numbers and surds -...

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Arithmetic is the study of numbers and operations on them. This short chapter reviews whole numbers, integers, rational numbers and real numbers, with particular attention to the arithmetic of surds and their approximations. Most of this material will be familiar from earlier years.

Numbers and surds

2

ISBN 978-1-108-46904-3 Photocopying is restricted under law and this material must not be transferred to another party.

© Pender et al. 2019 Cambridge University Press

2A Whole numbers, integers and rationals 33

Whole numbers, integers and rationals

Our ideas about numbers arise from the two quite distinct sources:• The whole numbers, the integers and the rational numbers are developed from counting.• The real numbers are developed from geometry and the number line.

This section very briefly reviews whole numbers, integers and rational numbers, with particular attention to percentages and recurring decimals.

The whole numbersCounting is the first operation in arithmetic. Counting things such as people in a room requires zero (if the room is empty) and then the successive numbers 1, 2, 3, . . ., generating all the whole numbers:

0, 1, 2, 3, 4, 5, 6, …

The number zero is the first number on this list, but there is no last number, because every number is followed by another number, distinct from all previous numbers. The list is therefore called infinite, which means that it never ‘finishes’. The symbol is generally used for the set of whole numbers.

A non-zero whole number can be factored, in one and only one way, into the product of prime numbers, where a prime number is a whole number greater than 1 whose only divisors are itself and 1. The primes form a sequence whose distinctive pattern has confused every mathematician since Greek times:

2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, …

The whole numbers greater than 1 and not prime are called composite numbers:

4, 6, 8, 9, 10, 12, 14, 15, 16, 18, 20, 21, 22, 24, 25, 26, 27, 28, 30, 32, …

The whole numbers 0 and 1 are special cases, being neither prime nor composite.

The integersAny two whole numbers can be added or multiplied, and the result is another whole number. Subtraction, however, requires the negative integers as well:

…, −6, −5, −4, −3, −2, −1

so that calculations such as 5 − 7 = −2 can be completed. The symbol (from German ‘Zahlen’ meaning numbers) is conventionally used for the set of integers.

2A

1 THE SET OF WHOLE NUMBERS

• The whole numbers are 0, 1, 2, 3, 4, 5, 6, … • Every whole number except 0 and 1 is either prime or composite, and every composite number

can be factored into primes in one and only one way. • When whole numbers are added or multiplied, the result is a whole number.

2 THE SET OF INTEGERS

• The integers are …, −5, −4, −3, −2, −1, 0, 1, 2, 3, 4, 5, … • When integers are added, subtracted or multiplied, the result is an integer.



Chapter 2 Numbers and surds34 2A

The rational numbersA problem such as, ‘Divide 7 cakes into 3 equal parts’, leads naturally to fractions, where the whole is ‘fractured’ or ‘broken’ into pieces. Thus we have the system of rational numbers, which are numbers that can be written as the ‘ratio’ of two integers. Here are some examples of rational numbers written as single fractions:

213

= 73 − 1

3= −1

3 30 ÷ 24 = 5

4 3.72 = 372

100 4 = 4

1The symbol for ‘quotient’ is conventionally used for the set of rational numbers.

Operations on the rational numbersAddition, multiplication, subtraction and division (except by 0) can all be carried out within the rational numbers.• Rational numbers are simplified by dividing top and bottom by their HCF (highest common factor).

For example, 21 and 35 have HCF 7, so

2135

= 21 ÷ 735 ÷ 7

= 35

• Rational numbers are added and subtracted using the LCM (lowest common multiple) of their denominators. For example, 6 and 8 have LCM 24, so

16

+ 58

= 1 × 424

+ 5 × 324

= 1924

16

− 58

= 1 × 424

− 5 × 324

= − 1124

• Fractions are multiplied by multiplying the numerators and multiplying the denominators, after first cancelling out any common factors. To divide by a fraction, multiply by its reciprocal.

1021

× 925

= 27

× 35

= 635

821

÷ 34

= 821

× 43

= 3263

The symbol for ‘quotient’ is conventionally used for the set of rational numbers.

Decimal notation — terminating or recurring decimalsDecimal notation extends place value to negative powers of 10. For example:

123.456 = 1 × 102 + 2 × 101 + 3 × 100 + 4 × 10−1 + 5 × 10−2 + 6 × 10−3

Such a number can be written as a fraction 123 4561000

, and so is a rational number.

3 THE SET OF RATIONAL NUMBERS

• The rational numbers are the numbers that can be written as fractions ab

, where a and b are

integers and b ≠ 0.

