Significant Figures 1

Uploaded by AlanEarhartChemistry on 19.12.2011


We are working our way up to doing calculations.
We’re going to do calculations using formulas,
using conversion relationships, using measurements.
And we want to come up with a set of rules that will help us figure out
how to quickly determine how many digits to write in our answer.
These are just real quick estimations on how to write it.
So before we can even talk about what to do in a calculation,
we have to learn how to count these things called “significant figures”
in your measurements.
So we have two broad categories of measurements,
relationships, and things like that. We have one type which is
an exact value that can come about from counting. So if I ask you to
the number of pennies on the screen, well, there are 10 and there is
no doubt about that. There are 10. If in the middle of lecture I were to
stop and count the number of students in my classroom, as long as I
am careful, I can know how many students there are and there
is no doubt about that. You can have exact numerical relationships.

These four are exact relationships. There is no doubt.
They don’t change! 100 cents in a dollar. That doesn’t change.
1 minute = 60 seconds... doesn’t change.
That’s a definition. That’s important and we’ll learn how to deal
with that later in calculations. We also have inexact values
or relationships and measurements are always inexact. You cannot
have an exact measurement. It does not exist.
If I take this as my measuring instrument,
my fake ruler, and then I have a piece of paper,
and I’m measuring the length of it. So I lay that down on my ruler
and at first glance, I see that it’s at least 1.7 cm long.
This measuring instrument is marked every tenth of a centimeter.
So if you look at the end of that piece of paper, it’s a little bit past
the 1.7. But, it’s not really appropriate for me to just write 1.7.

For metric measurements, unless you are told something
different about the measuring device, the rule-of-thumb
is that you go one digit after your markings.
So, this little ruler has markings every tenth of a centimeter
so I’m going to estimate to the one hundredth because that’s
in-between two markings. And to my best estimate, it looks like it’s
1.74 cm. Maybe 1.73 but that’s what I see.
Those digits are important. That last digit is really important, too.
The last digit in a measurement depends on
a few things. It depends upon the measuring instrument,
and how good it is as far as the markings. It also depends on the
person using the measuring instrument. We’ll come back to this.
You can have inexact numerical relationships.
1 mile is approximately 1.6 kilometers.
I can do better than that.
1 mile is approximately 1.61 kilometers.
I can do better than that.
One mile is approximately 1.609 kilometers. All three of those
are inexact. Which one you use
can affect how you write your answer if you were converting from
one distance unit to another one.

Uncertainty or Error
That is a mouthful!
Let me try it a different way.
Let’s go back to that measurement. I wrote it as 1.74 cm.
Now think about what’s known and what’s not known about this.
The “1” is known. That piece of paper is at least 1 cm but it’s not 2 cm.
So the “1” is fine. The “7”... it is at least 1.7 cm.
It is less than 1.8 but it is 1.7.
I estimated in-between two lines and I took that as about
4 tenths of the way between those tenths.
So, I estimate that to be 1.74 cm.
Anybody who would look at that measurement, would understand
that the uncertainty in the measurement
is in that digit unless you are told something special about the
measuring instrument. But you aren’t so you have to assume that that
last digit is where the uncertainty is. Don’t mischaracterize “error”
as being a mistake. It’s not. A lot of times we use
“error” and “uncertainty” as being synonymous with each other.
“uncertainty” is really the better term.
We need to learn how to count these significant figures
or digits because it can affect how we write our final answer.
When you count them you start at the very left and
work your way right.
You start your count as soon as you hit a nonzero digit.
Let’s go back to this 1.74 cm. I start at the left my first
digit on the left is a “1” which is not zero so I start my count.
That is significant. The “7” is significant. And the “4” is significant.
All three of those are significant
so I have three significant figures. Anytime you see
the abbreviation “SF”, assume it means “significant figure”.
Let me give you four more examples.
Ah. before I give those four, let me do one thing.
I’m going to take the units off. With significant figures there have to
be measurements involved somehow so you need units in there.
But, we’re just going to look at the numbers so pretend there are units.
Let me give you those four examples.
What do you notice about these four examples?
You noticed that there is at least one zero in all four of those
compared to the very first example. Correct! You have to be careful
as to how you handle the zeroes when you count significant figures.
So let’s do those rules.
Let’s pull up that 210.6 example.
I start at the left and work my way right, but I notice right away that
there’s a zero in the middle. Now is that significant or not?
It’s between two nonzero digits. It’s significant.
Think of it as just being part of the measurement and
just happened to be “0”. Since I start with the “2” and work my way
to the “6”, all four of those digits are significant.

So the next example I had up, 0.089.
Well, you start your count on the left, but you
don’t start the count really until you hit your first nonzero digit.
So those zeroes? I ignore. Those zeroes are what we call “placeholders”.
They are there to make sure you understand that the number
is 0.089 and not .89 or 89 or something like that.
But as far as significant figures go, we don’t count them.
You don’t start your count until you hit the “8”.
0.089 has two significant figures.

175.000. The balances in our lab are good to three decimal places.
So if I were to put something on the balance and it read
175.000, that’s what I would write down for my data.
I’d say that the mass of this is 175.000 g.
Don’t leave off the zeroes! It’s important. By including
the zeroes after the decimal place, you are telling someone
that you can actually record the mass to three decimal places.

So those zeroes are significant because they are at the end
of the number and they are after the decimal place.
End of the number. That’s important. Start at the “1” and start counting.
You have six significant figures there.
And then the last example I gave.

So that 6000. There’s no decimal place there.
So those three zeroes on the end, they are not significant.
They are there to make sure you understand that this is supposed
to be 6000 and not 6. When you count your significant figures,
start at the left, work your way right. There’s no decimal point.
“6” is the only one you have with one significant figure.
Let me summarize these.
Remember- pretend there are units all over these values.

Now, that last one. The 6000.
While there is one significant figure,
if that were the result of a calculation,
it’s possible there are more than one significant figure.
If it’s the result of a calculation. But, I’m getting ahead of myself a bit.
That will come later.