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3 Measurement, Calculation, and Approximate Numbers

# 3 Measurement, Calculation, and Approximate Numbers

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12

CHAPTER 1 Basic Algebraic Operations

This means that 38.3 - 12.91 -3.582 = 84.482.

Note in the display that the negative sign of -3.58 is smaller and a little higher to

distinguish it from the minus sign for subtraction. Also note the * shown for multiplication; the asterisk is the standard computer symbol for multiplication.

■ Some calculator keys on different models

are labelled differently. For example, on some

models, the EXE key is equivalent to the

ENTER key.

■ Calculator keystrokes will generally not be

shown, except as they appear in the display

screens. They may vary from one model to

another.

Looking back into Section 1.2, we see that the minus sign is used in two different

ways: (1) to indicate subtraction and (2) to designate a negative number. This is clearly

shown on a graphing calculator because there is a key for each purpose. The - key

is used for subtraction, and the ( - ) key is used before a number to make it negative.

We will first use a graphing calculator for the purpose of graphing in Section 3.5.

Before then, we will show some calculational uses of a graphing calculator.

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Most scientific and technical calculations involve numbers that represent a measurement or count of a specific physical quantity. A measurement represents an estimate

of the value of the physical quantity that exists in reality, and is usually accompanied

by an uncertainty or error in that measured value. To report a measurement in a

meaningful way, the units of measurement, which indicate a specific size or magnitude of a physical measurement, have to be expressed. For example, if the length of an

object is measured to be 12.5, it is critical to know if that is measured in centimetres,

metres, feet, or some other unit of length.

The definition and practical use of units of measurement has spawned many different systems of counting and units throughout human history. Many of the ancient systems invented were largely based on dimensions of the human body. Consequently,

measurements varied from place to place, and communication of the measured values

was inconsistent since each unit did not have a universally recognized size. The metric

system, first adopted in France in the late 1700s, incorporated the feature of standardization of units, wherein everyone using the system agreed to a specific size for each

unit. The SI metric system of units (International System of Units) has been agreed

upon by international committees of scientists and engineers and was established in

1960. Most scientific endeavours worldwide employ the SI system of units. It is important for scientists, engineers, and technologists to be able to communicate measurements to each other easily and without confusion.

The SI system consists of seven base units (from which all other units are constructed), supplementary units (used for measuring plane and solid angles), and derived

units (which are formed by multiplication and division of the seven base units).

Each unit measures a specific physical quantity, has a standard symbol, and has a

single spelling when written out in full. (Exception: The United States has different

spellings for deca, metre, and litre, writing them as deka, meter, and liter.)

Fig. 1.9 summarizes some SI physical quantities and common variable symbols,

their unit names and SI unit symbols, and any re-expression of a derived unit in terms

of more fundamental base units.

Among the units for time, for which the standard unit is the second, other units like

minute (min), hour (h), day (d), and year (y or yr) are also acceptable. For angles, divisions such as the degree, minute of arc, and second of arc are also permitted.

The kilogram is the SI unit for mass (not weight). It is different because it also contains an SI prefix kilo, which denotes a power of 103. Please note that weight and mass

are different: Mass is the amount of material in an object (in kg), and weight is the

gravitational force (in N) exerted on that mass. Weight changes with the local strength

of the gravity field, whereas mass remains constant.

Originally the metre was defined as one ten-millionth of the length along the globe

from the North Pole to the equator. Today it is defined as the distance travelled by light in

a vacuum in 1 >299 792 458 s. Similarly, the second was once defined as the fraction

1 >86 400 of the mean solar day. It is now defined as the time required for 9 192 631 770

cycles of the radiation corresponding to the transition between the two lowest energy

states of the cesium-133 atom.

1.3 Measurement, Calculation, and Approximate Numbers

13

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Base Units

Quantity

Length

Quantity

Symbol

s

Unit Name

metre

Unit Symbol

m

Mass

m

kilogram

Time

t

second

s

Electric current

I, i

ampere

A

Thermodynamic

temperature

Amount of substance

T

kelvin

K

n

mole

mol

Luminous intensity

I

candela

cd

Supplementary

Units

Derived Units

Plane angle

Solid angle

Area

q

q

A

Volume

V

sr

m2

Volume

V

litre

Velocity

v

kg

unitless

unitless

L

(1000 L = 1 m3)

