MASTERY: SIGNIFICANT FIGURES
Minimum Level: 93%
Significant figures are used to report information about the certainty of measurements
and calculations. With this method, a measurement of 3.56 m means that while the 3 in
ones place and the 5 in tenths place are certain (read directly from the scale on the
measuring instrument), the 6 in hundredths place is an estimate. When this
measurement is combined with several other measurements in a formula, there must
be some way of tracking the amount of uncertainty in each measurement and in the
final result. For example, using a calculator to find the volume of a cube that measures
3.56 m on a side, you get 45.118016 m3. This answer indicates far greater precision in
the volume measurement than is realistic. Remember that the 6 in 3.56 is an estimated
digit.
The rules and examples that follow will show you how to work with the uncertainty in
measurements to express your results with an appropriate level of precision.
A. Determining the Number of Significant Figures
In order to properly calculate a problem, you must first know how to tell if the digit is
significant or not. A measurement expressed to the appropriate number of
significant figures includes all digits that are certain and one digit in the
measurement that is uncertain.
• ALL digits are significant in numbers that have a decimal point in the middle.
(4 8 0 . 9 0 5 has 6 SF, 4 8 0 . 9 0 5 0 has 7 SF)
•
ALL non-zero digits (1, 2, 3, 4, 5, 6, 7, 8, 9) are ALWAYS significant.
•
0's always hold a place. However, they are not always a significant part of the
measurement or calculation. 0's are considered a significant figure ONLY if they
BOTH hold a place AND indicate part of the measurement (either read directly
from the scale or are in the estimated position).
•
0's that are to the right of non-zero digits in a whole number may or may not be a
significant part of the measurement. Their significance is shown by either
1. putting a bar over the 0 that is in the rightmost place of significance, (80,700)
OR
2. if the significant zero is in one's place, using a decimal point at the end of the
number. (80,700. )
Never use both the bar and the decimal point in the same number.
Never use more than one bar in the same number.
Never use the bar on zeroes to the right of the decimal point.
The following explains a method you may use to determine the number of significant
figures:
For decimal numbers only:
• Draw a vertical line through the first non-zero digit to the right of the decimal
point.
• 0’s to the left of the line are NEVER significant.
• 0’s to the right of the line are ALWAYS significant.
EXAMPLE:
0.006 050
NEVER | ALWAYS ANSWER: There are 4 significant figures…. 6, 0, 5, 0.
1. For whole numbers only:
• Draw a vertical line through the first non-zero digit to the left of the decimal
point.
• 0’s to the left of the line are ALWAYS significant.
• 0’s to the right of the line are may or may not be significant...DEPENDS on
whether or not they have been designated significant by EITHER a bar above
OR a decimal point after. If there is no bar over the zero or a decimal point
after the zero, then the zero in that position is NOT significant, it only holds a
place.
EXAMPLE:
605 000
NEVER | DEPENDS • No bar or decimal point = NOT significant
• Bar or decimal point = SIGNFICANT from the
designated significant zero.
ANSWER: There are 3 significant figures…. 6, 0, 5.
If the example had been 605 000., then there are 6 significant figures.
EXAMPLES:
DECIMAL POINT
PRESENT
0.032
0.000302
0.003020
0.00300200
2 SF
3 SF
4 SF
6 SF
DECIMAL POINT
"ABSENT"
302
3020
3020.
3 002 000
3 002 000.
3 SF
3 SF
4 SF
4 SF
7 SF
B. Calculating with Significant Figures
There are two sets of rules for calculations: one for addition/subtraction and one for
multiplication/division.
Addition/Subtraction
1. Line up the decimal points.
2.
The answer's least significant figure will be the PLACE where there was
significance in each of the numbers that were being added or subtracted.
EXAMPLE:
10.07
506.3
225.799
742.169 round to the nearest tenth = 742.2
Since tenths place is the PLACE of least significance in the problem (all
numbers to be added have significance in tenths place), the answer is
rounded to the nearest tenth.
Multiplication/Division
1. Multiply and/or divide.
2.
Count the NUMBER of significant figures in number that are being
multiplied/divided. The answer will have the same NUMBER of significant
figures as the least number that you counted in the original data.
EXAMPLE:
4.0037 X 6.23 = 2.28
10.955
•
4.0037 has 5 SF, 6.23 has 3 SF, 10.955 has 5 SF;
therefore the 6.23 with 3 SF determines that the answer will only have 3 SF