# Determining the Significant Figures of Measurements

To perform calculations with measurements, taking into account their limited accuracy,
we first need a way to determine the accuracy of a measurement.
This is done by counting the number of digits in the measurement that convey meaningful
information or the number of significant figures.

Significant figures are the digits that specify places in a measurement.

As an example, lets look at the measurement

10.043 g

.
Here we have five digits specifying from the tens to the thousandths place,
and therefore this measurement has 5 significant figures.
Another example would be 3.3476666666666666 g

which has 17
significant figures (sig figs). For some quick practice, count the sig figs in following
example measurements.
4.31 lbs

3

Simply count the number of digits in this measurement.
The 4 in the ones place, the 3 in the tenths place, and the 1
in the hundredth place.

1176 molecules

4

Again, just count the number of digits in this measurement.

Trailing zeros in an integer measurement are only place holders and aren't significant.

Trailing zeros could be an ambiguous edge case.
For instance with the measurement

Practice the trailing zeros rule with the following quick examples.

1300 in

we
know the thousands and hundreds places are specified by the 1 and the 3 respectively.
In contrast, we don't know if the 0s specify values or if they only serve as place holders.
In the later case we'd have 4 significant figures with the tens and ones place specified,
where as if they only serve as place holders we'd only have 2.
To prevent such ambiguities, we take any such trailing zeros in
integer measurements as place holders and they don't contribute to significant figures.
Therefore the measurement 1300 in

has two significant figures.
Practice the trailing zeros rule with the following quick examples.

200 tons

1

As before, count the number of digits, but now we know not to include trailing
zeros.

10350 m

4

Non-trailing zeros, i.e. the zero between 1 and 3, still contribute meaningful
information specifying a place in the measurement and therefore contribute to
significant figures.

In decimal measurements, trailing zeros are significant.

The rules are slightly different for numbers with decimal points, as trailing zeros
are not needed as place holders.
As an example, the trailing zero in the measurement

5.0 kg

is not needed as a place holder and its inclusion
shows that we do know the tenths place of this measurement to be zero.
This would be in contrast to 5 kg

which would show we
are uncertain of the tenths place.
Therefore 5.0 kg

has two significant figures.
Thus in decimal measurements, trailing zeros are significant.
What about a measurement such as

1300.05 ft

? Here we
have trailing zeros in the integer portion of the measurement as well as non-zero
digits in the decimals place.
Since the decimal portion specifies the measurement down to the accuracy of the hundredth
place, we must also know the tenths, ones, and tens places and these are all zero.
Therefore all of these digits are significant and the measurement
1300.05 ft

has 6 sig figs.
The general rule is that
zeros between non-zero digits are significant.
Go ahead and figure out the number of significant figures in the following decimal
measurements.

12.0 acres

3

Remember, trailing zeros in the decimals place still contribute meaning as
they aren't needed as place holders and thereby count in sig figs.

700.358 mg

6

Since we know the measurement down to the thousandths place, we must
also know for certain the tenths and ones place. Therefore these
zeros must carry significance.

Leading zeros in entirely decimal measurements are also place holders.

One last situation you'll encounter is the case of entirely decimal measurements
such as

0.0054 liters

.
For such measurements the leading zeros serve only as place holders,
just as the trailing zeros do in a decimal measurement.
Therefore 0.0054 liters

is a measurement with 2 significant
figures.
0.00076 cm

2

The leading zeros in this decimal measurement are only place holders and
thereby impart no significance.

0.0301 cm

3

Only the leading zeros are place holders. Zeros after a non-zero digit
still specify a value for a place in the measurement are significant.

In scientific notation, all digits before the multiplication sign are significant.

One way to resolve the ambiguity of trailing and leading zeros is to use scientific notation,
where a number is represented as a decimal with a single digit before the decimal place times
10 raised to a power.
Here

1300 in

would be represented as 1.3e3 in

and there is no need for trailing zeros as place holders.
Further, if you wanted to show that the zero in tenths place was not just a place holder and
does specify a value for the tens place, then you could report the measurement as 1.30e3 in

.
In scientific notation, all of digits before the multiplication sign contribute
significance, which makes this a convenient way to display the significance of a measurement.
The digits after the multiplication sign, the 10 and the power, are just place holders and
aren't significant.
1.7031e-5 kg

5

Remember that all of the digits before the multiplication sign -
and only these digits - contribute significance.

1.300e19 atoms

4

As with decimal notation, trailing zeros in scientific notation
specify a place and are significant.

Lastly, many software programs - including the Sig Figs Calculator - can accept scientific
notation using the e shorthand where e stands
for times 10 to the power. For instance

1.30e3 in

can be entered into the Sig Figs calculator as 1.30e3

.
The rules for determining the significant figures of a measurement can
be summed up as follows …

- All non-zero digits are significant.
- Trailing zeros in an integer (no decimal place) measurement are just place holders and are not significant.
- Leading zeros are also place holders and aren't significant.
- Trailing zeros following the decimal point and zeros between digits are significant.
- In scientific notation, all digits before the multiplication sign are significant.

Now that we know how to determine the significant figures of a measurement,
we can get ready to perform calculations by learning
how to round the results of calculations with sig figs.
→