Acceleration

a

Force

F

newton

Density

r

Pressure

p

pascal

Pa

E, W

joule

J

Power

P

watt

W

Frequency

f

hertz

Hz

Electric charge

q

coulomb

C

V, E

volt

V

Capacitance

C

F

Inductance

L

henry

H

Resistance

R

ohm

Ω

Heat

Q

joule

J

Temperature

T

degrees Celsius

Electric potential

m3

Energy, work

In Terms of

Other SI Units

m/s

m/s2

N

kg>m3

°C

kg # m>s2

N>m2 = kg> 1m # s2 2

N # m = kg # m2 >s2

J>s = kg # m2 >s3

1/s

A#s

J> 1A # s 2 = kg # m2 > 1A # s3 2

s> Ω = s # A>V

Ω # s = V # s>A

V>A

N # m = kg # m2 >s2

A change of 1°C = 1 K

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When writing units, there are several conventions that one must follow:

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OFXUPOT
TRVBSFNFUSFT
QBTDBMT



Exception: Degrees Celsius.

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/GPS

Isaac Newton, Pa for Blaise Pascal). Exception: The litre symbol is L, which is

not named for a person. It used to be l or l but it was easily confused with the

digit 1 (one) so it was altered. The l symbol still has some international acceptance. Both °C and L were added to the SI system due to their practical

importance.

r &OTVSFUIBUB # symbol appears between units that are multiplied (e.g., kg # m2 >s2

not kgm2 >s2). This will prevent confusion between units and SI prefixes, some of

which use the same symbol (e.g., mm is millimetres, but m # m is metres squared).

14

CHAPTER 1 Basic Algebraic Operations

r %POPUVTFNPSFUIBOPOFEJWJTJPOTZNCPMJOBTJOHMFVOJUFH
VTFN>s2 not

m>s>s).

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BOEQIZTJDBMRVBOUJUZWBSJBCMFTZNCPMT

are italicized (e.g., V is the quantity of electrical potential, and V is the unit volts).

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LH
OPULHT

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r 4QBDFTNBZCFVTFEUPTFQBSBUFUIPVTBOETUPBWPJEDPOGVTJPOXJUINBOZJOUFSOBtional practices of different interpretations of commas (e.g., 10 585 is acceptable, while 10,585 means 10.585 in some countries).

SI PREFIXES

In science, it is common to deal with measurements that consist of very large numbers,

or very small numbers. In order to avoid the problem of having to write many zeros in

a decimal (whether trailing or leading zeros), one can utilize some common unit prefixes allowing for a quick way to write a specific multiple of 10 applied to the unit.

These prefixes have specific names and symbols, just like units, but are written preceding the unit, as a normal prefix. There can never be more than one prefix for a single

unit. Scientific and engineering notations, which are used to report very large or very

small measurements using these prefixes, will be discussed in Section 1.5.

Fig. 1.10 SI Prefixes

Multiple of 10

1012, or trillion

109, or billion

106, or million

103, or thousand

102, or hundred

101, or ten

10 –1, or one tenth

10 –2, or one hundredth

10 –3, or one thousandth

10 –6, or one millionth

10 –9, or one billionth

10 –12, or one trillionth

Prefix

tera

giga

mega

kilo

hecto

deca

deci

centi

milli

micro

nano

pico

Symbol

T

G

M

k

h

da

d

c

m

m

n

p

E X A M P L E 2 6TJOH4*QSFGJYFT

(a) Using SI prefixes, we can rewrite the following measurements:

123 000 000 s

= 123 × 106 s

= 123 Ms, or 0.123 Gs

0.000 005 0 m

= 5.0 × 10 –6 m

= 5.0 mm

3

85 300 Ω

= 85.3 × 10 Ω

= 85.3 kΩ

(b) We also use the definitions of the SI prefixes to give the name and meaning of the

units corresponding to the following symbols:

ks

= kiloseconds

= 1000 s

mC

= millicoulombs

= 0.001 C

GHz

= gigahertz

= 1 000 000 Hz

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When using measurements, it may be necessary to convert from one set of units to

another. To change a given set of units into another set of units, we perform mathematical

operations with the units in the same manner as we do with any other algebraic symbol.

1.3 Measurement, Calculation, and Approximate Numbers

15

This process is more fully discussed in Sections 1.7 to 1.12, but it is important to discuss

the principle here, since measurements and units have a fundamental role in most subsequent applied problems.

