Calculations with significant figures

Calculations with significant figures So now you know how to determine the number of significant figures in a number and how to round numbers off, what good does that do?

Calculations with significant figures Well, let’s consider the rectangle below. If we want to determine the area of the rectangle, the easiest way would be to measure the length and the height, and then multiply these two numbers together. (For rectangles, Area = Length x Height.)

Calculations with significant figures First let’s measure the length: a correct reading might be something like 36.3 cm (…or 36.2 cm or 36.4 cm) 50 40 30 20 0 10 cm

20 0 10 cm Calculations with significant figures Now let’s measure the height: for height, a correct reading might be something like 6.7 cm (…or 6.6 cm or 6.8 cm).

Calculations with significant figures So let’s multiply these two numbers:

Calculations with significant figures So let’s multiply these two numbers: 36.3 x 6.7 = 243.21 (and cm x cm = cm2). So we have 243.21 cm2. But if we state the area to be 243.21cm2, we are stating a pretty high level of precision.

Calculations with significant figures But if we say the area is 243.21cm2, we are saying we know the area to a very high level of precision.

Calculations with significant figures But if we say the area is 243.21cm2, we are saying we know the area to a very high level of precision. We are saying that we are certain of the “243.2..” and that we are guessing the “1.”

Calculations with significant figures But if 36.2 cm long and 6.6 cm high were also correct measurements, then 36.2 cm x 6.6 cm = 238.92 cm2 would have to be a correct area for the same rectangle.

Calculations with significant figures But if 36.2 cm long and 6.6 cm high were also correct measurements, then 36.2 cm x 6.6 cm = 238.92 cm2 would have to be a correct area for the same rectangle. That implies we are certain of the “238.9…” and only guessing the “2.”

Calculations with significant figures And if 36.4 cm long and 6.8 cm high were also correct measurements, then 36.4 cm x 6.8 cm = 247.52 cm2 would also have to be a correct area for the same rectangle.

Calculations with significant figures And if 36.4 cm long and 6.8 cm high were also correct measurements, then 36.4 cm x 6.8 cm = 247.52 cm2 would also have to be a correct area for the same rectangle. That implies we are certain of the “247.5…” and only guessing the “2.”

Calculations with significant figures 243.21 cm2, 238.92 cm2 and 247.52 cm2

Calculations with significant figures 243.21 cm2, 238.92 cm2 and 247.52 cm2 These three values cannot all be correct.

Calculations with significant figures 243.21 cm2, 238.92 cm2 and 247.52 cm2 These three values cannot all be correct. The only digit that seems to be definite is the first “2” (in the hundreds place). After that the values are not at all consistent with one another.

Calculations with significant figures This would mean that our guess should be the second digit (in the tens place), and that the values should all be rounded there – to two significant figures.

Calculations with significant figures 243.21 cm2 rounds to 240 cm2 238.92 cm2 rounds to 240 cm2 and 247.52 cm2 rounds to 250 cm2.

Calculations with significant figures 240 cm2, 240 cm2 and 250 cm2.

Calculations with significant figures 240 cm2, 240 cm2 and 250 cm2. These are all consistent with one another.

Calculations with significant figures 240 cm2, 240 cm2 and 250 cm2. These are all consistent with one another. They all have two significant figures, and they show disagreement only in the guessed digit.

Calculations with significant figures Is there a way we could have known from the beginning that our answer needed to be rounded to only two significant figures?

Calculations with significant figures If we look at the original measurements that went into the calculation, we see a length of 36.3 cm, which has three significant figures, and a height of 6.7 cm, which has two significant figures.

Calculations with significant figures Imagine there is a chain that is made of only two links, and one link is able to hold 3 kg before it breaks and the other is able to hold 2 kg, how much weight can the entire chain hold? Strong enough to hold 2 kg Strong enough to hold 3 kg

Calculations with significant figures If you are thinking that the chain could hold 5 kg (3 kg + 2 kg), then think again! Strong enough to hold 2 kg Strong enough to hold 3 kg

Calculations with significant figures If you are thinking that the chain could hold 5 kg (3 kg + 2 kg), then think again! The chain would break at its weakest point. And so, as a whole, the chain would only be able to hold 2 kg before it broke. Together only strong enough to hold 2 kg

Calculations with significant figures There is an old expression that says: “A chain is only as strong as its weakest link.”

Calculations with significant figures There is an old expression that says: “A chain is only as strong as its weakest link.” If a chain were made of ten links, and nine of those links could hold 100 kg, but one could only hold 1 kg…

Calculations with significant figures How much weight would the entire chain be able to hold?

Calculations with significant figures How much weight would the entire chain be able to hold? Just 1 kg!

