>> Welcome to the video tutorial on significant figures.
So significant figures are a really, a very elegant
and subtle way to communicate how accurately we know the value
of a measurement.
So the rules of sig figs, these are just something you want
to memorize, all right?
So significant figures, they're the digits
in a measurement known with certainty,
plus one estimated digit.
So you could think of the estimated digit
as the uncertain digit.
And I like to symbolize it, right,
the estimated digit is the uncertainty
and that's the question mark, that's where we guess.
So, when we're reporting measurements using sig figs,
we're communicating the magnitude of the measurement,
but also how accurately we understand the measurement,
or how much we can trust it.
So when we're looking at measurements,
it's very straightforward, if we have nonzero digits,
we know that they're always significant.
And so remember this is the definition
of significant up here.
So nonzero digits are either the, are the digits we know
with certainty, or they're the estimated digit that we guess.
It's the zero that can cause the challenge.
The digit of zero, the zeros may nor may not be significant.
We have to look at the location of the zero
in the number, all right?
So, here, the leading zeros, this is a pretty good one,
I like this one, I don't get to say never very much.
The zeros located at the beginning
of a number are never significant.
And what I'm going to do is I'm going to go through the rules
and I'll do some examples.
and then we'll practice on the following pages.
Confined zeros.
Those are the zeros located between nonzero digits.
They're always significant.
And then trailing zeros.
So, trailing zeros you're going to have to depend
on who your instructor is.
In my class, we follow this rule for trailing zeros.
That the zeros located at the end of a number are significant
if the number has a decimal point,
or the zeros carry overbars.
All right?
Some professors and textbooks call this ambiguous and so,
you're going to need to just defer
to what your professor says about the rules.
So, now let's see how these rules work.
So for example if we had 2583, these are all nonzero digits.
So if we looked at this as a, as a measurement we would say
that we know the 2, the 5 and the 8, those are certain,
we know those absolutely.
The 3 is the uncertainty.
So the rule of sig fig say, that we take all the digits we know
for sure, plus the first estimated digit,
and that determines our significant figures.
So we would say that this measurement has 4 sig figs.
Now, significant figures is a mouthful,
they often get described as sig figs,
and I will reduce it just down to SF.
All right.
Now, now let's bring in those zeros to see how they work.
All right, so let's say that we have .002583, all right?
So these are the leading zeros.
Leading zeros never count.
So if we are looking at this number, we say we know the 2,
the five and the 8 with certainty,
and we're guessing on the 3.
So once again 1, 2, 3, 4, 4 sig figs.
Okay? We don't, leading zeros never count.
Confined zeros.
Zeros located between nonzero digits.
Let's say we had 2003 as our number.
These zeros are confined between the nonzero numbers of 2 and 3.
So this tells us we know the 2 and the zero and the zero
with certainty, and we guessed on the 3.
So once again we're looking at a number with 1, 2, 3, 4 sig figs.
Alrighty, now comes the last option for zeros
and that's the trailing zeros.
So this is where the decimal point is critical.
So, if we have 2500, and we put a decimal point,
then what we're telling everybody is we know the 2,
the 5 and the zero with certainty,
and we guessed this last digit, and we happened
to guess it to be a zero.
So according to the rules of sig figs, 1, 2, 3, 4,
this number would have 4 sig figs.
Now, if we write the number 2500, and we leave off the zero,
that's telling us that we have no idea what's going on here,
that our first guess, our guess is happening at the 5.
So we know the 2 with certainty, and we were guessing
in the hundreds place.
Now it's important, think about money.
So these zeros are here as important placeholders,
so the value is around 2500.
We do not know this measurement very accurately, it's,
we're guessing in the hundreds.
So we would describe this as having 2 sig figs.
What happens if we wanted to show 3 sig figs here?
Well, we can't put a decimal point,
this is where we use the overbar or sometimes the underbar.
We put a overbar here to let everybody know
that this is where we're guessing.
So, we know the 2 with certainty, we know the 5
with certainty, and we guessed zero here.
So this would be an example of 3 sig figs.
So the trail, the key with trailing zeros is
to focus your attention on the decimal point.
When the decimal point is present, then we say that we,
that these zeros are here as part of our measurement,
then the question mark is
in this position here, the ones place.