To convert a set of units, you multiply the measurement by a fraction equal to one, where

the fraction represents the equivalency ratio between the two units. You put the units you

want to eliminate on the opposite side of the fraction of the converting ratio from where they

are in the original measurement when you multiply. By multiplying by a fraction equal to

one, the measurement is not changing. To convert multiple units at the same time, just use

more than one conversion fraction multiplication. This is illustrated in Example 3.

E X A M P L E 3 Converting units

Convert the following units.

(a) 1350 m into km:

1 km

b = 1.35 km

1350 m * a

1000 m

(b) 25.2 kg into g:

1000 g

b = 25 200 g

1 kg

(c) 72.0 km>h into m>s:

25.2 kg * a

km

1000 m

1h

b * a

b = 20.0 m>s

* a

h

1 km

3600 s

(d) 8.75 g>cm3 into kg>m3:

72.0

1 kg

100 cm 3

b

a

b = 8750 kg>m3

*

1000 g

1m

cm3

(e) 62.8 kPa into N>cm2:

8.75

g

* a

1000 N/m2

1m 2

b * a

b = 6.28 N>cm2

1 kPa

100 cm

(f) 32 500 ft to km (1 ft = 0.3048 m):

62.8 kPa * a

32 500 ft * a

0.3048 m

1 km

b * a

b = 9.91 km

1 ft

1000 m

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Most numbers in technical and scientific work are approximate numbers, having

been determined by some measurement. Certain other numbers are exact numbers,

having been determined by a definition or counting process.

E X A M P L E 4 Approximate numbers and exact numbers

If a voltage on a voltmeter is read as 116 V, the 116 is approximate. Another voltmeter

might show the voltage as 115.7 V. However, the voltage cannot be determined exactly.

If a computer prints out the number of names on a list of 97, this 97 is exact. We

know it is not 96 or 98. Since 97 was found from precise counting, it is exact.

By definition, 60 s = 1 min, and the 60 and the 1 are exact.

Significant digits are digits in a measurement or result that you can confidently estimate. That is to say, those digits that are not swamped by the error or uncertainty in the

measurement are significant. The accuracy of a measurement refers to the number of

significant digits it has.

The measurements 5.00 m and 5.000 m may not seem to be very different, but to a

scientist, an engineer, or a technologist, they are not the same thing. The first measurement has been measured to the nearest centimetre and the second measurement to the

nearest millimetre. The precision of a measurement is defined as the last decimal place

16

CHAPTER 1 Basic Algebraic Operations

to which the measurement is expressed, or the decimal place corresponding to the last

measured (significant) digit in the measurement. For instance, 5.00 m has precision

0.01 m = 1 cm, and 5.000 m has precision 0.001 m = 1 mm. Therefore, the second

measurement is more precise (it has a smaller precision). The concept of precision is

important when finding the proper significant digits in a calculated result.

To find the number of significant digits in a single measurement, you start counting

at the first nonzero digit, and finish counting once the precision of the measurement is

reached. Some rules to remember are:

r "MMOPO[FSPEJHJUTare significant

r ;FSPTCFUXFFOOPO[FSPEJHJUTare significant

r ;FSPTUPUIFMFGUPGUIFGJSTUOPO[FSPEJHJUBSFnot significant

r 5SBJMJOH[FSPTBGUFSBEFDJNBMare significant

r 5SBJMJOH[FSPTCFGPSFBEFDJNBMBSFHFOFSBMMZnot significant

Measurement

# Significant Digits

23.0 m

55.001 cm

0.0034 m

125 000 s

3

5

2

3

0.0120 s

120 cm

3

2

In this case, you don’t know the precision—is it

1000 s, 100 s, 10 s, or 1 s? It is ambiguous, so

you must assume the worst, i.e., nearest 1000 s

ambiguous again

E X A M P L E 5 Accuracy and precision

(a) Suppose that an electric current is measured to be 0.31 A on one ammeter and

0.312 A on another ammeter. The measurement 0.312 A is measured to the nearest

thousandth ampere, so it is more precise than 0.31 A, which is measured to the

nearest hundredth ampere. 0.312 A is also more accurate, since it contains three

significant digits, whereas 0.31 A contains only two.

(b) If a concrete driveway is measured to be 135 m long and 0.1 m thick, the measurement

0.1 m (measured to the nearest tenth metre) is more precise than the measurement 135 m

(measured to the nearest metre). On the other hand, 135 m is more accurate, since it contains three significant digits, whereas 0.1 m contains only one.

E X A M P L E 6 Significant digits

■ To show that zeros at the end of a whole

number are significant, a notation that can be

used is to place a bar over the last significant

zero. Using this notation, 78 000 is shown to

have four significant digits.