Calculations with significant figures How much weight would the entire chain be able to hold? Just 1 kg! Essentially, the one weak link ruins it for the rest of the links.

Calculations with significant figures The same holds true for calculations involving measurements. Consider the calculation below. 23.40 cm x 0.47 cm x 6.05 cm = precise to 4 significant figures precise to 2 significant figures precise to 3 significant figures

Calculations with significant figures The calculator answer has 6 significant figures. 23.40 cm x 0.47 cm x 6.05 cm = 66.5379 cm3 precise to 4 significant figures precise to 2 significant figures precise to 3 significant figures

Calculations with significant figures The calculator answer has 6 significant figures. But the weakest measurement has only 2 significant figures. 23.40 cm x 0.47 cm x 6.05 cm = 66.5379 cm3 precise to 4 significant figures precise to 2 significant figures precise to 3 significant figures

Calculations with significant figures 67 cm3 The calculator answer has 6 significant figures. But the weakest measurement has only 2 significant figures. This means the answer must be rounded to only two significant figures: 23.40 cm x 0.47 cm x 6.05 cm = 66.5379 cm3 precise to 4 significant figures precise to 2 significant figures precise to 3 significant figures

Calculations with significant figures 67 cm3 When you report an answer to be something like “66.5379 cm3”(just because that is what showed up on your calculator), you are claiming a level of precision much higher than the measurements deserve. 23.40 cm x 0.47 cm x 6.05 cm = 66.5379 cm3 precise to 4 significant figures precise to 2 significant figures precise to 3 significant figures

Calculations with significant figures 67 cm3 An answer of 66.5379 cm3 means that the “66.537..” are definite, and only the “9” is a guess. 23.40 cm x 0.47 cm x 6.05 cm = 66.5379 cm3 precise to 4 significant figures precise to 2 significant figures precise to 3 significant figures

Calculations with significant figures 67 cm3 An answer of 66.5379 cm3 means that the “66.537..” are definite, and only the “9” is a guess. But if the 0.47 cm could have just as easily been read as 0.46 cm, consider how different the answer would be. 23.40 cm x 0.47 cm x 6.05 cm = 66.5379 cm3 precise to 4 significant figures precise to 2 significant figures precise to 3 significant figures

Calculations with significant figures So here is the rule: When multiplying or dividing two or more measurements, always round your answer off to the number of significant figures in the weakest measurement. (The weakest measurement is the one with the fewest significant figures)

Calculations with significant figures So here is the rule: When multiplying or dividing two or more measurements, always round your answer off to the number of significant figures in the weakest measurement. (The weakest measurement is the one with the fewest significant figures) This ensures that your answer will not be any more or less precise than it should be.

Calculations with significant figures If crude measurements were made, then only crude values can be calculated from them. If more precise measurements were made, then more precise values can be calculated.

Calculations with significant figures So let’s say a student is calculating the average speed of a car as it traveled down the road.

Calculations with significant figures So let’s say a student is calculating the average speed of a car as it traveled down the road. Speed is distance divided by time.

Calculations with significant figures So let’s say a student is calculating the average speed of a car as it traveled down the road. Speed is distance divided by time. The student measures the time with a very precise stop watch and records a time of 146.39 s.

Calculations with significant figures So let’s say a student is calculating the average speed of a car as it traveled down the road. Speed is distance divided by time. The student measures the time with a very precise stop watch and records a time of 146.39 s. Distance is measured rather crudely: 680 m. 0 100 200 300 400 500 600 700 800 m

680 m 146.39 s distance time Calculations with significant figures Speed = = = 0 100 200 300 400 500 600 700 800 m

680 m 146.39 s distance time Calculations with significant figures Speed = = = 4.645126033 m/s 0 100 200 300 400 500 600 700 800 m

680 m 146.39 s distance time Calculations with significant figures Speed = = = 4.645126033 m/s This answer is what appears on the calculator, but it is obviously way too precise. 0 100 200 300 400 500 600 700 800 m

680 m 146.39 s distance time Calculations with significant figures Speed = = = 4.645126033 m/s This answer is what appears on the calculator, but it is obviously way too precise. What should it be rounded to? 0 100 200 300 400 500 600 700 800 m

680 m 146.39 s distance time Calculations with significant figures Speed = = = 4.645126033 m/s The distance (680 m) has two significant figures 0 100 200 300 400 500 600 700 800 m

680 m 146.39 s distance time Calculations with significant figures Speed = = = 4.645126033 m/s The distance (680 m) has two significant figures… and the time (146.39 s) has five significant figures. 0 100 200 300 400 500 600 700 800 m

Calculations with significant figures