The number of significant digits recorded for a measurement includes all of those digits known with certainty plus the first digit about which there is some uncertainty. The first digit in which there is some uncertainty is the first digit which is estimated
The only time significant digits must be considered is when dealing with measured quantities. You will only deal with two types of numbers, those which are part of a measured quantity and pure numbers. An example of a pure number is the number 2 when it indicates the diameter of a circle is twice the radius of a circle. Another pure number would be the number 5 when you say five people. Many conversion facts are also pure numbers like 5280 ft = 1 mi or 1 in = 2.54 cm. Pure numbers have infinite number of significant digits. The fact that pure numbers have an infinite number of significant digits means they will never be the number which limits the numbers of significant digits in the result of a calculation using measured quantities.
All digits which are not zeros are significant digits.
Any zeros between nonzero digits are significant
Any zeros which simply hold the decimal point in position are not significant digits. A simple test for this kind of zero is to write the quantity in scientific notation. If the zeros disappear they are not significant. A further description of these kinds of zeros would be either -
"ending" zeros (for big numbers) which are to the right of any nonzero digit but to the left of the decimal point, or
"leading" zeros (for small, decimal numbers) which are to the right of a decimal point but to the left of any nonzero digit.
In a number such as 56,500 a special effort must be made to indicate the place value to which the quantity was recorded in the event the zeros are significant. A line above a zero or below it indicates that the zero is significant and it is the first estimated digit or the last significant digit.
"Trailing" zeros (for decimals) which are to the right of the decimal point and to the right of any nonzero digit are significant because they indicate the measurement has been carried to that degree of precision.
When adding or subtracting measurements you should first calculate the answer using all digits available. Then you should determine to which place value you should round your answer. To do this determine the estimated digit in each number used in the calculation (this would be the last significant digit in each number). Then, as you proceed from left to right, the first column in which you find an estimated digit should be the column or place value to which you should round off your answer.
In multiplication or division of measured quantities you should first perform all of the calculations involved. Then determine how many significant digits are in each of the quantities used in the calculations. Round off your answer so it has only as many significant digits as the quantity which contains the least number of significant digits.
When taking a logarithm the number of digits to the right of the decimal is equal to the number of significant digits in the number that you are taking the logarithm of.
I have gained information from a professor researching significant figures and he has provided me with the following information based upon his research. To find out more about this, his site can be found at http://www.angelfire.com/oh/cmulliss/index.html
Standard Rule for multiplication and division: as stated above.
The application of the standard rule can yield only three possible results: it can predict the correct number of significant digits, 1 digit too many, or 1 digit too few. It predicts the correct number of digits less than half of the time. On very rare occasions it predicts 1 digit too many, overstating the precision. Most of the time it predicts 1 digit too few, causing valuable information to be lost.
Alternate Rule for multiplication and division: as stated above plus 1 more significant digit.
The application of the alternate rule can yield only three possible results: it can predict the correct number of significant digits, 1 digit too many, or 2 digits too many. It predicts the correct number of digits over half of the time. It predicts 1 digit too many less than half of the time, overstating the precision. On very rare occasions it can predict 2 digits too many, overstating the precision.