# Adding & Subtracting with Sig Figs

When adding or subtracting with sig figs, the rounding rules are slightly different than for
multiplication & division. To understand this difference, we'll revisit uncertainty
in measurements in terms of which places are specified.

The least-significant place of a measurement is the smallest (right-most) place
specified in the measurement.

For instance, the the least-significant place in the measurement

15300

is the hundreds place and less significant places (such as the tens place)
are not known and could have any value. This will be important in understanding
the uncertainty in the results of addition & subtraction with sig figs, so practice
identifying the least-significant place in the following measurements.
14,350

tens

Use your knowledge of sig figs to determine whether each digit in this number
specifies a place (has significance) or is simply a place holder. Then simply
choose the least-significant (right-most) place that is specified.

0.06010

hundred-thousandths

Again, determine which places are specified in this number and then
select the least-significant (right-most).

2.37e4

hundreds

With scientific notation you'll have to figure which place each digit in the pre-exponential
part of the number corresponds. For this it may help to write out the number in normal
notation so you can see the place of each digit.

When adding or subtracting two measurements, the result should be rounded to the same place as the
least-significant place of the measurement with the greatest least-significant place.

To understand this statement, imagine we're adding

Let practice the addition & subtraction rounding rule with the following examples. Again, you're given the arithmetic result and you can practice working out the rounded answer.

15,300

(certain to the hundreds place) and
851

(certain to the ones place).
Enter
15300 + 851

into the calculator gives
us 16151

and
how certain should we be of this result?
If the answer is any more certain than the hundreds place, then we'd be ignoring
the uncertainty of the input measurement 15,300

which is only certain to the hundreds place.
Therefore the answer can only be certain to the hundreds place and we round
to this place to reflect the uncertainty.
Thus answer of this addition with proper
sig figs is 16,200

.
Let practice the addition & subtraction rounding rule with the following examples. Again, you're given the arithmetic result and you can practice working out the rounded answer.

16 + 1.7

width: 24em;

width: 16em;

You'll want to determine how the answer should be rounded using the rule for
addition & subtraction:
the answer should be rounded to the same place as the least-significant place
of the measurement with the greatest least-significant place.

So look at each of the two added numbers and determine the least-significant place of each. The one that has the largest least-significant place specifies the place that the answer should be rounded to.

Now you can simply round the arithmetic result to this place.

So look at each of the two added numbers and determine the least-significant place of each. The one that has the largest least-significant place specifies the place that the answer should be rounded to.

Now you can simply round the arithmetic result to this place.

12.037 - 3.93

width: 24em;

width: 24em;

Again, determine the least-significant place of each number in this subtraction.
The greatest least-significant place determines the place of rounding for
the result.

0.073 + 10.0037

width: 24em;

width: 20em;

This example is to demonstrate how the results of adding (or subtracting)
two measurements can have more sig figs than one of the two inputs, due to
the rule of adding & subtracting with sig figs.

3.78e4 - 2.3e3

width: 24em;

width: 18em;

With scientific notation it can be helpful to write down each
number in normal notation to determine the least-significant place
of each number in this subtraction.

Additionally, scientific notation can be helpful to specify the answer & you can enter such an answer using the e shorthand; i.e.

Additionally, scientific notation can be helpful to specify the answer & you can enter such an answer using the e shorthand; i.e.

1.03e5

=
1.03e5

.
When adding or subtracting a measurements with an exact number, the answer should be
rounded to the same place as the least-significant place of the measurement.

This is similar to the rule for multiplication & division in that the
exact number doesn't influence how the result is rounded.
For instance if you add the exact value

Use the following examples to practice adding & subtracting sig figs with exact values.

10x

to the measurement
1.073

the result is
11.073

as we round the result
to the thousandths place; the least-significant place of the measurement.
As before, exact quantities are denoted by a blue underline.
Use the following examples to practice adding & subtracting sig figs with exact values.

13.7 + 7x

width: 22em;

width: 17em;

Using this new rounding rule, we know the answer should be rounded to the same
place as the least-significant place of the measurement.
Simply round the arithmetic result to the right number of sig figs.

3.71 - 1.325x

width: 22em;

width: 17em;

Exact numbers can be decimals and the place of rounding is still determined
by the least-significant place of the other number - the measurement.

In summary…

- The least-significant place of a measurement is the smallest (right-most) place specified in the measurement.
- When adding or subtracting two measurements, the result should be rounded to the same place as the least-significant place of the measurement with the greatest least-significant place.
- When adding or subtracting a measurements with an exact number, the answer should be rounded to the same place as the least-significant place of the measurement.

Now that we've got down the basic arithmetic of sig figs, lets find out
how to apply mathematical functions to measurements.
→