Trigonometry
Scientific
Notation and
Significant Digits
Copyright
This publication © The Northern
Alberta Institute of Technology
2002. All Rights Reserved.
LAST REVISED June, 2007
Scientific Notation and Significant Digits:
Statement of Prerequisite Skills
Complete all previous TLM modules before completing this module.
Required Supporting Materials
Access to the World Wide Web.
Internet Explorer 5.5 or greater.
Macromedia Flash Player.
Rationale
Why is it important for you to learn this material?
Physical quantities resulting from measurements or calculations are an every day
occurrence in the technologies. Students must be familiar with the rules and guidelines
for recording and expressing these results. An error in rounding or significant digits can
have dramatic impacts on technologies projects resulting in wasted time and resources.
The rules learned in this module will be applied by the student in both their academic and
professional lives.
Learning Outcome
When you complete this module you will be able to…
… express measurements using several different means including significant digits,
rounding, scientific notation, and precision and accuracy.
Learning Objectives
1.
2.
3.
4.
5.
6.
Identify the number of significant digits in a number.
Perform the operation of rounding numbers.
Express numbers in scientific notation.
Identify a value as being exact or approximate.
Define the terms precision and accuracy and how they affect calculations.
Calculate percentage error.
Connection Activity
Image you are building a bookshelf. You measure a piece of wood to find out how long
it is. How precise a measurement can you achieve with the measuring instruments in
your garage? Is the measurement you achieve exact or approximate? If you can measure
to a high degree of precision, how much precision is adequate for your building plans. If
you are to round off the measurement where will you round and how? These are the
types of questions that will be addressed in this module.
1
Scientific Notation and Significant Digits
OBJECTIVE ONE
When you complete this objective you will be able to…
Identify the number of significant digits in a number.
Exploration Activity
Significant Digits
In recording a physical quantity as a result of some measurement or computation care
must be taken to record only those numbers that are consequential, in other words the
significant digits.
Definition:
The significant digits in an approximate value are those digits counting from left to right,
beginning with the first non-zero digit and ending with that digit which occupies the
decimal place denoting the precision of the value. Each of the digits 1, 2, 3 ... 9 is a
significant digit, and 0 may or may not be significant.
EXAMPLE 1
1. In the measurement of 168.3 m, the significant digits are 1, 6, 8 and 3. i.e. all digits
are significant.
2. Consider the measurement of 0.087 cm. the significant digits are 8 and 7. This is a
direct consequence of our definition but it is important to realize why the two zeros
are not significant. It is because in this case the zero digits are used only to fix the
position of the decimal point. They are not actually read on the measuring device
and hence have no significance to the measurement. This is not always the case.
3. Consider the measurement of 150.02 cm. In this case according to our definition
the significant digits are 1, 5, 0, 0, 2. Why are the zeros significant in this case?
This is because rather than being merely space fillers as in the previous example,
the zeros here actually arise from the reading of the measuring device.
4. Consider the measurement 168.300 cm. The significant digits are 1, 6, 8, 3, 0, 0,
since final zeros to the right of the decimal point are always significant. They
should not be included unless they are significant.
In any number the only digits that may not be significant are the "zeros". Zeros are not
significant when:
(a) they precede a non-zero digit; and
(b) they follow a non-zero digit and precede the decimal place. All other zeros are
significant.
Item b has exceptions to the rule as shown in example 3 below.
2
Scientific Notation and Significant Digits
EXAMPLES
1.
Number
4.362 P
32.705 m
0.000 4 F
76 436 219
km
32.00 kg
No. of Sig. Digits
4
5
1
8
376 404 cm
4 020
6
3
4
Comments
No zeros to worry about.
Item a above.
No zeros here either.
Item b does not apply because zeros
follow the decimal place
Item b above.
2. How many significant digits are there in 0.060 metres?
Comment
The answer is 2.
Note that the first two zeros establish the “position” of the digits but the last zero
implies that the person doing the measuring recorded to the nearest 0.001 metre,
and therefore that particular zero is significant.
NOTE: These two concepts are very important:
(a)
The leading zeros are not significant.
(b)
The last zero is significant because it occurs after the decimal place and
after a non-zero number.
