in order to eliminate the recurring part of the decimal number.

5

Solve the equation.

6

Simplify where appropriate. (Multiply numerator and denominator by 10 to obtain whole numbers.)

Let x = 0.636 363 . . .

[1]

[1] × 100: 100x = 63.636 363 . . .

[2]

[2] − [1]: 100x − x = 63.636 363 . . . −0.636 363 . . .

99x = 63 x=

63 -----99 7 -----11

b 0.63˙ = 0.633 333 3 . . .

Let x = 0.633 333 . . .

[1]

10x = 6.333 33 . . .

[2]

[2] − [1]:

10x − x = 6.333 33 . . . −0.6333 33 . . .

9x = 5.7 5.7 x = ------9 x=

57 -----90

x=

19 -----30

Similarly, for three repeating digits, multiply by 1000; for four repeating digits, multiply by 10 000; and so on. It is possible to do this using other multiples of 10.

Can you see why recurring decimal numbers are considered to be rational numbers?

remember remember 1. To convert a fraction to a decimal number, divide the numerator by the denominator.

2. To write a recurring decimal number, place a dot or line segment over all recurring digits.

3. Rational numbers are those numbers that can be written as a fraction with integers in both numerator and denominator. (The denominator cannot be zero.)

They include: integers, fractions, finite and recurring decimal numbers.

MQ10 VIC ch 01 Page 9 Tuesday, November 20, 2001 10:49 AM

Chapter 1 Rational and irrational numbers

History of mathematics S R I N I VA S A R A M A N U J A N ( 1 8 8 7 – 1 9 2 0 )

During his life . . .

The Sherlock Holmes stories are written.

X-rays are discovered.

The Wright brothers build their aircraft.

World War I is fought.

Srinivasa Ramanujan was an Indian mathematician. He was born in Madras into a very poor family. Although he was a selftaught mathematical genius, Ramanujan failed to graduate from college and the best job he could find was as a clerk. Fortunately some of the people he worked with noticed his amazing abilities — he had discovered

more than 100 theorems including results on elliptic integrals and analytic number theory.

Ramanujan was persuaded to send his theorems to Cambridge University in England for evaluation. Godfrey Hardy, a fellow of Trinity College who assessed the work, was very impressed. He organised a scholarship that enabled Ramanujan to come to Cambridge in 1914. The notebooks which Ramanujan brought with him to Cambridge displayed an obvious lack of formal training in mathematics and showed that he was unaware of many of the findings of other mathematicians. Remarkably he seemed to achieve many of his results by intuition.

While at Cambridge, Ramamujan published many papers, some in conjunction with Godfrey Hardy. He worked in several areas of mathematics including number theory, elliptic functions, continued fractions and prime numbers. Palindromes were also of interest to him. A palindrome reads the same backwards as forwards, such as 12321 or abcba. He was elected a fellow of Trinity in 1918 but poor health forced him to return to India. Ramanujan died of tuberculosis at the age of 32.

Questions 1. What had Ramanujan discovered before he went to Cambridge?

2. Name four areas of mathematics that Ramanujan worked in.

3. How old was he when he died?

4. Challenge: Ramanujan found a formula for π as below. Use a calculator or computer to see what value you get for this irrational number. ∞

8 1 ( 4n )! ( 1103 + 26 390n ) --- = ------------ ∑ ------------------------------------------------------π 9801 n = 0 ( n! ) 4 ( 396 4n )

9

MQ10 VIC ch 01 Page 10 Tuesday, November 20, 2001 10:49 AM

10

Maths Quest 10 for Victoria

Finite and recurring decimal numbers

1B 1.5

1 Express each of the following fractions as a finite decimal number.

SkillS

HEET

GC pr

ogram

Converting fractions to decimal numbers

3 --4

b

2 --5

c

9 -----10

d

5 --8

e

33 -----50

f

11 -----40

g

73 -----80

h

5 -----16

i

13 -----25

j

9 -----20

k

57 --------100

l

2 -----25

2 Write each of the following as an exact recurring decimal number. a 0.333 3 . . . b 0.166 66 . . . c 0.323 232 . . . d 0.785 55 . . . e 0.594 594 594 . . . f 0.125 125 151 51 . . . g 0.375 463 75 . . . h 0.814 358 14 . . .

1.6

SkillS

HEET

a

WORKED

Example

4

Mat

d hca

Finite and recurring decimal numbers

3 Express each of the following fractions as a recurring decimal number. a

2 --3

b

3 -----11

c

8 --9

d

5 -----18

e

5 --6

f

1 --7

g

11 -----12

h

1 -----15

i

10 -----11

j

7 -----24

k

17 -----30

l

7 -----27

4 multiple choice a

31 ---------------10 000

is equal to:

A 0.031 b

67 -----99

9 -----14

C 0.000 31

D 0.003

E 0.31

B 0.676

C 0.67

D 0.67

E 0.676˙

is equal to:

A 0.676 c

B 0.0031

is equal to:

A 0.642 857 142 D 0.642 857 1 d

1.7 e

EXCE

et et

EXCE

37 --------200

B

2 -----11

C

26 --------135

D

5 -----27

6 Convert each of the following to a fraction in simplest form. a 0.5˙ b 0.6˙ c 0.84 5 ˙ e 0.46 f 0.18 g 0.18˙ i 0.363˙ j 0.382 k 0.616

E

167 --------900

d 0.67 h 0.875 l 0.3625

WORKED

Example

Converting recurring decimals to fractions

C 0.123 456 78

5 Convert each of the following to a fraction in simplest form. a 0.8 b 0.3 c 0.14 e 0.95 f 0.75 g 0.12 i 0.675 j 0.357 k 0.884

Converting decimal numbers to fractions reads L Sp he

B 0.123 456 78 E 0.123 456 790

0.185 is equal to: A

reads L Sp he

C 0.642 857 1

is equal to:

A 0.123 456 79 D 0.123 456 79

SkillS

HEET

10 -----81

B 0.642 857 1 E 0.642 857 1

d 0.71 h 0.27˙ l 0.725

MQ10 VIC ch 01 Page 11 Tuesday, November 20, 2001 10:49 AM

Chapter 1 Rational and irrational numbers

b 0.0625 is equal to: 1 A ----B 16 c 0.32˙ is equal to: A

32 -----99

B

d 0.90 is equal to: 9 A ----B 10

58 -----99

C

29 -----50

D

58 -----10

E

43 -----90

5 --8

C

3 -----50

D

625 --------999

E

7 -----11

29 -----99

C

29 -----90

D

16 -----45

E

8 -----25

9 -----11

C

91 -----99

D

10 -----11

E

ET SHE

Work

7 multiple choice a 0.58 is equal to: A 5--8B

11

44 -----45

1 1 Simplify

64 ------ .

88

2 Evaluate 1 7--8- + 2 3--7- . ×

×

3 Evaluate

3 --8

4 Evaluate

2 ------ ÷ 2 --- .

1 11 21 7 1 3 2 --2- − --8- × 1 7--9- .

