### Video Transcript

In this video, we will learn about
the ways we express precision with numbers. We’ll look at the difference
between significant figures, decimal places, and the situations where we might use
one over the other.

We use numbers to express the
values we get from measurements. I’m one meter and 86 centimeters
tall, which can be expressed as 1.86 meters. But am I really? That value comes from a
measurement, one I took shortly before making this video, and I used a tape
measure. This tape measure had markings for
meters, centimeters — which are one hundredth of a meter — and millimeters — which
are thousandths of meters. So, one meter is equivalent to 100
centimeters or 1000 millimeters, and one centimeter is equivalent to 10
millimeters.

But what if the tape measure had
only had the markings every meter? I’d know I was taller than one
meter, but shorter than two meters. We can estimate that my height is
closer to two meters, but beyond that it’s just a guess. So, for now, we’d round to two
meters, if only that was all it took to make me taller. Instead, what if the tape measure
had the meter markings and markings every 10 centimeters? With this tape measure, I could see
my actual height was above 1.8 meters, below 1.9 meters, but actually a little bit
closer to 1.9 than 1.8. Using a tape measure with finer
increments means I can make a more precise measurement, and I get my height closer
to the true value.

Next, we include the one-centimeter
markings. So, we can measure my height to be
186 centimeters or 1.86 meters. This is even more precise than
before. You may have noticed that this is
also more accurate because it’s closer to my actual height. But accuracy is not the focus of
this video, but we could have even used the millimeters. If we go further, we should get
more precise, right? For cases like this, it’s hard to
measure someone’s height correctly to the nearest millimeter because we grow and
shrink with our breathing, people also slouch, or people could try to cheat and
stand on tiptoes. For this purpose, centimeters are
precise enough.

Let’s take a look back at the
measurements we made, two meters, 1.9 meters, and 1.86 meters. It would be helpful if we had a
clear way to describe the differences between these numbers. This way we can talk to other
people about how precise we’re being. One of the most important terms we
use when describing the precision of numbers is significant figure. Here is the long definition of a
significant figure. A significant figure is a digit of
a number that’s necessary to understand the value and precision of the number. However, that’s a little awkward to
start with. So, let’s go in with a few
examples.

Here are the numbers from the
previous example. The more significant digits there
are in the number, the more precise it is. So, let’s imagine for a minute that
I am exactly two meters tall. I could write it as two meters, 2.0
meters, or 2.00 meters. The fact we have zeros there is
still important because it means we’ve measured that value to that precision. Let’s take a closer look at the
example of 2.00. Two is in the units column, so it’s
describing two lots of a single unit. Zero is in the tenths column, so
it’s describing zero-tenths. And there’s a zero in the
hundredths column, so it’s describing zero hundredths.

If we were talking about meters,
this would mean we had two meters, zero-tenths of centimeters, and zero
centimeters. If we threw away these zeros, the
number we’ll be left with would be less precise. The two in this example gives us
both an indication of the value and the precision, and the zeros help us know we
know that value to a great degree of precision. So, we know a zero is a significant
digit If it was part of the measurement.

Now, let’s go a little bit
deeper. Here are the digits we established
to be significant. But what about this zero here? We know this number has no lots of
10, so we know we could put zero here and that would still be correct. We call this type of zero a leading
zero. It’s a zero to the left of the
first nonzero digit. We could write an unending number
of these and it wouldn’t change the value or precision of our number. The two is our first significant
digit going left to right because it’s the first one that starts to define the value
of the number. Leading zeros are not
significant.

There’s another type of zero you
might come across. Once we finished our measurement to
three significant figures, we didn’t actually measure any further. So we don’t know what those values
are. Quite often, you might see these
values written as zeros. We call these trailing zeroes. Just like leading zeros, trailing
zeros can theoretically go on forever. Now, the definition of a trailing
zero is simply any zero that’s to the right of the last nonzero digit. Since we only have one nonzero
digit in our number, all the zeros to the right of our two are trailing zeroes.

The two zeros that were part of our
measurement are trailing zeroes. But since they define the precision
of our measurement, they are significant, While the trailing zeros we added are not
significant. So, trailing zeros can be
significant if they’re part of the measurement; otherwise, they are not
significant. But there’s one more possibility we
need to discuss.

In this example, the two in the
ones position is both the first significant figure and the last nonzero digit. But what if, instead, we were
dealing with a measurement of 2.01 of something? The zero in the middle isn’t
leading because it’s not in front of the first nonzero digit, and it’s not trailing
because it’s not to the right-hand side of the last nonzero digit. We call this a sandwiched zero. It’s a zero between two nonzero
digits. And sandwiched digits are always
significant.

If we have a nonzero digit, that
must mean it was part of our initial measurement. Therefore, even if it’s a zero, any
numbers in between must have been part of the measurement as well. So, a zero between two nonzero
digits, a sandwiched zero, or a middle zero is always significant. And it doesn’t matter how many
zeros there are that are sandwiched or where they are in the sequence.

Now, we get to the important part
where we’re dealing with rounding according to a specific number of significant
figures. When we do a calculation on our
calculator, it doesn’t know how many significant figures we want, so it just gives
them all. Nine divided by eight in perfect
arithmetic is equal to 1.125. We can imagine the unimportant
leading zeros stretching off to the left forever. And because this is an exact
mathematical process, we get an exact answer going off into ∞. So, we could think of every single
trailing zero as being significant. But instead we’re going to think
about this as a measured number and assume that these are the significant
digits.

To work out how many significant
figures we have, we first need to find the first nonzero digit. That’s the one here. All the digits to the left are
zeros. The next thing we want to do is
find the last significant digit. In this example, that’s this one
here, the five to the far right. And then what we want to do is
round that off, and we do that by looking at the next digit along. If that digit is a zero, one, two,
three, or four, we round down. If it’s a five, six, seven, eight,
or nine, we round up.

