Significant Figures


1. Significant Digits

All digits ( non-zero and zero) are considered significant except
zeroes placed to the right of a decimal solely for spacing.(in red)

Examples: (significant figures in bold type)

3 sig figs 4 sig figs 5 sig figs
     
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
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:

6o 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

p = 3.14     (3 sig figs) This number is rounded off from 3.1415926535... because
                                    it is easier to work with

p = 3.14159     (6 sig figs)

Examples of inexact conversion factors:

0.6 miles / km     (1 sig fig)

0.62 miles / km     (2 sig figs)