Scientific Notation and Significant...

Trigonometry

Scientific Notation and Significant Digits

Copyright This publication © The Northern Alberta Institute of Technology 2002. All Rights Reserved.

LAST REVISED June, 2007


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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. Identify the number of significant digits in a number. 2. Perform the operation of rounding numbers. 3. Express numbers in scientific notation. 4. Identify a value as being exact or approximate. 5. Define the terms precision and accuracy and how they affect calculations. 6. 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.

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Scientific Notation and Significant Digits 2

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.


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EXAMPLES 1.

Number No. of Sig. Digits Comments 4.362 P 4 No zeros to worry about. 32.705 m 5 0.000 4 F 1 Item a above. 76 436 219 km

8 No zeros here either.

32.00 kg 4 Item b does not apply because zeros follow the decimal place

376 404 cm 6 4 020 3 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.


Experiential Activity One 1. State the number of significant digits in each of the following:

(a) 1.64 (b) 72 (c) 65.00 (d) 0.01 (e) 100.01 (f) 1.000 (g) 126.3 (h) 200.00 (i) 200 000 (j) 0.00300 (k) 136 000.00 (l) 0.00160

2. How many significant digits are there in each of the following measurements?

(a) 36 000 km (b) 36 020 km (c) 55.5 cm (d) 0.000 08 F (e) 136.42 P (f) 0.003 cm (g) 5.484 m/s (h) 297 000 km (i) 960 m (j) 237.4 K (k) 152.0003 H (l) 2.0000 A

Experiential Activity One Answers 1. 2.

(a) 3 (a) don't know (b) 2 (b) 4 (c) 4 (c) 3 (d) 1 (d) 1 (e) 5 (e) 5 (f) 4 (f) 1 (g) 4 (g) 4 (h) 5 (h) don't know (i) don't know (i) 2 (j) 3 (j) 4 (k) 8 (k) 7 (1) 3 (1) 5


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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

Rounded off to nearest tenth

24.637 24.6 24.641412767 24.6 24.650000001 24.7 24.649999999 24.6 24.662173512 24.7


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 Three significant Digits Nearest Tenth a) 1.61 1.6 b) 0.214 0.2 c) 30.5 30.5 d) 0.0328 0.0 e) 0.453 0.5 f) 91.4 91.4 g) 2 590 000 2 589 998.0 h) 4 050 4 046.9 i) 3.79 3.8 j) 61.0 61.0 k) 16.4 16.4 1) 35.3 35.3 m) 4.89 4.9 n) 0.205 0.2 0) 29.5 29.5 P) 1100 1 097.6 q) 9 240 000 9 235 000.0 r) 3 280 3 280.8 s) 3 000 3 000.0 t) 0.000 489 0.0


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

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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

1001

10101

=×

10 × 10 × 10 × 10 or 10000 and 10−2 means .

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.


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

TLM Input

7.614 × 107 7.614E07 4.173 x 100 4.173E00 2.140 x 10−6 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


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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) equatorial radius of earth, 6.378 140 × 106 m. (b) mean angular rotational velocity of the earth, 7.2 × 10−5 rad/s. (c) mean orbital speed of the earth, 2.977 × 104 m/s. (d) Bohr (first electron orbit) radius, 0.529 177 06 × 10−10 m. (e) 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.


Experiential Activity Three Answers 1.

(a) 6.94 × 1011 (b) 1.276 × 104 (c) 6.06 × 1023 (d) 9.45 × 1012 (e) 8.3 × 10−22 (f) 3.86 × 105 (g) 5.0 × 10−3 (h) 5.0 × 10−7 (i) 2.0 × 109 (j) 1.49 × 108

2.

(a) 6 378 140 (b) 0.000 072 (c) 29 770 (d) 0.000 000 000 052 917 706 (e) 299 792 458

3.

(a) 6.38 × 106 (b) 7.20 × 10−5 (c) 2.98 × 104 (d) 5.29 × 10−11 (e) 3.00 × 108


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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. Exact because it is counted 1 yard = 36 inches Exact definition There are 19 departments at NAIT 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.


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. Approximate 2. Approximate 3. Exact 4. Approximate 5. Approximate 6. Approximate 7. Approximate 8. Exact 9. Exact 10. Approximate 11. Exact 12. Exact


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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 the 3 is in tenths column. 0.087 cm is precise to a thousandth of a cm the 7 is in thousandths column. 7960 m is precise to the nearest ten metres 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 m 43.49 m 0.137 m 2924.627 m

The least precise value is 2881, thus the answer must be rounded off to the nearest unit. Thus, the answer equals 2925 m.


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.


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Experiential Activity Five State the precision of the following values: 1. 346.32 cm 2. 7.001 mm 3. 100 m 4. 0.0003 kg 5. 3.02 kg 6. 8.1032 g 7. 870 000 m 8. 1 000 000 N 9. 80 032 cm 10. 40.1 cm Exercise set 2 Perform the indicated operations in accordance with accepted procedures involving precision. 1. 237.4 + 3.097 + 84 + 9.97 2. 29.48 − 2.5754 + 59.1 + 1.94 + 306.8094 3. 49.9 + 287 + 3.984 + 20.793 4. 5.14 + 63.012 + 1.7849 + 0.3527 + 829.367 5. 917.222 + 9641.3 + 8764 6. 4.3 − 0.642318 7. 78.695 − 0.01121 8. 23.686 − 0.124 9. 28.59 − 0.3752 10. 233700000 − 7308271


Exercise set 3 State the accuracy of the following values: 1. 29.037 cm 2. 60.0 km 3. 2.00 cm 4. 3.8497 km 5. 0.83 cm 6. 8.0 cm 7. 248.37 m 8. 57.6 km 9. 0.0202 cm 10. 20.09 cm 11. 8.30 cm 12. 0.7 cm Exercise set 4 Evaluate:

0045.00000210.073200000×1.

2. 18 × 0.058576 3. 297.4 ÷ 1.3 4. 6.43 × 12048161 5. 2188.59 ÷ 24.0 6. 0.53 × 0.0088877 7. 29.7469 × 0.212 8. 479755000 ÷ 211000000 9. 93000000 ÷ 3.1415926 10. 0.0087 ÷ 3.6267

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Experiential Activity Five Answers 1. hundredth 2. thousandth 3. hundred 4. ten-thousandth 5. hundredth 6. ten-thousandth 7. ten-thousands 8. million 9. units 10. tenth Answers to exercise set 2. 1. 334 2. 394.8 3. 362 4. 899.66 5. 19323 6. 3. 7 7. 78.684 8. 23.562 9. 28.21 10. 226400000 Answers to exercise set 3. 1. 5 2. 3 3. 3 4. 5 5. 2 6. 2 7. 5 8. 3 9. 3 10. 4 11. 3 12. 1 Answers to exercise set 4. 1. 340 000 2. 1.1 3. 230 4. 77 500 000 5. 91.2 6. 0.004 7 7. 6.31 8. 2.27 9. 30 000 000 OR 3.0 × 107 10. 0.0024


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.

100valuetrue

value true valuemeasured Error % ×−

=In formula form:

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: true value = 6 A

measured value = 6.2 A To Find: % Error Solution:

%33.31000333.0

1002.6

100

=×=

×−

=

×−

=

A 6A 6 A

value truevalue true value measured Error %

Notice the answer is positive. If the measurement were less than the true value the answer would have been negative.

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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. −0.67% or −0.7% (four decimals -0.6667%) 2. 30.78 mA to 34.02 mA 3. −2.76% (four decimals -2.7624%) 4. 3.0472 cm 5. 0.01% (four decimals 0.0110%)


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.

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