Example: Suppose that you measure the length of a particular table in two steps. You first measure with a meter stick that the distance from one end of the table to a particular mark is 0.95 meters. Then you use a more precise ruler to determine that the distance from the mark to the other end of the table is 0.0153 meters. You would then think that the length of the table is
0.95 +0.0153 --------- 0.9653But the first measurement was only known to 2 places past the decimal point, so the final result can only be known to that precision. To get the closest answer to the sum above we round the answer to two places past the decimal. So the best value we can get for the length of the table is
0.97 metersYou can think of the first measurement as "0.95????" and the second measurement as "0.0153???", where the "?" indicates the digits we don't know. When you add the digits, you can treat the "?" as a zero, but that is just a convenience; you don't know for sure what the digit should really be. Any colum that has a "?" in it is "tainted" and should not be trusted. Thus:
0.95???? +0.0153?? --------- = 0.9653 = 0.97???? = 0.97Notice that we don't just truncate the result, we round. After all, the 0.0153 is closer to 0.02 than it is to 0.01, so the final result should be closer to 0.97 than 0.96.
You can easlily convince yourself that the same considerations apply to subtraction. Any column containing a "?" is tainted and so we round to the last untainted column of digits. Thus we are lead to the following rule for addition and subtraction:
Next consider multiplication. Following the previous example, suppose that we want to know the area of a particular rectangular region on the table-top. With the meter stick we again measure the length of one side of the rectangle to be 0.95 meters, and with the more precise ruler we measure the other side to be 0.217 meters. The area is then
0.95 x 0.217 ----------- = 0.20615The first number only has two digits, the second number has three, and the result has 5 digits. How precise is this result? Again, think of the missing digits is as "?". You can work out the multiplication of these two numbers the 'long' way (digit by digit), and keep track of which columns get tainted by "?"'s. In the end you'll find that the "?" in the shortest number will taint the result after the same number of digits. That is, when you multiply by "0.95?" there are two good digits in the result, and then the third one will be tainted by the "?". The result will therefore look like:
0.95?? x 0.215? ----------- = 0.20615 = 0.21???Again, notice that when we restricted the answer to the two good digits we round, not just truncate.
Similar consideration apply to division. If you work out your result by 'long' division and keep track of which colum is tainted by a "?", the result will be limited by the number with the fewest total digits. This leads us to the following rule for multiplication and division:
One final point is important: these rules are only approximations -- they are "rules of thumb", that work well for getting a good rough estimate of the precision of a numerical result. They are not a replacement for a careful analysis of the propagation of errors through a calculation, but they are usually enough for most practical purposes.