Scientific Notation
Now all that is left to do is multiply 0.0856 by 1000. To do this, you are just going to move the decimal point three places to the right: (0.0856 m) x (1000 mm / 1 m) = 85.6 mm Example: Convert 153 grams to centigrams. Solution: 153 grams x
= 15300 centigrams
Scientific Notation Scientific notation is a way to write very large or very small numbers; the decimal point is moved so that there is one digit in the unit’s position and all of the decimal places are held as a power of ten. This is important in chemistry because many of the measurements we make either involve very large numbers of atoms/molecules or very tiny measurements, such as masses of electrons or protons. For example, consider a number such as 839,000,000. While this number is not too difficult to write out, it is more conveniently 8
8
written in scientific notation. Written in scientific notation, this number becomes: 8.39 x 10 . The “10 ” means 8
that ten is multiplied by itself eight times: 10 x 10 x 10 x 10 x 10 x 10 x 10 x 10. As you can see, writing 10 is much easier!
Figure 4: Even using large distance units such as kilometers, you would still need to use scientific notation to measure the size of this galaxy. (Source: http://www.flickr.com/photos/pingnews/474783096/, License: CC-BY-SA) We can also use scientific notation to write very small numbers. Take a number such as 0.00000481. It is easy to make mistakes in counting the number of zeros in this number. Also, many calculators only let you enter in a certain number of digits. When we write this in scientific notation, it is important to notice that the measurement is less than one, therefore, the exponent on 10 will be negative: this number becomes 4.81 -6
x 10 . In this case, the decimal point was moved six places to the right. It is important that you know how to perform calculations using numbers written in scientific notation. For example, the following problem shows two numbers with exponents being multiplied together: 3
6
(2.90 x 10 )(1.60 x 10 ) = ? To solve this problem, you would multiply the terms (2.90 and 1.60) like you normally would; then you would add the exponents:
53
2.90 x 1.60 = 4.64 3
6
9
10 x 10 = 10
9
Therefore, combining these values gives the answer 4.64 x 10 .
Significant Figures The tool that you use determines the number of digits that will be in a measurement. For example, if you say an object has a mass of “5 kg”, that is not the same as saying it has a mass of “5.00 kg” since you must have measured the masses with two different tools – the two zeros in “5.00 kg” would not be written if the tool that was used could not measure to two decimal places. Even though the mass seems to be the same, the uncertainty of the measurement is not. When you say “5 kg”, that means you have measured the mass to within +/-1 kg. The actual mass could be 4 or 6 kg. For the 5.00 kg measurement, you have measured the mass to within +/-0.01 kg, so the actual mass is between 4.99 and 5.01 kg.
Using Significant Figures in Measurements How do you know how many significant figures are in a measurement? General guidelines are as follows: •
Any nonzero digit is significant [4.33 has three significant figures].
•
A zero that is between two nonzero digits is significant [4.03 has three significant figures].
•
All zeros to the left of the first nonzero digit are not significant [0.00433 has three significant figures].
•
Zeros that occur after the decimal are significant. [40.0 has three significant figures. The zero after the decimal point tells us that the value was measured to the tenths place].
•
Zeros that occur without a decimal are not significant [4000 has one significant figure since the zeros are holding the 4 in the thousands position].
Examples: 1) How many significant figures are in the number 1.680? Solution: There are three nonzero digits and one zero appears after the decimal point. Therefore, there are four significant figures. 2) How many significant figures are in the number 0.0058201? Solution: There are 4 nonzero digits and 1 zero between two numbers. Therefore, there are 5 significant figures. The first three zeros are not significant since they are simply holding the number away from the decimal point.
Lesson Summary •
The metric system is a decimal system; all magnitude differences in units are multiples of 10.
•
Unit conversions involve creating a conversion factor.
•
Very large and very small numbers are expressed in exponential notation.
•
Significant figures are used to express uncertainty in measurements.
Review Questions 1. Convert the following linear measurements:
54
(Beginning)
CK-12 Foundation is a non-profit organization with a mission to reduce the cost of textbook materials for the K-12 market both in the U.S. and worldwide. Using an open-content, webbased collaborative model termed the “FlexBook,” CK-12 intends to pioneer the generation and distribution of high-quality educational content that will serve both as core text as well as provide an adaptive environment for learning. Copyright © 2009 CK-12 Foundation, www.ck12.org
Except as otherwise noted, all CK-12 Content (including CK-12 Curriculum Material) is made available to Users in accordance with the Creative Commons Attribution/NonCommercial/Share Alike 3.0 Unported (CC-by-NC-SA) License (http://creativecommons. org/licenses/by-nc-sa/3.0/), as amended and updated by Creative Commons from time to time (the “CC License”), which is incorporated herein by this reference. Specific details can be found at http://about.ck12.org/terms.
