Significant Figures

Introduction to significant figures

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.

So if you don't see the decimal point, I would go with two.