5 Evaluate

2 --9

4 --- .

5

6 Write

13 -----40

7 Write

1 --6

as a finite decimal. as a recurring decimal.

8 Write 0.625 as a fraction in simplest form.

9 Write 0.7˙ as a fraction in simplest form.

10 Write 0.256 as a fraction in simplest form.

Irrational numbers Irrational numbers are those which cannot be expressed as fractions. These include (i) non-recurring, infinite decimal numbers (ii) the special numbers π and e (iii) surds or roots of numbers that do not have a finite, exact answer, for example, 5 and 3 6 .

A surd is an exact answer but the calculator answer is an approximation because it has been rounded.

roots Graphics Calculator tip! Calculating of numbers 1. To find the square root of 5 on your graphics calculator press 2nd [ number concerned (in this case, 5), close the brackets by pressing optional) and press ENTER .

], enter the

)

(this is

1.1

MQ10 VIC ch 01 Page 12 Tuesday, November 20, 2001 10:49 AM

12

Maths Quest 10 for Victoria

2. To find the cube root of 5 we need to use the MATH function. Press MATH , select 4: 3 , press 5 , close the brackets by pressing ) (this is optional) and then ENTER .

3. To find higher order roots we again use the MATH function. To find the 6th root of 32, first enter 6 then press MATH , select 5: x , press 32 and then ENTER .

WORKED Example 6 State whether each of the following numbers is a surd or not. a 3 b 0.49 c 38 d 5 15 THINK

WRITE

a

Write the number. Consider square roots which can be evaluated: 1 = 1 and 4 = 2.

Check on a calculator if necessary then state whether the number is a surd or not.

a

3 is a surd.

Write the number. Consider whether the number is a perfect square or not.

Check with a calculator if necessary and then write the exact answer if there is one.

b

0.49 = 0.7 so

c Write the number. Consider whether the cube root can be found by cubing small numbers and write the exact answer if there is one. 1 × 1 × 1 = 1; 2 × 2 × 2 = 8

c

3

8 = 2 so

d Write the number. Consider whether the 5th root can be found and write the exact answer if there is one. 15 = 1 is too small; 25 = 32 is too big.

d

5

15 is a surd.

1

2

b

1 2

3

0.49 is not a surd.

8 is not a surd.

MQ10 VIC ch 01 Page 13 Tuesday, November 20, 2001 10:49 AM

Chapter 1 Rational and irrational numbers

13

Rational approximations for surds When an infinite decimal number is rounded, the answer is not exact, but it is very close to the actual value of the number. It is called a rational approximation because once it is rounded it becomes finite and is therefore rational.

WORKED Example 7 Find the value of

56 correct to 2 decimal places.

THINK 1 Write the surd and use a calculator to find the answer.

2 Round the answer to 2 decimal places by checking the 3rd decimal place.

WRITE 56 ≈ 7.483 314 774 = 7.48 (2 decimal places)

Exact answers are the most accurate and should be used in all working. Irrational numbers that are rounded are close approximations to their true values and should be used in the final answer only when asked for.

History of mathematics RICHARD DEDEKIND (1831–1916) During his life . . .

Louis Braille invents the Braille system.

Edison invents the light bulb.

Lewis Carroll writes Alice in Wonderland.

Richard Dedekind was a German mathematician. He was the son of a professor and was the youngest of four children. He went to school in his home town of Brunswick where he initially studied science but became more interested in mathematics because he liked its logic. At the age of 21 he went to the University of Göttingen where he completed his doctorate under the famous mathematician Carl Gauss. Dedekind taught probability and geometry at Göttingen and then taught for a short while in Zürich before returning to Brunswick to teach at the Polytechnic.

He published significant material on number theory including continuity and irrational numbers. A remarkable achievement was his redefinition of irrational numbers by using what are known as Dedekind cuts.

However, one of his most important

contributions to mathematics was his ability to express mathematical concepts with great clarity and logic. This talent is obvious in his own work and also in his writings about the work of other mathematicians. He produced editions of the collected works of Peter Dirichlet, Carl Gauss and Georg Riemann.

Dedekind was elected to the Göttingen Academy in 1862, the Berlin Academy in 1880 and the Academy of Sciences in Paris in 1900. He received honorary doctorates from the universities of Oslo, Zürich and Brunswick. Dedekind never married and for most of his life he lived with his sister, Julie.

It was reported that Dedekind had died on September 4, 1899 but he had actually ‘passed the day in perfect health’.

Questions 1. Who did Dedekind work with while studying for his doctorate at Göttingen?

2. What did Dedekind use to redefine irrational numbers?

3. As well as his own work, what did Dedekind produce?

MQ10 VIC ch 01 Page 14 Tuesday, November 20, 2001 10:49 AM

14

Maths Quest 10 for Victoria

remember remember 1. Irrational numbers are those which cannot be expressed as fractions. These include: (i) non-recurring, infinite decimals (ii) the special numbers, π and e (iii) surds.

2. A surd is an exact value. π and e are also exact values.

3. Rounded decimal answers to surd questions are only rational approximations.

1C 1.8

WORKED

Example

6

SkillS

HEET

Irrational numbers

1 State whether each of the following numbers is a surd or not. a

7

b

e

5

64

f

6

i

7

– 2354

j

6

100

c

216

g

9

d

–1

h

3

7 --8

k

74 4

2401 16 -----25

l

2 multiple choice

Mat

d hca

Irrational numbers

a Which of the following is a surd?

B π A 28.09 C

48.84

D 0.9875

46

D

4.84

E

0

64

E

101

D 0.83˙

E

1 --4

0.9

E

b Which of the following is not a surd?

65

A c

56

B

C

Which of the following is a surd?

4.48

A

B 0.83

C

d Which of the following is not a surd?

5.44

A

B

3

82.511

C

4

108.8844 D

5

143.489 07

e Which of the following numbers is irrational?

A a square root of a negative number B a recurring decimal number C a fraction with a negative denominator D a surd E a finite decimal number 3 Classify each of the following numbers as either rational or irrational. a 5

b

5

c

1 --5

d 0.55

e

16

f

4.124 242 4 . . .

g 7 4--9-

h

3

i

5.0129

j

4

k −60

l

2.714 365 . . .