= 15300 centigrams
Scientific Notation Scientific notation is a way to write very large or very small numbers; the decimal point is moved so that there is one digit in the unit’s position and all of the decimal places are held as a power of ten. This is important in chemistry because many of the measurements we make either involve very large numbers of atoms/molecules or very tiny measurements, such as masses of electrons or protons. For example, consider a number such as 839,000,000. While this number is not too difficult to write out, it is more conveniently 8
8
written in scientific notation. Written in scientific notation, this number becomes: 8.39 x 10 . The “10 ” means 8
that ten is multiplied by itself eight times: 10 x 10 x 10 x 10 x 10 x 10 x 10 x 10. As you can see, writing 10 is much easier!
Figure 4: Even using large distance units such as kilometers, you would still need to use scientific notation to measure the size of this galaxy. (Source: http://www.flickr.com/photos/pingnews/474783096/, License: CC-BY-SA) We can also use scientific notation to write very small numbers. Take a number such as 0.00000481. It is easy to make mistakes in counting the number of zeros in this number. Also, many calculators only let you enter in a certain number of digits. When we write this in scientific notation, it is important to notice that the measurement is less than one, therefore, the exponent on 10 will be negative: this number becomes 4.81 -6
x 10 . In this case, the decimal point was moved six places to the right. It is important that you know how to perform calculations using numbers written in scientific notation. For example, the following problem shows two numbers with exponents being multiplied together: 3
6
(2.90 x 10 )(1.60 x 10 ) = ? To solve this problem, you would multiply the terms (2.90 and 1.60) like you normally would; then you would add the exponents:
53
2.90 x 1.60 = 4.64 3
6
9
10 x 10 = 10
9
Therefore, combining these values gives the answer 4.64 x 10 .
Significant Figures The tool that you use determines the number of digits that will be in a measurement. For example, if you say an object has a mass of “5 kg”, that is not the same as saying it has a mass of “5.00 kg” since you must have measured the masses with two different tools – the two zeros in “5.00 kg” would not be written if the tool that was used could not measure to two decimal places. Even though the mass seems to be the same, the uncertainty of the measurement is not. When you say “5 kg”, that means you have measured the mass to within +/-1 kg. The actual mass could be 4 or 6 kg. For the 5.00 kg measurement, you have measured the mass to within +/-0.01 kg, so the actual mass is between 4.99 and 5.01 kg.
Using Significant Figures in Measurements How do you know how many significant figures are in a measurement? General guidelines are as follows: •
Any nonzero digit is significant [4.33 has three significant figures].
•
A zero that is between two nonzero digits is significant [4.03 has three significant figures].
•
All zeros to the left of the first nonzero digit are not significant [0.00433 has three significant figures].
•
Zeros that occur after the decimal are significant. [40.0 has three significant figures. The zero after the decimal point tells us that the value was measured to the tenths place].
•
Zeros that occur without a decimal are not significant [4000 has one significant figure since the zeros are holding the 4 in the thousands position].
Examples: 1) How many significant figures are in the number 1.680? Solution: There are three nonzero digits and one zero appears after the decimal point. Therefore, there are four significant figures. 2) How many significant figures are in the number 0.0058201? Solution: There are 4 nonzero digits and 1 zero between two numbers. Therefore, there are 5 significant figures. The first three zeros are not significant since they are simply holding the number away from the decimal point.
Lesson Summary •
The metric system is a decimal system; all magnitude differences in units are multiples of 10.
•
Unit conversions involve creating a conversion factor.
•
Very large and very small numbers are expressed in exponential notation.
•
Significant figures are used to express uncertainty in measurements.
Review Questions 1. Convert the following linear measurements:
54
(Beginning)
CK-12 Foundation is a non-profit organization with a mission to reduce the cost of textbook materials for the K-12 market both in the U.S. and worldwide. Using an open-content, webbased collaborative model termed the “FlexBook,” CK-12 intends to pioneer the generation and distribution of high-quality educational content that will serve both as core text as well as provide an adaptive environment for learning. Copyright © 2009 CK-12 Foundation, www.ck12.org
Except as otherwise noted, all CK-12 Content (including CK-12 Curriculum Material) is made available to Users in accordance with the Creative Commons Attribution/NonCommercial/Share Alike 3.0 Unported (CC-by-NC-SA) License (http://creativecommons. org/licenses/by-nc-sa/3.0/), as amended and updated by Creative Commons from time to time (the “CC License”), which is incorporated herein by this reference. Specific details can be found at http://about.ck12.org/terms.