Without the decimal, we're letting everybody know
that we guessed in the hundreds place.
And, if the decimal doesn't work, then we can bring
in the overbar or the underline, underbar.
So then to wrap this, to kind of finish up this page here,
if we said how many sig figs are in this measurement, right?
Here are the leading zeros, they never count.
Here we have confined zeros, they always count.
And here we have trailing zeros, and we do have a decimal point,
so they're going to count.
So we say 1, 2, 3, 4, 5, 6, 7, 8, there would be 8 sig figs,
and really all significant figures are telling us,
is that this is where our guess is occurring,
in this last zero place.
So, now that we have an overview
of the sig fig rules let's give you a chance
to practice applying them.
[ Silence ]
So what happens is we'll do a calculation,
and our calculator is going to spit out a result that's going
to have a ton of digits, but we're going to need to round
to the correct number of sig figs.
But remember sig figs are all about where,
where is the question mark right,
we got to keep the value the same.
Alrighty, so if we look here, if our calculator spits
out the number 6478, but we only want to report 2 sig figs,
that tells us that this is where the uncertainty is in our data.
So we look to the right, and we see that we have a 7.
Right, so 7 is greater than 5, so we're going to round up.
So if we were going to report this measurement
to 2 sig figs, it would be 6500.
So the zeros are the important placeholders,
because they help us keep the overall value.
In this next example we have a bunch of leading zeros.
Leading zeros never count.
So if we want to do two sig figs, one, two,
so that's telling us this is where our question mark is.
So we look to the right and we see 3.
3 is less than 5 so we round down.
So we bring in, right leading zeros never count,
so we bring in all the leading zeros
and then we would put 7, 9.
So now we have rounded this measurement
to two significant figures.
Here we go, we now have a nice big number, so think money.
Imagine that you just won 456,924 dollars
in the lottery, all right?
So, we want to report this to three sig figs.
That's telling us 1, 2, 3.
It's letting us know that the 6 is where our uncertainty is.
We look to the right.
9 is greater than 5, so we round up.
So we would rewrite this number, 457,000.
Right, so sig figs is just about the details.
These are important placeholders right?
They keep the overall value.
All right, so here we have 1, 2, 3.
So sig figs are all the digits we know
with certainty, plus the first guess.
All right, so let's try a couple more of these
to make sure everybody's clear.
We're back to leading zeros.
Leading zeros never count.
1, 2, 3. So our uncertainty is here at the zero.
7 is greater than 5 so we're going
to round, whoops, round up.
So leading zeros never count so there's the zero,
now we start counting, 1, 2, 3.
So now we've reported our value to three sig figs,
recognizing that that's telling us where our guess is.
And then this last one is to remind us about the overbars.
So, 2 sig figs, 1, 2, that tells us this is where we're guessing,
we're guessing in the hundreds place.
So we would rewrite this as 4, zero with a overbar,
or we could use scientific notation,
and do 4.0 times ten to the third.
And if you're not sure how I'm jumping back and forth
between these two values, there is a video tutorial
on scientific notation to help you.
And the main thing to keep in mind is
that sig figs are telling us
where the data starts to be uncertain.
So, this is a huge important distinction right?
The zeros are important placeholders,
whether they're leading zeros or trailing zeros,
they preserve the overall value of the measurement, right?
The sig figs are about the accuracy
and the precision of the measurement.
So go ahead and take a few minutes
to round the numbers below to the correct number of sig figs
and then compare answers with me.
So, in this first example, right so we want 1, 2, 3, 4.
So this is telling us that the uncertainty is
in the fourth position.
So if we look to the right, here right,
3 is less than 5, so we round down.
So when we report our value, 689,500.
So we would take this huge number and we would,
we've got to maintain the overall value,
we're just letting everybody know where we can,
where we trust the results.
And then if we reduce this measurement to only 2 sig figs,
right, so then 2 sig figs tells us this is our question mark,
so now we have, we would look at the 9,
and the 9 is greater than 5.
So we round up, so we would report this value as 690,000.
And so, remembering that sig figs is just telling us how
trustworthy the measurement is,
and the zeros are the important placeholders
to keep the overall value.
And then a little bit of practice with numbers
with the leading zeros.
So here it's important to remember
that leading zeros never count.
So 1, 2, 3, 4.
