1 Chapter 1: Getting Started with Units and Measurement
The first chapter of our textbook deals with units, unit conversion, and significant figures. Hopefully you’ve seen these ideas before! You can use the summaries below as a refresher, and if you’d like more information, use the links to our textbook and/or videos that further explain the concepts.
When expressing a number in scientific notation, only the significant digits are included. Your number should always be expressed in the form X.XX * 10X, where the first significant digit (not a zero!) is followed by a decimal place.
A positive number in the exponent of the 10 means you need to move the decimal place that many times to the right in order to get the original number back. For example, 9.0320 * 103 can also be expressed as 9,032.0.
A negative number in the exponent means you need to move the decimal place that many times to the left to get back the original number. For example, 1.309 * 10-5 can also be written as 0.00001309.
A significant figure or significant digit represents a known quantity. The last significant digit of a number is the digit in which the uncertainty lies. For example, imagine you have a ruler which can measure quantities down to 1 mm. But what if the length of your object lies between 5.2 cm and 5.3 cm? Then you have to estimate the final digit between 0 and 9. Because you had to make an educated guess about the last digit, you cannot add any more significant digits after that one. Further digits have 100% uncertainty (we don’t have any idea what they are using our ruler).
To count how many significant figures your number has, begin with the first non-zero digit on the left. Stop counting at the last non-zero digit on the right. Your number has at least that many significant digits. Was that the entire number? Congratulations, you’re done. Examples of this include 5 (1 significant digit), 12.04 (4 significant digits), and 1,004 (4 significant digits).
What if your number starts with a zero? That zero is never significant. Only zeros at the end can possibly be significant, not zeros at the beginning. Examples of this include 0.005702 (4 significant digits) and 02.3 (2 significant digits).
What about zeros on the end? If there are zeros on the very end and you are to the left of the decimal place, those zeros are not significant unless they are followed by a decimal point (in which case all zeros immediately preceding them are also significant). Examples of this are 10,000 (1 significant digit), 56,700.00 (7 significant digits) and 340. (3 significant digits).
If there are zeros on the very end and you are to the right of the decimal point, all zeros following non-zero numbers are significant but no zeros that appear before any non-zero numbers start are significant. Examples of this are 0.0060200 (5 significant digits), 0040.52800 (7 significant digits), and 1.03920 (6 significant digits).
All zeros between non-zero numbers are significant. Examples include 750.023 (6 significant figures), 56,0092 (6 significant figures), and 0.004009 (4 significant figures).
OpenStax Textbook: Chapter 1, Significant Figures
No matter how complicated the problem, each problem can be broken down into steps that require addition, subtraction, multiplication, or division of two or more numbers at a time.
When multiplying or dividing numbers, your answer will have the same number of significant figures as the number going into the equation with the least number of significant figures. For example, if you are doing the problem (38 * 0.892) / (194), your answer (0.17433) should have two significant digits, because 38 only has two (whereas 0.892 and 194 each have three) so the answer you give should be 0.174.
Rounding Rules: if your answer ends in 5 or higher, round up. Otherwise, round down.
Hint: Keep all of your digits (significant or not significant) until you are ready to give an answer. Then determine how many significant digits the answer should have based on the problem and round off to that place.
When adding or subtracting two or more numbers, the uncertainty of your answer will be the same as the uncertainty of your least certain number. For example, if you are adding 7, 4.1, and 15.72, your answer (before determining significant figures) would be 26.82. Note the actual number of significant digits in each number you add or subtract doesn’t matter; it’s only the decimal place associated with the uncertainty. For 7, you know none of the decimal places. For 4.1, you know the decimal out to the tenth’s place. For 15.72, you know it out to the hundredth’s place. Because 7 has the most uncertainty (no decimal places) your answer must be given in that same uncertainty (no decimal places). So you would round up your answer to 27.
What about if we combine these steps? Follow the “order of operations” rules. First determine everything inside parentheses, then reduce those to the correct number of significant digits, then solve the rest of the problem.
Khan Academy: Addition and Subtraction with Significant Figures
Khan Academy: Multiplying and Dividing with Significant Figures
Units are extremely important in the sciences because they tell you what you’re dealing with. Just giving numbers is never[1] a correct answer without the appropriate units.