COMMON ERROR

All numbers in this example are assumed to be approximate.

34.7 has three significant digits.

0.039 has two significant digits. The zeros properly locate the decimal point.

706.1 has four significant digits. The zero is not used for the location of the decimal

point. It shows the number of tens in 706.1.

5.90 has three significant digits.

1400 has two significant digits, unless information is known about the number that

makes either or both zeros significant. (A temperature shown as 1400°C has two significant digits. If a price list gives all costs in dollars, a price shown as \$1400 has four

significant digits.) Without such information, we assume that the zeros are placeholders for proper location of the decimal point.

Other approximate numbers with the number of significant digits are 0.0005 (one),

960 000 (two), 0.0709 (three), 1.070 (four), and 700.00 (five).

Do not write trailing zeros if they are not significant. The measurement 15 m is different

from 15.0 m because the precision is different.

1.3 Measurement, Calculation, and Approximate Numbers

17

From Example 6, we see that all nonzero digits are significant. Also, zeros not used

as placeholders (for location of the decimal point) are significant.

The last significant digit of an approximate number is not exact. It has usually been

determined by estimating or rounding off. However, it is not off by more than one-half

of a unit in its place value.

E X A M P L E 7 .FBOJOHPGUIFMBTUEJHJUPGBOBQQSPYJNBUFOVNCFS

When we write the voltage in Example 4 as 115.7 V, we are saying that the voltage is

more than 115.65 V and less than 115.75 V. Any such value, rounded off to tenths,

would be expressed as 115.7 V.

In changing the fraction 23 to the approximate decimal value 0.667, we are saying

that the value is strictly between 0.6665 and 0.6675.

■ On graphing calculators, it is possible to set

the number of decimal places (to the right of

the decimal point) to which results will be

rounded off. Note that calculators round

half up.

The method of unbiased rounding (also known as round half to even) for rounding

off any measurement to a specific precision, or a number to a specified number of significant digits, consists of three simple rules. Locate the last significant digit (the digit

to be rounded). Then:

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PSBGPMMPXFECZPOFPSNPSFOPO[FSPEJHJUT


round up (increase the rounded digit by one, discard the rest);

r JGUIFOFYUEJHJUJTPSMFTT
round down (leave the rounded digit as is, discard the rest);

r JGUIFOFYUEJHJUJTBGPMMPXFECZOPNPSFEJHJUTPUIFSUIBO
round to the

nearest even (make the rounded digit the nearest even number and discard

the rest).

This last rule ensures proper statistical treatment of all the measurements falling

in this category, as half will round up, and half will round down. This technique

will not statistically bias your measurements to be consistently larger upon

rounding.

We will use unbiased rounding throughout the text. However, there are many different rules that can be followed when rounding. For example, in the common method of

round half up, if the first discarded digit is 5, then the number is always rounded up. It

can be seen that the two methods are identical except for their treatment of those numbers where the digit following the rounding digit is a five and has no nonzero digits

after it.

E X A M P L E 8 Rounding off

Practice Exercises

Round off each number to three significant

digits.

1. 2015 2. 0.3004

70 360 rounded off to three significant digits is 70 400. Here, 3 is the third significant

digit, and the next digit is 6. Since 6 7 5, we add 1 to 3 and the result, 4, becomes the

third significant digit of the approximation. The 6 is then replaced with a zero in order

to keep the decimal point in the proper position.

70 430 rounded off to three significant digits, or to the nearest hundred, is 70 400.

Here the 3 is replaced with a zero.

187.35 rounded off to four significant digits, or to tenths, is 187.4, because 4 is the

nearest even to 3.5.

187.349 rounded off to four significant digits is 187.3. We do not round up the 4 and

then round up the 3.

35.003 rounded off to four significant digits is 35.00. We do not discard the

zeros since they are significant and are not used only to properly place the decimal

point.

187.45 rounded off to four significant digits is 187.4 since 4 is the nearest even

to 4.5.

18

CHAPTER 1 Basic Algebraic Operations

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COMMON ERROR

When performing operations on approximate numbers or measurements, we must not

express the result to an accuracy or precision that is not valid. Measurement uncertainty

restricts how many significant digits can exist in a calculated result.

Consider the following examples.

E X A M P L E 9 Application of precision

16.3 m

0.927 m

17.227 m

smallest values

16.25 m

0.9265 m

17.1765 m

largest values

16.35 m

0.9275 m

17.2775 m

A pipe is made in two sections. One is measured as 16.3 m long and the other as

0.927 m long. What is the total length of the two sections together?