• Every integer a can be written as a fraction a1

, and so is a rational number.

• When rational numbers are added, subtracted, multiplied and divided (but not by zero), the result is a rational number.




If a rational number can be written as a fraction whose denominator is a power of 10, then it can easily be written as a terminating decimal:

325

= 12100

= 0.12 and 578 350

= 578 + 6100

= 578.06

If a rational number cannot be written with a power of 10 as its denominator, then repeated division will yield an infinite string of digits in its decimal representation. This string will cycle once the same remainder recurs, giving a recurring decimal.

23

= 0.666666666666 … = 0.6. (which has cycle length 1)

637

= 6.428571428571 … = 6.4.28571

. (which has cycle length 6)

4 722

= 4.31818181818 … = 4.31.8. (which has cycle length 2)

Conversely, every recurring decimal can be written as a fraction — such calculations are discussed in Year 12 in the context of geometric series.

PercentagesMany practical situations involving fractions, decimals and ratios are commonly expressed in terms of percentages. The symbol % evolved from the handwritten ‘per centum’, meaning ‘per hundred’ — interpret the symbol as ‘/100’, that is, ‘over 100’.

Many problems are best solved by the unitary method, illustrated below.

Example 1 2A

a A table marked $1400 has been discounted by 30%. How much does it now cost?b A table discounted by 30% now costs $1400. What was the original price?

SOLUTION

a 100% is $1400

÷ 10 10% is $140× 7 70% is $980

so the discounted price is $980.

b 70% is $1400

÷ 7 10% is $200× 10 100% is $2000

so the original price was $2000.

4 PERCENTAGES

• To convert a fraction to a percentage, multiply by 1001

%:

320

= 320

× 1001

% = 15%

• To convert a percentage to a fraction, replace % by × 1100

:

15% = 15 × 1100

= 320



Chapter 2 Numbers and surds36 2A

Exercise 2ANote: Questions 1–11 are non-calculator questions.

1 Write as a fraction in lowest terms:a 30% b 80% c 75% d 5%

2 Write as a decimal:a 60% b 27% c 9% d 16.5%

3 Write as a percentage:

a 14

b 25

c 625

d 1320

4 Write as a percentage:a 0.32 b 0.09 c 0.225 d 1.5

5 Factor into primes:a 35 b 18 c 90 d 220

6 Cancel each fraction down to lowest terms.

a 412

b 810

c 1015

d 2128

e 1640

f 2145

g 2442

h 4554

i 3660

j 5472

7 Express each fraction as a decimal by rewriting it with denominator 10, 100 or 1000.

a 12

b 15

c 35

d 34

e 125

f 720

g 18

h 58

8 Express each terminating decimal as a fraction in lowest terms.a 0.4 b 0.25 c 0.15 d 0.16e 0.78 f 0.005 g 0.375 h 0.264

9 Express each fraction as a recurring decimal by dividing the numerator by the denominator.

a 13

b 23

c 19

d 59

e 311

f 111

g 16

h 56

10 Find the lowest common denominator, then simplify:

a 12

+ 14

b 310

+ 25

c 12

+ 13

d 23

− 25

e 16

+ 19

f 512

− 38

g 710

+ 215

h 225

− 115

FOUNDATION




11 Find the value of:

a 14

× 20 b 23

× 12 c 12

× 15

d 13

× 37

e 25

× 58

f 2 ÷ 13

g 34

÷ 3 h 13

÷ 12

i 112

÷ 38

j 512

÷ 123

DEVELOPMENT

12 a Find 12% of $5.b Find 7.5% of 200 kg.c Increase $6 000 by 30%.d Decrease 11

2 hours by 20%.

13 Express each fraction as a decimal.

a 33250

b 140

c 516

d 2780

e 712

f 1 911

g 215

h 1355

14 a Steve’s council rates increased by 5% this year to $840. What were his council rates last year?b Joanne received a 10% discount on a pair of shoes. If she paid $144, what was the original price?c Marko spent $135 this year at the Easter Show, a 12.5% increase on last year. How much did he

spend last year?

CHALLENGE

15 Express each fraction in lowest terms, without using a calculator.

a 588630

b 4551 001

c 5001 000 000

16 a Use your calculator to find the the recurring decimals for 111

, 211

, 311

, 411

, …, 1011

. Is there a pattern?

b Use your calculator to find the recurring decimals for 17 , 2

7 , 3

7 , 4

7 , 5

7 and 6

7 . Is there a pattern?

