# Significant Figures in Measurements and Computations

**Uncertainty in Measurements:** The word accuracy has
various connotations depending on the

circumstances under which it is used. In the study of physics it is commonly
used to denote the

reliability of a number or the limiting accuracy of a measurement.

As applied to a measurement, the accuracy is expressed by
stating the uncertainty of the numerical

result, that is, the estimated maximum amount by which the result may differ
from the "true" or

accepted value.

For instance, if we have a keg advertised to contain
15,000 nails, we can assume that there are

between 14,000 and 16,000 nails. The uncertainty amounts to +/- 1 in the last
significant figure.

**Significant Figures: **The accuracy of a physical
measurement is properly indicated by the number

of figures used to express the numerical measure. Conventionally, only those
figures that are

reasonably trustworthy are retained. These are called significant figures.

In recording certain numbers, the location of the decimal
point requires zeros to be added to the

significant figures. These zeros serve only to locate the decimal point and are
never considered

significant.

**Examples:**

Length of a page 22.7 cm (3 significant figures)

Thickness of a page = 0.011 cm (2 significant figures)

Distance to the sun = 149,600,000 km (4 significant fig)

Speed of light = 299,790 km/sec (5 significant figures)

If each of these numbers is expressed in terms of
scientific notation, there is no doubt as to the

number of significant figures for only the significant figures are then
retained. Thus:

Length of page = 2.27 x 10^{1} cm

Thickness of a page = 1.1 x 10^{-2} cm

Distance to the sun = 1.496 x 10^{8} km

Speed of light = 2.9979 x 10^{5} km/sec

Do not be confused by the placement of a zero in a string
of numbers. For instance, say the land

speed record for a bicycle is 80.13 miles per hour. In this case the zero does
more than position the

8, 1, and 3. It indicates that the number is 80, not 79 or 81 miles per hour. A
zero imbedded in a

string of significant figures can never itself be considered non-significant.
The number 80.13 is

accurate to 4 significant figures.

Further, do not be confused by a zero that is said to be
of significance. Something may be measured to

an accuracy of 2.000 units. In this case we know that the measurement is
accurate to three decimal

places. That is, the measurement lay somewhere between 2.001 and 1.999.

There are some numbers that, by their definition, may be
taken to have an unlimited number of

significant figures. For example, the integer 2 and the irrational number ! in
the following

relationship for a circle: circumference = 2πr where r is the radius of a
circle.

In calculations there is frequent need to use data that
have been recorded without a clear indication of

the number of significant figures. For example, a textbook problem may refer to
a "two-kilogram

mass", or in an experiment a student may announce that she has measured a
certain distance as "3

meters". In such cases the values with the approximate number of significant
figures should be written

from what is know or assumed about the way in which the measurement were made.
If the distance

referred to were measured with an ordinary tape measure, the distance might
appropriately be written

as 3.0 meters. If the distance were carefully measured with a steel scale
accurate to the nearest

millimeter, the distance might be recorded as 3.000 meters. In academic problem
work a good rule to

follow is to retain three figures unless there is good reason to decide
otherwise.

**Recapping:** The significant figures in a quantity
are the meaningful digits in it. There are several

conventions to keep in mind when dealing with significant figures:

1) Any nonzero digit is a significant digit. Examples:
3.14 has three significant digits and 239.67

has five significant digits.

2) Zeros in between nonzero values are significant digits. Example 20.4 has
three significant digits.

3) Zeros to the left of the first nonzero digit are NOT significant. Example:
0.0012 has two

significant digits.

4) For numbers with decimal points, zeros to the right of nonzero digits are
significant. Example:

Both 2.0 and 0.0020 have two significant digits.

5) For numbers without decimal points, zeros to the right of nonzero digits may
or may not be

significant. Example: 350 may have two or three significant digits. To avoid
confusion, use a

decimal point: 350. has three significant digits, 350 has two.

6) The last significant digit in a number should have the same order of
magnitude as the

uncertainty. Examples: 3.1415±0.0001; 0.8±0.2

7) The uncertainty should be rounded to one or two significant digits.

**Rules for Computation: **There is always a pronounced
and persistent tendency on the part of

beginners to retain too many figures in a computation. This not only involves
too much work but,

worse still, leads to a fictitiously precise result.

The following rules are recommended and will save much
time that would otherwise be spent in

calculation. Furthermore, their careful use will result in properly indicated
accuracies.

1) In recording the result of a measurement or
calculation, one, and only one, doubtful digit is

retained. For instance, a caliper whose finest division is 100-thousandths can
produce a

measurement that is accurate to five decimal places at best. i.e. 3.02364.

2) In addition or subtraction, do not carry the result
beyond the first column that contains a

doubtful figure. Add all figures and then round (see #4 below) the result. For
example:

which becomes 33.66 | which becomes 0.8 |

3) In multiplication and division, carry the result to the
same number of significant figures that

there are in that term which has the least number of significant figures. For
example:

3.95 x 5.1 = 20.145, rounded to 2 significant figures = 20.

34.2/6.29934 = 5.429140196... becomes upon rounding 5.43

4) When “doing” mathematical manipulations with numbers
that have varying numbers of

significant digits, the final result cannot have more significant digits than
the least well known

digit. In practice, we will keep all the numbers to their known accuracy through
out the

calculation and then round to the number of significant figures at the end.
Examples:

2.0*3.14159 = 6.3; 25.001+32.0 = 55.0

5) In dropping figures that are not significant
(rounding), the last figure retained should be

unchanged if the following figures dropped are less than half the previous
digit. It should be

increased by 1 if the following figures dropped are greater than half the
previous digit. If the last

digit dropped is 5, the preceding digit should be left unchanged if it is an
even number but

increased by 1 if it is an odd number. For example:

24.5627 rounded to four significant figures becomes 24.56

34.2523 rounded to three significant figures becomes 34.3

3.485 rounded to three significant figures becomes 3.48

7.550 rounded to two significant figures becomes 7.6