Hi everyone,
As the title says, this is an NYU essay assignment for General Chemistry I for Professor John Halpin. If you're not in this school, but also have to write a response on significant digits, this is it.
The following is the essay question. My response is pasted right after.
Good luck!
Required Length: 3 - 3000 words
Directions: In a logical, grammatically correct essay, describe to the reader the nature of significant figures and the procedures to be used concerning them in this course. Be sure to address the items in the "guiding questions" section, but remember that you are writing an essay, not just answering a list of questions.
In a numerical value, how do you know which digits are significant and which are not?
When performing a measurement, how you know how many significant digits to record?
Comment on the nature of the last significant digit in any measured value.
When you are given a numerical value that is the result of a measurement, what do the number of significant digits tell you about the precision of the measurement? About its accuracy?
What rules are followed in determining the number of significant digits in the answer to a calculation involving measured values?
What is an "exact" number? How does it affect calculations?
In a multi-step calculation, how should significant figures be handled?
How does the course policy on significant figures differ from that of the textbook?
How should you round an numerical result so that it contains the proper number of significant digits?
Follow the links to the web sites found in the "Source Material" section and use your textbook ("Chemistry: The Molecular Nature of Matter and Change", by Martin S. Silberberg) to gather information about handling significant figures. Write a comprehensive, coherent essay concerning the nature of significant figures, how they are recognized, how the are assessed after performing a calculation involving measured data, and how numbers are rounded to the proper number of significant figures. The "Guiding Questions" below are intended to help you to determine what concepts should be included in your essay, but do not simply write a list of answers to those questions - write an informative essay on the subject of significant figures.
My response:
The idea behind significant figures, sometimes called significant digits, is to verify that the amount of precision and answer is not overrepresented when making a big calculation, and thus, is not more precise than the numbers used to compute that answer. In other words, this is just a method to get everyone to understand and consent to one universal way to write a measurement and level of precision behind that measurement.
There are five rules used to determine the amount of significant figures in a number. The first rule is that non-zero digits are always significant. For example, 678 has three significant figures, while 8.3762 has five.
But here’s the tricky part: how about zeros? Well, depending on where the zero is placed, sometimes it can be considered significant, and sometimes it can not. By the second rule, zeros sandwiched between non-zero digits are always significant. Let’s say we have the number 8067. We know that the 8, the 6, and the 7 are all significant digits. However, from the second rule, since the zero is wedged between non-zero digits, it, too, is significant. So there is a total of four significant figures in this value.
Then again, a value doesn’t have to have only one zero, and the zero doesn’t have to be in one place. For example, let’s say we are given the number 206008. Well, since the 2, the 6, and the 8 are significant and since the zeros are sandwiched between them, they, too, are significant. So there is a total of 6 significant figures in this number. The same goes for decimal points. We can have 3.007009, but as long as we have 3, 7, and 9 (significant figures), the zeros between them are all significant. So there is a total of 7 significant figures in this number.
The third rule is that zeros that come before all non-zero digits are never significant. For example, let’s say we are given the number 0.000876. All the zeros that are before, or to the left of, the non-zero digits (4, 9, and 1), are not significant. So there is a total of 3 significant figures in this number. In simple terms, it doesn’t matter how many zeros you have. None of those zeros before the non-zero digits are significant.
However, that rule is used when we have zeros that are before, or to the left of, non-zero digits. What about zeros that come after, or to the right of, non-zero digits? By the fourth rule, if we have zeros that come after non-zero digits, and there is no decimal place in that number, the zeros are not significant. For example, 66,000 has only 2 significant figures. But, if we have zeros that come after non-zero digits, and there is no decimal place in that number, the zeros are significant. For example 8800. has 4 significant figures, while 88.000 has 5 significant figures. In other words, count trailing zeros only if the value has a decimal point.
Lastly, the fifth rule is that for numbers in scientific notation, disregard the exponent and apply the previous rules to the value. For example, 5.6078 x 10^16 has 5 significant figures. Notice the different methods the textbook and lecture notes use to determine significant figures for trailing zeros. While the textbook says that placing a decimal point to the right of the zeros will make the zeros significant (for example, 300. has three significant figures), what we do is use scientific notation to show which zeros are significant (for example, 3.00 x 10^2 has three significant figures). The reason for this method is to avoid ambiguous zeros. For example, if we measure the length of a cabinet and obtain a measurement of 500 cm, who’s to say that the measurement isn’t 499 cm or 501 cm, or even 501.3 cm, long? Since the zeros of 500 cm (with one significant digit) appear ambiguous, we must write it in scientific notation to reduce this ambiguity.
