Significant Figures From Addition Or Subtraction:
When adding or subtracting, the number of decimal places in the answer is equal to the operand with the smallest number of decimal places.
Consider 12.346 + 102.34. If we use a calculator to evaluate this sum, we get:
The number of decimal places in our answer should be two, since the operand with the least decimal places is 102.34, and this has two decimal places. We should therefore round the result to two decimal places.
What Constitutes A Significant Figure
First, lets review these criteria that define sig figs. We can classify numbers as significant figures if they are:
- All digits comprising N are significant in accordance with the rules above
- Neither 10 nor x are significant
Specific amounts of precision, designated by significant figures, must appear in our mathematical calculations. These appropriate degrees of precision vary, corresponding to the type of calculation being completed.
To determine the number of sig figs required in the results of certain calculations, consult the following guidelines.
Counting Significant Digits Examples
1. 23.5 Three significant digits as all are non-zero numbers .
2. 23.50 Four significant digits. The final zero is significant because the number contains a decimal place .
3. 402 Three significant digits. The zero is between non-zero numbers .
4. 5,200 Two significant digits. There is no decimal place so the trailing zeros are simply placeholders and not-significant .
5. 0.030 Two significant digits. Leading zeros are never significant . The trailing zero is significant because the number contains a decimal place .
6. 0.0070080 Five significant digits. Leading zeros are never significant . The two zeros between 7 and 8 are significant because they are between non-zero numbers . The trailing zero is significant because the number contains a decimal place .
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How To Use Significant Figures In Calculations
When adding or subtracting numbers, the answer must be expressed with the same number of decimal places as the participating number with the least number of decimal places.
When multiplying or dividing numbers, the answer must be expressed with the same number of significant figures as the participating number with the least number of significant figures.
Exercise 15: 2.187 + 10.2 = ?
The calculator gives the answer 12.387. Were adding here, so the rule of least number of decimal places applies. One number has 3 decimal places, and the other has 1 decimal place. The answer should, therefore, have 1 decimal place, so the answer is 12.4.
Exercise 16: 178.1 2.08 + 15 = ?
The calculation involves addition/subtraction, so we are applying the rule of least number of decimal places. The three participating numbers, 178.1, 2.08 and 15, have 1, 2 and 0 decimal places, respectively, so the answer should have no decimal place. The calculators answer is 191.02, but were rounding it off to 191.
Exercise 17:/31.0= ?
Here, the calculation involves multiplication and division, so the answer has the same sig figs as the number with the least number of sig figs. The numbers 0.081, 1090 and 31.0 have 2, 4 and 3 sig figs, respectively. The answer should have only 2 sig figs. The calculators answer is 2.848064516. We need to round off the answer to 2.8.
Exercise 18: What is the perimeter of a square with a side measuring 10.5 inches?
Exercise 19: /1.25= ?
Significant Figures And Units
Overview: In reporting numerical results, it is important to include the correct number of significant digits. While determining the correct number of digits to include is a straightforward process, beginning students often overlook this important detail. Here we outline the rules involved in determining the appropriate number of digits to include when reporting results of calculations and experimental measurements.
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Calculations Involving Multiplication/division And Addition/subtraction
In practice, chemists generally work with a calculator and carry all digits forward through subsequent calculations. When working on paper, however, we often want to minimize the number of digits we have to write out. Because successive rounding can compound inaccuracies, intermediate rounding needs to be handled correctly. When working on paper, always round an intermediate result so as to retain at least one more digit than can be justified and carry this number into the next step in the calculation. The final answer is then rounded to the correct number of significant figures at the very end.
What Are Significant Figures
In chemistry, Significant figures are the digits of value which carry meaning towards the resolution of the measurement. They are also called significant figures in chemistry.
All the experimental measurements have some kind of uncertainty associated with them. In order to ensure precision and accuracy in measurements and get real data, a fixed method to compensate for these uncertainties was required and this led to the significant figures. Before moving on to significant figures, lets discuss the difference between precision and accuracy.
Significant Figures Chemistry
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Rules For Rounding Off
Before we move to addition, subtraction, multiplication, and division, it is necessary to understand the rounding off numbers. There are three rules which are explained below.