15

8

MQ10 VIC ch 01 Page 15 Tuesday, November 20, 2001 10:49 AM

15

Chapter 1 Rational and irrational numbers

L Spread XCE

E

a

67

b

82

c

147

d

5.22

e

6.9

f

0.754

g

2534

h

1962

i

607.774

j

8935.0725

k

12.065

l

355.169

Square roots DIY

5 Find approximate answers to each of the following surds, rounded to 4 significant figures. a

3

23

b

3

– 895

c

f

8

1.5

g

3

2.8856

h

5 9

1048

d

6

45 867

e

4

654.8

– 54 988

i

5

– 84.848 484

j

4

0.7882

6 Calculate each of the following, correct to the nearest whole number. a

546

b

4

54 637

c

5

d

697 643

3

– 2116

e

7

8 564 943

7 multiple choice a b c d

e

43.403 correct to 4 decimal places is: A 6.5881 B 6.5880 C 6.5889

D 6.5888

E 6.589

65 – 55 + 25, rounded to 3 decimal places is: A 5.916 B −21.938 C 28.162 D 25.646

E 15

56 × 68 – 42 ÷ 8 rounded to 2 decimal places is: A 59.42 B 61.67 C 494.02 D 59.28

E 61.66

456 × 5 – 456 – 4 456 rounded to the nearest whole number is: A 5 B 22 C 31 D −22 E −31 3

56.6 + 65.5 ----------------------------------- rounded to 3 decimal places is: 56.6 – 65.5 A 49.583 B −19.389 C −6.624

D −27.402

E 5.236

8 Calculate each of the following, correct to 2 decimal places: a

67 + 54 × 43

b

c

8.3 – 5.7 × 8.3 – 5.7

d

e

6.7 × 4.9 ------------------------6.7 ÷ 4.9

f

3

HEET

768 – 564 + 4 684 5.86 ÷ 8.64 ÷ 3 4.23

58.8 – 21.7 ----------------------------------58.8 – 21.7

9 Rali’s solution to the equation 3x = 13 is x = 4.33, while Tig writes his answer as x = 4 1--3- . When Rali is marked wrong and Tig marked right by their teacher, Rali complains. a Do you think the teacher is right or wrong?

The teacher then asks the two students to compare the decimal and fractional parts of the answer. b Write Rali’s decimal remainder as a fraction. c Find the difference between the two fractions. d Multiply Rali’s fraction by 120 000 and multiply Tig’s fraction by 120 000. e Find the difference between the two answers. f Compare the difference between the two fractions from part C and the difference between the two amounts in part d. Comment.

1.9

SkillS

7

4 Find the value of each of the following, correct to 3 decimal places.

sheet

Example

HEET

SkillS

WORKED

1.10

MQ10 VIC ch 01 Page 16 Tuesday, November 20, 2001 10:49 AM

16

Maths Quest 10 for Victoria

10 Takako is building a corner cupboard to go in her bedroom and she wants it to be 10 cm along each wall. a Use Pythagoras’ theorem to find the exact length of timber required to complete the triangle. b Find a rational approximation for the length, rounding your answer to the nearest millimetre.

11 Phillip uses a ladder which is 5 metres long to reach his bedroom window. He cannot put the foot of the ladder in the garden bed, which is 1 metre wide. If the ladder just reaches the window, how high above the ground is Phillip’s window?

Plotting irrational numbers on the number line We know that it is possible to find the exact square root of some numbers, but not others. For example, we can find 4 exactly but not 3 or 5 . Our calculator can find a decimal approximation of these, but because they cannot be found exactly they are called irrational numbers. There is a method, however, of showing their exact location on a number line.

1 Using graph paper draw a right-angled triangle with two equal sides of length 1 cm as shown below.

0

1

2

3

4

5

6

7

8

2 Using Pythagoras’ theorem, the length of the hypotenuse of this triangle is 2 units. Use a compass to make an arc that will show the location of 2 on the number line.

0

1 2 2

3

4

5

6

7

8

3 Draw another right triangle using the hypotenuse of the first triangle as one side and make the other side 1 cm in length.

0

1 2 2

3

4

5

6

7

8

4 The hypotenuse of this triangle will have a length of find the location of 3 on the number line.

3 units. Draw an arc to

5 Repeat steps 3 and 4 to draw triangles that will have sides length units, etc.

4,

5,

6

MQ10 VIC ch 01 Page 17 Tuesday, November 20, 2001 10:49 AM

17

QUEST

GE

S

EN

M AT H

Chapter 1 Rational and irrational numbers

CH

AL

L

1 Find three numbers, w, x and y, none of which are perfect squares or zero and that make the following relationship true. w+ x = y 2 a If 132 = 169, 1332 = 17 689 and 13332 = 1 776 889, write down the answer to 13 3332 without using a calculator or computer. b If 192 = 361, 1992 = 39 601 and 19992 = 3 996 001, write down the answer to 19 9992 without using a calculator or computer. c Can you find another number between 13 and 19 where a similar pattern can be used?

Simplifying surds Some surds, like some fractions, can be reduced to simplest form.

Only square roots will be considered in this section.

Consider: Now, Taking

9 and

36 = 6 36 = 9 × 4, so we could say: 9×4 =6 4 separately: 9× 4 =3×2=6

If both 9 × 4 = 6 and 9 × 4 = 6, then 9 × 4 = 9 × 4 .

This property can be stated as: ab = a × b and can be used to simplify surds.

8 = =

4×2 4× 2

= 2 × 2 which can be written as 2 2 .

A surd can be simplified by dividing it into two square roots, one of which is the highest perfect square that will divide evenly into the original number.

WORKED Example 8 Simplify each of the following. a 40 b 72 THINK a 1 Write the surd and divide it into two parts, one being the highest perfect square that will divide into the surd.

2 Write in simplest form by taking the square root of the perfect square.

WRITE a 40 =

b

b

1

2

Write the surd and divide it into two parts, one being the highest perfect square that will divide into the surd.

Write in simplest form by taking the square root of the perfect square.

4 × 10

= 2 10 72 =

36 × 2

=6 2

MQ10 VIC ch 01 Page 18 Tuesday, November 20, 2001 10:49 AM

18

Maths Quest 10 for Victoria

22 can not be simplified because no perfect square divides exactly into 22.

If a smaller perfect square is chosen the first time, the surd can be simplified in more than one step.

72 =

4 × 18

= 2 18 = 2× 9× 2 = 2×3 2 =6 2 This is the same answer as found in worked example 8(b) but an extra step is included.

When dividing surds into two parts, it is critical that one is a perfect square.

For example, 72 = 24 × 3 is of no use because an exact square root can not be found for either part of the answer.

WORKED Example 9 Simplify 6 20 .

THINK

WRITE 6 20 = 6 × 4 × 5

1

Write the expression and then divide the surd into two parts, where one square root is a perfect square.

2

Evaluate the part which is a perfect square.

= 6×2 5

3

Multiply the whole numbers and write the answer in simplest form.

= 12 5

Sometimes it is necessary to change a simplified surd to a whole surd. The reverse process is applied here where the rational part is squared before being placed back under the square root sign.

WORKED Example 10 Write 5 3 in the form

a.

THINK

WRITE

1

Square the whole number part, then express the whole number as a square root.

52 = 25 so 5 =

2

Write the simplified surd and express it as the product of 2 square roots, one of which is the square root in step 1 .

5 3 =

3

Multiply the square roots to give a single surd.

25

25 × 3

=

25 × 3

=

75

MQ10 VIC ch 01 Page 19 Tuesday, November 20, 2001 10:49 AM

Chapter 1 Rational and irrational numbers

19

WORKED Example 11 Ms Jennings plans to have a climbing frame that is in the shape of a large cube with sides 2 metres long built in the school playground.

H a Find the length of material required to join the opposite vertices G of the face which is on the ground.

