Significant figures
Significant Figures Introduction to significant figures
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- Let's see if we can learn a thing or two about significant figures, sometimes called significant digits.
- And the idea behind significant figures is just to make sure that when you do a big computation and you have a bunch of digits there,
- that you're not over representing the amount of precision you have
- that your result isn't more precise than the things that you actually measured - that you usually use to get that result.
- But before we go into the depths of it and how you use it with computation
- let's just do a bunch of examples of identifying significant figures, then we'll try to come up with some rules of thumb.
- But the general way to think about it is - "Which digits are really giving me information about how precise my measurement is?"
- So on this first thing right over here, the significant figures are this seven-zero-zero.
- So over here you have three significant figures.
- And it might make you a little uncomfortable that we're not including these zeros that are after the decimal point and before this seven.
- That we're not including those - because that does help define the number.
- And that is true but it's not telling us how precise our measurement is.
- And to try to understand this a little bit better,
- imagine if this right over here was a measurement of kilometers.
- So, if we measured zero point zero zero seven zero zero kilometers.
- That same measurement we could have - this would have been the exact same thing as seven point zero zero meters.
- Maybe in fact we just used a meter stick. And we said it's exactly seven point zero zero meters.
- So we measured to the nearest centimeter.
- And we just felt like writing it in kilometers.
- These two numbers are the exact same thing - they're just different units. But I think when you look over here
- it makes a lot more sense why you only have three significant figures.
- These zeroes are just kind of telling you - are just shifting it based on the units of measurement that you're using.
- But he numbers that are really giving you the precision are the seven, the zero and the zero.
- And the reason why we're counting these trailing zeros is that whoever wrote this number didn't have to write them down.
- They wrote them down to explicitly say "Look, I measured this far."
- If they didn't measure this far they would have just left these zeros off, and they would have just told you seven meters - not seven point zero zero.
- Let's do the next one - so based on the same idea we have the five and the two - the non zero digits are going to be significant figures.
- You don't include this leading zero by the same logic that if this was point zero five two kilometers
- this would be the same thing as fifty two meters, which clearly has two significant figures.
- So you don't want to count leading zeros before the first non zero digit, I guess we could say.
- You don't on't want to include those. You just want to include all the non zero digits and everything in between.
- and - and trailing zeros - trailing zeros if a decimal point is involved.
- I'll make those ideas a little bit more formal. So over here, the person did three hundred seventy,
- and then they wrote the decimal point.
- If they didn't write the decimal point it would be a little unclear on how precise this was.
- But becuase they wrote the decimal point it means they measured it to be exactly three hundred seventy.
- They didn't get three hundred seventy two and then round down or they didn't have a kind of roughness only to the nearest 10s place.
- This decimal tells you that all three of these are significant.
- So this is three significant figures over here.
- Then on this next one, once again, this decimal tells us that not only did we get to the nearest one,
- but then we put another trailing zero here which means we got to the nearest tenth.
- So in this situation once again we have three significant figures.
- Over here - the seven is in the hundreds but we got all the way down - the measurement went all the way down to the thousandths place
- And even though there are zeros in between, those zeros are part of our measurement,
- because they are in between non zero digits.
- So in this situation every digit - the way it's written - is a significant digit. So you have six significant digits.
- Now this last one is ambiguous. The thirty seven thousand - it's not clear whether you measured exactly thirty seven thousand.
- Maybe you measured to the nearest one, and you got an exact number - you got
- exactly thirty seven thousand. Or, maybe you only measured to the nearest thousand.
- So it depends on what - there's a little bit of ambiguity here - if you've just seen something written exactly like this,
- you'd probably say if you had to guess - or not guess - but if there wasn't any more information, you would say that there's just two significant figures or significant digits.
- For this person to be less ambiguous they would want to put a decimal point right over there.
- And that let's you know that there was actually five digits of precision - that we actually go to five significant figures.
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