Significant Figures



Significant Figures

1. Significant Digits

All digits ( non-zero and zero) are considered significant except

zeroes placed to the right solely for spacing. (in red, italics and underlined)

zeroes at the end of a number that does not have a decimal. (in red, italics, underlined and bolded)

Examples: (significant figures in bold type)

|3 sig figs |4 sig figs |5 sig figs |

|3150 m/s  |74850 m/s  |  |

|250. m/s |3150. m/s |74850. m/s |

|25.0 m/s |315.0 m/s |7485.0 m/s |

|2.50 m/s |31.50 m/s |748.50 m/s |

|0.250 ml |3.150 ml |74.850 ml |

|0.0250 ml |0.3150 ml |7.4850 ml |

|0.00250 ml |0.03150 ml |0.74850 ml |

|0.000250 ml |0.003150 ml |0.074850 ml |

|700. m/s |7 000. m/s |70 000. m/s |

|151 000 m/s |1 515 000 m/s |15 151 000 |

|8.07 x 106 m/s |8.007 x 106 m/s |8.0007 x 106 m/s |

2. Multiplying / Dividing / Trigonometric Functions

    a) First perform all the operations, even if changing from one formula to another.

    b) Round off the result so that it has the same number of sig figs as the least of all those used

        in your calculation.

Example: (2.5 m) x (2.01 m) x (2.755 m) = 13.843875 m

Answer = 14 m     (2 sig figs)

3. Addition / Subtraction

    a) First perform all the operations.

    b) Round off your result so that you include only 1 uncertain digit.

The last digit of any measurement is considered uncertain.

When an uncertain digit is added to (or subtracted from)

a certain digit, the result is an uncertain digit.

(UNCERTAIN DIGITS ARE HIGHLIGHTED)

Example: 153. ml + 1.8 ml + 9.16 ml = 163.96 ml

Answer = 164 ml (3 sig figs; only 1 uncertain digit)

Notice that the answer is rounded to the same precision as the least precise measurement, which was 153. ml

4. Multiplication / Division combined with Addition / Subtraction

First, follow the order of operations that you learned in math. Use the appropriate sig fig rules, as stated above, depending on which operation you are performing at that time. (Example: 1. multiply/divide/trigonometric functions; or 2. add/subtract functions) At the end of each step, you must ask yourself,

"What is the next operation that I will perform on the number that I just calculated?"

If the next operation is in the same group of operations that you just used, (Example:

1. multiply/divide/trigonometric; or 2. add/subtract) then do NOT round off yet.

If the next operation is from the other group, then you must round off that number before

moving on to the next operation.

5. Exact Values

All exact values or conversion factors have an infinite (never ending) number of significant figures.

They are called exact values because they are measured in complete units and are not divided into smaller parts. You might count 8 people or 9 people but it is not possible to count 8.5 people.

Examples of exact values:  12 complete waves ;  17 people ;  28 nails

Examples of exact conversion factors:  60 s / minute ;  1000 m / km ;  12 eggs / dozen;  7 days / week

There are exactly:

60 seconds in one minute

1000 meters in one kilometer [this is the definition of kilo (k)]

12 eggs in one dozen

7 days in one week

6. Inexact Values

All inexact conversion factors or constants will be treated like measurements.

They are called inexact because they are not exact like above. This means that there isn't an exact number to work with. It requires a fraction that creates a number with several digits after the decimal.

Examples of inexact constants:

c = 3.00 x 108 m/s     (3 sig figs) This number is rounded off from 2.99876... because

                                                   it is easier to work with

π = 3.14     (3 sig figs) This number is rounded off from 3.1415926535... because

                                    it is easier to work with

π = 3.14159     (6 sig figs)

Examples of inexact conversion factors:

0.6 miles / km     (1 sig fig)

0.62 miles / km     (2 sig figs)

7. Rules Specific for Zeroes

| | |

|Rule |Examples |

| | |

|Zeros appearing between nonzero digits are significant | |

| |40.7 L has three sig figs |

| |87 009 km has five sig figs |

| | |

|Zeros appearing in front of nonzero digits are not significant | |

| |0.095 987 m has five sig figs |

| |0.0009 kg has one sig fig |

| | |

|Zeros at the end of a number and to the right of a decimal are significant | |

| |85.00 g has four sig figs |

| |9.000 000 000 mm has 10 sig figs |

| | |

|Zeros at the end of a number but to the left of a decimal may or may not be | |

|significant. If such a zero has been measured, or is the first estimated digit, |2000 m may contain from one to four sig figs, depending on how many zeros are |

|it is significant. On the other hand, if the zero has not been measured or |placeholders. For measurements given in this text, assume that 2000 has one sig |

|estimated but is just a placeholder, it is not significant. A decimal placed |fig. |

|after the zeros indicates that they are significant. |2000. m contains four sig figs, indicated by the presence of the decimal point |

| | |

|Scientific notation - All digits expressed before the exponential term are | |

|signicant. |5.060 x 10-3 m has four sig figs. |

| |9.00 x 102 g has three sig figs. |

Basic Rules for Significant Figures:

1- All non-zero digits are significant

2- All zeroes between significant figures are significant

3- All zeros which are both to the right of the decimal AND to the right of all non-zero significant digits are themselves significant.

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