If the digit to be dropped is less than 5 , drop it and all the digits right to it. The preceding digit to dropped digit remains unchanged. The numbers which are dropped before the decimal point are replaced with zeros.
Round off number 864.73192 to 4 significant figures. To make the number to four significant number, we must drop number 3 and all the numbers right to it . Since number 3 is less than 5, the preceding number remains unchanged as per rule. Therefore, the round off figure is 864.7.
If the same number is round off to 2 significant figures, we must drop 4 and all the numbers following it . Since number 4 is less than 5, the preceding number remains unchanged. Also, 4 is replaced by 0, because it is before the decimal. The new round off number is 860.
If the digit to be dropped is more than 5 , drop it and all the digits right to it. And importantly, the preceding number is increased by one. The numbers which are dropped before the decimal point are replaced with zeros.
Consider rounding off the same number, 864.73192, to 6 significant figures. We have to drop number 9 and all the numbers following it. Since 9 is more than 5, preceding number 1 is up by one. Thus, the round off number is 864.732.
For 1 significant figure, the round off number will be 900.
Rules To Identify Significant Figures In A Number
Note that identifying the significant figures in a number requires knowing which digits are reliable since only reliable digits can be significant e.g., 3 and 4 in 0.00234 g are not significant if the measurable smallest weight is 0.001 g.
- Non-zero digits within the given measurement or reporting resolution are significant.
- 91 has two significant figures if they are measurement-allowed digits.
- 123.45 has five significant digits if they are within the measurement resolution. If the resolution is 0.1, then the last digit 5 is not significant.
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Rounding And Truncating Numbers
There are different methods which may be used to round numbers. The usual method is to round numbers with digits less than 5 down and numbers with digits greater than 5 up .
Example:If you are subtracting 7.799 g – 6.25 g your calculation would yield 1.549 g. This number would be rounded to 1.55 g because the digit ‘9’ is greater than ‘5’.
In some instances, numbers are truncated, or cut short, rather than rounded to obtain appropriate significant figures. In the example above, 1.549 g could have been truncated to 1.54 g.
Limitations And Complications Of Significant Digits
Using Significant Digits can be a good simple way to introduce students to the idea of measurement errors. It allows us to begin to relate the measurement scale to measurement quality, and does not require much math to implement. However, Significant Digits are only an approximation to true error analysis, and it is important to avoid getting bogged down in trying to make Significant Digits work well when they really don’t.
One type of difficulty with Significant Digits can be seen with reading a scale to the nearest “tenth”. In this case, 1.1 and 9.1 are both proper measurements. If we assume for simplicity that each measurement is good to +/- 0.1, the uncertainty in the first measurement is about 10% and the uncertainty in the second measurement is about 1%. Clearly, simply saying that both numbers are good to two Significant Digits is only a rough indication of the quality of the measurement.
Further, Significant Digits does not convey the magnitude of the reading uncertainty for any specific scale. The common statement, which I used in the previous paragraph, is that readings are assumed to be good to 1 in the last place shown. But on some scales, it would be much more realistic to suggest that the uncertainty is 2 or even 5 in the last place shown. A similar problem can occur when the errors from many numbers are accumulated in one calculation. Example 10 illustrated this.
How Many Significant Digits Does 30800 Have
five significant figures1) 3.0800 five significant figures.
How many significant digits does 0.0100 have?
0.0100 contains three significant figures. Therefore, the decimal portion of the log answer contains three significant figures.
What are three significant digits?
The third significant figure of a number is the digit after the second significant figure. This is true even if the digit is zero, and so on. Hence the third significant figure of 20,499 is 4 and the fourth is 9, as are the third and fourth significant figures of 0.0020499.
What makes a number a significant number in chemistry?
Rules For Determining If a Number Is Significant or Not. All non-zero digits are considered significant. For example, 91 has two significant figures , while 123.45 has five significant figures . Zeros appearing between two non-zero digits are significant.
How to count significant figures in chemistry rule 1?
How to Count Significant Figures Rule 1 : All non-zero digits are significant. This rule should be simple, if the digit is not zero, count it as significant.
How to determine the number of significant digits?