So, in this first example, it's telling us
that the uncertainty appears at the 6 in the number here,
this digit here, represented with the 6.
So if we rewrite it, it would be .0 right,
leading zeros never count, so then it would be 278,
we would round up because of the 9.
Whoops. Actually I meant a 7, I'll rewrite that.
Oh I think it's clear.
So, leading zeros never count, 2787.
So we round up, we round the 6 up to a 7 because of the 9.
Okay? Or, if we only want to report two sig figs we're going
to stop here at the 7.
So leading zeros never count.
So that would be 0.002.
The 7 is next, there's an 8, so we round up to zero, 0.0028.
Alrighty. Now let's, now that we've had some practice
with significant figures,
let's look at how they're used in calculations.
Okay. So, the guiding principle here is the worst data, right,
the least accurate, the least precise, limits the result.
So, with multiplication and division, the result is limited
by the term with the smallest number of significant figures.
So, if we look here, we see 1, 2, 3,
so we have three significant figures
in this first measurement,
but we only have two significant figures
in the second measurement right?
So significant figures are all
about where we're guessing the values.
So, if we multiply these two measurements,
our calculator spits out 18.33.
Okay and then here's our units, that would be feet squared.
And so once we have our results with our units we look back,
and this is our worst data, two sig figs.
So now we need to round our answer to two sig figs.
So that would be 18 feet squared.
So we're letting everybody know
that this is the guess, in the ones place.
For this next example, 5.0, we have a trailing zero
with a decimal, so that's two sig figs.
And here we have 3 nonzero digits, 3 sig figs.
So, and in this one we see that we have milliliters on the top
and milliliters on the bottom
so we have some units canceling in our calculation.
So we're left with grams.
If we punch the, this data into our calculator,
the calculator tells us that we would get 69.5 grams.
And once we have our final result, take a moment
to look back, the worst data is two sig figs.
So now we're going to round this to two sig figs, 1, 2.
Notice there's the 5, so we round up.
So when we round 69.5 up to 2 sig figs,
it's going to be 70 grams.
But remember that we need to show 2 sig figs
and we have a trailing zero, so to let everybody know
that the uncertainty is here at this zero,
we add our decimal point, so that we have a 2 sig fig answer.
The vast majority of the calculations that we do
in chemistry involve multiplication and division.
It's very rare that we need to do addition and subtraction,
mostly addition and subtraction show
up in calculating molar masses, which we'll learn in the future.
Or, doing temperature con versions.
So the applications for addition and subtraction
with sig figs are pretty limited, but there are a couple
of places where we need to use the rules,
so we'll practice them.
So when we're doing addition
and subtraction the rules are different.
The result is limited by the term with the least precision,
so how do we find the least precision?
We look at which digit place has the uncertainty.
So when we're looking at this first calculation,
we see that the first measurement,
the uncertainty is in the hundredths.
Right, it's in the hundreds place, 2 to the right,
.01 right there, that's where we're not sure,
in the hundredths place.
With the second data, we're uncertain in the tenths place.
This is where we're uncertain right?
So which data is worse?
Definitely right, tenths are bigger right?
So tenths are greater than hundreds,
so that tells us this is the worst data.
So we do our calculation.
So we get 9.65 feet, so we're just adding here.
All right.
So now we want to round our answer so,
because we have uncertainty in the tenths position,
and that's the worst data, we limit our results.
So we would round this to 9.7, 5s will round up, 9,7 feet.
And then here's one more example without decimals.
When we look at this first value here there's no decimal,
so the uncertainty is in the hundreds place.
[ Writing on Paper ]
Right, so we're in the hundreds, that's where we're not sure.
When we look at the second measurement,
it's uncertain in the ones place.
Right? Okay, so 100 is bigger than 1,
so this is our worst data.
So we go ahead and do the addition.
Okay, so we get this result, but now we're going to round, right,
we're going to round to the hundreds place,
because that is our worst data.
So this final answer would be reported
as 63,600 grams because, our uncertainty is right here,
and the 6 is greater than 5 so we round up.
So our final answer would be 63,600 grams.
Alrighty, so that concludes the significant figures tutorial.
Please take the time now to work a few practice problems
and do some calculations so that you're comfortable working
with sig figs, especially in calculations.