## Significant Digits

Significant figures are digits that are statistically significant. There are two kinds of values in Science:

1. Measured Values

2. Computed Values

The way that we identify the proper number of significant figures in science are different for these two types.

### Measured Values

Let's first take up measured values. Identifying a measured value with the correct number of significant digits requires that the instrument's calibration be taken into consideration.

THE LAST SIGNIFICANT DIGIT IN A MEASURED VALUE WILL BE THE FIRST ESTIMATED POSITION.

For example, a metric ruler is calibrated with numbered calibrations equal to 1 cm. In addition, there will be ten unnumbered calibration marks between each numbered position.

These would be referred to as "minor calibration marks"

Then one could with a little practice estimate between each of those markings.

That first estimated position would be the last significant digit reported in the measured value. Let's say that we were measuring the length of a tube, and it extended past the fourteenth numbered calibration half way between the third and fourth unnumbered mark. The metric ruler was a meter stick with 100 numbered calibrations.

Another example: If a thermometer had ten degree numbered calibration markings and ten equal spaced unnumbered markings in between each numbered calibration,

### Computed Values

The other type of value is a computed value. THE PROPER NUMBER OF SIGNIFICANT FIGURES THAT A COMPUTED VALUE SHOULD HAVE IS DECIDED BY A SET OF CONVENTIONAL RULES.

However before we get to those rules for computed values we have to consider how to determine how many significant digits are indicated in the numbers being used in the math computation. The rules for expressing the correct number of significant digits in a computed result speak of the significant digits in each number or value involved in the computation. So what are these rules?

### Rules for Determining the Number of Significant Digits In Numbers With Indicated Decimals

1. Non-zero digits (1-9) are always significant.
2. Any zeros between two significant digits are significant.
3. A final zero or trailing zeros ONLY in the decimal portion are significant.

For example: 0.0040000

The 4 is obviously to be counted significant (Rule 1), but what about the zeros? The first three zeros would not be considered significant since they are not surrounded by any significant digits and are not trailing in the decimal portion. (Rule 2&3) The last four zeros would all be considered significant since each of them are trailing in the decimal portion (Rule 3) Therefore the number has a total of five significant digits.

Here is another example:

120.00420

The digits 1, 2, 4, and 2 are all considered significant (Rule 1) The middle three zeroes are significant because they are between the first 2 and 4 (Rule 2). The last zero is significant because it is trailing in the decimal portion

### Determining the Number of Significant Digits In Numbers Not Having an Indicated Decimal

The decimal indicated in a number tells us to what position of estimation the number has been indicated. But what about 1,000,000 ??

Notice that there is no decimal indicated in the number. In other words, there is an ambiguity concerning the estimated position. This ambiguity can only be clarified by placing the number in exponential notation.

For example, if I write the number above in this manner:

1.00 X 10^6

I have indicated that the number has been recorded with three significant digits. On the other hand, if I write the same number as:

1.0000 X 10^6

I have identified the number to have 5 significant digits. Once the number has been expressed in exponential notation form then

THE DIGITS THAT APPEAR BEFORE THE POWER OF TEN WILL ALL BE CONSIDERED SIGNIFICANT.

So for example

2.0040 X 10^4 will have five significant digits.

Question 5 How many significant digits does the following number have?

3.010 X 10^57

### Rule For Expressing Proper Number Of Significant Digits In An Answer From Multiplication or Division

For multiplication AND Division there is the following rule for expressing a computed product or quotient with the proper number of significant digits:

THE PRODUCT OR QUOTIENT WILL BE REPORTED AS HAVING AS MANY SIGNIFICANT DIGITS AS THE NUMBER INVOLVED IN THE OPERATION WITH THE LEAST NUMBER OF SIGNIFICANT DIGITS.

For example:

0.000170 X 100.40

The product could be expressed with no more than three significant digits since 0.000170 has only three significant digits, and 100.40 has five. So according to the rule the product answer could only be expressed with three significant digits.

Another example:

2.000 X 10^4 / 6.0 X 10^-3

The quotient could be expressed with no more than two significant digits since the least digited number involved in the operation has two significant digits.

Sometimes this would require expressing the answer in exponential notation.

For example:

(3.0) (800.0)

The number 3.0 has two significant digits and the number 800.0 has four. The rule states that the answer can have no more than two digits expressed. However the answer as we can all see would be 2400. How do we express the answer 2400 while obeying the rules? the only way is to express the answer in exponential notation so 2400 could be expressed as

2.4 X 10^3

Question 6: Indicate the number of significant digits the answer to the following would have.(I don't want the actual answer but only the number of significant digits the answer should be expressed as having.)

