Worksheet 1 Chapter 2 You Must Show Your Work For Credit

Worksheet 1 Chapter 2you Must Show Your Work For Credit

Worksheet 1 – Chapter 2 You must show your work for Credit Part 1: Significant Figures Let’s start off with scientific notation … Large numbers (numbers for which the absolute value is greater than 1) will always have a positive exponent when in scientific notation. When converting to scientific notation, you move the decimal point until there is a single digit to the left. The number of places that the decimal spot moved becomes the exponent and the “x10â€. Example: –450000 –4.5x105. The decimal point was moved 5 times to the left, so the exponent is 5.

Example: .x108. The decimal was moved to the left by 8 spots so the exponent is 8. Example: 57293.264 5.x104 since the decimal was moved 4 times to the left. Small numbers (numbers between 1 and –1) will always have a negative exponent when in scientific notation. When converting to scientific notation, you move the decimal point until there is a single digit to the left.

The number of places that the decimal spot moved becomes the exponent and the “x10–â€. Example: 0..528x10–4. The decimal moved 4 times to the right, so the exponent become –4. Example: –0. –5.8500x10–15. The decimal point was moved 15 times to the right, so the exponent became –15.

Example: 0.002 2x10–3 since the decimal was moved 3 times to the right. You try: 1a) 54,670,000,b) –5526.c) 0.d) 100.00 How many significant figures in a number : First and foremost, you need to be able to tell how many sig. figs. are in a number. Here are three rules that you can use: 1) If the number is in scientific notation: The number of digits shown, excluding the order of magnitude, is equal to the number of sig. figs. Examples: 6.626x10–34 has 4 significant figures (6.626x10–.30x104 has 3 significant figures (8.30x.0x101 has 2 sig. figs. (3.0x) If the number has a decimal in it: Start at the RIGHT of the number and count to the left until you get to the last NONZERO number, this is the number of sig. figs.

Examples: 195.3040 has 7 sig. figs. (195..003081 has 4 sig. Figs. (0..00 has 8 sig. figs. (180048.. has 1 sig. fig. (0.. has 2 sig. fig. (10.) 3) If the number does NOT have a decimal in it: Start at the LEFT of the number and count to the right until you get to the last NONZERO number, this is the number of sig. figs. Examples: 160 has 2 sig. figs. ( has 1 sig. figs ( has 3 sig. figs. ( has 6 sig. figs. ( has 1 sig. fig. (10) You try: 2a) 6200 2e) 0.h) 23.b) 1.032 2f) 1x104 2i) 100.c) 420. 2g) j) d) 3.750x10–6

Precision There are a lot of different ways of thinking about precision. The definition of precision is “how close a series of measurements are to one another or how reproducible they areâ€. Huh? Basically, this means that as you use an instrument to make measurements, the closer those measurements are to each other, the more precise the instrument is. Wait… huh?? Try this.

You can think about precision in terms of the size of the graduations on the instrument. The smaller the graduations are, the more precise the instrument is. This is directly related to the above definition because the smaller the graduations are, the smaller the space is that you have to “guess†in and the smaller the space that you have to “guess†in, the more likely that your “guesses†will be the same. BAM, definition of precision. When we are trying to determine the precision of a measurement, we can use significant figures. (In fact, another definition of precision is related to significant figures.) In order to determine the precision of a measurement, we will need to find the last significant figure in the measurement.

We learned above how to count how many sig figs there are in a value and this is related, but slightly different. The last sig fig in a measurement is always the one that is furthest to the right , regardless of the presence or absence of a decimal place. Example: has two significant figures (the 6 and the 2), but we want to know the precision in this case. Again, the precision of this measurement lies in the last sig fig (the sig fig that lies furthest to the right). For , the last sig fig is the 2, which lies in the hundreds–place.

Therefore this measurement is precise to the hundreds–place. Example: 1..032 has four significant figures (all of the digits in this one happen to be significant). Again, the precision of this measurement lies in the last sig fig (the sig fig that lies furthest to the right). For 1.03 2 , the last sig fig is the 2, which happens to lie in the 3rd decimal place. Therefore this measurement is precise to the 3rd decimal place (or the thousandths place).

Example: 420. 420. has three significant figures (the 0 is significant in this case because of the decimal place). For 42 0 ., the last sig fig is the 0, therefore this measurement is precise to the ones–place. Example: 3.750x10–6 This one is actually a bit tricky because it is written in scientific notation. If you’re not careful, you will report the precision of this measurement as the 3rd decimal place, which is incorrect.