F E b Find the exact length of material required to strengthen the frame by joining a vertex on the ground to the vertex which is in the air D C and which is furthest away. c Find an approximate answer rounded to the nearest cm.

A B THINK

WRITE

a

a D

1

Draw a diagram of the face, mark in the diagonal, the appropriate measurements and label the vertices.

C 2m

A 2

B

2m 2

2

AC = AB + BC

Use Pythagoras’ theorem to find the length of the diagonal.

2

= 22 + 2 2 =8

AC =

8

=2 2

b

3

Answer the question in a sentence.

1

Draw a diagram of the triangle required, label the vertices and mark in the appropriate measurements.

2 2 metres of material is required. b

G 2m

A 2

Use Pythagoras’ theorem to find the length of the diagonal.

2 2m 2

2

AG = CG + AC

Simplify the surd.

4

Write your answer in a sentence.

c Round the answer to 2 decimal places.

2

= 22 + ( 2 2 ) 2 = 12

AG = 3

C

12

=2 3 The length of material required is 2 3 metres. c The approximate length to the nearest cm is 3.50 metres.

MQ10 VIC ch 01 Page 20 Tuesday, November 20, 2001 10:49 AM

20

Maths Quest 10 for Victoria

remember remember 1. To simplify a surd, divide it into two square roots, one of which is a perfect square.

2. Not all surds can be simplified.

3. ab = a Ă— b 4. Some perfect squares to learn are: 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144 . . .

1D 1.11

SkillS

HEET

WORKED

Example

8

Mat

d hca

Simplifying surds WORKED

Example

EXCE

et

reads L Sp he

Simplifying surds

9

WORKED

Example

10

GC pr

ogram

Surds

Simplifying surds

1 Simplify each of the following. a

20

b

8

c

18

d

49

e

30

f

50

g

28

h

108

i

288

j

48

k

500

l

162

m

52

n

55

o

84

p

98

q

363

r

343

s

78

t

160

6 64

d 7 50

e

10 24

i

j

12 242

d 4 5

e

8 6

i

j

13 2

2 Simplify each of the following. a 2 8

b 5 27

c

f

g 4 42

h 12 72

5 12

3 Write each of the following in the form

a.

a 2 3

b 5 7

c

f

g 4 2

h 12 5

3 10

9 45

6 3

10 6

4 multiple choice 1000 is equal to: a A 31.6228 B 50 2 C 50 10 D 10 10 80 in simplest form is equal to: b A 4 5 B 2 20 C 8 10 D 5 16 c Which of the following surds is in simplest form?

147 A 60 B C 105 D 117 d Which of the following surds is not in simplest form?

110 A 102 B C 116 D 118 e 6 5 is equal to: 30 A 900 B C 150 D 180 f Which one of the following is not equal to the rest?

A 128 B 2 32 C 8 2 D 4 8 g Which one of the following is not equal to the rest?

A 4 4 B 2 16 C 8 D 16 h 5 48 is equal to: A 80 3 B 20 3 C 9 3 D 21 3

E 100 10 E 10 E

432

E

122

E 13.42 E 64 2 E

64

E 15 16

MQ10 VIC ch 01 Page 21 Tuesday, November 20, 2001 10:49 AM

21

Chapter 1 Rational and irrational numbers

5 Challenge: Reduce each of the following to simplest form. a

675

b

1805

c

1792

d

578

e

a2c

f

bd 4

g

h 2 jk 2

h

f3

6 A large die with sides measuring 3 metres is to be placed in front of the casino at Crib Point. The die is placed on one of 11 its vertices with the opposite vertex directly above it. a Find the length of the diagonal of one of the faces. b Find the exact height of the die. c Find the difference between the height of the die and the height of a 12-metre wall directly behind it. Approximate the answer to 3 decimal places.

WORKED

Example

7 A tent in the shape of a tepee is being used as a cubby house.

The diameter of the base is 220 cm and the slant height is 250 cm. a How high is the tepee? Write the answer in simplest surd form. b Find an approximation for the height of the tepee in centimetres, rounding the answer to the nearest centimetre.

250 cm

220 cm

Career profile PETER RICHARDSON — Analyst Programmer I use basic mathematical skills throughout the day to calculate screen heights and check whether all necessary fields and labels will fit. More advanced mathematics such as working with formulas and other secondary school mathematics are used in Excel spreadsheets for statistics and data manipulation.

During a typical day, all my work is done on computer, usually using a software package to write code in Java or Cobol. I create screens for use by clients, and the supporting code to ensure screens react as expected.

Qualifications: Bachelor of Applied Science (Computer Science and Software Engineering) I entered this field as a change of career and find it to be interesting and diverse.

Questions 1. What computer language does Peter use to write code?

2. Name one aspect of Peter’s job.

3. Find out what courses are available to become an analyst programmer.

MQ10 VIC ch 01 Page 22 Tuesday, November 20, 2001 10:49 AM

22

Maths Quest 10 for Victoria

Braking distances At the start of the chapter, a formula was given to calculate the speed of a car before the brakes are applied to bring it to stop in an emergency. The formula given was v = 20d where v is the speed in m/s and d is the braking distance in m.

1 What is the speed of a car before braking if the braking distance is 32.50 m?

2 Explain why your answer to part 1 is an irrational number.

3 State your answer to part 1 as an exact irrational number in simplest form and as a rational approximation.

4 Convert the speed from m/s to km/h.

5 Calculate the speed of a car before braking if the braking distance is 31.25 m.

6 Is your answer to part 5 rational or irrational?

7 State your answer to part 5 in km/h. Is this number rational or irrational?

The effect of speed Research using data from actual road crashes has estimated the relative risk for cars travelling at or above 60 km/h becoming involved in a casualty crash (a car crash in which people are killed or hospitalised). It was found that the risk doubled for every 5 km/h above 60 km/h. So a car travelling at 65 km/h was twice as likely to be involved in a casualty crash as one travelling at 60, while the risk for a car travelling at 70 km/h was four times as great.

We will consider two elements which affect the distance travelled by a car after the driver has perceived danger — the reaction time of the driver and then the braking distance of the car.

Let’s consider the total distance travelled to bring cars travelling at different speeds to a stop after the driver first perceives danger. Assume a reaction time of 1.5 seconds. (This means that the car continues to travel at the same speed for 1.5 s until the brakes are applied.)

8 Complete the following table. (Remember to convert speed in km/h to m/s before substituting into a formula to find the distance in m.)

Distance travelled to bring a car to a complete stop (metres)

Speed km/h

m/s

Reaction distance

Braking distance

Total stopping distance

60 65 70 9 Compare the difference between the total stopping distance travelled at each of the given speeds.

10 Give an example to explain how the difference between these stopping distances could literally mean the difference between life and death.

11 What other factors could affect the stopping distance of a car?

MQ10 VIC ch 01 Page 23 Tuesday, November 20, 2001 10:49 AM

23

Chapter 1 Rational and irrational numbers

Place the expressions in simplest surd form and use the code section to match up the letter beside each expression with a number.

