Significant Figures 1.1

In this lesson we're going to learn about significant figures.

When students first learn about significant figures in chemistry or physics class

you always think it's like the most useless thing in the world.

and just a big waste of time.

Most people hate significant figures when they first learn them.

The problem is, the one group of people who don't hate significant figures

are teachers, which love them.

And teachers also, it seems, love to take points off of tests

when people don't get their significant figures quite right.

So no matter what stage you are, in learning chemistry or physics,

even if you're in the middle of the year,

it's a great help to refresh significant figures.

Because there are plenty of points that can be earned

on homework assignments, quizzes and lab reports

by brushing up on this very important topic.

The other thing is - significant figures

just aren't that hard when you really spend a little bit of time practicing them.

One of the most difficult things about significant figures, I think,

is that students don't really understand what they're for -

what the purpose of them is.

Here, let me show you. Here's an example.

Let's imagine that we want to do a pretty simple calculation.

We want to take 62 - divided by 41.

We plug this into our calculator and we get a really ugly answer.

The answer that we get, what the calculator is going to spit out,

is going to be 1.512195122.

How's that for an ugly answer?

Now, here's the thing. You obviously don't want to put this whole thing down on your sheet,

so you're going to ask your teacher - how many numbers do we need to round it to?

Well, that's the thing about significant figures.

If you understand significant figures, you don't need to ask your teacher that question.

Because the answer to that question - how many numbers do we round it to -

is already answered by the numbers you divided in the first place.

Let me show you what I mean.

I'll give you a demonstration first, and then I'll explain the details.

62 has two significant figures in it.

A 6 and a 2. Two significant figures.

41 has a 4 and a 1 - also two significant figures.

That means that I'm going to round the answer that I got to two numbers - simple as that.

So that 1 is going to stay and the 5 is going to stay.

But remember, I'm going to be rounding.

So I'm going to have to look at the number next to the 5 to determine whether I keep the 5 the same,

or if it's a big number, if I'm going to round up to 6.

In this case it's a 1 so I don't round up.

I keep it as 1.5.

And that's my final answer.

The question of how many numbers I need to round it to -

is already answered by the number of significant figures in the numbers that I divided together.

Now, the question you may ask is - how did I know that there were two significant figures in the 62 and the 41?

The answer is pretty simple.

Any number that isn't zero is always a significant figure.

We'll talk about zero in the next lesson because it can get a little bit tricky.

Let's take a look at another example.

Let's do 782 divided by 231.

Put that into the calculator again. We get a really ugly answer out.

3.385 blah blah blah.

I don't want to put that whole thing on my sheet either.

So I look at the numbers that I divided together.

How many significant figures in 782? Well, none of them are zero, so - 3 significant figures here.

They're all significant.

231? No number zero - 3 significant figures here.

What does that mean? It means that I'm going to round my answer to 3 significant figures.

This is going to stay, this is going to stay, and this is going to stay.

As before, I'm rounding, so I look at the number to the right of this 8.

It's a 5, which means that I'm going to have to round up.

That 8 is going to become a 9.

And my final answer is going to be 3.39.

Again - it's 3.39 because that 5 there makes me round the 8 up.

Let's go to one more example.

1219 divided by 3462. Another pretty ugly number.

Dot 352109.

How many numbers do I round it to?

I'm sure you've already gotten the hang of it.

Four significant figures here, four significant figures here -

That means I round my answer to - you guessed it - four significant figures.

This stays, this stays, this stays, and this stays.

And I look at the number just to the right of the 1. It's a zero so I certainly don't round it up.

My final answer is going to be .3521.

There you go. Rounded to four significant figures.

Now, we can do this exact same thing with multiplication.

All we have to do is round, and then we add a couple zeros if we need to.

Let me show you what I mean.

When I take 56 and multiply it by 28, the answer that I get is 1568.

But as with division, I look at the significant figures in the two numbers I multiplied together -

two here and two here -

which means that I'm going to round my final answer to two numbers.

This one stays and this one stays,

and I look to the right of it to decide whether I round up or down.

It's a 6, so I round up.

My final answer becomes 1,6 - because these two numbers stay -

and I just fill everything else in with zeros. So I get 1,600.

If I multiply these guys together, 4833 times 1741, I get a really big number -

8 million, 414 thousand, 253.

Again - how can I round this using significant figures?

4 significant figures here and 4 significant figures here, so -

You guessed it - I round my answer to 4 digits.

I look here to find out whether I round up or down.

It's a 2 so it stays the same - 8,414 - and I fill the rest of the number in with zeros.

I don't need them.

Round it to four digits.

Now, if you're smart, you've already realized that I'm doing something pretty simple.

All the numbers that I have been multiplying together or dividing have the same number of significant figures.

So maybe you're asking yourself - OK, this seems easy enough,

but what would I do if I had to multiply together or divide two numbers that didn't have the same number of significant figures?

What if, like for example, one had 5 and one had 2 -

How would I round then? Let's take a look.