17 The numbers you obtain in this question may vary depending on the calculator used.a Use your calculator to express 1

3 as a decimal by entering 1 ÷ 3.

b Subtract 0.33333333 from this, multiply the result by 108, and then take the reciprocal.c Show arithmetically that the final answer in part b is 3. Is the answer on your calculator also equal

to 3? What does this tell you about the way fractions are stored on a calculator?



Chapter 2 Numbers and surds38 2B

Real numbers and approximations

This section introduces the set of real numbers, which are based not on counting, but on geometry — they are the points on the number line. They certainly include all rational numbers, but as we shall see, they also include many more numbers that cannot be written as fractions.

Dealing with real numbers that are not rational requires special symbols, such as √ and π , but when a real number needs to be approximated, a decimal is usually the best approach, written to as many decimal places as is necessary.

Decimals are used routinely in mathematics and science for two good reasons:• Any two decimals can easily be compared with each other.• Any quantity can be approximated ‘as closely as we like’ by a decimal.

Every measurement is only approximate, no matter how good the instrument, and rounding using decimals is a useful way of showing how accurate it is.

Rounding to a certain number of decimal placesThe rules for rounding a decimal are:

For example,

3.8472 ≑ 3.85, correct to two decimal places. (look at 7, the third digit)3.8472 ≑ 3.8, correct to one decimal place. (look at 4, the second digit)

Scientific notation and rounding to a certain number of significant figuresThe very large and very small numbers common in astronomy and atomic physics are easier to comprehend when they are written in scientific notation:

1 234 000 = 1.234 × 106 (there are four significant figures) 0.000065432 = 6.5432 × 10−5 (there are five significant figures)

The digits in the first factor are called the significant figures of the number. It is often more sensible to round a quantity correct to a given number of significant figures rather than to a given number of decimal places.

To round, say to three significant figures, look at the fourth digit. If it is 5, 6, 7, 8 or 9, increase the third digit by 1. Otherwise, leave the third digit alone.

3.0848 × 109 ≑ 3.08 × 109, correct to three significant figures. 2.789654 × 10−29 ≑ 2.790 × 10−29, correct to four significant figures.

The number can be in normal notation and still be rounded this way:

31.203 ≑ 31.20, correct to four significant figures.

2B

5 RULES FOR ROUNDING A DECIMAL

To round a decimal to, say, two decimal places, look at the third digit. • If the third digit is 0, 1, 2, 3 or 4, leave the second digit alone. • If the third digit is 5, 6, 7, 8 or 9, increase the second digit by 1.

Always use ≑ rather than = when a quantity has been rounded or approximated.



2B Real numbers and approximations 39

Unfortunately, this may be ambiguous. For example, when we see a number such as 3200, we do not know whether it has been rounded to 2, 3 or 4 significant figures. That can only be conveyed by changing to scientific notation and writing,

3.2 × 103 or 3.20 × 103 or 3.200 × 103.

There are numbers that are not rationalAt first glance, it would seem reasonable to believe that all the numbers on the number line are rational, because the rational numbers are clearly spread ‘as finely as we like’ along the whole number line. Between 0 and 1 there are 9 rational numbers with denominator 10:

0 1102

103

104

105

106

108

109

107

10 1

Between 0 and 1 there are 99 rational numbers with denominator 100:

0 50100 1

Most points on the number line, however, represent numbers that cannot be written as fractions, and are called irrational numbers. Some of the most important numbers in this course are irrational, such as √2 and π, and the number e that will be introduced in Chapter 9.

The square root of 2 is irrationalThe number √2 is particularly important, because by Pythagoras’ theorem, √2 is the diagonal of a unit square. Here is a proof by contradiction that √2 is an irrational number — regard this proof as extension.

Suppose that √2 were a rational number.

Then √2 could be written as a fraction ab

in lowest terms.

That is, √2 = ab

, where a and b have no common factor.

We know that b > 1 because √2 is not a whole number.

Squaring, 2 = a2

b2 , where b2 > 1 because b > 1.

Because ab

is in lowest terms, a2

b2 is also in lowest terms,

which is impossible, because a2

b2= 2, but b2 > 1.

This is a contradiction, so √2 cannot be a rational number.

The Greek mathematicians were greatly troubled by the existence of irrational numbers. Their concerns can still be seen in modern English, where the word ‘irrational’ means both ‘not a fraction’ and ‘not reasonable’.

1

1

2




The real numbers and the number lineThe whole numbers, the integers, and the rational numbers are based on counting. The existence of irrational numbers, however, means that this approach to arithmetic is inadequate, and a more general idea of number is needed. We have to turn away from counting and make use of geometry.

