When working with number quantities in the sciences, particularly in Physics and Chemistry, it is important to follow the conventions about how to round answers. In order to do this correctly, you have to know how many digits in a number are significant.
Rules for Deciding How Many Digits in a Number are Significant:
All nonzero digits in a number are significant
Zeros between other digits are significant
Zeros at the end of a decimal value are significant
Zeros at the beginning of a number are NOT significant
Zeros at the end of a whole number are ambiguous

314.27
17
2007
6.0005
17.3400
2.5000
0.0031
0.00005
036
17300
150
1300

five significant digits
two significant digits
four significant digits
five significant digits
six significant digits
five significant digits
two significant digits
one significant digit
two significant digits
You don't know! These numbers should have been written in scientific notation.

In a whole number ending in zeros like 1300, it is impossible to tell if the zeros are significant. The number of significant digits in 1300 is at least two, but could be three or four. To avoid uncertainty, scientific notation should have been used to indicate how many digits were significant. For example:
1.3 x 10^{3} has two significant digits
1.30 x 10^{3} has three significant digits
1.300 x 10^{3} has four significant digits

Using Significant Digits in Calculations:
In any calculation, the number of significant digits in the answer can only be as large as the smallest number of significant digits in any one of the numbers in the calculation.
For example:

0.1326

+ 0.68571

+ 0.947

= 1.76531

<<< This is the answer you get by just doing the calculation.

Using the significant digits rule:
since 0.947 has only three significant digits, then the answer can have only three significant digits. The correct solution would be 1.77
Another example:

3.275

+ 1.03

+ 16

= 20.305

<<< This is the answer you get by just doing the calculation.

Using the significant digits rule:
since 16 has only two significant digits, then the answer can have only two significant digits. The correct solution would be 20, but this is ambiguous (it's a whole number ending in zero), so we use scientific notation: 2.0 x 10^{1}, which has two significant digits.
Additional Notes:
 Sometimes a whole number in a calculation can be considered to have an unlimited number of significant digits ... which means essentially that you ignore it.
For example, if an object's mass is 2.5 kg (two significant digits), and you have seven of them, the calculation for total mass would look like this:
7 x 2.5 = 17.5 kg = 18 kg (rounded to two significant digits to match the 2.5). The number 7's significant digits are ignored.
 When doing multistep calculations, keep at least one more significant digit in intermediate results than you will need in your final answer.
For example, if a final answer requires two significant digits, then carry at least three significant digits in intermediate calculations. If you roundoff all your intermediate answers to only two digits, you are discarding the information contained in the third digit, and as a result the second digit in your final answer might be incorrect. (This is known as 'roundoff error')
Remember:
 You can't have more significant digits in an answer (intermediate or final) than there were in any one of the data numbers.
Try a QUIZ on significant digit rules
