There are three rules on determining how many significant figures are in a number:
Please remember that, in science, all numbers are based upon measurements (except for a very few that are defined). Since all measurements are uncertain, we must only use those numbers that are meaningful. A common ruler cannot measure something to be 22.4072643 cm long. Not all of the digits have meaning (significance) and, therefore, should not be written down. In science, only the numbers that have significance (derived from measurement) are written.
Rule 1: Non-zero digits are always significant.
Hopefully, this rule seems rather obvious. If you measure something and the device you use (ruler, thermometer, triple-beam balance, etc.) returns a number to you, then you have made a measurement decision and that ACT of measuring gives significance to that particular numeral (or digit) in the overall value you obtain.
Hence a number like 26.38 would have four significant figures and 7.94 would have three. The problem comes with numbers like 0.00980 or 28.09.
Rule 2: Any zeros between two significant digits are significant.
Suppose you had a number like 406. By the first rule, the 4 and the 6 are significant. However, to make a measurement decision on the 4 (in the hundred's place) and the 6 (in the unit's place), you HAD to have made a decision on the ten's place. The measurement scale for this number would have hundreds and tens marked with an estimation made in the unit's place. Like this:
Rule 3: A final zero or trailing zeros in the decimal portion ONLY are significant.
This rule causes the most difficulty with students. Here are two examples of this rule with the zeros this rule affects in boldface:
0.00500
0.03040
Here are two more examples where the significant zeros are in boldface:
2.30 x 10¯^{5}
4.500 x 10^{12}
Here are the first two numbers from just above with the digits that are NOT significant in boldface:
0.00500
0.03040
These zeros serve only as space holders. They are there to put the decimal point in its correct location. They DO NOT involve measurement decisions. Upon writing the numbers in scientific notation (5.00 x 10¯^{3} and 3.040 x 10¯^{2}), the non-significant zeros disappear.
Zero Type #2: the zero to the left of the decimal point on numbers less than one.
When a number like 0.00500 is written, the very first zero (to the left of the decimal point) is put there by convention. Its sole function is to communicate unambiguously that the decimal point is a deciaml point. If the number were written like this, .00500, there is a possibility that the decimal point might be mistaken for a period. Many students omit that zero. They should not.
Zero Type #3: trailing zeros in a whole number.
200 is considered to have only ONE significant figure while 25,000 has two.
This is based on the way each number is written. When whole number are written as above, the zeros, BY DEFINITION, did not require a measurement decision, thus they are not significant.
However, it is entirely possible that 200 really does have two or three significnt figures. If it does, it will be written in a different manner than 200.
Typically, scientific notation is used for this purpose. If 200 has two significant figures, then 2.0 x 10^{2} is used. If it has three, then 2.00 x 10^{2} is used. If it had four, then 200.0 is sufficient. See rule #2 above.
How will you know how many significant figures are in a number like 200? In a problem like below, divorced of all scientific context, you will be told. If you were doing an experiment, the context of the experiment and its measuring devices would tell you how many significant figures to report to people who read the report of your work.
Zero Type #4: leading zeros in a whole number.
00250 has two significant figures. 005.00 x 10¯^{4} has
three.
There are 100 years in a century.
2 molecules of hydrogen react with 1 molecule of oxygen to form 2 molecules of water.
There are 500 sheets of paper in one ream.
Interestingly, the speed of light is now a defined quantity. By definition,
the value is 299,792,458 meters per second.
In mathematical operations involving significant figures, the answer
is reported in such a way that it reflects the reliability of the least
precise operation. Let's state that another way: a chain is no stronger
than its weakest link. An answer is no more precise that the least precise
number used to get the answer. Let's do it one more time: imagine a team
race where you and your team must finish together. Who dictates the speed
of the team? Of course, the slowest member of the team. Your answer cannot
be MORE precise than the least precise measurement.
For addition and subtraction, look at the decimal portion (i.e., to the right of the decimal point) of the numbers ONLY. Here is what to do:
1) Count the number of significant figures in the decimal portion of each number in the problem. (The digits to the left of the decimal place are not used to determine the number of decimal places in the final answer.)
