### Significant Figure Rules

There are three rules on determining how many significant figures are in a number:

1. Non-zero digits are always significant.
2. Any zeros between two significant digits are significant.
3. A final zero or trailing zeros in the decimal portion ONLY are significant.

Focus on these rules and learn them well. They will be used extensively throughout the remainder of this course. You would be well advised to do as many problems as needed to nail the concept of significant figures down tight and then do some more, just to be sure.

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.

If you're not convinced significant figures are important, you may want to read this Significant Figure Fable.

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: These are sometimes called "captured zeros."

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 and underlined:

0.00500

0.03040

Here are two more examples where the significant zeros are in boldface:

2.30 x 10¯5

4.500 x 1012

### What Zeros are Not Discussed Above

Zero Type #1: Space holding zeros on numbers less than one.

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 decimal 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 102 is used. If it has three, then 2.00 x 102 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.

### Exact Numbers

Exact numbers are counting up how many of something are present, they are not measurements made with instruments. The only involve your eye seeing and your brain counting. I counted 15 people in the room. There are 10 eggs in that container.

Another example of exactness is what is called a defined number, such as 1 foot = 12 inches. There are exactly 12 inches in one foot and it is by definition. If a number is exact, it DOES NOT affect the accuracy of a calculation nor the precision of the expression. Exact numbers will have no effect on how many significant figures will be used in an answer.

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.
There are 2.54 cm in one inch, by definition. This one will be used often, when you are learning about how to convert to/from an English unit (foot, inch, pound) and a metric unit (such as meter, centimeter, gram).

Identify the number of significant figures in:

Example #1: 3.0800

Solution:

There are five significant figures. All the rules are illustrated by this example. Rule one: the 3 and the 8. Rule Two: the zero between the 3 and 8. Rule three: the two trailing zeros after the 8.

Example #2: 0.00418

Solution:

There are three significant figures: the 4, the 1, and the 8. This is a typical type of problem where the student errs by giving five significant figures as the answer. The zero in the tenths place and the zero in the hundredths place are not significant.

Example #3: 7.09 x 10¯5

Solution:

There are three significant figures. When a number is written in scientific notation, only the significant figures are placed into the numerical portion. If this number were taken out of scientific notation, it would be 0.0000709.

Here's another with three sig figs: 8.00 x 105

Example #4: 91,600

Solution:

There are three significant figures. The last two zeros are usually considered to be not significant. This is because of rule #3. It is ONLY trailing zeros in the decimal portion that are considered significalt (normall).

Suppose you had information that showed the zero in the tens place to be significant. How would you show it to be different from the zero in the ones place, which is not significant? The answer is scientific notation. Here is how it would be written: 9.160 x 104. This CLEARLY indicates the presence of four significant figures.

Look again at this, from the answer to example #3: 8.00 x 105

If I took it out of scientific notation, it would be 800000. That has, by rule 3, only one significant figure (the 8). However, we know that the number has three sig figs. The ONLY way to show three SF is by using scientific notation.

Example #5: 0.003005

Solution:

There are four significant figures. No matter how many zeros there are between two significant figures, all the zeros are to be considered significant. A number like 70.000001 would have 8 significant figures.

By the way, the zeros in the tenth and hundredth places are NOT significant.

Example #6: 3.200 x 109

Solution:

There are four significant figures. Notice the use of scientific notation to indicate that there are two zeros which should be significant. If this number were to be written without scientific notation (3,200,000,000) the significance of those two zeros would be lost and you would - wrongly - say that there were only two significant figures.

Example #7: 250

Solution:

There are two significant figures. Suppose 250 was known to have three sig figs. How would you show that? One answer is by using scientific notation (2.50 x 102. Another technique would be to use an explicit decimal point. In this case, writing 250. would indicate three SF.

Example #8: 780,000,000

Solution:

There are two significant figures. If more sig figs were known to exist, you would turn to scientific notation: 7.80 x 108 in the case of three sig figs, 7.800 x 108 in the case of four sig figs.

Example #9: 0.0101

Solution:

There are three significant figures. The two 'ones' as well as the zero trapped between them. A real common wrong answer, if this is on a test, would be four fig figs.

Example #10: 0.00800

Solution:

There are three sig figs. Here are the sig figs underlined: 0.00800. A real common wrong answer on a test would be 5 SF.

If you're STILL not convinced significant figures are important, you may want to read this Significant Figure Fable.