It may appear that we simply add the numbers as shown at the left. However, both

numbers are approximate, and adding the smallest possible values and the largest possible values, the result differs by 0.1 (17.2 and 17.3) when rounded off to tenths.

Rounded off to hundredths (17.18 and 17.28), they do not agree at all since the tenths

digit is different. Thus, we get a good approximation for the total length if it is rounded

off to tenths, the precision of the least precise length, and it is written as 17.2 m.

E X A M P L E 1 0 Application of accuracy

207.54 m

2

16 900 m

0.005 m

Fig. 1.11

0.05 m

81.4 m

We find the area of the rectangular piece of land in Fig. 1.11 by multiplying the length,

207.54 m, by the width, 81.4 m. Using a calculator, we find that 1207.542

181.42 = 16 893.756. This apparently means the area is 16 893.756 m2.

However, the area should not be expressed with this accuracy. Since the length and

width are both approximate, we have

1207.535 m 2 181.35 m 2 = 16 882.972 25 m2

1207.545 m 2 181.45 m 2 = 16 904.540 25 m

2

least possible area

greatest possible area

These values agree when rounded off to three significant digits (16 900 m2) but do not

agree when rounded off to a greater accuracy. Thus, we conclude that the result is accurate only to three significant digits, the accuracy of the least accurate measurement, and

that the area is written as 16 900 m2.

Following are the rules used in expressing the result when we perform basic operations on approximate numbers. They are based on reasoning similar to that shown in

Examples 9 and 10.

Operations with Approximate Numbers

1. When approximate numbers are added or subtracted, the result is expressed

with the precision of the least precise number.

2. When approximate numbers are multiplied or divided, the result is expressed

with the accuracy of the least accurate number.

3. When the root of an approximate number is found, the result is expressed

with the accuracy of the number.

4. When approximate numbers and exact numbers are involved, the accuracy of

the result is limited only by the approximate numbers.

LEARNING TIP

Always express the result of a calculation with the proper accuracy or precision.

When using a calculator, if additional digits are displayed, round off the final result

(do not round off in any of the intermediate steps).

1.3 Measurement, Calculation, and Approximate Numbers

19

E X A M P L E 1 1 Adding approximate numbers

■ When rounding off a number, it may seem

difficult to discard the extra digits. However, if

you keep those digits, you show a number with

too great an accuracy, and it is incorrect to

do so.

Find the sum of the approximate numbers 73.2, 8.0627, and 93.57.

Showing the addition in the standard way and using a calculator, we have

73.2

least precise number (expressed to tenths)

8.0627

93.57

174.8327

final display must be rounded to tenths

Therefore, the sum of these approximate numbers is 174.8.

E X A M P L E 1 2 Combined operations

Practice Exercises

Evaluate using a calculator.

3275

(Numbers are

3. 40.5 +

- 60.041 approximate.)

In finding the product of the approximate numbers 2.4832 and 30.5 on a calculator, the

final display shows 75.7376. However, since 30.5 has only three significant digits, the

product is 75.7.

In Example 1, we calculated that 38.3 - 12.91 -3.582 = 84.482. We know that

38.3 - 12.91 -3.582 = 38.3 + 46.182 = 84.482. If these numbers are approximate,

we must round off the result to tenths, which means the sum is 84.5. We see that when

there is a combination of operations, we must examine the individual steps of the calculation and determine how many significant digits can carry through to the final result. ■

E X A M P L E 1 3 Operations with exact numbers and approximate numbers

LEARNING TIP

A note regarding the equal sign ( = )

is in order. We will use it for its

defined meaning of “equals exactly”

and when the result is an approximate number that has been properly

rounded off. Although 127.8 ≈ 5.27,

where ≈ means “equals approximately,” we write 127.8 = 5.27, since

5.27 has been properly rounded off.

Using the exact number 600 and the approximate number 2.7, we express the result to

tenths if the numbers are added or subtracted. If they are multiplied or divided, we

express the result to two significant digits. Since 600 is exact, the accuracy of the result

depends only on the approximate number 2.7.

600 + 2.7 = 602.7

600 * 2.7 = 1600

600 - 2.7 = 597.3

600 , 2.7 = 220

You should make a rough estimate of the result when using a calculator. An estimation may prevent accepting an incorrect result after using an incorrect calculator

sequence, particularly if the calculator result is far from the estimated value.

