Embedded Files
For this assignment (adopted from The Chem Team website), I want you to complete your answers by creating a google doc at docs.dunmorestudents.net.
Change the title to "Significant Digits- ___ period" (where the blank is your period; first or third) and share it with me (bennettj@dunmorestudents.net).
1. Read the fable below and answer the questions that follow:
From the Chem Team website:
A student once needed a cube of metal which had to have a mass of 83 grams. He knew the density of this metal was 8.67 g/mL, which told him the cube's volume. Believing significant figures were invented just to make life difficult for chemistry students and had no practical use in the real world, he calculated the volume of the cube as 9.573 mL. He thus determined that the edge of the cube had to be 2.097 cm. He took his plans to the machine shop where his friend had the same type of work done the previous year. The shop foreman said, "Yes, we can make this according to your specifications - but it will be expensive."
"That's OK," replied the student. "It's important." He knew his friend has paid $35, and he had been given $50 out of the school's research budget to get the job done.
He returned the next day, expecting the job to be done. "Sorry," said the foreman. "We're still working on it. Try next week." Finally the day came, and our friend got his cube. It looked very, very smooth and shiny and beautiful in its velvet case. Seeing it, our hero had a premonition of disaster and became a bit nervous. But he summoned up enough courage to ask for the bill. "$500, and cheap at the price. We had a terrific job getting it right -- had to make three before we got one right."
"But--but--my friend paid only $35 for the same thing!"
"No. He wanted a cube 2.1 cm on an edge, and your specifications called for 2.097. We had yours roughed out to 2.1 that very afternoon, but it was the precision grinding and lapping to get it down to 2.097 which took so long and cost the big money. The first one we made was 2.089 on one edge when we got finshed, so we had to scrap it. The second was closer, but still not what you specified. That's why the three tries."
"Oh!"
1a. What is your reaction to the student's actions?
1b. Give another real-life example where this scenario could occur?
2. Rules for Rounding Off
Now that "everyone" has a calculator that will give a result to six or eight (or more) figures, it is important that we know how to round the answer off correctly. The typical rule taught is that you round up with five or more and round down with four or less.
THIS RULE IS WRONG!
However, please do not rush off to your elementary school teacher and read 'em the riot act!
The problem lies in rounding "up" (increasing) the number that is followed by a 5. For example, numbers like 3.65 or 3.75, where you are to round off to the nearest tenth.
OK, let's see if I can explain this. When you round off, you change the value of the number, except if you round off a zero. Following the old rules, you can round a number down in value four times (rounding with one, two, three, four) compared to rounding it upwards five times (five, six, seven, eight, nine). Remember that "rounding off" a zero does not change the value of the number being rounded off.
Suppose you had a very large sample of numbers to round off. On average you would be changing values in the sample downwards 4/9ths of the time, compared to changing values in the sample upward 5/9ths of the time.
This means the average of the values AFTER rounding off would be greater than the average of the values BEFORE rounding.
This is not acceptable.
We can correct for this problem by rounding "off" (keeping the number the same) in fifty percent of the roundings-even numbers followed by a 5. Then, on average, the roundings "off" will cancel out the roundings "up."
The following rules dictate the manner in which numbers are to be rounded to the number of figures indicated. The first two rules are more-or-less the old ones. Rule three is the change in the old way.
When rounding, examine the figure following (i.e., to the right of) the figure that is to be last. This figure you are examining is the first figure to be dropped.
- If it is less than 5, drop it and all the figures to the right of it.
- If it is more than 5, increase by 1 the number to be rounded, that is, the preceeding figure.
- If it is 5, round the number so that it will be even. Keep in mind that zero is considered to be even when rounding off.
Example #1 - Suppose you wish to round 62.5347 to four significant figures. Look at the fifth figure. It is a 4, a number less than 5. Therefore, you will simply drop every figure after the fourth, and the original number rounds off to 62.53.
Example #2 - Round 3.78721 to three significant figures. Look at the fourth figure. It is 7, a number greater than 5, so you round the original number up to 3.79.
Example #3 - Round 726.835 to five significant figures. Look at the sixth figure. It is a 5, so now you must look at the fifth figure also. That is a 3, which is an odd number, so you round the original number up to 726.84.
Example #4 - Round 24.8514 to three significant figures. Look at the fourth figure. It is a 5, so now you must also look at the third figure. It is 8, an even number, so you simply drop the 5 and the figures that follow it. The original number becomes 24.8.
When the value you intend to round off is a five, you MUST look at the previous value ALSO. If it is even, you round down. If it is odd, you round up. A common question is "Is zero considered odd or even?" The answer is even.
Here are some more examples of the "five rule." Round off at the five.
3.075
3.85
22.73541
0.00565
2.0495
This last one is tricky (at least for high schoolers being exposed to this stuff for the first time!). The nine rounds off to a ten (not a zero), so the correct answer is 2.050, NOT 2.05.
Would your teacher be so mean as to include problems like this one on a test? In the ChemTeam classroom, the sufferers (oops, I mean students) have learned to shout "YES" in unison to such easy questions.
Lastly, before we get to the problems. Students, when they learn this rule, like to apply it across the board. For example, in 2.0495, let's say we want to round off to the nearest 0.01. Many times, a student will answer 2.04. When asked to explain, the rule concerning five will be cited. However, the important number in this problem is the nine, so the rule is to round up and the correct answer is 2.05.
Practice Problems
Practice Problems
Round the following numbers as indicated.
To four figures: To the nearest 0.1: To the nearest whole number:
1) 2.16347 x 105 13) 3.64 25) 56.912
2) 4.000574 x 106 14) 4.55 26) 3.4125
3) 3.682417 15) 7.250 27) 251.7817
4) 7.2518 16) 0.0865 28) 112.511
5) 375.6523 17) 0.5182 29) 63.541
6) 21.860051 18) 2.473 30) 7.555
To two figures: To one decimal place:
7) 3.512 19) 54.7421
8) 25.631 20) 100.0925
9) 40.523 21) 1.3511
10) 2.751 x 108 22) 79.2588
11) 3.9814 x 105 23) 0.9114
12) 22.494 24) 0.2056
3. Using the internet as a resource
Being a student in the 21st century means there is no such thing as "I don't know". This is the because knowledge is so easy to come by using the internet. What I have noticed is there is a disparity between what teachers think students know about the internet and what students actually know. I am always surprised by the response I get when I ask this question, "How many of you have used the internet to learn a new math or science concept?" With this in mind I want you to see what I found in a few quick searches using google's search engine and also the delicious social bookmarking site.
Click on each of the three links below and try the practice quizzes for significant figures. After completing the quizzes, copy and paste this sentence with a few of your answers: Mr. Bennett, I tried the quizzes you assigned and I will think of the internet the next time I need help with a topic in any of my classes.
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