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Significant Figures { Precision in Measurements Significant Figures When you make a measurement in a laboratory, there is always some uncertainty that comes with that measurement. By reporting the number with a particular number of digits, you show how precise your measurement was. For example – 2 people determine the mass of a block of iron. One is able to report it as 23 grams. The other is able to tell us that it is 23.965 grams. Which number was more precise? Significant Figures and Precision The 2nd report of 23.965 grams is more precise because it has more digits shown. The number of digits that you can use depends on what you are measuring with. Some instruments are very precise (and therefore let us report more digits) where as some instruments are not precise (and therefore let us report fewer digits) Reporting Precision How precise your number is depends entirely on what instrument you use. For example, think of using a ruler to measure a piece of string. The more markings on our ruler, the more precise our measurements can be. What are the measurements from the following rulers? decimetres centimetres millimetres Precision and Significance So we know that the more digits that are present, the more precise our number is. This doesn’t mean that we can just add more digits to make it more precise. We have to use rules to determine which numbers are significant and how many digits we can use Significant Figures In a reported number, there are rules to use to tell which numbers are significant and which aren’t. All numbers that are not ZERO are significant. (1-9) 1. 2. Example – the number 421 has 3 significant digits because it has a 4, a 2, and a 1 Zeros that are in between significant figures are also significant. Example – in the number 4021, the zero is significant because it is in between a 4 and a 2. This number has 4 significant figures. In the number 0421, the zero is not significant. This number has 3 significant figures. Examples How many significant figures in each number? 1. 967 5. 0967.05 2. 9.67 6. 0032.004 3. 90.67 7. 0709.08 4. 9.607 Significant Figures 3. Zeros after (to the right of) a decimal point and after (to the right of a number) are significant. 4. Example – in the number 42.0, the zero is significant. This number has 3 significant figures. In the number 42.10000, all of the zeros are significant. This number has 7 significant figures A zero before a decimal is not significant Example – in the number 0.421, the zero is not significant. This number has 3 significant figures. Examples How many sig figs? 1. 3.20 5. 00.67 2. 60.0320 6. 0.4080 3. 06.200 7. 0.400000100 4. 0.5607 8. 0.010 Significant Figures 5. Zeros to the right of a decimal but before the number are not significant Example – In the number 0.000421, all of the zeros are not significant. This number has 3 significant figures. 6. Zeros after the number but before the decimal can be confusing as they might be significant and they might not be significant. Example – In the number 421000, it could be read as having 3, 4, 5, or even 6 significant digits. It is impossible to tell. Numbers such as this should be written in scientific notation to avoid confusion Examples How many sig figs? Rewrite these numbers into proper scientific notation. 1. 0.0097 5. 07.00420 2. 0.0200 6. 0.90007 3. 4300 7. 0.003420 4. 52.0067 8. 0.001 Adding and Subtracting Rules As well as having rules for reporting the number of significant figures, we also have rules to tell us how to add and subtract using significant figures. When adding and subtracting using significant figures, your answer should have the same number of decimal places as the number with the least number of decimal places. Example 3.461728 14.91 0.980001 + 5.263 24.614729 If you are adding (or subtracting) 2 or more numbers, first look at the number of decimal places for each. Next, add or subtract normally. Finally, round off your number to the same number of decimals as the lowest number of decimals used at the start. → 6 numbers after decimal → 2 numbers after decimal → 6 numbers after decimal → 3 numbers after decimal Round off to 2 decimal places because 14.91 has the least number of decimal places. 24.61 The answer is 24.61 Examples What is the solution? How many sig figs does the solution have? 1. 3.298 + 0.14536 5. 3.298 - 0.14536 2. 645.95 + 273.7 6. 645.95 - 273.7 3. 50.72334 + 13.214 7. 50.72334 - 13.214 4. 2.00 + 1.0300 8. 2.00 - 1.0300 Multiplying and Dividing Rules When you multiply and divide, your final solution should have the same number of significant figures as the number used to calculate the solution with the lowest number of significant figures. Example First, determine how many significant figures are in the numbers provided. Next, multiply or divide normally. Finally, round off your solution so that it has the same number of significant figures as the number with the smallest amount of significant figures. 3.6 → 2 significant figures 7.63 → 3 significant figures 0.245 → 3 significant figures x 4.671 → 4 significant figures 31.43424186 Because 3.6 has the least significant figures (2), our answer must have the same number of significant figures (2). 31 The answer is 31. Examples What is the solution? How many sig figs does it have 1. 13.2 x 4.l 5) 13.2 ÷ 4.1 2. 6.5 x 9.0321 6) 6.5 ÷ 9.0321 3. 0.0023 x 611.59 7) 0.0023 ÷ 611.59 4. 94.60 x 0.0900 8) 94.60 ÷ 0.0900 BEDMAS Rules When you must evaluate an expression with both addition and multiplication, the rules of BEDMAS still apply, but you must determine the correct number of significant digits at every step along the way to the final answer Although you need to keep track of the sig figs along the way, you must not round off your any number until you reach your final answer! BEDMAS Examples 1) 150.0 x 14.5 - 1.0032 2) 750 ÷ 2.0 + 8 3) 4) (14.32 + 12.5) x (2.77 ÷ 0.9910) (105.20 - 92.5) x 5500.3