When you make a calculation, you use the right number of significant figures in order to present information about how precise it is. But if in doubt about whether a number is significant, assume that it isn't. For example, if a question asked for 800 L of water, assume the answer to that question is known to one significant figure. In the end, it’s up to you to determine how many significant figures are left, since what you put down presents not only the measurement but also information regarding its quality.
In many measurements, the way to calculate the number of significant figures is to include one estimated digit beyond the values shown on the scale. Why? For example, if we have to measure the length of a table with a ruler, and our essential measurement is 105 inches, we would have to estimate an extra digit since “105” tells us that the random uncertainty is a few parts in the value. That is, the larger the number of significant figures, the more precise the result.
So, this brings in the question whether the idea of significant figures link with precision or accuracy. Well, imagine you were asked to measure the width of a bed several times. Since the ruler you use doesn’t change, you will most likely end up with similar measurements again and again. This is precision, which refers to how close your measurements are to each other. What about accuracy? Well, you don't know if the ruler you used was accurate or not, unless you compare it against a ruler you knew was accurate. For all you know, the ruler may be damaged or stretched, and therefore, impossible for you to know if the values you measured agree with the correct value. Through this, you can see that significant figures are associated only with precision.
Another point worth mentioning is that there are two types of numbers: exact and inexact. Basically, exact numbers do not affect calculations of significant figures, while inexact numbers do. For example, there are exactly 12 eggs in a dozen. An inch has exactly 2.54 cm. And, there are exactly 20 people in a room, not 20.5 or 19.567 people. On the other hand, an inexact number is any measurement. For example, if I measure the length of a piece of paper, I might get 340 mm, or if I am more precise, 341 mm, or even 341.2 mm. In other words, the values I get will never be exact.
This leads to rounding. When rounding with significant figures, there are a few rules used to help retain precision in the final measurement. First, round only at the very end since rounding early means losing significant figures for use in the next calculation. Second, while the textbook wants us to drop the trailing digits followed by a 5 if the last significant digit is even, and round up if the last significant digit is odd, we are not. For our case, if the last significant figure is followed by a 5, 6, 7, 8, or 9, round up. For example, if we have to round 7.688 to to three significant digits, the result will be 7.69. Likewise, if the last significant digit is followed by a 0, 1, 2, 3, or 4, then drop the trailing digits. For example, if we had to round 65,103 to three significant digits, the result will be 65,100, namely, 6.51 x 10^4.
Now that you are familiar with measuring one value, what if we add, subtract, multiply, or divide multiple significant figures? A simple answer is that there are also rules you must use to compute these numbers.
When you only add/subtract measurements, your result can only have as many significant digits as the least significant decimal place of the measurement. For example, if we have to calculate 12.037 - 3.93, our answer can only have 2 significant figures after the decimal point, which in this case, is 8.11.
When you only multiply/divide measurements, your result, without consideration of the decimal point, can only have as many significant digits as the smallest significant number that was used in the calculation. For example, if we have to calculate 2.55 x 4.5, our answer can only have 2 significant figures, which in this case, is 11.
Now what about a multi-step calculation? How would it be handled then? Well, if you are calculating a problem that involves both addition/subtraction and multiplication/division, carry out the arithmetic as you normally would, but keep track of your significant figures during each calculation. In the end, round using the multiplication/division rule. For example, let’s say we have to calculate 3.489 x (5.67 - 2.3). If you can remember from basic math, PEMDAS (parentheses, exponents, multiplication, division, addition, and subtraction) says you have to do the parentheses first. Hence, solving for 5.67 - 2.3, we get 3.37. However, we do not round to two significant digits now because rounding to 3.4 might produce a round off error. Although we use all of our numbers during the calculation, we keep track of the number of significant figures, in this case 2, to use for round off at the very end. Now, solving for 3.489 x 3.37, we get 11.7579. But this isn’t the final answer. Going back to the calculation, 3.489 has 4 significant figures, while 3.37 is supposed to have 2 (as we have mentally marked before). Taking the least number of total significant figures (multiplication/division rule), 11.7579 rounded to two significant figures become 12.
Once again, it’s crucial to remember to only round off at the very end of a calculation to avoid a round off error. Keep at least two extra digits during your calculation for better results.