3. How many significant digits are there in 150 000 000?
Comment
The answer is that you don't know. One must say “tell me more”. If the number
represented the distance to the sun, the measurement would likely to be to the
nearest million kilometres and therefore the significant digits would be 1, 5 and 0.
If the number represented the population of a country, they might expect that the
count would be to the nearest thousand and therefore the significant digits would
be 1, 5, 0, 0, 0, and 0. The zeros in the hundreds and tens are just there to establish
the positions of the other numbers. You might ask, how can you tell? We would
have to reply, you would just have to know, otherwise you can't tell. This is one
reason Scientific Notation was devised which we will discuss later.
3
Scientific Notation and Significant Digits
Experiential Activity One
1. State the number of significant digits in each of the following:
(a)
(b)
(c)
(d)
(e)
(f)
(g)
(h)
(i)
(j)
(k)
(l)
1.64
72
65.00
0.01
100.01
1.000
126.3
200.00
200 000
0.00300
136 000.00
0.00160
2. How many significant digits are there in each of the following measurements?
(a)
(b)
(c)
(d)
(e)
(f)
(g)
(h)
(i)
(j)
(k)
(l)
36 000 km
36 020 km
55.5 cm
0.000 08 F
136.42 P
0.003 cm
5.484 m/s
297 000 km
960 m
237.4 K
152.0003 H
2.0000 A
Experiential Activity One Answers
1.
(a) 3
(b) 2
(c) 4
(d) 1
(e) 5
(f) 4
(g) 4
(h) 5
(i) don't know
(j) 3
(k) 8
(1) 3
2.
(a) don't know
(b) 4
(c) 3
(d) 1
(e) 5
(f) 1
(g) 4
(h) don't know
(i) 2
(j) 4
(k) 7
(1) 5
4
Scientific Notation and Significant Digits
OBJECTIVE TWO
When you complete this objective you will be able to…
Perform the operation of rounding numbers.
Exploration Activity
Rules For Rounding
1. Determine the least significant digit.
2. (a) If the digit to the right of the least significant digit represents a half or more,
round the least significant digit up.
(b) If the digit to the right represents less than half, leave the number unchanged.
(NOTE: You may have learned a more complicated rule for rounding when
the digits exactly equal a half. We have chosen the above rule because it is
straight forward and it is the way our calculators do it.)
Given
Number
24.637
24.641412767
24.650000001
24.649999999
24.662173512
Rounded off to nearest
tenth
24.6
24.6
24.7
24.6
24.7
5
Scientific Notation and Significant Digits
Experiential Activity Two
1. Round off each of the following approximate values to three significant digits and to
the nearest tenth.
(a) 1.609 35
(b) 0.213 7
(c) 30.480 1
(d) 0.032 808
(e) 0.453 499
(f) 91.440 2
(g) 2 589 998
(h) 4 046.873
(i) 3.785 33
(j) 61.025 0
(k) 16.387 2
(l) 35.314
(m) 4.885 01
(n) 0.204 500
(o) 29.523
(p) 1 097.61
(q) 9 235 000
(r) 3 280.837 5
(s) 3 000.000 6
(t) 0.000 489
Experiential Activity Two Answers
a)
b)
c)
d)
e)
f)
g)
h)
i)
j)
k)
1)
m)
n)
0)
P)
q)
r)
s)
t)
Three significant Digits
1.61
0.214
30.5
0.0328
0.453
91.4
2 590 000
4 050
3.79
61.0
16.4
35.3
4.89
0.205
29.5
1100
9 240 000
3 280
3 000
0.000 489
Nearest Tenth
1.6
0.2
30.5
0.0
0.5
91.4
2 589 998.0
4 046.9
3.8
61.0
16.4
35.3
4.9
0.2
29.5
1 097.6
9 235 000.0
3 280.8
3 000.0
0.0
6
Scientific Notation and Significant Digits
OBJECTIVE THREE
When you complete this objective you will be able to…
Express numbers in scientific notation.
Exploration Activity
Scientific Notation
Scientific notation requires that the number be rewritten as a single non-zero digit (from 1
to 9) followed by a decimal point, times the appropriate power of 10 to retain the original
value. For example:
Conventional Scientific
Notation
12345
1.2345 × 104
0.012345
1.2345 × 10−2
A convenient rule is:
Place the decimal point immediately to the right of the first non-zero digit, and count the
places to the original position of the decimal point to determine the exponent of 10, using
a positive exponent for numbers greater than 1 and a negative exponent for numbers less
than 1.
