[narrator] Now that we know how to take measurements with the correct number of significant figures
how do we use significant figures in calculations to propagate uncertainty and proper level
of precision and accuracy? With addition and subtraction we re going to treat this like
elementary school and we re going to line everything up. For
perimeter it s the sum of all of the sides. We ve measured two side of a rectangle, assuming
that opposite sides are the same, and so we want to line these up. 1.73, 1.801--line them
up by place and we can do this again. 1.73, 1.801. We re going to add these up. Since
we've lined up by place we notice that there is a point at which a couple of our measurements,
here, or one measurement of--1.73 does not have a measurement to the thousandths place.
Remember, we've got ones place, we've got tenths place, hundredths place, thousandths
place. It s been a while since we've talked in these terms and so what we've got here
are question marks. We don t know what these two numbers are to the thousandths place.
What we do is we add up the number, without those pieces. 6, and then 14 plus 16 is 30,
3, 4, 5, 6, 7--so 7.062. But, any measurement that has uncertainty in it we have to round
off at the end. This 2, we re not sure if it s 2, a little bit more than 2, exactly
2--we don t know. We round it off and our final answer can only be what we are certain
of, after the round off. It s 7.06 and centimeters are our units. What did we do? We lined it
up and then we rounded at the end. Multiplication and division; little bit different, not too
bad. If we want to find the area of a rectangle, well area is length times width. We can tectonic
our 1.73 centimeters and take our 1.801 centimeters. Multiply those together and so with multiplication
and division the number of sig figs is going to be equal to the number of least sig figs
in any factor or divisor. Here, we have a number that has three sig figs. One, two,
three digits. Here we have one that has one, two, three four. Our final answer can only
have three. We multiply this out. 1.73 times 1.801, we get 3.11573 centimeters squared.
All this stuff is what my old high school teacher used to call calculator garbage, Mr.
Thomlinson. We have to say here three significant figures only. There s our--let s use the same
color here--there s our dividing line. Only three sig figs so we round
to 3.12 centimeters squared, for our area. We can t be any more accurate than the hundredths
place because of the uncertainty in the measurement. What about order of operations in sig figs?
Suppose we have a complex calculation such as this. You all know--please excuse my dear
aunt Sally--and that goes parentheses, exponents--parentheses, then exponents, multiplication and division
next, addition and subtraction last. So we need to do parentheses first. Here, we have
addition so we have to line them up and we notice here that we can only go to the hundredths
place because there s uncertainty in the second measurement. That s a question mark there
so we ll just make it an x so we can see it easier. We don t know what that is, okay 0.222,
so we have to round it off. Our new calculation, then, after parentheses is 0.22, whatever
the units are not given so we don t worry about those, 140--divided by 142. Now we re
down to division and so we have to go by the least number of sig figs. Here s two, here
s three. Our final answer can only have two sig figs. 0.22 times--or divided by 142 is
going to give us calculator garbage, 0.001549--keep going--296. All that junk, we can only have
two significant figures so we round off two. Whereas our two sig fig line, there it is,
15. Do we round up or down? No, we don t. We round down because it s a four so our final
answer is 0.0015. There will be a deduction of -1 if your significant figures are wrong
for a long calculation or a long problem, but several of these problems are going to
show up that may be four and five points apiece where the specific goal is proper significant
figures, so be very careful about significant figures.