Significant Figures1
- Defined relations
(constants such as p
or c and conversion factors) are invariable. They do not have
sigfigs per se and will not effect calculations. For this reason,
always use the maximum given digits for these - they should never be
the limiting factor in your final sigfigs!
- Measurements can be precise but never perfect.
There are uncertainties associated with the measured values, and
so only those digits which are significant are recorded.
- Recording measured values:
- include all of the digits which are certain
according to the measurements;
- include the first uncertain digit, which
is an estimate.
RULES FOR COUNTING SIGFIGS
- Counting sigfigs means
identifying those digits which "count" (i.e.
are valid for the calculation /measureent) and discarding those
which do not count.
- Begin with the first non-zero digit on the left
and continue to the last "includable" one (the first uncertain
one).
- Determining whether or not a zero in a number
is significant:
Case | Examples
| |
| value
| # sigfigs |
|
Zeros between other significant digits are
| 405 |
3 |
significant | 807.105
| 6 |
|
Zeros on the right of the first non-zero digit
| 0.03800
| 4 |
are significant | 3,6000,000
| 7 |
|
Zeros on the left of the first non-zero digit
| 0.006 |
1 |
are not significant
| 0.0352
| 3 |
|
Zeros on the right of the last non-zero digit,
| 47,000
| 2-5 |
but to the right of the decimal are ambiguous,
| 380 |
2-3 |
but can be clarified with scientific notation
| 1.00x107
| 3 |
RULES FOR CALCULATIONS WITH SIGNIFICANT FIGURES
- addition & subtraction:
a sum of sigfigs will have no more digits to the right of the
decimal than does any addend. That is, the sum is limited in its
right-of-decimal digits by the number in the calculation with
the fewest digits:
126.7 | |
| | |
| |
7439.575 |
| | | 4639.753
| | |
833 | |
| | -1542.61
| | |
7574.605 | ->
| 7574.6 |
| 3097.143 | ->
| 3097.14 |
- multiplication & division: the product will have
no more sigfigs than any factor in the calculation, and the quotient
not more than any divisor in the calculation (these exclude defined
relations):
32x87.1x2.6666= | 7432.3475
| | 167.90/22=
| 7.6311818 |
| 7.4x103
| | | 7.6
|
- rounding off
- When the unwanted digits are more than 5, round up: (to 3
sigfigs) 2.45633->2.46
- If the unwanted digits are less than 5, round down: (to 4
sigfigs) 37,913->37,910 or 3.791x104
- If the unwanted digits are exactly 5:
- if the last digit is odd, round up to the next even: (to 3
sigfigs) 23.3544->23.4
- if the digit is even, do nothing: (to 4 sigfigs) 0.07236581->0.07236
- Carry all of the figures through the calculation and round
off to the appropriate number of sigfigs only at the very end
to avoid "round off error."
1Adapted from Coop,
D.W., The Camelot Chemistry Primer, Kenndon, Kraistins
& Gould Publishers; Temecula, CA. 1992. P.29-31.