E X A M P L E 1 4 Estimating results

In Example 1, we found that

38.3 - 12.91 -3.582 = 84.482

using exact numbers

When using the calculator, if we forgot to make 3.58 negative, the display would be

-7.882, or if we incorrectly entered 38.3 as 83.3, the display would be 129.482.

However, if we estimate the result as

40 - 101 -42 = 80

we know that a result of -7.882 or 129.482 cannot be correct.

When estimating, we can often use one-significant-digit approximations. If the calculator result is far from the estimate, we should do the calculation again.

E XE R C ISE S 1 .3

In Exercises 1–4, make the given changes in the indicated examples of

this section, and then solve the given problems.

3. In the first paragraph of Example 12, change 2.4832 to 2.483 and

then find the result.

1. In Example 6, change 0.039 (the second number discussed) to

0.390. Is there any change in the conclusion?

4. In Example 14, change 12.9 to 21.9 and then find the estimated

value.

2. In the next-to-last paragraph of Example 8, change 35.003 to

35.303 and then find the result.

20

CHAPTER 1 Basic Algebraic Operations

In Exercises 5–8, give the symbol and the meaning for the given unit.

5. megahertz

6. kilowatt

7. millimetre

8. picosecond

In Exercises 9–12, give the name and the meaning for the units whose

symbols are given.

9. kV

10. GΩ

11. mA

12. pF

43. A typical electric current density in a wire is 1.2 * 106 A>m2.

Express this in milliamperes per square centimetre.

44. A certain car travels 24 km on 2.0 L of gas. Express the fuel consumption in litres per 100 kilometres.

In Exercises 45–48, determine whether the given numbers are

approximate or exact.

In Exercises 13–44, make the indicated changes in units.

45. A car with 8 cylinders travels at 55 km>h.

13. 1 km to centimetres.

46. A computer chip 0.002 mm thick is priced at \$7.50.

14. 1 kg to milligrams.

47. In 24 h there are 1440 min.

15. 20 s to megaseconds.

48. A calculator has 50 keys, and its battery lasted for 50 h of use.

16. 800 Pa to kilopascals.

17. 250 mm2 to square metres.

In Exercises 49–54, determine the number of significant digits in each

of the given approximate numbers.

18. 1.75 m2 to square centimetres.

49. 107; 3004

50. 3600; 730

51. 6.80; 6.08

52. 0.8735; 0.0075

53. 3000; 3000.1

54. 1.00; 0.01

19. 80.0 m3 to L.

20. 0.125 L to millilitres.

21. 45.0 m>s to centimetres per second.

22. 1.32 km>h to metres per second.

23. 9.80 m>s2 to centimetres per minute squared.

24. 5.10 g>cm3 to kilograms per cubic metre.

25. 25 h to milliseconds.

26. 5.25 mV to watts per ampere.

In Exercises 55–60, determine which of the pair of approximate

numbers is (a) more precise and (b) more accurate.

55. 30.8; 0.01

56. 0.041; 7.673

57. 0.1; 78.0

58. 7040; 0.004

59. 7000; 0.004

60. 50.060; 8.914

In Exercises 61–68, round off the given approximate numbers (a) to

three significant digits and (b) to two significant digits.

61. 4.936

62. 80.53

63. 50 893

64. 7.005

28. Determine how many metres light travels in one year.

65. 9549

66. 30.96

67. 0.9445

68. 0.9999

29. Determine the speed (in km>h) of the earth moving around the

sun. Assume it is a circular path of radius 150 000 000 km.

In Exercises 69–76, assume that all numbers are approximate. (a)

Estimate the result and (b) perform the indicated operations on a

calculator and compare with the estimate.

27. 15.0 mF to millicoulombs per volt.

30. At sea level, atmospheric pressure is about 101 300 Pa. How many

kilopascals is this?

69. 12.78 + 1.0495 - 1.633

70. 3.64117.062

31. A car’s gasoline tank holds 56 L. What is this capacity in cubic

centimetres?

71. 0.0350 -

32. A hockey puck has a mass of about 0.160 kg. What is its mass in

milligrams?

73.

33. The velocity of some seismic waves is 6800 m>s. What is this

velocity in kilometres per hour?

75.

34. The memory of a 1985 computer was 64 kB (B is the symbol for

byte), and the memory of a 2012 computer is 1.50 TB. How many

times greater is the memory of the 2012 computer?