0 1 2−1−2

At this point, geometry replaces counting as the basis of arithmetic.

An irrational real number cannot be written as a fraction, or as a terminating or recurring decimal. In this

course, such as a number is usually specified in exact form, such as x = √2 or x = π, or as a decimal approximation correct to a certain number of significant figures, such as x ≑ 1.4142 or x ≑ 3.1416. Very occasionally, a fractional approximation is useful or traditional, such as π ≑ 31

7 .

The real numbers are often referred to as the continuum, because the rationals, despite being dense, are scattered along the number line like specks of dust, but do not ‘join up’. For example, the rational multiples of √2, which are all irrational, are just as dense on the number line as the rational numbers. It is only the real line itself that is completely joined up, to be the continuous line of geometry rather than falling apart into an infinitude of discrete points.

Open and closed intervalsAny connected part of the real number line is called an interval.• An interval such as 1

3≤ x ≤ 3 is called a closed interval because it contains

all its endpoints.• An interval such as −1 < x < 5 is called an open interval because it does not

contain any of its endpoints.• An interval such as −2 ≤ x < 3 is neither open nor closed (the word half-closed

is often used).In diagrams, an endpoint is represented by a closed circle • if it is contained in the interval, and by an open circle ◦ if it is not contained in the interval.

Bounded and unbounded intervalsThe three intervals above are bounded because they have two endpoints, which bound the interval. An unbounded interval in contrast is either open or closed, and the direction that continues towards ∞ or −∞ is represented by an arrow.• The unbounded interval x ≥ −5 is a closed interval because it contains all its

endpoints (it only has one).• The unbounded interval x < 2 is an open interval because it does not contain

any of its endpoints (it only has one).• The real line itself is an unbounded interval without any endpoints.

3 x13

−1 5 x

–2 3 x

−5 x

2 x

6 DEFINITION OF THE SET OF REAL NUMBERS

• The real numbers are defined to be all the points on the number line.

• All rational numbers are real, but real numbers such as √2 and π are irrational.




A single point is regarded as a degenerate interval, and is closed. The empty set is sometimes also regarded as a degenerate interval.

An alternative notation using round and square brackets will be introduced in Year 12.

7 INTERVALS

• An interval is a connected part of the number line. • A closed interval such as 1

3≤ x ≤ 3 contains its endpoints.

• An open interval such as −1 < x < 5 does not contain its endpoints. • An interval such as −2 ≤ x < 3 is neither open nor closed. • A bounded interval has two endpoints, which bound the interval. • An unbounded interval such as x ≥ −5 continues to ∞ or to −∞ (or both).

Exercise 2B

1 Classify these real numbers as rational or irrational. Express those that are rational in the form ab

in lowest terms, where a and b are integers.a −3 b 11

2c √3 d √4 e √3 27

f √4 8 g √49 h 0.45 i 12% j 0.333k 0.3

.l 31

7m π n 3.14 o 0

2 Write each number correct to one decimal place.a 0.32 b 5.68 c 12.75d 0.05 e 3.03 f 9.96

3 Write each number correct to two significant figures.a 0.429 b 5.429 c 5.029d 0.0429 e 429 f 4290

4 Use a calculator to find each number correct to three decimal places.

a √10 b √47 c 916

d 3748

e π f π2

5 Use a calculator to find each number correct to three significant figures.

a √58 b √3 133 c 622

d 145 e √4 0.3 f 124−1

6 To how many significant figures is each of these numbers accurate?a 0.04 b 0.40 c 0.404d 0.044 e 4.004 f 400

FOUNDATION




DEVELOPMENT

7 a Classify each interval as open or closed or neither (that is, half-closed).i 0 ≤ x ≤ 7 ii x > 5 iii x ≤ 7iv 5 < x ≤ 15 v x < −1 vi −4 < x < 10vii x ≥ 6 viii −4 ≤ x < −3

b Classify each interval in part a as bounded or unbounded.

8 Write each interval in symbols, then sketch it on a separate number line.a The real numbers greater than −2 and less than 5.b The real numbers greater than or equal to −3 and less than or equal to 0.c The real numbers less than 7.d The real numbers less than or equal to −6.