General Rules for Determining Number of Significant Figures 1 All nonzero digits are significant. 2 Zeros are also significant with two exceptions: zeros preceding the decimal point. zeros following the decimal point and preceding the first nonzero digit. 3 Terminal zeros preceding the decimal point in amounts greater than one is an ambiguous case.
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Rounding: What To Do With A Final 5
There are two points to be made here. The first is to make sure that the final 5 really is a final 5. And then, if it is, what to do.
Is the final 5 really a final 5? This might seem to be simple enough, but with common calculators it is easy to be misled. Calculators know nothing about Significant Digits how many digits they display depends on various things, including how you set them. It is easy for a calculator to mislead you about a final 5. For example, imagine that the true result of a calculation is 8.347, but that the calculator is set to display two decimal places . It will show 8.35. If you want 2 Significant Digits, you would be tempted to round to 8.4. However, that is clearly incorrect, if you look at the complete result 8.347, which should round to 8.3 for 2 Significant Digits. How do you avoid this problem? If you see a final 5 that you want to round off, increase the number of digits displayed before making your decision.
What to do if you really have a final 5. There are two schools of thought on this.
- Some people will suggest that you always round a final 5 up.
- Others will suggest that you round it up and down each half of the time the usual way to do this is to round a final 5 to make the previous digit an even number. For example, 0.35 becomes 0.4 and 0.65 becomes 0.6.
Rules For Addition And Subtraction Calculations:
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Scale Reading: Digital Instruments
Some modern measuring instruments have a digital scale. Electronic balances are particularly common. How do you know how many Significant Digits to write down from a digital scale? Good question. Most such instruments will display the proper number of digits. However, you should watch the instrument and see if that seems reasonable. Remember that we usually estimate one digit beyond what is certain. With a digital scale, this is reflected in some fluctuation of the last digit. So if you see the last digit fluctuating by 1 or 2, that is fine. Write down that last digit you should try to write down a value that is about in the middle of the range the scale shows.
If the fluctuation is more than 2 or so in the last digit, it may mean that the instrument is not working properly. For example, if the balance display is fluctuating much, it may mean that the balance is being influenced by air currents — or by someone bumping the bench. Regardless of the reason, a large fluctuation may mean that a displayed digit is not really significant.
Significant Figures And Rounding
- Give an example of a measurement whose number of significant digits is clearly too great, and explain why.
- State the purpose of rounding off, and describe the information that must be known to do it properly.
- Round off a number to a specified number of significant digits.
- Explain how to round off a number whose second-most-significant digit is 9.
- Carry out a simple calculation that involves two or more observed quantities, and express the result in the appropriate number of significant figures.
The numerical values we deal with in science represent measurements whose values are never known exactly. Our pocket-calculators or computers don’t know this they treat the numbers we punch into them as “pure” mathematical entities, with the result that the operations of arithmetic frequently yield answers that are physically ridiculous even though mathematically correct. The purpose of this unit is to help you understand why this happens, and to show you what to do about it.
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Rules For Determining If A Number Is Significant Or Not
- All non-zero digits are considered significant. For example, 91 has two significant figures , while 123.45 has five significant figures .
- Zeros appearing between two non-zero digits are significant. Example: 101.12 has five significant figures: 1, 0, 1, 1, and 2.
- Leading zeros are not significant. For example, 0.00052 has two significant figures: 5 and 2.
- Trailing zeros in a number without a decimal are generally not significant . For example, 400 has only one significant figure . The trailing zeros do not count as significant.
- Trailing zeros in a number containing a decimal point are significant. For example, 12.2300 has six significant figures: 1, 2, 2, 3, 0, and 0. The number 0.000122300 still has only six significant figures . In addition, 120.00 has five significant figures since it has three trailing zeros. This convention clarifies the precision of such numbers. For example, if a measurement that is precise to four decimal places is given as 12.23, then the measurement might be understood as having only two decimal places of precision available. Stating the result as 12.2300 makes it clear that the measurement is precise to four decimal places .
- The number 0 has one significant figure. Therefore, any zeros after the decimal point are also significant. Example: 0.00 has three significant figures.
- Any numbers in scientific notation are considered significant. For example, 4.300 x 10-4 has 4 significant figures.