To find the precision of this value, you will need to convert it from scientific notation to decimal notation. 3.750x10–6 = 0. has four significant figures, the last of which is the 0 at the right of the value. For 0. , the precision of this measurement is in the 9th decimal place. Example: 1x104 Again, watch out for the scientific notation, you will need to convert it from scientific notation to decimal notation. 1x104 = 10000 which has only one significant figure (the 1).

For 1 0000, the precision of this measurement is in the ten–thousands–place. Your turn: 3a) e) 3.20xb) 6.341x10–4 3f) 9900. 3c) 90 3g) 43.d) 0.h) Significant figures in calculations One of the most missed rules in significant figures is that you are not allowed to round a value until you are completely finished with a calculation. This is especially important in situations where you are mixing operations or when the answer to one question is then used in the next question. In both of these situations, you will be tempted to round values before you get to the answer to the questions you are working on.

We will see how to avoid this mistake with mixed operations later in this exercise, so for now, let’s focus on the second of these two mistakes. Let’s say that you worked through “part a†of a questions and your calculator spit out the answer 52. and you knew that your answer had to have three significant figures. What would happen if you simply wrote down 52.5 as your answer and got halfway down the page before realizing that you needed to use that answer in “part fâ€? Three options. One, just use the rounded value of 52.5 to start the calculation in “part fâ€.

This is completely wrong. Two, put the calculation from “part a†back into your calculator so that you have the unrounded value to work with in “part fâ€. This is a complete waste of time. Three, write both the unrounded and rounded values when you complete “part aâ€?. This is by far the best option as it takes next to no time and it provides easy access to the unrounded value that you might need later.

Now, you do NOT need to write down every digit that your calculator displays. A general rule of thumb is to keep two digits more than sig figs would allow. calculation calculator write and write Example: 845. / 16.1 = 52..486 52.5 Rules: There are two distinct rules that you need to be able to use and keep straight. Addition and/or subtraction: The rule for addition and subtraction is based on the precision of the values being added and/or subtracted. When adding and/or subtracting values, the resulting answer MUST have the same precision as the LEAST precise value used in the calculation. For each value that you are adding or subtracting, you will need to determine the precision of the measurement.

The value that is least precise (has its precision furthest to the left) dictates the precision of your answer. (Note: Your book puts this rule in terms of decimal places, which works fine as long as all values have decimal places. This [incomplete] version of the addition/subtraction rule requires that you count the number of decimal places in each of the values. The answer must have the same number of decimal places as the value in the problem that has the FEWEST DECIMAL PLACES . What does that mean? If you add 5. decimal places), 12.123 (3 decimal places), and 0.12 (2 decimal places), your answer must have 2 decimal places.) Example: First, write the digits vertically with the decimal points lined up and find the number of decimal places for each value (this will help until you get more comfortable with the process).

The answer must have the same precision as the least precise value in the problem. 2500.0 is precise to the 1st decimal place, 1.236 to the 3rd decimal place, and 367.01 to the 2nd decimal place. The precision allowed in the answer is dictated by the first value (because that value is the least precise measurement), so you must round to that digit (the 2 in 2868.246 here). The answer is 2868.2 Example: So the answer is 0.0209 Example: The precision of the answer can only be to the 2nd decimal place. The answer is 0.00 Example: Again, your answer must have the same precision as the least precise value in the question; (3rd decimal place in this case).

The answer is 1.000 Example: When adding or subtracting, you need to first convert values in scientific notation into decimal notation, so this problem is actually – 28000 Again, your answer must have the same precision as the least precise value in the question; (the thousands–place for 28000 in this case), so the answer is or 7.188x106 Multiplication and/or division: The rule for multiplication and division is all about how many sig. figs. a number has. The value in the calculation that has the FEWEST number of SIGNIFICANT FIGURES determines the number of sig. figs. in your answer. If you are multiplying 3 different numbers, one has 4 s.f., one has 2 s.f. and one has 7 s.f., your answer can only have 2 s.f.

Example: 0.01116 x 23.44600 = 0..01116 has 4 s.f. and 23.44600 has 7 s.f. Therefore the answer is limited to 4 s.f. The answer would be rounded to 0.2617 Example: 26.375 x 3790 = 99961..375 has 5 s.f. and 3790 has 3 s.f., so the answer is again limited to 3 s.f. This is a fairly large number, so put it into scientific notation before rounding. It becomes 9.996125x104.