9 Use a calculator to evaluate each expression correct to three decimal places.

a 67 × 2943

b 67 + 2943

c 6743 × 29

d 6743 + 29

e 67 + 2943 + 71

f 67 + 7143 × 29

10 Use Pythagoras’ theorem to find the length of the unknown side in each triangle, and state whether it is rational or irrational.a

8

6

b

4

5

c 17

15

d

3

2

e 13

45

f

108 117




11 Calculate, correct to four significant figures:

a 10−0.4 b 1240 − 13 × 17

c √6.5 + 8.32.7

d √3 10.57 × 12.83 e 3.5 × 104

2.3 × 105f 20 000 × (1.01)25

g 11.3

√19.5 − 14.7h

3 23

+ 5 14

4 12

+ 6 45

i (87.3 × 104) ÷ (0.629 × 10−8)

j √3 + √3 4

√4 5 + √5 6k

(25)

4

× (34)

5

(67)

2

+ (23)

3l √

36.41 − 19.5723.62 − 11.39

CHALLENGE

12 a Identify the approximation of π that seems to be used in 1 Kings 7:23, ‘He made the Sea of cast metal, circular in shape, measuring ten cubits from rim to rim and five cubits high. It took a line of thirty cubits to measure around it.’

b Many centuries later, in the 3rd century BC, Archimedes used regular polygons with 96 sides to prove that 223/71 < π < 22/7. To how many significant figures is this correct?

c What is the current record for the computation of π?d It is well known that a python has a length of about 3.14159 yards. How many pythons can be lined

up between the wickets of a cricket pitch (22 yards)?

Use a calculator to answer questions 13 and 14. Write each answer in scientific notation.

13 The speed of light is approximately 2.997 925 × 108 m/s.a How many metres are there in a light-year (which is the distance that light travels in a year)? Assume

that there are 36514 days in a year and write your answer in metres, correct to three significant figures.

b The nearest large galaxy is Andromeda, which is estimated to be 2 560 000 light-years away. How far is that in metres, correct to two significant figures?

c The time since the Big Bang is estimated to be 13.8 billion years. How long is that in seconds, correct to three significant figures?

d How far has light travelled since the Big Bang? Give your answer in metres, correct to two significant figures.

14 The mass of a proton is 1.6726 × 10−27 kg and the mass of an electron is 9.1095 × 10−31 kg.a Calculate, correct to four significant figures, the ratio of the mass of a proton to the mass of an electron.b How many protons, correct to one significant figure, are there in 1 kg?

15 Prove that √3 is irrational. (Adapt the given proof that √2 is irrational.)



Chapter 2 Numbers and surds44 2C

Surds and their arithmetic

Numbers such as √2 and √3 occur constantly in this course because they occur in the solutions of quadratic equations and when using Pythagoras’ theorem. The last three sections of this chapter review various methods of dealing with them.

Square roots and positive square rootsThe square of any real number is positive, except that 02 = 0. Hence a negative number cannot have a square root, and the only square root of 0 is 0 itself.

A positive number, however, has two square roots, which are the opposites of each other. For example, the square roots of 9 are 3 and −3.

Note that the well-known symbol √x does not mean ‘the square root of x’. It is defined to mean the positive square root of x (or zero, if x = 0).

For example, √25 = 5, even though 25 has two square roots, −5 and 5. The symbol for the negative square root of 25 is −√25.

Cube rootsCube roots are less complicated. Every number has exactly one cube root, so the symbol √3 x simply means ‘the cube root of x’. For example:

√3 8 = 2 and √3 −8 = −2 and √3 0 = 0.

What is a surd?The word surd is often used to refer to any expression involving a square or higher root. More precisely,

however, surds do not include expressions such as √49 and √3 8, which can be simplified to rational numbers.

The word ‘surd’ is related to ‘absurd’ — surds are irrational.

2C

8 DEFINITION OF THE SYMBOL √x

• For x > 0, √x means the positive square root of x. • For x = 0, √0 = 0. • For x < 0, √x is undefined.

9 SURDS

An expression √n x, where x is a rational number and n ≥ 2 is an integer, is called a surd if it is not itself a rational number.



2C Surds and their arithmetic 45

Example 2 2C

Simplify these expressions involving surds.

a √108 b 5√27 c √216

SOLUTION

a √108 = √36 × 3 = √36 × √3

= 6√3

b 5√27 = 5√9 × 3 = 5 × √9 × √3 = 15√3

c √216 = √4 × √54 = √4 × √9 × √6

= 6√6

Example 3 2C

Simplify the surds in these expressions, then collect like terms.

a √44 + √99 b √72 − √50 + √12

SOLUTION

a √44 + √99 = 2√11 + 3√11 = 5√11

b √72 − √50 + √12 = 6√2 − 5√2 + 2√3 = √2 + 2√3

Simplifying expressions involving surdsHere are some laws from earlier years for simplifying expressions involving square roots. The first pair restate the definition of the square root, and the second pair are easily proven by squaring.

