Which Digits are Significant Figures in a Given Number Dr. Yau
Numerically 3.0, 3.00, 3.000 are of the same value, but 3.000 shows that it was measured with the more precise instrument. The zeroes in all three numbers are considered "significant figures". They are shown to indicate the precision of the measurements. If we take away the zeroes, the value does not change. The measurement is still "three".
On the other hand, the zero in ".03" is not a significant figure. It is important though, because if we leave it out and write .3 then the value is completely different from .03 (it is 10 times bigger). Thus, such zeroes are said to "place the decimal", and not considered "significant".
Sometimes .03 is written as 0.03. The first zero also does not change the value of the number, but neither does it indicate more precision. It is generally included to stress the location of the decimal point, and its inclusion is never essential.
The general rule is as follows:
All digits are significant with the exception that:
1. leading zeroes are NOT significant (0.0005 has only one sig. fig.)
2. tailing zeroes in numbers without decimal points are ambiguous. (zeroes in 700, not zeroes in 700.0)
Such tailing zeroes are assumed not significant.
They must be expressed in scientific notation to remove the ambiguity.
5200 as stated is assumed to have 2 sig. fig.
If it were to have 3 sig. fig., it should have been expressed as 5.20 x 10 3.
If it were to have 4 sig. fig., it should have been expressed as 5.200 x 103,
or it could have been expressed as 5200. where the decimal indicates the tailing zeroes to be significant. (Remember tailing zeroes are assumed not significant only when there is no decimal point. Tailing zeroes in numbers with decimal points are significant.)
In the following examples, the significant figures are
underlined in the [ ]:
30 is assumed to have one sig.fig.[ 30 ].
If you have to report such a number, you MUST express it in scientific notation.
30. has 2 sig. fig. [ 30. ] (The number has a decimal point, so all tailing zeroes are significant.)
30.0 has 3 sig. fig. [ 30.0 ] (Again, the number has a decimal point, so all tailing zeroes are significant.)
0.0050200 has 5 sig. fig. [0.0050200 ] (Leading zeroes are not significant, but the tailing zeroes are significant, because the number has a decimal point.)
12.00 has 4 sig. fig. [ 12.00 ]
32.0 x 102 has 3 sig. fig. [ 32.0 x 102 ]
Do not confuse the number of sig. fig. with the number of decimal places. The number of decimal places refer to the number of digits to the right of the decimal point. Thus 30.0 has three sig. fig. but only one decimal place.
Practice on the following questions.
1. Give the number of sig. fig. and the number of decimal places in each of the number below.
a) 72.32
b) 10.002
c) 0.003
d) 0.00170
e) 3,000
f) 3,000.
g) 3,000.00
Write down your answers and click here to see the correct answers to Question #1.
2. These numbers have ambiguous zeroes. Remove the
ambiguity by expressing them in scientific notation.
a) 42000 in 2 sig. fig.
b) 42000 in 3 sig. fig.
c) 42000 in 4 sig. fig.
d) 2100 in 3 sig. fig.
e) 790,000 in 4 sig. fig.
f) 3800 x 10-7 in 3 sig. fig.
Write down your answers and click here to see the correct answers to Question #2.
[Extra practice: Brady & Senese 5th Ed, p.30 #36 & 37]
|
|
# of sig. fig. |
# of decimal places |
|
a) 72.32 |
4 |
2 |
|
b) 10.002 |
5 |
3 |
|
c) 0.003 |
1 |
3 |
|
d) 0.00170 |
3 |
5 |
|
e) 3,000 |
assume 1 |
0 |
|
f) 3,000. |
4 |
0 |
|
g) 3,000.00 |
6 |
2 |
Return to do Question #2.
Return to Beginning.
|
|
Answers |
|
a) 42000 in 2 sig. fig. |
4.2 x104 |
|
b) 42000 in 3 sig. fig. |
4.20 x 104 |
|
c) 42000 in 4 sig. fig. |
4.200 x 104 |
|
d) 2100 in 3 sig. fig. |
2.10 x 103 |
|
e) 790,000 in 4 sig. fig. |
7.900 x 105 |
|
f) 3800 x 10-7 in 3 sig.fig. |
3.80 x 10-4 |