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Multiplying / Dividing / Trigonometric Functions a) First perform all the operations, even if changing from one formula to another. b) Round off the result so that it has the same number of sig figs as the least of all those used in your calculation. Example: (2.5 m) x (2.01 m) x (2.755 m) = 13.843875 m Answer = 14 m (2 sig figs) 3. Addition / Subtraction a) First perform all the operations. b) Round off your result so that you include only 1 uncertain digit. The last digit of any measurement is considered uncertain. When an uncertain digit is added to (or subtracted from) a certain digit, the result is an uncertain digit. (UNCERTAIN DIGITS ARE HIGHLIGHTED) Example: 153. ml + 1.8 ml + 9.16 ml = 163.96 ml Answer = 164 ml (3 sig figs; only 1 uncertain digit) Notice that the answer is rounded to the same precision as the least precise measurement, which was 153. ml 4. Multiplication / Division combined with Addition / Subtraction First, follow the order of operations that you learned in math. Use the appropriate sig fig rules, as stated above, depending on which operation you are performing at that time. (Example: 1. multiply/divide/trigonometric functions; or 2. add/subtract functions) At the end of each step, you must ask yourself, "What is the next operation that I will perform on the number that I just calculated?" If the next operation is in the same group of operations that you just used, (Example: 1. multiply/divide/trigonometric; or 2. add/subtract) then do NOT round off yet. If the next operation is from the other group, then you must round off that number before moving on to the next operation. 5. Exact Values All exact values or conversion factors have an infinite (never ending) number of significant figures. They are called exact values because they are measured in complete units and are not divided into smaller parts. You might count 8 people or 9 people but it is not possible to count 8.5 people. Examples of exact values: 12 complete waves ; 17 people ; 28 nails Examples of exact conversion factors: 60 s / minute ; 1000 m / km ; 12 eggs / dozen; 7 days / week There are exactly: 60 seconds in one minute 1000 meters in one kilometer [this is the definition of kilo (k)] 12 eggs in one dozen 7 days in one week 6. Inexact Values All inexact conversion factors or constants will be treated like measurements. They are called inexact because they are not exact like above. This means that there isn't an exact number to work with. It requires a fraction that creates a number with several digits after the decimal. Examples of inexact constants: c = 3.00 x 108 m/s (3 sig figs) This number is rounded off from 2.99876... because it is easier to work with p = 3.14 (3 sig figs) This number is rounded off from 3.1415926535... because it is easier to work with p = 3.14159 (6 sig figs) Examples of inexact conversion factors: 0.6 miles / km (1 sig fig) 0.62 miles / km (2 sig figs) 7. Rules Specific for Zeroes Rule Examples Zeros appearing between nonzero digits are significant 40.7 L has three sig figs 87 009 km has five sig figs Zeros appearing in front of nonzero digits are not significant 0.095 987 m has five sig figs 0.0009 kg has one sig fig Zeros at the end of a number and to the right of a decimal are significant 85.00 g has four sig figs 9.000 000 000 mm has 10 sig figs Zeros at the end of a number but to the left of a decimal may or may not be significant. If such a zero has been measured, or is the first estimated digit, it is significant. On the other hand, if the zero has not been measured or estimated but is just a placeholder, it is not significant. A decimal placed after the zeros indicates that they are significant. 2000 m may contain from one to four sig figs, depending on how many zeros are placeholders. 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