Taking out square divisorsA surd such as √500 is not regarded as being simplified, because 500 is divisible by the square number 100, so √500 can be written as 10√5:

√500 = √100 × 5 = √100 × √5 = 10√5.

10 LAWS CONCERNING SURDS

Let a and b be positive real numbers. Then:

√a2 = a

(√a )2

= a and

√a × √b = √ab

√a√b

= √ab

11 SIMPLIFYING A SURD

• Check the number inside the square root for divisibility by one of the squares

4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144, …

• Continue until the number inside the square root sign has no more square divisors (apart from 1).



Chapter 2 Numbers and surds46 2C

Exercise 2C

1 Write down the value of:a √16 b √36 c √81 d √121e √144 f √400 g √2500 h √10 000

2 Simplify:

a √12 b √18 c √20 d √27e √28 f √40 g √32 h √99i √54 j √200 k √60 l √75m √80 n √98 o √800 p √1000

3 Simplify:

a √3 + √3 b 5√7 − 3√7 c 2√5 − √5d −3√2 + √2 e 4√3 + 3√2 − 2√3 f −5√5 − 2√7 + 6√5

g 7√6 + 5√3 − 4√6 − 7√3 h −6√2 − 4√5 + 3√2 − 2√5 i 3√10 − 8√5 − 7√10 + 10√5

DEVELOPMENT

4 Simplify:

a 3√8 b 5√12 c 2√24 d 4√44

e 3√45 f 6√52 g 2√300 h 2√96

5 Write each expression as a single square root. For example, 3√2 = √9 × √2 = √18.a 2√5 b 5√2 c 8√2 d 6√3

e 5√5 f 4√7 g 2√17 h 7√10

6 Simplify fully:

a √8 + √2 b √12 − √3 c √50 − √18

d √54 + √24 e √45 − √20 f √90 − √40 + √10

g √27 + √75 − √48 h √45 + √80 − √125 i √2 + √32 + √72

CHALLENGE

7 Simplify fully:

a √600 + √300 − √216 b 4√18 + 3√12 − 2√50 c 2√175 − 5√140 − 3√28

8 Find the value of x if:

a √63 − √28 = √x b √80 − √20 = √x c 2√150 − 3√24 = √x

FOUNDATION



2D Further simplification of surds 47

Further simplification of surds

This section deals with the simplification of more complicated surdic expressions. The usual rules of algebra, together with the methods of simplifying surds given in the last section, are all that is needed.

Simplifying products of surdsThe product of two surds is found using the identity

√a × √b = √ab .

It is important to check whether the answer needs further simplification.

Using binomial expansionsAll the usual algebraic methods of expanding binomial products can be applied to surdic expressions.

2D

Example 4 2D

Simplify each product.

a √15 × √5 b 5√6 × 7√10

SOLUTION

a

√15 × √5 = √75 = √25 × 3 = 5√3

b 5√6 × 7√10 = 35√60 = 35√4 × 15 = 35 × 2√15 = 70√15

Example 5 2D

Expand these products and then simplify them.

a (√15 + 2 )(√3 − 3 ) b (√15 − √6 )2

SOLUTION

a (√15 + 2 )(√3 − 3 ) = √15(√3 − 3 ) + 2(√3 − 3 ) = √45 − 3√15 + 2√3 − 6

= 3√5 − 3√15 + 2√3 − 6

b

(√15 − √6)2 = 15 − 2√90 + 6 (using the identity (A − B)2 = A2 − 2AB + B2) = 21 − 2 × 3√10

= 21 − 6√10



Chapter 2 Numbers and surds48 2D

Exercise 2D

1 Simplify:

a (√3 )2 b √2 × √3 c √7 × √7 d √6 × √5e 2 × 3√2 f 2√5 × 5 g 2√3 × 3√5 h 6√2 × 5√7

i (2√3 )2 j (3√7 )2 k 5√2 × 3√2 l 6√10 × 4√10

2 Simplify:

a √15 ÷ √3 b √42 ÷ √6 c 3√5 ÷ 3 d 2√7 ÷ √7

e 3√10 ÷ √5 f 6√33 ÷ 6√11 g 10√14 ÷ 5√2 h 15√35 ÷ 3√7

3 Expand:

a √5(√5 + 1) b √2(√3 − 1) c √3(2 − √3 )d 2√2(√5 − √2 ) e √7(7 − 2√7 ) f √6(3√6 − 2√5 )

