We must understand the use of significant figures when we are using numbers that represent measurements.  Measurements are made with limited precision.  Anything we calculate from numbers that have limited precision also has limited precision.  Note that numbers that do not come from measurements (like the number 8 in "I repeated the experiment 8 times," or the number 2 in "The diameter is 2 times the radius.") are pure numbers and have infinite precision (there is no uncertainty in how many times I repeated the experiment or in how many times the radius is the diameter).  The most unambiguous way of expressing sig figs is to use scientific notation.

When numbers from measurements are combined in calculations (multiplied, divided, added, subtracted), the imprecision carries along through the calculation.  When we report the final  answer, we must indicate (through use of the convention of significant figures) the precision of the answer.  See the video lilnks given for examples of how we determine the number of sig figs in final answer from a calculation involving a measurement.

Getting the sig figs right is an important aspect of communicating technical information.  If you write your materials with too many sig figs, you risk making greater claims of certainty that you can back up with your work; you may also signal to your reader that you do not understand your calculation and measurements.  See this interview of Nobel Prize winners on the BBC The World Debate (presented by Zeinab Badawi, 10 December 2011) where at minute 38:30 the discussion turns to the inevitibility of uncertainty in scientific matters:  www.bbc.co.uk/iplayer/episode/p00lzhqc/The_World_Debate_Nobel_Minds/


Useful Videos on Significant Figures:
After multiplying and dividing numbers, how many sig figs are in the answer? (keep all digits in intermediate calculations)
http://www.khanacademy.org/video/multiplying-and-dividing-with-significant-figures?playlist=Pre-algebra

After adding and subtracting numbers, how many sig figs are in the answer? (more tricky; don't over-represent precision)
http://www.khanacademy.org/video/addition-and-subtraction-with-significant-figures?playlist=Pre-algebra

Pure numbers and sig figs
http://www.khanacademy.org/video/pure-numbers-and-significant-digits?playlist=Pre-algebra


Sig Fig Basics:
How many sig figs are there in this number?
http://www.khanacademy.org/video/significant-figures?playlist=Pre-algebra

What are the rules of thumb for how many sig figs there are in a number? (intro to sci notation)
http://www.khanacademy.org/video/more-on-significant-figures?playlist=Pre-algebra