Daffynitions!

Code L = 50 =

D = 18 =

Dandelion: 1

2

3

4

5

6

7

4

8

9

10

1

11

11

12

1

11

13

9

1

14

15

16

1

7

10

17

10

8

18

11

12

9

15

14 20 ——– 49

X=

C = 48 =

V = 270 =

T = 2 45 =

F = 63 =

=

I = 7 12 =

R = 90 =

H = 40 =

O = 2 63 =

Cheap:

J=

U = 54 =

56 —– 14

G=

P = 140 =

B = 3 125 =

2 20 —–– 16

S = 108 =

E = 2 96 =

W = 160 =

=

Expand: 9

15

6

9

23

15

9

14

1

8

19 11 18 22

15

6

20

8

1

17

3

7

7

1

3

10

3

1

4

1

15 5

2

14 3

3

5

4

2

5

3 6

6

5 7

7

5 2

8

8 6

9

4 3

10

6 5

11

2 10

12

4 10

13

3 10

14

19

A = 120 =

Y=3 8 =

=

14

3

2

9

23

3

15

7

9

22

21

9

7

3

N = 175 =

7

1

17

2

9

19

1

20

19

7

6

18

15

1

2 30

15 13 12 18

14 18

6

22

6 3

15

3 7

16

6 2

17

6 7

18

3 2

19

13 2

4 5

20

21

2 35

22

3 30

23

M = 338 =

MQ10 VIC ch 01 Page 24 Tuesday, November 20, 2001 10:49 AM

24

Maths Quest 10 for Victoria

Addition and subtraction of surds Operations with surds have the same rules as operations in algebra.

1. Like surds are those which contain the same surd when written in simplest form.

2. Like surds can be added or subtracted after they have been written in simplest form.

WORKED Example 12

Simplify each of the following. a 6 3 + 2 3 + 4 5 – 5 5

b 3 2–5+4 2+9

THINK

WRITE

a

a 6 3+2 3+4 5–5 5 = 8 3– 5

1 2

b

1 2

Write the expression.

All surds are in simplest form, so collect like surds.

Write the expression.

All surds are in simplest form, so collect like terms.

b 3 2–5+4 2+9 3 2+4 2–5+9 = 7 2+4

We need to check that all surds are fully simplified before we can be sure whether or not they can be added or subtracted as like terms.

WORKED Example 13

Simplify 5 75 – 6 12 + 2 8 + 4 3 .

THINK 1 Write the expression.

2 Simplify all surds.

WRITE 5 75 – 6 12 + 2 8 + 4 3 = ( 5 × 25 × 3 ) – ( 6 × 4 × 3 ) + ( 2 × 4 × 2 ) + 4 3 = 25 3 – 12 3 + 4 2 + 4 3

3

= 17 3 + 4 2

Collect like surds.

remember remember 1. Only like surds can be added or subtracted.

2. All surds must be written in simplest form before adding or subtracting.

1E WORKED

Example

d hca

Mat

12 Addition and subtraction of surds

Addition and subtraction of surds

1 Simplify each of the following. a 6 2+3 2–7 2

b 4 5–6 5–2 5

c

–3 3 – 7 3 + 4 3

d –9 6 + 6 6 + 3 6

e

10 11 – 6 11 + 11

f

g 4 2+6 2+5 3+2 3

7+ 7

h 10 5 – 2 5 + 8 6 – 7 6

MQ10 VIC ch 01 Page 25 Tuesday, November 20, 2001 10:49 AM

Chapter 1 Rational and irrational numbers

5 10 + 2 3 + 3 10 + 5 3

j

12 2 – 3 5 + 4 2 – 8 5

k 6 6+ 2–4 6– 2

l

16 5 + 8 + 7 – 11 5

m 10 7 – 4 – 2 7 – 7

n 6+2 2+5–3 2

i

o

13 + 4 7 – 2 13 – 3 7

p 8 6–4 3+2 6–7 6

q 5 2+ 7–3 7–4 7 Example

13

r

1+ 5– 5+1

b

45 – 80 + 5

2 Simplify each of the following. a

8 + 18 – 32

c

– 12 + 75 – 192

d

7 + 28 – 343

e

24 + 180 + 54

f

12 + 20 – 125

g 2 24 + 3 20 – 7 8

h 3 45 + 2 12 + 5 80 + 3 108

i

6 44 + 4 120 – 99 – 3 270

j

2 32 – 5 45 – 4 180 + 10 8

k

98 + 3 147 – 8 18 + 6 192

l

2 250 + 5 200 – 128 + 4 40

m 5 81 – 4 162 + 6 16 – 450

n

108 + 125 – 3 8 + 9 80

3 multiple choice a

2 + 6 3 – 5 2 – 4 3 is equal to: A –5 2 + 2 3 B –3 2 + 23 E −3 D –4 2 + 2 3

C 6 2+2 3

b 6 – 5 6 + 4 6 – 8 is equal to: A –2 – 6 D –2 – 9 6 c

B 14 – 6 E 14 + 6

C –2 + 6

4 8 – 6 12 – 7 18 + 2 27 is equal to: A –7 5 D –13 2 + 6 3

B 29 2 – 18 3 E cannot be simplified

C –13 2 – 6 3

d 2 20 + 5 24 – 54 + 5 45 is equal to: A 19 5 + 7 6 D –11 5 – 7 6

B 9 5–7 6 E 12 35

C –11 5 + 7 6

4 Elizabeth wants narrow wooden frames for three different-sized photographs, the smallest frame measuring 2 × 2 cm, the second 3 × 3 cm and the largest 4 × 6 cm. If each frame is made up of four pieces of timber to go around the edge of the photograph and one diagonal support, how much timber is needed to make the three frames? Give your answer in simplest surd form.

School 5 Harry and William walk to school each day. If the ground is not wet and boggy they can cut across a vacant block, otherwise they must stay on the paths. a Find the distance that they walk when it is wet and they follow the path.

Vacant block b Find the distance that they walk on a fine day when Home they follow the shortest path across the vacant block.

16 m 24 m Give your answer in simplest surd form. c Exactly how much further do they walk when it is a wet day? d Approximately how much further do they walk when it is a wet day?

7m

20 m ET SHE

Work

WORKED

25

1.2

MQ10 VIC ch 01 Page 26 Tuesday, November 20, 2001 10:49 AM

26

Maths Quest 10 for Victoria

When 3 people fell in the water, why did only 2 of them get their hair wet?

The answer to each question and the letter beside it give the puzzle answer code.