Exponents will be covered in more detail in a later module, but for the present, it is
sufficient to understand that 104 means
10 × 10 × 10 × 10 or 10000 and 10−2 means
1
1
=
.
10 × 10 100
The exponent is the count of the number of times 10 is multiplied by itself. If it is
negative, it is one over that value. Also, remember that 100 equals 1.
More examples:
Conventional Scientific
Notation
0.000147
1.47 x 10−4
1.643
1.643 x 100
773
7.73 x 102
3001.6
3.0016 x 103
Scientific notation was devised to assist in working with very large and very small
numbers but it also helps to avoid confusion that occurs with significant figures.
Consider the number 150 000 000. The number of significant digits in this number is not
clear.
7
Scientific Notation and Significant Digits
Converted to scientific notation, it becomes 1.5000 ... × 108. Now the user has the ability
to add just the number of zeros to clearly identify the number of significant digits.
For example, an astronomer may choose an accuracy of 1.50 × 108 m for a distance while
a statistician recording the population of a country may be sure of the count to 1.500000
× 108 people. (i.e. accurate to within 100 people).
NOTE FOR TLM
When you ask for a question from the system, we can print a value in scientific notation,
taking two lines and printing it such as 2.613 × 104. However, when you answer a
question, if we ask you to give us such a number, two lines are not practical so we
expect you to use a convention common to many scientific computer languages.
EXAMPLE 1
2.613 × 104 would be written 2.613E04
where E stands for exponential and the 04 is the power of the product 10.
NOTE:
This does not mean 2.6134!!
A few more examples:
Scientific
Notation
7.614 × 107
4.173 x 100
2.140 x 10−6
TLM Input
7.614E07
4.173E00
2.140E−06
Alternate TLM Format
Some questions have the following format for Scientific Notation:
Each answer will have two parts:
(a) A number between 1 and 10
(b) The appropriate power of 10 - Do not type the 10
ANSWER 1 = __________ ANSWER 2 = __________
If the answer were 1.23 × 103 we would have:
ANSWER 1: 1.23
ANSWER 2: 3
If the answer were 6.43 × 10−6 we would have:
ANSWER 1: 6.43
ANSWER 2: −6
8
Scientific Notation and Significant Digits
Experiential Activity Three
1. Express each of the following quantities in scientific notation:
(a) the age of the earth is estimated at 694 000 000 000 days.
(b) the diameter of the earth is 12 760 km.
(c) the number of atoms in 1.008 g of hydrogen is estimated to be
606 000 000 000 000 000 000 000.
(d) one light-year, the distance light travels in one year is 9 450 000 000 000 km.
(e) the mass of a water molecule is estimated to be 0.000 000 000 000 000 000 000
83 grate.
(f) the distance from the earth to the moon is 386 000 km.
(g) the diameter of the smallest visible particle is about 0.005 cm.
(h) the thickness of a film of oil is about 0.000 000 5 cm.
(i) the diameter of the Universe according to Einstein's Theory of Relativity is
2 000 000 000 light years.
(j) the distance of the earth from the sun is 149 000 000 km.
2. Express each of the following quantities in decimal notation:
(a)
(b)
(c)
(d)
(e)
equatorial radius of earth, 6.378 140 × 106 m.
mean angular rotational velocity of the earth, 7.2 × 10−5 rad/s.
mean orbital speed of the earth, 2.977 × 104 m/s.
Bohr (first electron orbit) radius, 0.529 177 06 × 10−10 m.
velocity of light in vacuum, 2.997 924 58 × 108 m/s.
3. Express each of the quantities in 2, above, in Scientific Notation rounded to 3
significant digits.
9
Scientific Notation and Significant Digits
Experiential Activity Three Answers
1.
(a)
(b)
(c)
(d)
(e)
(f)
(g)
(h)
(i)
(j)
6.94 × 1011
1.276 × 104
6.06 × 1023
9.45 × 1012
8.3 × 10−22
3.86 × 105
5.0 × 10−3
5.0 × 10−7
2.0 × 109
1.49 × 108
(a)
(b)
(c)
(d)
(e)
6 378 140
0.000 072
29 770
0.000 000 000 052 917 706
299 792 458
2.