In Exercises 77–80, perform the indicated operations. The first

number is approximate, and the second number is exact.

35. The recorded surface area of a DVD is 112 cm2. What is this area

in square metres?

0.0450

1.909

72.

0.3275

1.096 * 0.500 85

23.962 * 0.015 37

10.965 - 8.249

74.

0.693 78 + 0.049 97

257.4 * 3.216

3872

2.056 * 309.6

503.1

395.2

76.

1.00

3.6957

+

0.5926

2.935 - 1.054

77. 0.9788 + 14.9

78. 17.311 - 22.98

36. A solar panel can generate 0.024 MW # h each day. Convert this

to joules.

79. 3.1421652

80. 8.62 , 1728

37. The density of water is 1000 kg>m3. Change this to grams per litre.

81. The manual for a heart monitor lists the frequency of the ultrasound wave as 2.75 MHz. What are the least possible and the

greatest possible frequencies?

38. Water flows from a kitchen faucet at the rate of 8500 mL>min.

What is this rate in cubic metres per second?

39. The speed of sound is about 332 m>s. Change this speed to kilometres per hour.

40. Fifteen grams of a medication are to be dissolved in 0.060 L of

water. Express this concentration in milligrams per decilitre.

41. The earth’s surface receives energy from the sun at the rate of

1.35 kW>m2. Reduce this to joules per second per square

centimetre.

42. The moon travels about 2 400 000 km in about 28 d in one rotation about the earth. Express its velocity in metres per second.

In Exercises 81–84, answer the given questions.

82. A car manufacturer states that the engine displacement for a certain model is 2400 cm3. What should be the least possible and

greatest possible displacements?

83. A flash of lightning struck a tower 5.23 km from a person. The thunder was heard 15 s later. The person calculated the speed of sound

and reported it as 348.7 m>s. What is wrong with this conclusion?

84. A technician records 4.4 s as the time for a robot arm to swing

from the extreme left to the extreme right, 2.72 s as the time for

the return swing, and 1.68 s as the difference in these times. What

is wrong with this conclusion?

1.4 Exponents

In Exercises 85–100, perform the calculations on a calculator.

85. Evaluate: (a) 2.2 + 3.8 * 4.5

(b) 12.2 + 3.82 * 4.5

86. Evaluate: (a) 6.03 , 2.25 + 1.77

(b) 6.03 , 12.25 + 1.772

87. Evaluate: (a) 2 + 0 (b) 2 - 0 (c) 0 - 2 (d) 2 * 0 (e) 2 , 0

Compare with operations with zero in Section 1.2.

88. Evaluate: (a) 2 , 0.0001 and 2 , 0 (b) 0.0001 , 0.0001 and

0 , 0 (c) Explain why the displays differ.

89. Enter a positive integer x (five or six digits is suggested) and then

rearrange the same digits to form another integer y. Evaluate

1x - y2 , 9. What type of number is the result?

90. Enter the digits in the order 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, using

between them any of the operations 1 +, - , * , , 2 that will

lead to a result of 100.

91. Show that p is not equal exactly to (a) 3.1416, or (b) 22>7.

92. At some point in the decimal equivalent of a rational number,

some sequence of digits will start repeating endlessly. An irrational number never has an endlessly repeating sequence of digits. Find the decimal equivalents of (a) 8>33 and (b) p. Note the

repetition for 8>33 and that no such repetition occurs for p.

93. Following Exercise 92, show that the decimal equivalents of the

following fractions indicate they are rational: (a) 1>3 (b) 5>11 (c)

2>5. What is the repeating part of the decimal in (c)?

21

95. In 3 successive days, a home solar system produced 32.4 MJ,

26.704 MJ, and 36.23 MJ of energy. What was the total energy

produced in these 3 days?

96. Two jets flew at 938 km>h and 1450 km>h, respectively. How

much faster was the second jet?

97. If 1 K of computer memory has 1024 bytes, how many bytes are

there in 256 K of memory? (All numbers are exact.)

98. Find the voltage in a certain electric circuit by multiplying the

sum of the resistances 15.2 Ω, 5.64 Ω, and 101.23 Ω by the current 3.55 A.

99. The percent of alcohol in a certain car engine coolant is found by

100140.63 + 52.962

performing the calculation

. Find this

105.30 + 52.96

percent of alcohol. The number 100 is exact.

100. The tension (in N) in a pulley cable lifting a certain crate was

50.4519.802

found by calculating the value of

, where the

1

+ 100.9 , 23

1 is exact. Calculate the tension.