E=6 7– 4 7 = A=3 5+ 5 =

H=3 6–2 6 =

B=5 2+3 2 =

L=7 5–5 5 =

C= 3+ 3 =

M = 7 6 – 54 =

A= 8+3 2 =

N = 45 – 20 =

F = 3 3 + 12 =

4 7

8 2

S= 5+3 5– 3 =

D = 2 + 2 5 +3 2 – 5

T = 8 + 18 – 2 =

E = 200 – 147 =

U = 12 – 32 + 6 2 =

H=5–2 2+ 9 =

E = 50 + 27 – 5 2 =

O = 75 + 4 5 + 12 =

W= 3+ 4+ 5 =

S = 8 5 – 45 – 20 =

A = 48 – 2 3 + 20 =

E = 150 + 2 6 – 96 =

=

D = 18 – 2 2 =

E=2 7+ 7 =

3 6

B = 108 – 5 3 =

E = 10 3 – 4 3 =

D=2 6+3 6 =

3

A=5 2+3 3– 2 =

O = 6 7 – 28 = 2 3

4 2+3 3

2 2+2 3

5 3

4 2

8–2 2

2 3+2 5

2 5

4 2+ 5

2 7

6

3 7

4 5– 3

4 6

7 3+4 5

2+ 3+ 5

10 2 – 7 3

5 2

4 5 5 6

5

6 3

3 5 3 3

2

MQ10 VIC ch 01 Page 27 Tuesday, November 20, 2001 10:49 AM

Chapter 1 Rational and irrational numbers

27

Multiplication and division of surds Surds can be multiplied and divided in the same way as pronumerals are in algebra.

The multiplication rule, when simplifying surds.

a× b =

ab , was used in the form

ab =

a× b

This rule can be extended to: c a × d b = cd ab . a The division rule is ------- = b

a --- . b 36 6 ---------- = --3 9 =2

An example of this is:

36 so ---------- = 9

36 ------ = 9

while

4

=2 36 -----9

All answers should be written in simplest form.

WORKED Example 14 Simplify each of the following. a

3¥ 6

b

7¥ 7

c –4 5 ¥ 7 6

THINK

WRITE

a

a

1

Write the expression and multiply the surds.

2

Simplify if appropriate.

3× 6 = =

18 9× 2

=3 2 b

c

1

Write the expression and multiply the surds.

2

Simplify if appropriate.

(Note that a × a = a , so the answer could have been found in one step.)

1

Write the expression, multiply whole numbers and multiply the surds.

2

Simplify if appropriate.

b

7× 7 =

49

=7

c –4 5 × 7 6 = –4 × 7 × 5 × 6 = – 28 30

When dividing surds, it is easier if both the numerator and denominator are simplified before dividing. If this is done we can then simplify the fraction formed by the rational and irrational parts separately.

MQ10 VIC ch 01 Page 28 Tuesday, November 20, 2001 10:49 AM

28

Maths Quest 10 for Victoria

WORKED Example 15 40 Simplify each of the following. a ---------2

40 b --------2

16 15 c ---------------24 75

THINK

WRITE

a

40 2 10 a ---------- = ------------2 2

b

c

1

Write the expression and simplify the numerator.

2

Write the surds under the one square root sign and divide.

1

Write the expression and simplify the numerator.

2

Divide numerator and denominator by 2, which is the common factor.

1

Write the expression and simplify the denominator. (The numerator is already fully simplified.)

2

Simplify the fraction formed by the rational and irrational parts separately.

10 = 2 -----2 =2 5 40 2 10 b ---------- = ------------2 2 =

10

16 15 16 15 c ---------------- = ---------------24 75 120 3 2 15 = ------ Ă— -----15 3 2 5 = ---------15

A mixed number under a square root sign must be changed to an improper fraction and then simplified.

WORKED Example 16 Simplify

1

3 --2- .

THINK

WRITE

1

Write the expression.

2

Change the mixed number to an improper fraction. Neither the numerator nor the denominator are perfect squares so both the numerator and denominator are written as surds.

3 1--2=

7 --2

7 = ------2

The same algebraic rules apply to surds when expanding brackets. Each term inside the brackets is multiplied by the term immediately outside the brackets.

MQ10 VIC ch 01 Page 29 Tuesday, November 20, 2001 10:49 AM

Chapter 1 Rational and irrational numbers

29

WORKED Example 17 Expand each of the following, simplifying where appropriate.

7(5 – 2)

a

b 5 3( 3 + 2 6)

THINK

WRITE

a

a

b

7(5 – 2)

1

Write the expression.

2

Remove the brackets by multiplying the surd outside the brackets by each term inside the brackets.

1

Write the expression.

2

Remove the brackets by multiplying the term outside the brackets by each term inside the brackets.

= 5 9 + 10 18

3

Simplify as appropriate.

= ( 5 × 3 ) + ( 10 × 9 × 2 )

= 5 7 – 14

b 5 3( 3 + 2 6)

= 15 + ( 10 × 3 × 2 ) = 15 + 30 2 Binomial expansions are completed by multiplying the first term from the first bracket with the entire second bracket, then multiplying the second term from the first bracket by the entire second bracket.

WORKED Example 18

Expand ( 2 + 6 ) ( 2 3 – 6 ) .

THINK 1 Write the expression.

WRITE ( 2 + 6)(2 3 – 6) =

3

Multiply each term in the first bracket by each term in the second bracket.

Remove the brackets.

4

Simplify surds.

= 2 6 – ( 4 × 3) + (2 × 9 × 2) – 6

2

2(2 3 – 6) + 6(2 3 – 6)

= 2 6 – 12 + 2 18 – 36 = 2 6 – 2 3 + (2 × 3 × 2) – 6 = 2 6–2 3+6 2–6

remember remember 1. To multiply and divide surds, use the following rules. (i)