Unit conversion factors (to move from one type of unit to another) are given in forms such as 1 m = 3.281 ft. You’ll find common unit conversions in the front cover of this manual. To convert from one unit to another, use the following notation. Remember that the goal is to get rid of one kind of a unit, and you do that by canceling one on the top with one on the bottom. Here’s an example converting the distance to Vega (25.04 light years, ly) to miles.
![\[\Large 25.04 {\rm \, ly} = \left( \frac{25.04 {\rm \, ly}}{1} \right) \times \left( \frac{9.461 \times 10^{15} {\rm \, m}}{1 {\rm \, ly}} \right) \times \left( \frac{3.281 {\rm \, ft}}{1 {\rm \,m}} \right) \times \left( \frac{1 {\rm \,mi}}{5,280 {\rm \,ft}} \right) = 1.472 \times 10^{14} {\rm \, mi}\]](https://rwu.pressbooks.pub/app/uploads/quicklatex/quicklatex.com-9b76c0a1b50e9c08652f08775c89a98a_l3.svg "Rendered by QuickLaTeX.com")
How many significant figures should our answer have? In this case, the original number had 4, and so did each of our conversion factors. While some conversions are exact and therefore can be considered to have an infinite number of significant digits (like 60 seconds in a minute) others are rounded to 3 or 4 figures, and those you must take into account when doing your conversion.
What if you need to change the units on the top and bottom of the equation? No problem, just keep multiplying by your conversion factors. Here, we’ll convert 17.2 ft/min into m/s.
![\[\left( \frac{17.2 {\rm \,ft}}{1 {\rm \,min}} \right) \times \left( \frac{1 {\rm \,m}}{3.281 {\rm \, ft}} \right) \times \left( \frac{1 {\rm \,min}}{60 {\rm \,s}} \right) = 0.0874 {\rm \, m/s}\]](https://rwu.pressbooks.pub/app/uploads/quicklatex/quicklatex.com-c57a52fdb883996127c7ddd8e5a4010b_l3.svg "Rendered by QuickLaTeX.com")
How about when a factor is squared or cubed? Let’s look at how we would convert 5.9 cubic feet into cubic meters. When a factor is squared, cubed, or further, you need to do the unit conversion two, three, or more times.
![\[5.9 {\rm \, ft}^3 = \left( \frac{5.9 {\rm \,ft}^3}{1} \right) \times \left( \frac{1 {\rm \,m}}{3.281 {\rm \, ft}} \right)^3 = 1.7 {\rm \, m}^3\]](https://rwu.pressbooks.pub/app/uploads/quicklatex/quicklatex.com-81b46271e6a932ea1c453f5a041a3419_l3.svg "Rendered by QuickLaTeX.com")
Note that the 5.9 ft isn’t cubed again; it’s only the unit (ft) that’s cubed, not the 5.9. But since our conversion is only for one meter to 3.281 feet, we need to do it three times (entire conversion cubed) to convert all three feet to meters.
OpenStax Textbook: Chapter 1, Units and Standards
OpenStax Textbook: Chapter 1, Unit Conversion
OpenStax Textbook: Chapter 1, Dimensional Analysis
Units are important. Missing units will always result in a loss of points. However, we’re not all that picky about significant digits. Here are the general rules for when to worry about sig figs:
(1) During lab. This is the only time we’ll be picky about them.
(2) Don’t give fewer than 2 sig figs with any answer, and 3 is usually best.
(3) Don’t just copy the 10 digits your calculator gives you! Your answer will never be that accurate.
(4) On the few occasions the online homework tells you to pay attention to sig figs, take them seriously! The rest of the time they don’t care as long as your answer is within about 2% of theirs.
- Except in the case of values that don't have units, such as coefficients of friction and indices of refraction. ↵
A way of abbreviating a number. The format is X.XXX times 10^(something), where you include all the significant figures in the X.XXX number starting with one digit, a decimal place, followed by all the other significant digits.
The last digit of a significant figure tells you how well we know what this number really is.