1. 2020

2. 0.300

3. -14.0

94. Following Exercise 92, show that the decimal equivalent of the

fraction 124>990 indicates that it is rational. Why is the last digit

different?

1.4

Exponents

1PTJUJWF*OUFHFS&YQPOFOUT t ;FSPBOE

/FHBUJWF&YQPOFOUT t 0SEFSPG

0QFSBUJPOT t &WBMVBUJOH"MHFCSBJD

Expressions

In mathematics and its applications, we often have a number multiplied by itself several times. To show this type of product, we use the notation an, where a is the number

and n is the number of times it appears. In the expression an, the number a is called the

base, and n is called the exponent; in words, an is read as “the nth power of a.”

E X A M P L E 1 .FBOJOHPGFYQPOFOUT

(a) 4 * 4 * 4 * 4 * 4 = 45

the fifth power of 4

(b) 1 -22 1 -22 1 -22 1 -22 = 1 -22 4

(c) a * a = a2

1 1 1

1 3

(d) a b a b a b = a b

5 5 5

5

the fourth power of - 2

the second power of a, called “a squared”

the third power of 15 , called “15 cubed”

We now state the basic operations with exponents using positive integers as exponents. Therefore, with m and n as positive integers, we have the following operations:

■ Two forms are shown for Eq. (1.4) in order

that the resulting exponent is a positive integer.

We consider negative and zero exponents after

the next three examples.

am * an = am + n

am

= am - n (m 7 n, a ≠ 0)

an

1am 2 n = amn

a n

an

1ab2 n = anbn

a b = n

b

b

(1.3)

am

1

n = n-m

a

a

(m 6 n, a ≠ 0)

(1.4)

(1.5)

(b ≠ 0)

(1.6)

22

CHAPTER 1 Basic Algebraic Operations

E X A M P L E 2 Illustrating Eqs. (1.3) and (1.4)

Using Eq. (1.3):

■ In a3, which equals a * a * a, each a is

called a factor. A more general definition of

factor is given in Section 1.7.

Using the meaning of exponents:

8 factors of a

(3 factors of a)(5 factors of a)

a3 * a5 = a3 + 5 = a8

Using first form Eq. (1.4):

■ Here we are using the fact that a (not zero)

divided by itself equals 1, or a>a = 1.

a3 * a5 = 1a * a * a2 1a * a * a * a * a2 = a8

Using the meaning of exponents:

5 7 3

1

5

Using second form Eq. (1.4):

a3

a5

=

1

=

a5 - 3

1

1

1

1

a * a * a * a * a

a5

=

= a2

3

a * a * a

a

a

= a5 - 3 = a2

a3

1

Using the meaning of exponents:

1

a3

a2

a5

=

1

1

1

1

1

a * a * a

1

= 2

a * a * a * a * a

a

1

5 7 3

E X A M P L E 3 Illustrating Eqs. (1.5) and (1.6)

Using Eq. (1.5):

Using the meaning of exponents:

multiply exponents

1a5 2 3 = 1a5 2 1a5 2 1a5 2 = a5 + 5 + 5 = a15

1a5 2 3 = a5132 = a15

Using first form Eq. (1.6):

LEARNING TIP

When an expression involves a product or a quotient of different bases,

only exponents of the same base

may be combined.

Using the meaning of exponents:

1ab2 3 = a3b3

1ab2 3 = 1ab2 1ab2 1ab2 = a3b3

Using second form Eq. (1.6):

Using the meaning of exponents:

a 3

a3

a b = 3

b

b

a 3

a a a

a3

a b = a ba ba b = 3

b

b b b

b

E X A M P L E 4 Other illustrations of exponents

Practice Exercises

Use Eqs. (1.3)–(1.6) to simplify the given

expressions.

1. ax3 1 - ax2 2

2.

12c2 5

13cd2 2

(a) 1 -x2 2 3 = 3 1 -12x2 4 3 = 1 -12 3 1x2 2 3 = -x6

exponent

of 1

(b) ax2 1ax2 3 = ax2 1a3x3 2 = a4x5

(c)

(d)

COMMON ERROR

13 * 22

13 * 52

1ry3 2 2

2 4

r 1y 2

4

3

=

4 4

=

4

32

3 * 2

=

3 3

35

53

r 2y 6

ry8

=

r

y2

Note that ax 2 means a times the square of x and does not mean a 2x 2, whereas 1ax 2 3

does mean a3x 3.

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