a× b =

ab

(ii) c a × d b = cd ab

a (iii) ------- = b

a --b

2. Leave answers in simplest surd form.

3. To remove a bracket containing surds, multiply each term outside the bracket by each term inside the bracket.

4. To expand two brackets containing surds, multiply each term in the first bracket by each term in the second bracket.

MQ10 VIC ch 01 Page 30 Tuesday, November 20, 2001 10:49 AM

30

Maths Quest 10 for Victoria

Multiplication and division of surds

1F WORKED

Example

14

Mat

d hca

Multiplication and division of surds

1 Simplify each of the following. a 5× 5

b

5× 5

c

– 5× 5

d

5× 7

e

6 × – 11

f

32 × 2

g

25 × – 4

h

30 × 2

i

7× 8

j

12 × 6

k – 90 × – 5

l

3 2×4 2

m –5 5 × 6 5

n 3 10 × 2 8

o 7 3 × – 4 12

p 2 3× 6

q – 10 5 × – 5 125

r

s

8 16 × 10 50

t

7 × 4 49

3 8×6 9

u –2 5 × –3 2 × 6

2 multiple choice a 2 6 × 5 4 × 6 6 is equal to: A 13 12

B 60 12

C 132

D 156

E 720

C 48 3

D – 48 3

E 4 3

C 14 5

D 100 5

E 500

b – 3 8 × – 4 6 is equal to: A – 7 48 c

6 5 + 4 5 × 2 5 is equal to: A 6 5 + 40

WORKED

Example

15

B – 12 48

B 6 5 + 30

3 Simplify each of the following.

6 a ------2

10 b ---------5

c

20 ---------4

32 d ---------16

e

75 ---------5

f

30 ---------10

4 5 g ---------4

i

– 6 10 ---------------3 2

j

18 18 ---------------2 6

k

15 15 m ---------------20 45

n

3 200 ---------------2 2

16 125 o ------------------– 10 5

– 14 49 q ------------------– 10 81

r

5 3×3 3 --------------------------2 2×8 2

s

– 24 6 ---------------6 12

2 5×3 6 -----------------------------4 10 × 2 3

h

4 5 ---------5

l

5 6 ------------10 3

6 p ---------6 6 t

2 2× 5×6 2 ---------------------------------------5 8×2 5

4 multiple choice – 75 a ------------- is equal to: 3 A −5

–5 3 B ------------3

C 5

– 25 3 D ---------------3

E −25

MQ10 VIC ch 01 Page 31 Tuesday, November 20, 2001 10:49 AM

Chapter 1 Rational and irrational numbers

31

10 12 b ---------------- is equal to: 20 2 A 2 6 c

2 B ------6

6 C ------2

D 3

E

1 --3

3 C ---------2 3

4 D ---------2 3

E

1 --4

C 4 3 + 6 10

28 D 4 3 + 3 5 E ------2

6 20 × 4 2 --------------------------------- is equal to: 16 3 × 2 10 4 3 A ---------3

3 3 B ---------4

8 6 + 6 10 d ------------------------------ is equal to: 2 2 4 3 A 6 3 + 4 5 B ------- + ------3 5

16

WORKED

Example

17

WORKED

Example

18

5 Simplify each of the following. a

2 7--9-

b

-----1 13 36

c

2 1--4-

d

1 3 ----16

6 Expand each of the following, simplifying where appropriate. a 3( 2 + 5)

b 5( 6 – 2)

d 8( 2 + 3)

e

g 7(6 + 7)

h

3( 2 + 5)

i

10 ( 2 + 2 )

5( 5 + 2)

l

6( 6 – 5)

4( 7 – 5)

c

6 ( 5 + 11 )

f

2(5 – 2)

j

14 ( 3 – 8 )

k

m

8( 2 + 8)

n 6 5(2 5 – 3)

o 2 7(3 8 + 4 5)

p 3 5 ( 2 20 – 5 5 )

q 5 2(5 2 – 3)

r

b ( 7 + 2)(3 5 – 2) c

d ( 5 + 3)( 5 – 3)

e

g ( 5 – 3 )2

h ( 2 + 3 )2

(2 2 + 5)(3 2 – 5) f i

1.12

4 3(2 2 – 5 3)

7 Expand each of the following. a ( 5 + 3)(2 2 – 3)

HEET

SkillS

Example

( 2 + 3)( 2 – 3) (3 2 + 3)(5 2 – 3) ( 2 6 – 3 2 )2

8 A tray, 24 cm by 28 cm, is used for cooking biscuits. Square biscuits, measuring 4 cm by 4 cm are placed on the tray. a What is the greatest number of biscuits that would fit on the tray if it was not necessary to allow for expansion in the cooking? b If each biscuit had a strip of green mint placed along its diagonal, how much mint would be required for each biscuit? Give an exact answer in simplest surd form. c How many centimetres of mint would be necessary for all the biscuits to be decorated in this way? d If the dimensions of the tray were 12 6 cm and 14 3 cm, find the area of the tray in simplest surd form. e Use approximations for the lengths of the sides of the tray to find how many of the 4 × 4 biscuits would fit on the new tray.

HEET

SkillS

WORKED

1.13

MQ10 VIC ch 01 Page 32 Tuesday, November 20, 2001 10:49 AM

32

Maths Quest 10 for Victoria

9 The material in the front face of the roof of a house has to be replaced. The face is triangular in shape.

GAM

me E ti

Rational and irrational numbers — 002

a If the vertical height is half the width of the base and the slant length is 6 metres, find the exact vertical height of this part of the roof. b Find the exact area of the front face of the roof.

Recurring surds Consider the expression x =

6 + 6 + 6 + 6 + … . We will call this a

recurring surd. Although 6 is irrational, this recurring surd actually has a rational answer. To find it we form a quadratic equation.

1 Find an expression for x2.

2 In your expression for x2, you should be able to find the original expression for x. Substitute the pronumeral x for this expression.

3 You should now be able to form a quadratic equation to solve. You will get two solutions but you need consider only the positive solution.

4 Now use the same method to find the value of x = 5 Evaluate the following recurring surds. a

x =

12 + 12 + 12 + 12 + …

b

x =

20 + 20 + 20 + 20 + …

c

x =

12 – 12 – 12 – 12 – …

6– 6– 6– 6–…

d x = 20 – 20 – 20 – 20 – …

6 Try writing a few recurring surds of your own. Some will not have a rational answer. Can you find the condition for a recurring surd to have a rational answer?

MQ10 VIC ch 01 Page 33 Tuesday, November 20, 2001 10:49 AM

Chapter 1 Rational and irrational numbers

33

2 1 Express 2 1--4- as a finite decimal.

2 Express

5 -----11

as a recurring decimal.

3 Convert 0.63 to a simple fraction.

4 Which of the following is irrational?

5 Calculate

81 ,

99 ,

169

16.44 correct to 2 decimal places.

6 Evaluate

72 × 2 ÷ 36 .

7 Simplify

90 .

8 Simplify 5 2 + 8 + 3 18 .

9 Simplify 4 5 × 40 .

2 6 10 Simplify ---------- .

72

Writing surd fractions with a rational denominator 1 1 ------- is a fraction with a surd in the denominator. If we multiply ------- by 1, its value will 2 2 remain unchanged. If the numerator and the denominator are both multiplied by the same number, the value of the fraction stays the same because we are multiplying by 1.

1 2 2 ------- × ------- = ------2 2 2 The value of the fraction has not changed but the denominator is now rational.

WORKED Example 19 Express each of the following fractions in simplest form with a rational denominator.

1 a ------5

5 2 b ---------4 5

THINK a

1

Write the fraction.

2

Multiply the numerator and the denominator by the surd in the denominator.

WRITE 1 a ------5 1 5 = ------- × ------5 5 5 = ------5

Continued over page

MQ10 VIC ch 01 Page 34 Tuesday, November 20, 2001 10:49 AM

34

Maths Quest 10 for Victoria

THINK

WRITE

b

5 2 b ---------4 5

1

Write the fraction.

2

Multiply the numerator and the denominator by the surd in the denominator and simplify.

5 2 5 = ---------- × ------4 5 5 5 10 = ------------4×5 5 10 = ------------20

3

Simplify by cancelling.

10 = ---------4

If there is a binomial denominator (two terms) such as (3 + 2 ) then the fraction can be written with a rational denominator by multiplying numerator and denominator by the same expression with the opposite sign. That is, ( 3 – 2 ) because: (3 + 2)(3 – 2) = 9 – 3 2 + 3 2 – 2 =9−2 =7 Using the difference of two squares rule removes the surd.

WORKED Example 20 5 Express ---------------- in simplest form with a rational denominator.

2+ 3 THINK

WRITE

1

Write the fraction.