Now do your rounding and you get 10.0x104. There can only be one digit to the left of the decimal, so the final answer is 1.00x105 . Example: 3.14159 has 6 s.f. and 502000 has 3 s.f. so the answer can only have 3 s.f. The answer is 0. or 6.26x10–6 Examples: 536 has 3 s.f., 0.3301 has 4 s.f., 60.002 has 5 s.f., 0.0048 has 2 s.f., and 12.1 has 3 s.f., so the answer can only have 2 s.f. This is a large number, so put it into scientific notation BEFORE rounding 1.x105.

SINCE this is a very long and detailed worksheet, keep in mind the key points: Properly convert numbers in scientific notation, correctly determine significant figures based on rules for scientific notation, decimal presence, and measurement, and apply addition/subtraction rules based on decimal places while applying multiplication/division rules based on significant figures. Be consistent with the order of operations and delay rounding until the very end of calculations, ensuring accuracy in all steps.

Part 2: Dimensional analysis YOU MUST SHOW ALL OF YOUR WORK TO GET CREDIT!!!!

Use prior conversions and methodically cancel units to answer the following questions:

  1. The mass of a proton is 1.673x10-27 kg. What is the mass of a proton in grains? (1 grain = 64.79891 mg)
  2. The volume of an average gas chromatography injection is 1.5x10-6 L. What is that volume in drops? (1 drop = 0.0500 mL)
  3. Jules Verne wrote the book Twenty Thousand Leagues Under the Sea. If 1 league = 5.556 km and 1 furlong = 660.0 feet, how many furlongs did the Nautilus travel?
  4. The maximum depth of Lake Tahoe is 99.6 rod. How deep is Lake Tahoe in fathoms if 1 fathom = 0.364 rod?
  5. Red light has a wavelength of roughly 700 nm. What is the wavelength in meters?
  6. An epoch is a period of time related to the alignment of the sun and the moon. If 1 epoch = 19 years, how long is our 125-minute class period in epochs?
  7. It is 90.123 km from here to Stockton. Convert this distance using appropriate units, showing all steps.

Remember to show all work for each calculation, cancel units properly, and apply correct significant figure rules where applicable.

Paper For Above instruction

Understanding significant figures and scientific notation forms the foundation for precise scientific communication and data analysis. Accurate handling of these concepts ensures that measurements and calculations reflect the true precision of the instruments used and the limitations inherent in measurement processes. This comprehensive understanding is essential for anyone involved in scientific research, engineering, or any field requiring precise quantitative data. This paper explores the key principles of scientific notation, significant figures, precision, and the application of dimensional analysis.

Scientific Notation and Significant Figures

Scientific notation provides an efficient way to express very large or very small numbers succinctly. Converting between standard notation and scientific notation involves moving the decimal point until only a single non-zero digit remains to the left of the decimal. For large numbers, such as 450,000, we move the decimal five places left, leading to 4.5 x 105. Conversely, for small numbers like 0.000528, the decimal moves five places right, resulting in 5.28 x 10–4. These transformations clarify the scale of the number while maintaining precision.

Significant figures (sig figs) denote the digits in a number that carry meaningful contribution to its precision, including all non-zero digits, zeros between non-zero digits, and trailing zeros in decimal numbers. For example, 6.626 x 10–34 has four significant figures, while 8.30 x 10–1 has three because of the trailing zero in the decimal. The rules for sig fig counting differ slightly depending on the presence of decimals and scientific notation, requiring careful attention to detail.

Determining the Precision of Measurements

The precision of a measurement relates to the reproducibility of the measurement results. It depends on the instrument's smallest graduated unit, with smaller graduations providing higher precision. The position of the last significant figure indicates the measurement's precision. For instance, 1.032 has four sig figs, and its precision extends to the thousandths place because the last sig fig is in that position. Similarly, in scientific notation, the number of decimal places in the decimal form indicates precision, as seen in 3.750 x 10–6, which is precise to the ten-millionths place.

Rules for Calculations with Significant Figures

Significant figure rules guide how to handle precision during arithmetic operations. When adding or subtracting, the result must be rounded to match the least precise measurement (the one with the fewest decimal places). For example, adding 2500.0 (1 decimal place), 12.123 (3 decimal places), and 0.12 (2 decimal places) yields a result with 2 decimal places, the least among the inputs. For multiplication or division, the number of significant figures in the final answer is limited by the input with the fewest sig figs.

Handling Mixed Operations and Rounding

In complex calculations involving both addition/subtraction and multiplication/division, the order of operations must be respected, and rounding should be delayed until the final answer. Throughout calculations, intermediate results should be recorded with extra digits to avoid accumulation of rounding errors, and only at the

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