Significant figures and rounding worksheet answers chemistry

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Significant figures and rounding worksheet answers chemistry

Provide an example of a measurement with a clearly too many active digits for a learning goal and explain why. Describe the purpose of the rounding and describe the information that must be known in order to do so properly. Round the number to the specified number of consecutive digits. Describes how to round a number who has the second most important number of 9. Perform a simple calculation that includes two or more observations, and express the results with an appropriate number of significant numbers. The numbers we deal with in science (and many other aspects of life) represent measurements who their values are not exactly known. Our pocket calculators and computers don't know this. They treat the numbers we put into them as pure mathematical entities, and arithmetic often gives us physically ridiculous answers, even though they are mathematically correct. The purpose of this unit is to help you understand why this happens and show you what to do about it. Consider two statements: Our city has a population of 157,872. As of January 1, there were 27,833 registered voters. Which of these will be fired soon? certainly not the second one, it probably comes from a database containing one record for each voter, so the number is found simply by counting the number of records. The first statement may be correct. Even if you could define the population of a city in an accurate way (permanent resident? warm body?), how can you explain the minute-by-minute changes that occur when people are born and die, or when they move and move? What is the difference between the two population numbers above? It is very likely that the last census accurately got a record of 157,872, which could be the population of the city for legal purposes, but it is certainly not a true population. To better reflect this fact, we may list the population (e.g. Atlas) as 157,900 or 158,000. These two quantities are rounded to four significant numbers and three significant numbers, respectively, and have the following meaning: 157900 (where the important numbers are underlined) means that the population is considered to be in the range of about 157850 to about 157950. That is, the population is 157?50. When plus or minus 50 is added to this numerical value, the absolute uncertainty of the population measurement is considered to be 50 to (-50) = 100. Also, the relative uncertainty is 100/157900, which can be expressed as part 1 in 1579, 1579 = 0.000633, or about 0.06%. A value of 158000 means that the population is likely to be between158500, or 158000?500. An absolute uncertainty of 1000 translates into 158 1000/158000 or one part, or about 0.6% relative uncertainty. Which of these two values you report as population depends on your confidence in the original census numbers. If the census is completed last week, you might round it to four digits, but if it's a year or so ago, rounding it to three places might be a wiser choice. In such cases, there is no real objective way to choose between the two options. This makes an important point: the concept of significant numerals has less to do with mathematics than our confidence in measurement. This confidence is often expressed numerically (for example, the height of the liquid in the measuring tube can be read up to ?0.05 cm), but as in the population example, we must rely on personal experience and judgment. So what is an important number? At 0 (4 consecutive digits), the leftest 3 digits are known exactly, while the 4th 9 can be 8 if the true value is in the implicit range of 15855 to 155. At 158000 (3 digits), the leftest two digits are exactly what they are, but if the true value is within the implicit range of 157500 to 158500, the third digit can be either 7 or 8. Rounding always results in the loss of numerical information, but what we are getting rid of can be thought of as numerical noise that does not contribute to the quality of the measurement. The purpose of rounding is to avoid expressing values with rather than matching the uncertainty of the measurement. Implicit uncertainty: If you know that the balance is accurate within 0.1 mg, the uncertainty in the measurement of mass made in this balance will be ?0.1 mg. However, the object is 0.42 cm long and does not show accuracy. In this case, the number of digits in the data is required. Therefore, the quantity 0.42 cm specifies 0.01 units as 042 or 1 part of 42. The implicit relative uncertainty of this figure is 1/42, or about 2%. Therefore, the accuracy of numeric answers calculated from this value is limited to the same amount. It is important to understand that the number of significant digits in a value provides only a rough display of its accuracy, and that information is lost when rounding occurs. For example, you might measure the weight of an object on balance at 3.28 g.? that the data is accurate within 0.05 grams. The resulting value of 3.28?.05 grams indicates that the true weight of the object will be anywhere between 3.23g and 3.33g. The absolute uncertainty here is 0.1 g (?0.05 g), and the relative uncertainty is 1 part at 32.8, or about 3%. How many consecutive digits should be in the reported measurement? Since only the left 3 of 3.28 is certain, you can probably hope to round the value to 3.3 g. So far so good. But when you see it in your report, is someone else going to make this number? the value 3.3 g suggests an implicit uncertainty of 3.3?0.05 g, meaning that the true value is likely to be between 3.25 g and 3.35 g. This range is less than or less than 0.02g associated with the original measurement, so rounding introduces this amount of bias into the results. This is less ? 