5 ---------------2+ 3

2

Multiply both numerator and denominator by ( 2 – 3 ) .

5 2– 3 = ---------------- × ---------------2+ 3 2– 3 5(2 – 3) = -----------------------------------------(2 + 3)(2 – 3)

3

Expand the denominator.

5(2 – 3) = -----------------------4–3

4

Simplify if applicable.

= 5(2 – 3)

MQ10 VIC ch 01 Page 35 Tuesday, November 20, 2001 10:49 AM

Chapter 1 Rational and irrational numbers

35

remember remember To express fractions in simplest form with a rational denominator: 1. If the fraction has a single surd in the denominator, multiply both numerator and denominator by the surd.

2. If the fraction has an integer multiplied by a surd in the denominator, multiply both numerator and denominator by the surd only.

3. Simplify the denominator before rationalising.

4. If the fraction’s denominator is the sum of 2 terms, multiply numerator and denominator by the difference of the 2 terms.

5. If the fraction’s denominator is the difference of 2 terms, multiply numerator and denominator by the sum of the 2 terms.

1G

Writing surd fractions with a rational denominator

1 Express each of the following fractions in simplest form with a rational denominator.

1 1 1 1 19a a ------b ------c ------d ------3 5 6 7 Rationalising denominators 2 5 3 6 e ---------f ------g ---------h ---------10 5 15 30 2 Express each of the following fractions in simplest form with a rational denominator.

3 5 2 6 a ------b ------c ------d ---------5 6 3 10 8 12 18 3 e ------f ---------g ---------h ------3 7 5 2 5 6 2 3 3 5 5 7 i ---------j ---------k ---------l ---------5 2 6 10 WORKED 3 Express each of the following fractions in simplest form with a rational denominator.

Example 6 5 14 6 4 3 5 2 19b a ---------b ------------c ---------d ------------7 3 3 7 5 2 4 10 WORKED

Example

6 Express each of the following fractions in simplest form with a rational denominator.

5 2 4 6 20 a ---------------b ---------------c ---------------d ---------------2– 3 1+ 2 5+2 3– 7

SkillS

1.14

HEET

1.15

SkillS

5 Find half of each of the following fractions by first expressing each one with a rational denominator.

24 20 a ---------b ---------32 50

HEET

cad

4 Express each of the following fractions in simplest form with a rational denominator.

2 4 3 5 3 a ------b ---------c ---------d ---------8 12 18 20

Math

WORKED

Example

e

3 3 -------------------5– 2

f

2 5 -------------------5+ 3

5 2 g -------------------7– 2

h

6 6 --------------------------3 6–5 2

Work

ET SHE

1.3

MQ10 VIC ch 01 Page 36 Tuesday, November 20, 2001 10:49 AM

36

Maths Quest 10 for Victoria

summary Copy the sentences below. Fill in the gaps by choosing the correct word or expression from the word list that follows.

1

To express a fraction as a finite denominator.

2

To express a fraction as a recurring decimal number, divide the numerator by the denominator and write the decimal number with signs over the recurring decimal pattern.

3

A

4

Finite and

5

To express a recurring decimal number as a fraction, eliminate the repeating decimal digits by multiplying by an appropriate of 10, then subtract the original decimal number and write the remainder as a fraction.

6

Numbers that cannot be expressed as

7

Any roots of numbers that do not have finite answers are called and are irrational.

8

When calculating surds on the calculator, the resultant answer is only an .

9

Some surds can be simplified by dividing the original surd into the product of two other surds, one of which is a square which can be calculated exactly.

, divide the numerator by the

number is one that can be expressed as a fraction. decimal numbers are rational.

10

Surds which do not have a perfect square fied.

11

Only

12

Surds can be

WORD repeater like factor

are irrational.

cannot be simpli-

surds can be added or subtracted. and divided.

LIST decimal number multiple surds

multiplied fractions perfect

rational recurring approximation

MQ10 VIC ch 01 Page 37 Tuesday, November 20, 2001 10:49 AM

Chapter 1 Rational and irrational numbers

37

CHAPTER review 1 Evaluate the following. a

1 --4

+

1 --3

b

−

1 --4

1 --3

c

1 --4

×

1 --3

d

1 --4

÷

1 --3

2 Two-fifths of students at Farnham High catch a bus to school, 3--8- walk to school and the rest come by car or bike. If there are 560 students at the school, how many come by car or bike?

3 Express each of the following as a decimal number, giving exact answers. a

2 -----25

b

13 -----16

c

2 --7

d

1B

5 --9

4 multiple choice a

11 -----14

1B

as a recurring decimal is:

A 0.785 714 285 D 0.785 71 (to 5 d.p.) b 0.30 is equal to: 3 A ----B 1--310

B 0.785 714 2 C 0.785 714 2 E cannot be written as a recurring decimal C

11 -----30

D

3 -----11

5 Convert each of the following to a fraction in simplest form. a 0.8 b 0.8˙ c 0.83 d 0.83˙ 6 Explain why

15 is a surd and

16 is not a surd.

7 Calculate each of the following, rounding the answer to 1 decimal place. a

62

b

72 + 27

c

7– 7 ---------------7+7

6× 5 d -------------------6– 5

c

6 32

d 4 90

8 Simplify each of the following. a

99

b

175

1A 1A

E

10 -----33

e

0.83

1B 1C 1C 1D

MQ10 VIC ch 01 Page 38 Tuesday, November 20, 2001 10:49 AM

38 1D

Maths Quest 10 for Victoria

9 multiple choice 96 written in simplest form is: A 4 6

B 2 24

C 8 12

1D

10 Express each of the following in the form

1E

11 Simplify each of the following.

a 5 6

1E

1F

c

E 12 3

a. d 3 2

11 5

6+3 7–4 7+3 6

a c

b 6 5

D 16 6

b

12 + 243 – 108

5 28 + 2 45 – 4 112 + 3 80

12 multiple choice 27 + 50 – 72 + 300 is equal to: A 30 3 – 30 2

B 13 3 + 11 2

D 13 3 – 2

E

C 13 3 + 2

305

13 Simplify each of the following.

5 × 10

a

– 16 12 d ------------------8 2

b 4 3×6 7

c

13 × 13

35 32 ---------------20 8

f

2 5×6 6 -----------------------------4 3 × 3 12

e

1F

14 Expand and simplify each of the following.

1F

15 multiple choice

b ( 4 3 – 5 )2

a 6 5 ( 2 5 + 3 20 )

15 48 ---------------- written in simplest form is: 20 6 3 8 A ---------4

4 8 B ---------3

3 2 C ---------2

4 2 D ---------3

E 6

16 multiple choice

1G

2 ------- written with a rational denominator in simplest form is: 5 A

1G CHAPTER

test yourself

1

2 --5

2 5 B ---------5

C

5 --2

5 D ------2

5 E ------5

17 Express each of the following fractions in simplest form with a rational denominator.

1 a ---------2 7

5 2 b ---------2 3

c

1 ---------------5+2

6 d --------------------------2 5–3 2