Here, we can see an example where
the digit following our last significant digit is either a one or a seven. With the one, we round down; with
the seven, we round up. In this case, we have a zero. So, we round 1.25000 and so on to
simply 1.125. But what if we wanted to round to
three significant figures? Well, that would make our last
significant digit the two because we’ve got one, one, and two as our three
significant figures. But when it comes to rounding, we
have this five. So, we round up to 1.13. If we only wanted two significant
figures, it would be even easier. The digit after our last
significant digit is this two. So, we round down to 1.1.

And finally, if we only wanted one
significant digit, we would round to one because the digit after the one is one. We round down and we’re left with
one to one significant figure. Remember, none of the other digits
count when you’re rounding, just the one immediately after your last significant
digit.

Of course, talking about
significant figures, we’d probably need to know about decimal places as well. When we express a round number,
it’s easy to do. We write the ones, the tens, the
hundreds, and so on. When we write a number that’s not a
whole number, we have to use the decimal point. After the decimal point, we get the
tenths, the hundredths, the thousandths, and so on. But we can also have numbers that
are smaller than one that may need leading zeros. These leading zeros are never
significant, but we do need to write them to put the nonzero digit in the right
place. If we remove them, we wouldn’t be
expressing the right value.

This number has three decimal
places in total. We don’t see any digits after the
two, so we stop counting. But this number is also to three
decimal places. We’ve got a one in the units
column, a three in the tenths column, and a four in the hundredths column. When we express something to a
fixed number of decimal places, we just keep on writing digits after the decimal
place until we’ve hit that many digits.

If the number goes beyond that,
like in this example, we just need to round in the same way we did with significant
figures. We find the third decimal place
because we’re rounding to three decimal places. And we round the number up if that
digit is a five, six, seven, eight, or nine. And we round it down if it’s a
zero, one, two, three, or four. So, in this case, we round to 1.343
rather than 1.342 because the digit five came after the two and we were rounding to
three decimal places.

The last thing we need to deal with
is how we know how to round if we’re doing a calculation. There’s a fairly simple golden rule
for rounding. When you round your final answer,
you do it to the same number of significant figures as the least precise value in
the calculation. Well, let’s have a think about what
this really means by trying to calculate the density of a penny. Maybe, we use some cheap kitchen
scales accurate to the nearest gram. So, we got a value of three grams
accurate to one significant figure. This isn’t particularly
precise. But we’ve used a piece of equipment
that tells us the volume of the penny very, very accurately.

So, we have the value of 0.360
cubic centimeters to three significant figures. The trailing zero at the end is
significant because it was part of our measurement. We calculate the density by
dividing the mass by the volume. So, we divide three grams by 0.360
centimeters cubed, and our calculator tells us that the answer is 8.3 repeating. However, this is more precision
than we can honestly say we have because the mass is accurate to only one
significant figure. Therefore, we round our final
answer to the same precision, one significant figure.

If we use less precise numbers, we
get a value we’re less confident in. We should never claim to know
something to a greater precision than we actually do. That’s why we round so that other
people know how precise our measurements were. There are occasional exceptions to
the golden rule, but we won’t be dealing with those in this video. Instead, let’s have some
practice.

To how many significant figures is
the value 0.0023 kilograms given?

We can think of significant figures
of a number as the digits that determine its value and precision. The first three digits in the
number are zero, zero, and zero. None of these zeros contribute
directly to the value or precision of the number to come. It’s the number two that’s the
first nonzero digit. So, it’s the two that’s our first
significant digit. The three has also been given, so
that’s a significant digit as well. And we could write zeros after the
three, but we haven’t actually been told what those digits are. The number could have been rounded,
so these trailing zeroes are not significant. So, the value 0.0023 kilograms has
been given to two significant figures.

Next, let’s look at an example
based in the lab.

A student needs to use the mass
balance apparatus shown below to measure out a certain number of moles of a
substance. Using a calculator, the student
determines that they need to weigh out 12.108225 grams of the substance. Why is this amount not a suitable
value to measure using the mass balance apparatus?

Well, the first thing we need to do
is look at the mass balance apparatus. From the display, we can see it
measures in grams. The student is looking out to weigh
a mass in grams, which helps, but that’s not quite enough. The balance gives a value to a
maximum of four decimal places. A gram mass balance accurate to
four decimal places will be accurate to the nearest 0.0001 grams, one ten
thousandths of a gram. But the mass the student calculated
is to six decimal places. This is 100 times more precise than
the balance is capable of.

The balance would be able to read
the mass of 12.1082 grams. And if the student added a little
bit more, it would be able to read 12.1083 grams. But the number produced by the
student’s calculator is much more precise than the mass measurements this mass
balance is capable of. So, this amount is not a suitable
value to measure using the mass balance apparatus because the mass balance apparatus
only measures to four decimal places.

What mass should the student try to
weigh out instead? The number from the calculator was
to six decimal places, but the balance only reads to four. So, to make sure we get as close to
the calculated value as possible, we want to round that number to four decimal
places. You may think you could just chop
off the last two digits, but sometimes that won’t be as good as rounding. We know the first three decimal
digits are one, zero, eight. And the digit in the fifth decimal
place is a two. So, we round down. So, our final answer is 12.1082
grams.

So, to finish up with the key
points, significant figures or digits affect the value and precision of a
number. Leading zeros are never
significant. Trailing zeroes may be
significant. And sandwiched zeros are always
significant. Decimal places are the positions
after the decimal point. And when you give your absolute
final answer, you should round it to the same number of significant figures as the
least precise value you used in the calculation.