3.
(a)
(b)
(c)
(d)
(e)
6.38 × 106
7.20 × 10−5
2.98 × 104
5.29 × 10−11
3.00 × 108
10
Scientific Notation and Significant Digits
OBJECTIVE FOUR
When you complete this objective you will be able to…
Identify a value as being exact or approximate.
Exploration Activity
Exact Values
Certain data values are exact, having been arrived at through some definition or counting
process.
We can determine whether or not a value is approximate or exact if we know how the
value was determined.
1. If an instructor counts the number of students at a work bench and states that there
are 3, this 3 is exact. We know the number of students was not 2 or 4. Since 3 was
determined through a counting process, it is exact.
2. When we say that 60 s = 1 minute, the 60 is exact, since this is a definition.
Similarly,
1 inch = 25.4 mm is exact by definition.
EXAMPLE 1
This garage has 2 cars in it.
1 yard = 36 inches
There are 19 departments at NAIT
Exact because it is counted
Exact definition
Exact by counting
Approximate Values
Some values have to be considered as approximate. This is because they result from
measurements. And it is physically impossible to measure with absolute accuracy. Hence
all measures are approximations.
EXAMPLE 2
1. The mass of an object is determined to be 23 kg.
2. The height of a statue is 37.4 cm.
These are approximations to the true value because the measuring device is not perfectly
constructed; also changes in temperature, dust, moisture, wear in bearings, etc., all affect
the reading.
Any value that results from “reading” an instrument is immediately subject to question
because of errors in the measuring device and possible inaccuracies by the person taking
the reading.
11
Scientific Notation and Significant Digits
Experiential Activity Four
State whether the values in the following statements are Approximate or Exact.
1. A cup of whole milk contains 166 calories.
2. The Vernal Falls in Yosemite National Park are 96.6 m in height.
3. Only 350 students graduated from the Southern Alberta Institute of Technology with
an Honours Diploma last year.
4. The lowest temperature recorded in Edmonton last winter was 34° below zero.
5. The Statue of Liberty weighs 204 metric tons.
6. The elevation of Mount Rundle is 2891 metres.
7. The area of Iceland is 102 790 km2.
8. In the Lions Gate Suspension Bridge in Vancouver, each cable is made up of 8066
wires.
9. The area of a circle is π r 2.
10. The number π = 3.14159
11. The standard metre is defined as 1650 763.73 wave lengths of the orange-red line of
krypton-86.
12. 1 inch = 25.4 mm.
Experiential Activity Four Answers
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
11.
12.
Approximate
Approximate
Exact
Approximate
Approximate
Approximate
Approximate
Exact
Exact
Approximate
Exact
Exact
12
Scientific Notation and Significant Digits
OBJECTIVE FIVE
When you complete this objective you will be able to…
Define the terms precision and accuracy and how they affect calculations.
Exploration Activity
Definition:
The precision of a measurement shall be defined to be the smallest placed value in which
it is expressed.
EXAMPLE 1
76.3 kg is precise to a tenth of a kg Å the 3 is in tenths column.
0.087 cm is precise to a thousandth of a cm Å the 7 is in thousandths column.
7960 m is precise to the nearest ten metres Å the 6 is in tens column.
RULE In any calculation involving addition and subtraction, the answer cannot be any
more precise than the LEAST precise value in the calculation.
EXAMPLE 2
Add:
2881
43.49
0.137
2924.627
m
m
m
m
The least precise value is 2881, thus the answer must be rounded off to the nearest unit.
Thus, the answer equals 2925 m.
13
Scientific Notation and Significant Digits
Accuracy
The accuracy of a measurement is related to the number of significant digits in the
number.
EXAMPLE 3
168.3 kg is accurate to 4 significant digits.
0.087 cm is accurate to 2 significant digits.
150.02 cm is accurate to 5 significant digits.
1.200 × 105 km is accurate to 4 significant digits.
The measurements 0.0063 cm, 48 cm, 7.0 cm, and 7.3 × 102 cm all have the same
accuracy, i.e. to 2 significant digits.
RULE: In any calculation involving multiplication and division the answer cannot be any
more accurate than the LEAST accurate value in the calculation.
EXAMPLE 4
371 m × 8216 m = 3 048 136 m2
The least accurate factor is 371. Hence the answer must be rounded off to 3 significant
figures which is the accuracy stated in the number 371.
