An answer is no more precise that the least precise number used to get the answer. When adding or subtracting, we go by the number of decimal places i. Identify the quantity having the smallest number of decimal places, and use this number to set the number of decimal places in the answer. The result must contain the same number of significant figures as in the value having the least number of significant figures.
Express the base logarithm of a value using the same number of significant figures as is present in the normalized form of that value. Similarly, for antilogarithms numbers expressed as powers of 10 , use the same number of significant figures as are in that power. The following examples will illustrate the most common problems you are likely to encounter in rounding off the results of calculations.
They deserve your careful study! A certain book has a thickness of mm; find the height of a stack of 24 identical books:. The last of the examples shown above represents the very common operation of converting one unit into another.
There is a certain amount of ambiguity here; if we take "9 in" to mean a distance in the range 8. The relative uncertainty in the answer must be the same, since all the values are multiplied by the same factor, 2. When the appropriate number of significant digits is in question, calculating the relative uncertainty can help you decide. Chem1 Virtual Textbook. Learning Objectives 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. Digits: Significant and otherwise Consider the two statements shown below: "The population of our city is , These two quantities have been rounded off to four and three significant figures, respectively, and the have the following meanings: 00 the significant digits are underlined here implies that the population is believed to be within the range of about 50 to about Implied Uncertainty If you know that a balance is accurate to within 0.
Rounding Error It is important to understand that the number of significant digits in a value provides only a rough indication of its precision, and that information is lost when rounding off occurs. Rules for Rounding The standard rules for rounding off are well known. The most significant digit is the left most digit not counting any leading zeros which function only as placeholders and are never significant digits.
If you are rounding off to n significant digits, then the least significant digit is the n th digit from the most significant digit.
The least significant digit can be a zero. Rounding-off rules If the first non-significant digit is less than 5, then the least significant digit remains unchanged. If the first non-significant digit is greater than 5, the least significant digit is incremented by 1.
If the first non-significant digit is 5, the least significant digit can either be incremented or left unchanged see below! All non-significant digits are removed. Fantasies about fives Students are sometimes told to increment the least significant digit by 1 if it is odd, and to leave it unchanged if it is even. Rounding up the Nines Suppose that an object is found to have a weight of 3. Rounding the Results of Calculations When carrying out calculations that involve multiple steps, you should avoid doing any rounding until you obtain the final result.
Instead, I just replace the final four digits with zeroes, to get:. To round 0. The zero between the decimal point and the 7 is not significant, as it serves only to "place" the 7 into the hundreds place. It provides no information about the accuracy of the following digits.
There are only three more digits, so all of them will be included in my answer. Since no digit follows the 4 , there is no information about rounding, so I'll just leave the 4 as it is.
When rounding 0. Since 4 is less than 5 , I won't round up. Because these sig-digs are after the decimal point, I will not replace the 4 with a zero. If I tacked a zero after the 8 , I would be adding an improper digit, as it would be "significant" according to the rules but wrong according to the original number they gave me.
Instead, I drop that last digit entirely. Since the 2 is followed by an 8 , I'll round the 2 up to 3 ; I'll drop everything that follows. The first significant digit in The next three digits are 3 , 1 , and 4. Since the 4 is followed by a 5 , I'll round the 4 up to 5. Because the original 5 came after the decimal point, I'll drop that digit and place from my answer.
To round Because the 1 is followed by a 4 , I won't round the 4 up. I'll drop everything after the decimal point. Zeros after non-zero digits within a number without decimals are not significant. Zeros after non-zero digits within a number with decimals are significant. Count from the first significant figure to the specified number.
If the decider is 5 or above, increase the previous value by 1. This is best explained with an example. The following numbers are rounded to 2 s.
Katie Ross. Sign up now. All rights reserved. Interested in a subscription?
0コメント