Significant Figures Addition and Subtraction while minding the “sig figs.”

A quick review  Recall that all measurements have inherent uncertainty due to  a) limits in the accuracy of the measuring device and  b) variations in the estimations made by those doing the measuring.  Also recall how to count significant figures, based on whether or not a decimal point is present.  Significant figures apply to MEASURED VALUES!

A problem…  Suppose that you are told to find the perimeter of a rectangular room.  Further suppose that the dimensions of the room are measured by two different methods (for some reason), which are reported as 5.1 meters and 8.27 meters, respectively.  What are the estimated values in each of these numbers? Which number expresses more confidence?  The ‘1’ and the ‘7’ are estimated values and the 8.27 measurement expresses more confidence than the 5.1 measurement.  5.1 m   8.27 m  Perimeter?

Problem Continued…  So what’s the perimeter of the room?  If you punch into your calculator , your calculator will report a perimeter of meters.  But you are much smarter than your calculator, which doesn’t understand a thing about significant figures.  You know that the ‘7’ in 8.27 m is estimated, which is to say that the length could be as low as 8.26 or as high as  Likewise, “5.1 meters” represents a range from 5.0 meters to 5.2 meters.  5.1 m   8.27 m  Perimeter? (sum of the sides)

Why not to trust your calculator  If you report your perimeter as meters, then you are expressing more confidence in you sum of numbers than you are in either of your starting numbers (5.1 and 8.27).  You would be stating that the perimeter of the room is certainly between meters and meters.  This is wrong, though. It could actually be as low as meters or as high as meters. How did I get those numbers?  I simply added the low ends and high ends of the ranges together, respectively.

So what to report?  When adding and subtracting measured values, we need to be mindful of the least certain starting number re: decimal position.  In other words, we need to round our final answer to the same number of decimal places as there are in the measurement with the smallest number of decimal places.  Doing so allows us to ‘hedge our bets’ about the accuracy of the number we report. It casts a bigger net in an attempt to capture the true/actual answer.  In the case of the perimeter of the room, we would report a value of 26.7 meters because it is expressed to the tenths place, as was 5.1 meters.

Some examples…  Give the final answer of the following addition/subtraction problems, assuming the values represent measurements.  = ?  The least certain position in our numbers is the tenths position in 2.1. Therefore, our answer needs to be expressed to the tenths place.  15.2 (not , as your calculator reports)  What about 1, ?  1300 (because rounded to the same level as the least certain starting number, is 1300).

The same rule applies to subtraction!  Try these… answers on next slide  Q1: 23.5 – 18 – 1.46 =  Q2: =  Q3: =  Q4: 2,100 – 189 – 12 =

Answers to the previous slide…  A1: 4 (because of the 18)  A2: (because of the 72.1)  A3: (because of the )  A4: 1,900

Some trickier ones… again, answers are on next slide.  Q5: – 8.2 =  Q6: – 3.1 x 10 2 =  Q7: 12.2 – – =

More answers…  A5: 4027 (note that there is a decimal point present in 4010., which means that all four figures are significant.)  A6: 1000  A7: -1.3  How fun was that?!?!?