2) Add or subtract in the normal fashion.
3) Round the answer to the LEAST number of places in the decimal portion
of any number in the problem.
In mathematical operations involving significant figures, the answer
is reported in such a way that it reflects the reliability of the least
precise operation. Let's state that another way: a chain is no stronger
than its weakest link. An answer is no more precise that the least precise
number used to get the answer. Let's do it one more time: imagine a team
race where you and your team must finish together. Who dictates the speed
of the team? Of course, the slowest member of the team. Your answer cannot
be MORE precise than the least precise measurement.
The following rule applies for multiplication and division:
The LEAST number of significant figures in any number of the problem determines the number of significant figures in the answer.
This means you MUST know how to recognize significant figures in order
to use this rule.
Example #1: 2.5 x 3.42.
The answer to this problem would be 8.6 (which was rounded from the calculator reading of 8.55). Why?
2.5 has two significant figures while 3.42 has three. Two significant
figures is less precise than three, so the answer has two significant figures.
Example #2: How many significant figures will the answer to 3.10 x 4.520 have?
You may have said two. This is too few. A common error is for the student to look at a number like 3.10 and think it has two significant figures. The zero in the hundedth's place is not recognized as significant when, in fact, it is. 3.10 has three significant figures.
Three is the correct answer. 14.0 has three significant figures. Note that the zero in the tenth's place is considered significant. All trailing zeros in the decimal portion are considered significant.
Another common error is for the student to think that 14 and 14.0 are the same thing. THEY ARE NOT. 14.0 is ten times more precise than 14. The two numbers have the same value, but they convey different meanings about how trustworthy they are.
Four is also an incorrect answer given by some ChemTeam students. It is too many significant figures. One possible reason for this answer lies in the number 4.520. This number has four significant figures while 3.10 has three. Somehow, the student (YOU!) maybe got the idea that it is the GREATEST number of significant figures in the problem that dictates the answer. It is the LEAST.
Sometimes student will answer this with five. Most likely you responded
with this answer because it says 14.012 on your calculator. This answer
would have been correct in your math class because mathematics does not
have the significant figure concept.
Example #3: 2.33 x 6.085 x 2.1. How many significant figures in the answer?
Answer - two.
Which number decides this?
Answer - the 2.1.
Why?
It has the least number of significant figures in the problem. It is,
therefore, the least precise measurement.
Example #4: (4.52 x 10¯^{4}) ÷ (3.980 x 10¯^{6}).
How many significant figures in the answer?
Answer - three.
Which number decides this?
Answer - the 4.52 x 10¯^{4}.
Why?
It has the least number of significant figures in the problem. It is, therefore, the least precise measurement. Notice it is the 4.52 portion that plays the role of determining significant figures; the exponential portion plays no role.
WARNING: the rules for add/subtract are different from multiply/divide. A very common student error is to swap the two sets of rules. Another common error is to use just one rule for both types of operations.
1) 3.0800
2) 0.00418
3) 7.09 x 10¯^{5}
4) 91,600
5) 0.003005
6) 3.200 x 10^{9}
7) 250
8) 780,000,000
9) 0.0101
10) 0.00800
11) 3.461728 + 14.91 + 0.980001 + 5.2631
12) 23.1 + 4.77 + 125.39 + 3.581
13) 22.101 - 0.9307
14) 0.04216 - 0.0004134
15) 564,321 - 264,321
16) (3.4617 x 10^{7}) ÷ (5.61 x 10¯^{4})
17) [(9.714 x 10^{5}) (2.1482 x 10¯^{9})] ÷ [(4.1212) (3.7792 x 10¯^{5})]. Watch your order of operations on this problem.
18) (4.7620 x 10¯^{15}) ÷ [(3.8529 x 10^{12}) (2.813 x 10¯^{7}) (9.50)]
19) [(561.0) (34,908) (23.0)] ÷ [(21.888) (75.2) (120.00)]