College Mathematics Notes
Section 1.3
Page 1 of 5
Chapter 1: Pre-Algebra
Section 1.3: Significant Digits
Big Idea for this section: The way we write a number from a measurement carries information about the
precision of the measurement. That, in turn, tells us how to round answers from calculations that involve
measurements.
Big Skills for this section: You should be able to identify the number of significant digits in a measurement
number, write the result of a measurement with the correct number of significant digits, and round calculations
involving measurements to the correct number of significant digits.
Section 1.3.1: Identifying and Writing Significant Digits
 There is a shortcut that scientists and engineers use to state the precision of a measurement when writing
down the measurement.
 For example, if you measure the width of a box of chalk with a school-type ruler, you might get an
answer of 6.1 cm. Since the ruler is only marked in increments of 0.1 cm, you really can’t measure any
finer than half that increment, which is 0.05 cm. Thus, the best way to state your measurement is as
6.1 cm  0.05 cm. That way, people know exactly how precise your measurement is, and they know not
to expect more precision (like 6.139 cm), and that a number of 6 cm is neglecting the full precision of
the measurement. Note: 6.1 cm  0.05 cm = 61 mm  0.5 mm…
 If you measure the width with a high-precision instrument like a digital caliper, then you might discover
the answer to be 61.22 mm. Again, the uncertainty would be  one-half of the smallest measurement
possible, which would be  0.005 mm. Thus, we’d write the measurement as 61.22 mm  0.005 mm.
 However, people have found a more efficient way to write a measurement like 6.1 cm  0.05 cm.
Instead, they just write 6.1 cm, and it is understood that the precision is  one-half of the farthest-right
digit. In this case, significant digits are being used to indicate the precision of the measurement.
 The significant digits (also called significant figures and abbreviated sig figs) of a number are those
digits that carry meaning contributing to its precision.
Practice:

Suppose that someone tells you the width of a piece of paper is 21.59 cm. However, you know
that they measured it with a ruler only marked in increments of 0.1 cm. Write the measurement
correctly using significant digits.

5
Suppose you measure the length of a board to be 13 inches, and your tape measure is marked in
8
eighths of an inch. Correctly write the measurement as a decimal using significant digits.
College Mathematics Notes

Section 1.3
Page 2 of 5
Suppose you measured the length of a race to be exactly 100 meters using a tape that measures to
the nearest centimeter (i.e., hundredth of a meter). Write the measurement correctly using
significant digits.

What if the tape only measures to the nearest decimeter (i.e., tenth of a meter)?

What if the tape only measures to the nearest meter?

What if the finest increment of the tape is ten meters?

What if the finest increment of the tape is 100 meters?

That last practice problem embodies one of the most confusing parts of working with significant digits:
measurements that have a lot of zeros in them.
o When you want to show that a zero is significant, you go to the extra effort of writing extra stuff
that you normally wouldn’t write, like putting extra zeros after the decimal point to show the
finest increment of measurement:
 Example: 100.00 m
o Continuing the theme of writing something a little extra to show what zeros are significant, an
old-fashioned trick for showing which trailing zeros are significant is to put a bar over every
significant zero:
 Like 100m or 100m for the last two examples above.
o A more widely used trick to show when a measurement has been made to the nearest ones place
is to put a decimal after the ones palce:
 100 m  100. m

The rule for identifying the significant digits in a measurement can be stated in two ways:
o All digits are significant except leading zeros after a decimal point and trailing zeros before it.
Both types of zeros merely serve as placeholders.
o “point right, otherwise left”:
 If there is a decimal point present, find the left-most nonzero digit, and then count digits
toward the right. If there is no decimal point in the number, find the right-most nonzero
digit and count toward the left. In both cases, keep counting digits until you reach the
other end of the number.
College Mathematics Notes
Section 1.3
Page 3 of 5
Practice: State the number of significant digits in each measurement below
 467.24 ft.

0.006020 m

1250 lb.

0.00003 g

93,000,000 mi

800 in.

800. in.

1250. lb.
Section 1.3.2: Significant Digits after Adding or Subtracting
 When you add or subtract measurements, then you have to round your answer to the precision of the
least accurate measurement. That’s because the uncertainty in the least precise measurement
overwhelms the precision of the other measurements.
28.25 cm
+15.67 cm
Practice:
114.37 cm
3.080 cm
+27.3
cm

22.85 g – 13.35 g =

13 ft – 5.811 ft =

1250.27 mi + 3367.7 mi + 2257 mi + 4800 mi =
College Mathematics Notes
Section 1.3
Page 4 of 5
Section 1.3.3: Significant Digits after Multiplying or Dividing
 Rule: Find the measurement with the fewest significant figures. Round your answer to that many
significant figures.
 The rule has to do with how the uncertainties can combine in the final answer:
 Example: if the floor of a room is measured at 14.12 feet long and 9.8 feet wide, our calculators
would give us an area of Area = 14.12 ft.  4.8 ft. = 67.776 sq. ft.
 However, 14.12 ft.  14.12 ft.  0.005 ft.  the length could range from 14.115 ft. to 14.125 ft.
 Also, 4.8 ft.  4.8 ft.  0.05 ft.  the width could range from 4.75 ft. to 4.85 ft.
 So, the largest possible area under these uncertainties would be 14.125 ft.  4.85 ft. = 68.50625
sq. ft.
 The smallest possible area under these uncertainties would be 14.115 ft.  4.75 ft. = 67.04625 sq.
ft.
 The average of those two areas is 67.77625 sq. ft., which is  0.73 sq. ft. from either extreme.
 Thinking of uncertainty as  one-half of a finest increment of measurement, we would round the
uncertainty to 0.5 sq. ft., so we would have to round the answer to 68 sq. ft. to reflect this
uncertainty.
 Or, we could have used the rule; the measurement of 4.8 ft. has the least number of significant
digits (two sig. figs.), so the final answer can only have two significant digits.
 A good way to show the work is: Area = 14.12 ft.  4.8 ft. = 67.776 sq. ft.  68 sq. ft.
 One exception to the rule: numbers without a unit of measurement attached are considered exact
numbers that have an infinite number of significant digits. Ignore them when determining the number of
sig. figs. in your answer.
 Example: 100  22.85 cm = 2285 cm
Practice:
1) 2.54 in.  3 in. =
2) 310.2 cm x 51.05 cm x 0.100 cm =
3) 22.85 ft2 / 4 ft. =
4) 358.9 m3 / 22 m2 =
5) 75.40 g / 7.1 mL =
College Mathematics Notes
6) 52.5 mi/hr  3.8752 hr =
Section 1.3
Page 5 of 5