Significant Digits
All measurements are approximations because no measuring device is absolutely perfect.
Be convention, a mass of 30.2 grams indicates that the mass is measured to the nearest tenth of a
gram and a mass of 30.20 grams indicates that the mass is measured to the nearest hundredth of a
gram. The number 30.2 contains 3 significant digits (sig. figs.) while the number 30.20 contains
four significant digits. It is important to note that the last digit recorded is always an estimate and
is, therefore, uncertain.
RULES FOR DETERMINING SIGNIFICANT DIGITS
1.
2.
3.
All non-zero digits are significant.
Zero to the right of the decimal AND at the end of the number of significant.
Zeros between two other significant digits are significant.
Now for some examples…
1.
All non-zero digits are significant.
3.76 contains 3 significant digits.
37.6 contains 3 significant digits.
175.624 contains 6 significant digits.
2.
Zeros to the right of the decimal AND at the end of the number of significant.
1.0 contains 2 significant digits.
5.200 contains 4 significant digits.
25.0 contains 3 significant digits.
3.
Zeros between two other significant digits are significant.
4008 contains 4 significant digits.
30.9 contains 3 significant digits.
8.100 contains 4 significant digits.
What about a number like 0.010? How many significant digits are in this number? The
answer is 2. Let’s look at one digit at a time to figure this out. The first zero (0.010) is not a
significant digit because is not to the right of the decimal nor is it between two other significant
digits. The second zero (0.010) is not a significant digit because even though it is to the right of
the decimal it is not at the end of the number. Also, it is not between two other significant digits.
The one (0.010) is a significant digit because all non-zero digits are significant. The final zero
(0.010) is a significant digit because it is both to the right of the decimal and at the end of
thenumber.
What about a number like 12,000? There are 2 sig. figs. Let’s take some time to figure
this out. Both the one and the two (12,000) are sig. figs. because all non-zero digits are
significant. None of the zeros (12,000) are sig. figs. because none of them are to the right of the
decimal and none of them are between two other sig. figs.
In science classes we commonly obtain numerical data in the lab and then use these
numbers in calculations. This begs the question: How are significant digits handled in
calculations? The answer depends on whether you are (a) adding and subtracting or (b)
multiplying and dividing.
ADDITION AND SUBTRACTION
The answer should be rounded off so as to contain the same number of decimal places as
the least accurate value.
Examples:
17.15
-4.170
12.98
12.7
4.716
+13.85
31.3
2 digits after the decimal
3 digits after the decimal
2 digits after the decimal
1 digit after the decimal
3 digits after the decimal
2 digits after the decimal
1 digit after the decimal
MULTIPLICATION AND DIVISION
The answer should be rounded off so as to contain the lowest number of significant
digits, regardless of the decimal point.
Examples:
3.17
x 5.186
16.4
18.35
0.717
= 25.6
3 significant digits
4 significant digits
3 significant digits
4 significant digits
3 significant digits
3 significant digits
ROUNDING OFF
Rounding off a number is much the same as you have always done with one exception.
When the number being dropped is a 5 we round to the even number (we don’t always round
up.)
7.185 is rounded to 7.18 (round down to the even number)
0.3735 is rounded to 0.374 (rounded up to the even number)