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Significant Figures 
Exact and Inexact Numbers and Significant Figures Numbers which are associated with measurements of any kind are uncertain to an extent. For example, if you weigh yourself on a bathroom scale that reads weights in increments of 0.1 lbs, and your weight is 132.5 lbs, you actually only know that your weight lies between 132.4 and 132.6 lbs (assuming that there is no systematic error in the scale). (Another way to think of you weight is 132.5 ± 0.1 lbs.) To take this example a little further, consider that you have just signed a lucrative contract as a point guard with the WNBA. Since you have had weight problems before, one of the clauses in your contract awards you $177.00 for each pound of your weight, as long as you remain under 135.0 lbs (stranger contract clauses exist). You pull out your trusty calculator and multiply 132.5 by 177.00 and get 23452.50 (assuming you have your calculator set to two figures to the right of the decimal point). When you receive the check for having met this stipulation of your contract, the amount is $23450 and you want to know why you were cheated out of $2.50. The answer is that you were not cheated. The number 177.00 and 132.5, are both inexact numbers. Your weight contains four significant figures, and as we shall see below, when multiplying numbers, the result can have no more significant figures than the number in the calculation with the fewest significant figures. Since 177.00 contains five significant figures, the result of the multiplication is limited to the four significant figures of 132.5. Looking at the result, 23452.50, we see that it contains six significant figures (we'll look at this in more detail below). We need to round this number to four significant figures, hence the check for $23450. How do we decide how many significant figures a number has? This is an area of some confusion if we write the numbers in standard notation such as 23450. Are there five significant figures, or is the last zero there only to show that the leftmost digit is in the ten thousands place? For example, if you say that your home is 1200 miles from your dorm at Wellesley, does this number have 2, 3 or 4 significant figures? (Note that since it is a measurement, it can't be exact, and since there are no numbers to the right of the (understood) decimal point, it can at best mean that the distance is between 1199.5 and 1200.5.) The way to avoid this confusion is to use scientific notation. If we write the number as 1.2*103, we have only two significant figures. If we know the mileage to the nearest mile the we would write 1.200*103. In other words, trailing zeros are significant when numbers are expressed in scientific notation whereas they can create confusion if the number is not written in scientific notation. (Thus the best way to write the amount of your check is $2.345*104). We also frequently encounter exact numbers in calculations. Consider the kinetic energy of a moving object. The definition of kinetic energy is (mv2)/2. The 2 here (or ½, if you write it ½ mv2) is an exact number and need not be considered when counting significant figures. Likewise, if you say you have three brothers, that is an exact number (saying you have three brothers and a half-brother is not the same as saying you have three and a half brothers). Before looking at specific examples we should also take care of the question of leading zeros. Zeroes before a number do not count as significant figures. The number 0.004 has one significant figure. As we will see below, for adding and subtracting, it is not significant figures that count, it is the number of decimal places. In that case, 0.004 has 3 digits after the decimal place.
Significant Figures in Calculations: Once you have determined how many significant figures various numbers contain, the question that arises is how to deal with significant figures when these numbers are combined in a calculation. There are a few simple rules that when followed make this process relatively painless. They are:
If a calculation involves a combination of mathematical operations having different significant figures, it is customary practice to carry out the calculation using all figures, and then go back and figure out how many significant figures the final result should have. For example, if your calculation is (all numbers inexact) ((6.378+0.0025)/42.6) - 1.414*10-3 you could perform the calculation on your calculator and get 0.148362995… (depending on how many figures your calculator shows). Now break the calculation into pieces. Here, the insignificant figures are in (). 6.378 + 0.0025 = 6.380(5) (since the sum is limited to the thousandths place by 6.378) 6.380(5) / 42.6 = 0.149(777) (since the quotient is limited to 3 significant figures by 42.6) 0.149(777) - 0.001414 = 0.148(363) (since the difference is limited to the thousandths place by 0.149 so the answer you should report is 1.48*10-1. The fact that our 0.148(363) ended up as 0.148 seems to imply that we just threw the insignificant figures away. That is not strictly the case - we rounded 0.148(363) to 0.148. The rules for rounding numbers are quite simple.
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