0.05 g of uncertainty in weighing, so it is not a very serious problem in itself. However, combining multiple values rounded in this way into a calculation can result in a large rounding error. The standard rules for rounding are well known. Before setting them up, let's agree on what to call the different components of numbers. The top digit is the leftest number (it only functions as a placeholder and does not count leading zeros that are not valid numbers). If you round to n active digits, the lowest digit is the nth digit of the top digit. The lowest digit can be zero. The first non-valid number is the n+1 digit. Rounding rule If the first non-active digit is less than 5, the lowest digit is not changed. If the first non-significant digit is greater than 5, the lowest digit increases by 1. If the first non-valid digit is 5, the lowest digit can be incremental or left unchanged (see below). All non-valid numbers are deleted. Students are sometimes told to increase the lowest digits by 1 for odd numbers and leave them unchanged if they are even. I wonder if this reflects the idea that even numbers are somehow better than the odd one! (Ancient superstitions are the exact opposite: only odd numbers are lucky.) In practice, you can increment only even numbers and do the same. It doesn't matter what you do if you just round a single number. However, if you round the set of numbers used in the calculation, processing the first insignificant 5 in the same way will over-or underestimate the value of the rounded number and accumulate rounding errors. Because even and odd numbers are equal, incrementing only one type does not result in this type of error being built.Of course, you can do the same by flipping the coin! Table \(\PageIndex{1}\): The example of a number of valid digits rounding is simply truncated because the result comment 34.216 3 34.2 The first non-active digit (1) is less than 5. 2.252 2 6.2 or 6.3 The first non-valid number is 5, so the minimum sig. number can remain unchanged or be incremental. 39.99 3 40.0 Crosses the hen-10 boundary, so all numbers change. 85,381 3 85,400 Two zeros are just placeholders 0.04597 3 0.0460 Two preceding zeros are not valid numbers. The object has a weight of 3.98 ? 0.05 g. It places its true weight somewhere in the range of 3.93 g to 4.03 g. When judging how to round this number, count the exact number of 3.98 digits and find nothing! 4 is the most left-hand number who who has an indeterrud value, which means that you should round the result to one significant number and report it as simply 4 g. Alternatively, you can bend the rule to round the two consecutive digits to generate 4.0 g. How can I decide what to do? Table \(\PageIndex{2}\) Rounded value Implicit minimum absolute uncertainty Relative uncertainty 3.98 g 3.985 g 3.985 g 3.975 g ?.005 g or 0.01 g 1 400, Or 0.25% 4g4.5g 3.5gg?.5g or 1g 1 4, 25% 4.0gg 4.05g 3.95g ?.05g or 0.1g 1 40, 2.5% obvious, Rounding to double digits is the only reasonable course in this example. Observations should be rounded to the number of digits that most accurately convey the uncertainty of the measurement. Typically, this means rounding to the number of consecutive digits in the quantity. In other words, the number of digits that are accurately recognized (counted from left) plus another number. If this cannot be applied (as in the example where the addition of absolute uncertainty subtraction ties a power of 10), round so that the relative implicit uncertainty of the result is as close as possible to that of the observation. If you are performing a calculation that contains multiple steps, you should not round until you have the final result. Using a calculator with rectangular areas calculator, the relative implicit uncertainty of rounded values 1.58 1 part 158, or 0.6% 1.6 1 part 16, or 6% Comment: Your calculator is of course correct as far as pure numbers go, but it is wrong to write down 1.57676 cm2 as an answer. Two options for rounding calculator answers appear on the right. It's clear that neither option is entirely satisfactory. If you round to three precision numbers, the answer is more accurate, but the following valuesRounding a rule to two digits of .42 has the effect of discarding some precision. In this case, it can be claimed that rounding to three digits is justified because the implicit relative uncertainty of the answer, 0.6%, more matches the uncertainty of the two factors. Rounding rules are generally useful and useful guidelines, but they do not always produce the most desirable results. If in doubt, it is better to resort to relative implicit uncertainty. Operations that contain important numbers report responses in a way that reflects the reliability of the least accurate operations. The answer is not the same as the most accurate number used to get the answer. For addition or subtraction, use the number of decimal places (the number of digits to the right of the decimal point) instead of the number of consecutive digits. Identifies the smallest number of decimal places and uses this number to set the number of decimal places in the answer. The result should contain the same number of significant numbers as the value with the fewest significant numbers. If a number is expressed in the form of ? 10b (scientific notation) with an additional limit of 1 to 10 coefficients a, the number is in a normalized format. Represents the bottom 10-number of values using the same number of significant numbers that exist in the normalized form of that value. Similarly, a counter-number (a number represented as a power of 10) uses the same number of significant numbers as that power. Example \(\PageIndex{1}\) The following example shows the problems that are most likely to occur when rounding calculation results. They deserve your careful study! rounded remarks 1.6 Rounded to two key numbers, which can create implicit uncertainty of 1/16 or 6%, three times the most accurately unknown factor. A diagram that shows how rounding can lead to loss of information. The 1.9E6 3.1 factor is specified as part 1 of 31, or 3%. In answer 1.9, the value is expressed in 1 part, or 5%, at 19. Since these accuracies are equal, the rounding rules have obtained reasonable results. The thickness of a book is 117mm. Find the stack height of 24 different books: 2810 mm 24 and 1 are accurate, so the only uncertain value is the thickness of each book given to the three effective numbers. The only 0 at the end of the answer is placeholders. 10.4 For addition or subtraction, look for the term with the smallest number of decimal places and round the answer to the same number of digits. The end of the example shown above 23cm [see below] represents a very common operation to convert one unit to another. There's a certain ambiguity.If you use 9 in to mean a distance of 8.5 to 9.5 inches, the implicit uncertainty is ?0.5 inches (1 part of 18, or about 6% ?). Because all values are multiplied by the same factor of 2.54 cm/in, the relative uncertainty of the answers must be the same. In this case, describing the answers to the two numbers creates ?1 cm of uncertainty. If you use the answer 20 cm (single digit), the implicit uncertainty is ?5 cm, or ?25%. If the appropriate number of consecutive digits is an issue, calculating relative uncertainty can help you make a decision. Contributor and Attribution Professor Emeritus Steven Lowery (Simon Fraser U.) Chem1 Virtual Textbook Textbooks

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