DEVELOPMENT

4 Simplify fully:

a √6 × √2 b √5 × √10 c √3 × √15

d √2 × 2√22 e 4√12 × √3 f 3√8 × 2√5

5 Expand and simplify:

a √2(√10 − √2 ) b √6(3 + √3 ) c √5(√15 + 4)d √6(√8 − 2) e 3√3(9 − √21 ) f 3√7(√14 − 2√7 )


a (√3 + 1)(√2 − 1) b (√5 − 2)(√7 + 3) c (√5 + √2 )(√3 + √2 )d (√6 − 1)(√6 − 2) e (√7 − 2)(2√7 + 5) f (3√2 − 1)(√6 − √3)

7 Use the special expansion (a + b)(a − b) = a2 − b2 to expand and simplify:

a (√5 + 1)(√5 − 1) b (3 − √7 )(3 + √7 ) c (√3 + √2 )(√3 − √2 )d (3√2 − √11 )(3√2 + √11 ) e (2√6 + 3)(2√6 − 3) f (7 − 2√5 )(7 + 2√5 )

8 Expand and simplify the following, using the special expansions (a + b)2 = a2 + 2ab + b2 and (a − b)2 = a2 − 2ab + b2.

a (√3 + 1)2 b (√5 − 1)2 c (√3 + √2 )2

d (√7 − √5 )2 e (2√3 − 1)2 f (2√5 + 3)2

g (2√7 + √5 )2 h (3√2 − 2√3 )2 i (3√5 + √10 )2

FOUNDATION



2D Further simplification of surds 49

CHALLENGE

9 Simplify fully:

a √40√10

b √18√50

c 2√6 × √5√10

d 5√7 × √3√28

e √15 × √20√12

f 6√3 × 8√2√32 × √27

10 Use Pythagoras’ theorem to find the hypotenuse of the right-angled triangle in which the lengths of the other two sides are:

a √2 and √7 b √5 and 2√5

c √7 + 1 and √7 − 1 d 2√3 − √6 and 2√3 + √6

11 Simplify by forming the lowest common denominator:

a 1

√3 + 1+ 1

√3 − 1b 3

2√5 − √7− 3

2√5 + √7

12 a Write down the expansion of (a + b)2.

b Use the expansion in part a to square √6 + √11 − √6 − √11.c Hence simplify √6 + √11 − √6 − √11.



Chapter 2 Numbers and surds50 2E

Rationalising the denominator

When dealing with surdic expressions, it is usual to remove any surds from the denominator, a process called rationalising the denominator. There are two cases.

The denominator has a single termIn the first case, the denominator is a surd or a multiple of a surd.

The denominator has two termsThe second case involves a denominator with two terms, one or both of which contain a surd. The method uses the difference of squares identity

(A + B)(A − B) = A2 − B2

to square the unwanted surds and convert them to integers.

2E

13 RATIONALISING A BINOMIAL DENOMINATOR

• In an expression such as 3

5 + √3 , multiply top and bottom by 5 − √3 .

• Then use the difference of squares.

12 RATIONALISING A SINGLE-TERM DENOMINATOR

In an expression such as √72√3

, multiply top and bottom by √3 .

Example 6 2E

Simplify each expression by rationalising the denominator.

a √72√3

b 55

√11

SOLUTION

a √72√3

= √72√3

× √3√3

= √212 × 3

= √216

b 55

√11= 55

√11× √11

√11

= 55√1111

= 5√11



2E Rationalising the denominator 51

Exercise 2E

1 Rewrite each fraction with a rational denominator.

a 1

√3b 1

√7c 3

√5d 5

√2

e √2√3

f √5√7

g 2√11√5

h 3√7√2


a 1

√3 − 1b 1

√7 + 2c 1

3 + √5d 1

4 − √7

e 1

√5 − √2f 1

√10 + √6g 1

2√3 + 1h 1

5 − 3√2

FOUNDATION

Example 7 2E

Rationalise the denominator in each expression.

a 3

5 + √3b 1

2√3 − 3√2

SOLUTION

a 3

5 + √3= 3

5 + √3× 5 − √3

5 − √3

= 15 − 3√325 − 3

= 15 − 3√322

Using the difference of squares:

(5 + √3 )(5 − √3 ) = 52 − (√3 )2

= 25 − 3

b 1

2√3 − 3√2= 1

2√3 − 3√2× 2√3 + 3√2

2√3 + 3√2

= 2√3 + 3√24 × 3 − 9 × 2

= − 2√3 + 3√26

Using the difference of squares:

