So what's it all about with significant figures?
I mean, really, you learned all these rules about how to multiply and divide with them,
how to do addition and subtraction, when is 0 significant and when it's not and how to
deal with decimal places.
Okay, that's great but it's such a pain in the neck.
Really!
So what is the point of significant figures?
If you're wondering, I don't blame you.
For one thing, significant figures are useful because they tell us how to round, so we don't
have to worry about whether we should round to one or two or three decimal places.
So that’s kind of useful.
But the real point is more important than that.
The real point of significant figures is to make sure that the answer you get, like from
a math problem, isn't more precise than the numbers you started out with in the first
place.
This is a little bit tricky so let me tell you what I mean by it.
Most of the math we do in science is with numbers that are measurements, right?
So it's like the amount of liquid in a beaker that a measurement or the mass of a bunch
of chemicals, also a measurement or maybe like the amount of pressure in a gas sample.
All these are measurements.
So when we're taking measurements, some measurements can be really precise and others can be not
so precise.
Let me show you what I mean.
Let's say that I have a rock here, a generic rock, and I want to find out what its mass
is.
I can weigh it in a way that's going to give me a precise answer or in a way that’s going
to give me a not so precise answer.
Here's what I mean.
I could take this rock and I could put it on a plastic toy scale from my little sister's
Barbie scientist play set, okay?
I do that and I get the answer of 82 grams.
That's the mass of the rock as determined by this scale.
Or I could take the rock and put it on an expensive NASA rocket scientist laser scale
that cost $10 million dollars but the laser scale gives me this answer.
It's says 82, similar to the Barbie playset, .1039217 grams.
All these extra numbers are here because the laser scale is much more sensitive and gives
me a much more precise number than I get from over here.
So I can say my Barbie scale number, 82, not very precise as compared to the NASA laser
scale which is super precise.
Okay?
Now I've got these two values for the mass of the rock.
Let's say that I want to use the mass of the rock to determine its density.
To do density, I’ll do mass divided by volume.
Let me take the mass from the Barbie playset here.
Okay, so I'll take 82 grams, that's the mass, and then I’ll only need to find out the
volume of the rock, so maybe I'll use like a graduated cylinder also from the Barbie
playset and find out that the volume of the rock is 82 cubic centimeters.
I do this math and the answer that I get is 1.952380952 grams per cubic centimeter.
Now check out your answer and check out the numbers that I started with.
This is a problem and here's why it's a problem… because I started with these two numbers that
were not very precise.
I measured both of these with plastic science equipment from the Barbie playset but then
the answer that I got is super, super precise or at least it looks super precise.
This answer here looks like it's so precise that I only could have measured it with this
super expensive NASA laser instrument, okay?
If you think about it, this doesn't make sense.
It doesn't make sense to start out with this really imprecise numbers and end up with a
number that is so precise I only could have gotten it with really sophisticated sensitive
equipment.
So this is where significant figures come in.
If I'm using significant figures or I’ll do the same calculations but I’ll realize
that there are two significant figures in both of the numbers I'm starting with which
means that my final answer here can't have more than two significant figures.
So instead of 1 point all this stuff, I'm only going to get 2.0 grams per cubic centimeter.
And now in this case, the answer that I get isn't more precise than what I started with.
So this is what I get without significant figures and now when I use significant figures,
I get an answer that is just as precise but not more precise than what I started with
so it actually makes sense.
Now on the other hand, if we started with two super precise numbers, if we measured
both the mass and volume with super-sensitive equipment, it would be totally fine first
to get an answer that itself is very precise but that's only because this starting information
here was precise so the answer too can be precise.
This is with the correct number of digits based on significant figures.
I hope you're beginning to get the hang of this and to help you I want to give you an
analogy to the everyday world that might really make this all come together for you.
Okay.
so imagine that you're at home and you’re trying to meet up with two friends were hanging
out somewhere in the city.
You give them a call and that's the first one.
“Hey, where are you?
I want to come meet up with you.”
Your first friend is not particularly helpful and they say, “Aw man, I don't really know
where we are…
I think we're somewhere downtown.”
They give you an answer that's not very precise.
So you're other buddy is hanging out with him and so you call him and the second friend
is also pretty unhelpful and they say, “Yeah man, I think we're on some street corner.”
Okay, this is also an answer that's not very precise.
So here is the two piece of information you're starting with, that your friends are somewhere
downtown and they’re on some street corner.
Now based on these two pieces of information, would it be possible for you to come up with
this answer?
Would it be possible for you to hear these two pieces of information and think, “AHA”,
you must mean the southwest corner of Mott Street and Bleecker Street.
Alright, this is an incredibly precise answer.
You know exactly where they are.
No you couldn't do this, right?
Based on these two very not precise starting pieces of information, it would be impossible
for you to come to this very, very precise conclusion.
Alright?
And so that is exactly what's going on when you take two imprecise numbers to start with
and you end up with a number that seems more precise than either one of them.
It's why a number like this doesn't make sense why you can't use it and why you have to use
significant figures to make sure that it's not too precise, alright?
So just to review, significant figures are useful because they help us to make sure that
that answer you get isn't more precise than the numbers you started with.
So even though the rules of significant figures can be real pain, there is an important point
to them.