(20.04) ( 16.0) (4.0 X 10^2)

### Rule For Expressing The Correct Number of Significant Digits in an Additon or Subtraction

The rule for expressing a sum or difference is considerably different than the one for multiplication or Division. The rule expresses that the answer should have so many digits to the RIGHT of the decimal instead of focusing in on the number of significant digits as the other rule did. The focus is upon the number of positions to the RIGHT of the decimal. This rule states :

THE SUM OR DIFFERENCE CAN BE NO MORE PRECISE THAN THE LEAST PRECISE NUMBER INVOLVED IN THE MATHEMATICAL OPERATION.

In other words the answer can have no more numbers to the right of the decimal than the number involved in the operation with the least number of positions to the right of its decimal. Precision has to do with the number of positions to the RIGHT of the decimal. The more positions to the right of the decimal, the more precise the number. So a sum or difference can have no more indicated positions to the right of the decimal as the number involved in the operation with the LEAST indicated positions to the right of it's decimal.

For example:

160.45 + 6.732

The answer could be expressed to the nearest hundredths position (ie: two positions to the right of the decimal) since 160.45 is the least precise.

Another example:

45.621 + 4.3 - 6.41

The answer could be expressed only to the nearest tenth of a unit (ie:one position to the right of the decimal) since the number 4.3 is the least precise number (ie: having only one position to the right of its decimal). Notice we aren't really determining the total number of significant digits in the answer with this rule

Question 7 How precise can the answers to the following be expressed to?

17.142 + 2.0013 + 24.11

However what is the answer to this question:

Question 8 How many total significant digits should the answer to the above addition have?

### Rules For Rounding Off Digits

One other topic that needs to be covered is how do we round off non-essential digits according to these rules. There are a set of conventional rules for rounding off.

1. Determine according to the rules what the last reported digit should be

2. Consider the digit to the right of the last reported digit.

3. If the digit to the right of the last reported digit is less than 5 round it and all digits to its right off.

4. If the digit to the right of the last reported digit is greater than 5 round it and all digits to its right off and increase the last reported digit by one.

5. If the digit to the right of the last reported digit is a 5 follwed by either no other digits or all zeros, round it and all digits to its right off and if the last reported digit is odd round up to the next even digit. If the last reported digit is even then leave it as is.

For example if we wish to round off the following number to 3 significant digits:

18.3682

The last reported digits would be the 3. The digit to its right is a 6 which is greater than 5 and Rule 4 above. therefore the digit 3 is increased by one and the annswer is:

18.4

Another example:

Round off 4.565 to three significant digits.

The last reported digit would be the 6. the digit to the right is a 5 followed by nothing. Therefore according to Rule 5 above since the 6 is even it remains so and the answer would be 4.56.

Click here if you would like to have a little more practice on this lesson.

This concludes the reading assignment for significant figures. If you have any questions on the reading, click on the mailbox icon below and leave e-mail for me.

R. H. Logan, Instructor of Chemistry, Dallas County Community College District, El Centro College.

Acknowledgements

Send Comments to R.H. Logan: rhlogan@ix.netcom.com

### Question 1

Which would each of those unnumbered marks be worth?

Answer: 0.1 cm for each minor calibration

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### Question 2

What decimal position would that represent?

Answer: 0.01 cm or hundredths position of estimation

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### Question 3

What would the reported measured length be in centimeters?

Answer: 14.35 cm Explaination: Being past the 14 th numbered calibration and each of those worth 1 cm that would be 14 cm. Since the measurement extends past the third minor unnumbered marking halfway to the fourth minor marking and each of those is worth a tenth of a cm (0.1 cm), then the measurement would be halfway between 14.30 and 14.40 cm or 14.35 cm.

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### Question 4

To what position of first estimation could you report the temperature?

Answer: Since each major numbered calibration are worth 10 degrees each and there are ten unnumbered markings which must be worth one degree each, then the decimal position of estimation is the tenths position (0.1 degree)

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### Question 5

How many significant digits does the following number have?

3.010 X 10^57

Answer: It has four significant digits since the digits 3,0,1, and 0 all appear to the left of the 10.

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### Question 6

Indicate the number of significant digits the answer to the following would have.(I don't want the actual answer but only the number of significant digits the answer should be expressed as having.)

(20.04) ( 16.0) (4.0 X 10^2)

Answer: There would be two significant digits indicated in the product answer. The 20.04 has 4, the 16.0 has 3, and the 4.0 X 10^2 has only 2 so according to the rule the answer can have only two.

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### Question 7

How precise can the answers to the following be expressed to?

17.142 + 2.0013 + 24.11

Answer: The answer could have two positions to the right of the decimal since the least precise term, 24.11, has only two positions to the right.

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### Question 8

• How many total significant digits should the answer to the above addition have?

Answer: four significant digits in the answer. When you total the three numbers you find the answer has two digits to the left of the decimal in addition to the two digits to the right. All of these must be considered significant digits.

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```All contents copyrighted (c) 1995
R.H. Logan, Instructor of Chemistry,DCCCD