(2√3 − 3√2 )(2√3 + 3√2 ) = (2√3 )2

− (3√2 )2

= 4 × 3 − 9 × 2



Chapter 2 Numbers and surds52 2E

DEVELOPMENT

3 Simplify each expression by rationalising the denominator.

a 2

√2b 5

√5c 6

√3d 21

√7

e 3

√6f 5

√15g 8

√6h 14

√10


a 1

2√5b 1

3√7c 3

5√2d 2

7√3

e 10

3√2f 9

4√3g √3

2√10h 2√11

5√7


a 3

√5 + 1b 4

2√2 − √3c √7

5 − √7d 3√3

√5 + √3

e 2√72√7 − 5

f √5√10 − √5

g √3 − 1√3 + 1

h √5 + √2√5 − √2

i 3 − √73 + √7

j 3√2 + √53√2 − √5

k √10 − √6√10 + √6

l 7 + 2√117 − 2√11

CHALLENGE

6 Simplify each expression by rationalising the denominator.

a √3 − 12 − √3

b 2√5 − √2√5 + √2

7 Show that each expression is rational by first rationalising the denominators.

a 3

√2+ 3

2 + √2b 1

3 + √6+ 2

√6

c 4

2 + √2+ 1

3 − 2√2d 8

3 − √7− 6

2√7 − 5

8 If x = √5 + 12

, show that 1 + 1x

= x.

9 The expression √6 + 1√3 + √2

can be written in the form a√3 + b√2. Find a and b.

10 a Expand (x +1x)

2

.

b Suppose that x = √7 + √6.

i Show that x + 1x

= 2√7.

ii Use the result in part a to find the value of x2 + 1x2

.



Chapter 2 Review 53

Revi

ew

Chapter review exercise

1 Classify each of these real numbers as rational or irrational. Express those that are rational in the

form ab

, where a and b are integers.

a 7 b −214

c √9 d √10

e √3 15 f √4 16 g −0.16 h π

2 Use a calculator to write each number correct to:

i two decimal places ii two significant figures.

a √17 b √3 102 c 1.167

d 4964

e 7.3−2 f π 5.5

3 Evaluate, correct to three significant figures:

a 7.938.22 − 3.48

b −4.9 × (−5.8 − 8.5) c √4 1.6 × 2.6

d 135

116 + 174e

49

− 27

58

− 310

f √2.4−1.6

g √1.347

2.518 − 1.679h 2.7 × 10

−2

1.7 × 10−5i √

12

+ √3 13√4 14 + √5 15

4 Simplify:

a √24 b √45 c √50

d √500 e 3√18 f 2√40

5 Simplify:

a √5 + √5 b √5 × √5 c (2√7 )2

d 2√5 + √7 − 3√5 e √35 ÷ √5 f 6√55 ÷ 2√11

g √8 × √2 h √10 × √2 i 2√6 × 4√15

Chapter 2 Review

Review activity • Create your own summary of this chapter on paper or in a digital document.

Chapter 2 Multiple-choice quiz • This automatically-marked quiz is accessed in the Interactive Textbook. A printable PDF worksheet

version is also available there.



Revi

ewChapter 2 Numbers and surds54

6 Simplify:

a √27 − √12 b √18 + √32

c 3√2 + 3√8 − √50 d √54 − √20 + √150 − √80

7 Expand:

a √7(3 − √7 ) b √5(2√6 + 3√2 )c √15(√3 − 5) d √3(√6 + 2√3 )


a (√5 + 2)(3 − √5 ) b (2√3 − 1)(3√3 + 5) c (√7 − 3)(2√5 + 4)d (√10 − 3)(√10 + 3) e (2√6 + √11 )(2√6 − √11 ) f (√7 − 2)

2

g (√5 + √2 )2 h (4 − 3√2 )2

9 Write with a rational denominator:

a 1

√5b 3

√2c √3

√11

d 1

5√3e 5

2√7f √2

3√10

10 Write with a rational denominator:

a 1

√5 + √2b 1

3 − √7c 1

2√6 − √3

d √3√3 + 1

e 3

√11 + √5f 3√7

2√5 − √7

11 Rationalise the denominator of each fraction.

a √7 − √2√7 + √2

b 3√3 + 53√3 − 5

12 Find the value of x if √18 + √8 = √x.

13 Simplify 3

√5 − 2+ 2

√5 + 2 by forming the lowest common denominator.

14 Find the values of p and q such that √5√5 − 2

= p + q√5.

15 Show that 2

6 − 3√3− 1

2√3 + 3 is rational by first rationalising each denominator.



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