Thus the answer = 3 050 000 m2.
14
Scientific Notation and Significant Digits
Experiential Activity Five
State the precision of the following values:
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
346.32 cm
7.001 mm
100 m
0.0003 kg
3.02 kg
8.1032 g
870 000 m
1 000 000 N
80 032 cm
40.1 cm
Exercise set 2
Perform the indicated operations in accordance with accepted procedures involving
precision.
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
237.4 + 3.097 + 84 + 9.97
29.48 − 2.5754 + 59.1 + 1.94 + 306.8094
49.9 + 287 + 3.984 + 20.793
5.14 + 63.012 + 1.7849 + 0.3527 + 829.367
917.222 + 9641.3 + 8764
4.3 − 0.642318
78.695 − 0.01121
23.686 − 0.124
28.59 − 0.3752
233700000 − 7308271
15
Scientific Notation and Significant Digits
Exercise set 3
State the accuracy of the following values:
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
11.
12.
29.037 cm
60.0 km
2.00 cm
3.8497 km
0.83 cm
8.0 cm
248.37 m
57.6 km
0.0202 cm
20.09 cm
8.30 cm
0.7 cm
Exercise set 4
Evaluate:
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
73200000× 0.0000210
0.0045
18 × 0.058576
297.4 ÷ 1.3
6.43 × 12048161
2188.59 ÷ 24.0
0.53 × 0.0088877
29.7469 × 0.212
479755000 ÷ 211000000
93000000 ÷ 3.1415926
0.0087 ÷ 3.6267
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Scientific Notation and Significant Digits
Experiential Activity Five Answers
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
hundredth
thousandth
hundred
ten-thousandth
hundredth
ten-thousandth
ten-thousands
million
units
tenth
Answers to exercise set 2.
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
334
394.8
362
899.66
19323
3. 7
78.684
23.562
28.21
226400000
Answers to exercise set 3.
1.
3.
5.
7.
9.
11.
5
3
2
5
3
3
2.
4.
6.
8.
10.
12.
3
5
2
3
4
1
Answers to exercise set 4.
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
340 000
1.1
230
77 500 000
91.2
0.004 7
6.31
2.27
30 000 000 OR 3.0 × 107
0.0024
17
Scientific Notation and Significant Digits
OBJECTIVE SIX
When you complete this objective you will be able to…
Calculate percentage error.
Exploration Activity
To find the percentage error: take the difference between the true value and the measured
value, divide this difference by the true value, and then multiply this fraction by 100.
In formula form: % Error =
measured value − true value
× 100
true value
EXAMPLE
The current in a circuit is to be 6 A. It was measured to be 6.2 A. What is the percent
error in this reading?
Given:
To Find:
Solution:
true value = 6 A
measured value = 6.2 A
% Error
measured value − true value
× 100
% Error =
true value
6.2 A − 6 A
=
× 100
6A
= 0.0333 × 100
= 3.33%
Notice the answer is positive. If the measurement were less than the true value the answer
would have been negative.
18
Scientific Notation and Significant Digits
Experiential Activity Six
1. The true value of the length of an airport runway was supposed to be 3000 m. Upon
measurement it is found to be 2980 m. What is the percentage error in its length?
2. A meter indicates an allowable error of ±5% on any of its scales. If it is used to
produce a reading of 32.4 mA, what is the range of acceptable answers?
3. A bank calculated the interest on a savings account to be $352.00, when the correct
value was $362.00. What was the percent error in this case?
4. A measurement was taken as 3.23 cm. This represents a 6% deviation from the true
value. What was the true value? Show Me.
5. A watch loses 10 seconds in a 24 hour period. What is the percent error in the time
after a 48 hour period?
Experiential Activity Six Answers
1.
2.
3.
4.
5.
−0.67% or −0.7% (four decimals -0.6667%)
30.78 mA to 34.02 mA
−2.76% (four decimals -2.7624%)
3.0472 cm
0.01% (four decimals 0.0110%)
19
Scientific Notation and Significant Digits
Practical Application Activity
Complete the Measurement: Part II assignment in TLM.
Summary
This module introduced significant digits, rounding off, scientific notation, and the terms
precision and accuracy.
20
Scientific Notation and Significant Digits