Continuing on with my ever-popular 'significant figures' series...

In the intro to significant digits, I talked about how the whole 'significant digits' method of representing a number is a shorthand for representing both the number, and the uncertainty associated with the number. This is useful in science, where every measured quantity has an uncertainty associated with it.

So, for example, the value "600 grams", when written just like that by a scientist, actually means 600±50 grams. You're supposed to infer the uncertainty from the way the measurement is written.

There are 3 different notions associated with significant digits; that of the least and most significant digits, and the number of significant digits. For example, for the measurement 30040 lbs., the 3 is the most significant digit, the 4 is the least significant digit, and the number has 4 significant digits (because every digit between the 3 and the 4 is also significant).

If you check your kid's science book, or even go to an online resource, you'll probably find that the rules for adding and subtracting values are expressed none too clearly. So if you get confused by a sum like this:

1500 g + 5.07 g

... then remember that it's really a sum like this:

(1500±50 g) + (5.07±.005 g). The intro post tells you how to read the uncertainty given the number.

That first number, 1500, has an uncertainty of plus or minus 50; so it really could mean any number between 1450 and 1550. The second number could lie anywhere between 5.065 and 5.075, which is a tiny number compared to the uncertainty in the first number. So, when we add them together, how much uncertainty do we have?

Well, the second number is small enough, by comparison with the first number, that the whole range of its possible values lies within the uncertainty of the first number. If we take a number that could lie anywhere between 1450 and 1550, and add a number that is within 1/100th of 5, the result will probably lie somewhere between 1455 and 1555.

So, just using common sense, we get (1500±50 g) + (5.07±.005 g) = 1505±50 g. This result has the same uncertainty as the first addend we started with. In general, the uncertainty associated with the sum doesn't get any better than the worst uncertainty we started with. Here's the principle we gather from this exercise:

When adding (or subtracting) two measurements, the least significant digit of the sum should be the same as the least significant digit in the least accurate summand.

example 1

To finish this example:

1500 g + 5.07 g

I add it normally to get 1505.07. Since the uncertainty in that first number is ±50 g, the uncertainty in the result has to be the same; so I round 1505.07 to the nearest 100 to get 1500 g. The answer is no different from that first summand! This reflects the fact that the second summand is so small that it disappears into the first summand's uncertainty.

example 2

If I add 15000+28.1, then the least accurate summand is the 15000; it has an uncertainty of ±500. Its uncertainty swallows up that whole second summand! So I start by adding normally:

15000+28.1 = 15028.1,

and then I remember that I should round to the least significant digit in the least accurate summand. Rounding gives

15028.1 -> 15000,

so in the peculiar arithmetic of scientific measurement, 15000+28.1 = 15000.

example 3

To calculate 14600 + 164.3, first I add the sum normally: 14600 + 164.3=14764.3.

Then I note that the least accurate summand is 14600, and that the least significant digit in that summand is the 6; so 14600 has an uncertainty of ±50. I therefore round the answer to the nearest 100, in order to express it in such a way that its uncertainty is also ±50 (the answer is 14800).

example 4

To calculate 13.754 + 1.5, first I add the sum normally, to get: 15.254.

Then I check that the least accurate summand is 1.5 (because it has the fewest decimal places), and that its least significant digit is the 5. In other words, it has an uncertainty of ±.05. I therefore round the answer to the nearest .1, in order to express it in such a way that its uncertainty is also ±.05. I get:

13.754 + 1.5 = 15.3.

final question

Should I finish up and do multiplication and division rules to be complete?

-- CarolynJohnston - 27 Oct 2005

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