Sigfig guide

Guide to sigfigs

The big idea behind significant figures: precision. Of the digits of a measurement, which digits were actually measured? The more digits you measure, the more precise your measurement is. And sigfigs let you say, very precisely, which digits you've measured- which digits are significant.

Say we have an elephant that we weigh to the nearest thousand pounds. If we report 6,000 pounds, we only measured the 6. The zeroes are there because that's how place value works.

But say we measure the elephant to the nearest pound, and report 6001 pounds. The zeroes are measured this time- we measured pounds, tens of pounds, hundreds of pounds, and thousands of pounds, and the tens and hundreds of pounds turned out to be zero.

And so when we count sigfigs, we're only concerned about zeroes, and whether we've actually measured a zero, or if it's there to make the 6 a 6,000 for place value. Any other digit is significant, because has to have been measured- how else could it have gotten there?

The simplest way to remember which, if any, zeroes to count as significant is to think of the United States. (Sorry, Europe.) Look for a decimal point- is it absent or present?

If it's absent, count all the digits starting with the first non-zero digit from the right- the Atlantic Ocean is to the right of the US. A is for absent; A is for Atlantic.

6,000 has one sigfig. There's no decimal point, and the first non-zero digit from the right is the 6.
5,040 has three sigfigs. The first non-zero digit from the right is the 4, and then we continue on, through the entire number, regardless of whether or not any other digits are zeroes.

If the decimal point is present, count from the left- the Pacific side. You get the drill.

0.050030 has five sigfigs. The present decimal point means we count from the left: we start at the 5, the leftmost non-zero digit, and count all the digits afterwards.

Now- on to the logic. I'll go though a bunch of cases, explaining why the sigfig rules work, and why they make sense (at least to me.)

ABSENT:

423 - no zeroes
Three sigfigs. No zeroes; everything else must be significant. Duh.

4,007 - zeroes between non-zeroes
Four sigfigs. We've measured the thousands place; we've measured the ones place. How can we stop measuring for two place values? The zeroes must have been measured. They're significant.

6,000 - zeroes to the right
One sigfig. Convention states that we've only measured the 6, and that the zeroes are there for place value. This is useful mainly in the context of huge numbers, like 602,214,129,000,000,000,000,000, Avogadro's constant. It would be a freaking huge coincidence if we actually measured all those place values and they all turned out to be zero, so we can safely assume they're for place value.

000,003 - zeroes to the left
One sigfig. Why would you write it like that? 000,003 is 3. One sigfig. End of transaction.

PRESENT:

1.6 - no zeroes
Two sigfigs. Same deal. What were you expecting?

1.06 - zeroes between non-zeroes
Three sigfigs. Use your diddly-darned gray matter.

9.00 - zeroes in the rightmost position
Three sigfigs. The assumption here is that the zeroes are written because they're measured, and so we measured to the hundredth. Why?

When a decimal point isn't present, you can't add zeroes to the right. 90 is hardly 9000000. But when you have a decimal point, you can: 9.00 is the same value as 9.000000. In fact, you might as well not bother writing it: 9.00 is the same as just 9, after all. And so you can assume that whenever rightmost zeroes are written, they're there for a purpose: namely, they've been measured, and they're significant. Why else would you write them? If you don't need to, just write the 9 and get on with your stupid life.

A few other examples of when this principle works:
- 5.10 - Three sigfigs. It doesn't matter that we have nonzeroes after the decimal.

- 70. - Two sigfigs. That's right, it's a decimal point with nothing after it, and there isn't a thing you can do about it. This is a very useful trick: without the decimal point, 70 would have one sigfig and would imply a measurement to the tenths. However, with the decimal point, we effectively communicate that the measurement has been taken down to the ones, all the way to the end. Combined with scientific notation, this is freaking powerful. If you've measured 4200 kilograms to the nearest ten kilograms, just write 420. * 10^1. Easy as pie. The beauty of this trick is that there's absolutely nothing exceptional- it's simply a clever application of the same sigfig rules we've been using all along.

When you write or report measurements, it's your responsibility (is that too big of a word? too bad, you still gotta do it, you little punk) to make sure that what the sigfig conventions implied you measured what you actually did measure. It's important when precision is an issue- and in science, precision had better darn well be an issue. Use scientific notation. Use that trailing decimal point trick. Do whatever you need to do.

There's one case I haven't covered: Say I measure 0.008. Sigfig conventions imply I've measured only the 8. But what if I've measured the 08?

Who cares? If you measured 008, you'd just say 8 and move on. Why does it make any difference when you scale it down? What matters is the most precise digit to which you measure. The less precise ones you can safely assume are zeroes. The more precise ones, you can't, of course, unless you measure to that precision, but the less precise ones are guaranteed*.

*I'm not entirely happy with this explanation. If you can give me a hand, let me know. Thanks.