by D. Panko, amended by G. Battaly

In scientific work all numbers are assumed to be derived from measurements and therefore the last digit in each number is uncertain.  All certain digits plus the first uncertain digit are significant. Only numbers determined by definition or by counting are exact.  These are said to have an infinite number of significant figures.

Scientific Notation: To express a number in scientific notation it must have one, and only one, non-zero digit to the left of the decimal point to be followed by the appropriate power of ten.  x.xx(10^k).  For example:   3.00(10⁷)  and 2.59(10^-5) are in scientific notation, but numbers such as 0.25(10³) and 63.0(10^-6) are NOT.

How many significant figures does it have?
   I.  If it is a number in scientific notation:  All digits, zero or non-zero are significant.
    II.  If it is a number in decimal notation:
            A. All non-zero digits are significant.
            B.   Crossed out zeros ARE NOT significant.  To cross out zeros:
                1)   If the number has an expressed decimal point:  draw a line from the left, crossing out all zeros until the first non-zero digit.   For example:  0.002020 contains 4 significant figures.
                2)   If the number has an understood decimal point:  draw a line from the right, crossing out all zeros until the first non-zero digit.  For example:   202000 contains 3 significant figures.

How to do Calculations using Significant Figures:
    I.  Addition or Subtraction
        A.  If powers of 10 are given, convert all numbers to the same power of 10.
        B.  Arrange addends in a column, lining up the decimal point.
        C.  Add normally.
        D.  Locate the uncertain digit in each number.
        E.  Round off the answer to the same decimal place as the uncertain (estimated) digit farthest to the left (the least significant addend).
    II.  Multiplication or division
        A.  The answer is expressed in the same number of significant figures as the factor with the fewest number of significant figures.
        B.  In a series of multiplications or divisions:  intermediate answers, and original factors, may be rounded off to one more significant figure than will be eventually retained in the final answer.