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When working with scientific data, we only want to show as many figures as carry accurate meaning, called significant figures. When adding or subtracting two numbers, we round to the same number of decimal places as the term with the fewest decimal places. When multiplying or dividing numbers we round to the same number of figures as the term with the lowest number of significant figures. In scientific notation, the digit term, not the exponential term counts as significant.
This segment let's go ahead and discuss significant figures or as you've probably heard they're more likely called sig figs. This is something that's going to come up throughout your Scientific career as long as you're doing Chemistry and Physics or anything of that nature. So you're going to want to go ahead and do your best to get a handle on understanding of sig figs. The beautiful part about this is that there are actually some rules and very few exceptions so that if you do it with practice over and over it should become second nature for you. So we do significant figures because we need to account for the degree of uncertainty in a final numerical result. So if you're in a laboratory and you've got a bunch of measurements and you want to add them together, to get a final result, you want to make sure that your final result is as accurate as your initial measurements were. So let's start with the rules.
The first rule applies to non zeros integers so non zero integers always count as sig figs so if I have a number 1, 2, 3, 4 all four of those numbers will be significant figures. Secondly zeros, zeros are going to be the largest source of complication. So there are 3 classes of zeros when it comes to counting sig figs. The first applies to leading zeros, so those are the zeros that proceed non zero digits and those guys never count as sig figs they just indicate the position of the decimal point. So for instance 0.00123 these zeros are all leading zeros they count as significant figures so this number only has 3 significant figures, so only the non zero integers count.
The second are captive zeros, those are the zeros that fall in between non zero digits and those guys always count as sig figs, for instance if you had the number 1.0012 you're going to have 5 significant figures. So all of these numbers are significant because these zeros are captivated basically by the non zero integers on either end.
The third class of zeros are trailing zeros so these guys are zeros at the end of the number and they're significant only if the number is written with a decimal point. So for instance if you have one thousand just written 1000 the only number that's significant here is the non zero integer 1. However if it is written 1000 decimal place then you have 4 significant figures. So I guarantee that the zeros whether they be on the right hand side or the left hand side are going to be 2 of your pitfalls. So it's really important to understand how the zeros on the front end work and how the zeros on the back end work.
Last but not least, for sig figs we have exact numbers, so these are the numbers that were determined by counting and not by doing experimental procedure. For instance if you had 6 apples or 10 pens or something of that nature those guys are just the number that they are, so they're assumed to have an unlimited number of significant figures. These numbers can also arise from definitions for instance if you're doing a calculation and you're saying that 1 inch is equal to 2.254 centimeters. Your significant figure value, your final result will not be limited by either of those values. And it also applies to Scientific notation, so if you've got one hundred written 100 that's got 3 sig figs if you wrote 100 in Scientific notation which will be 1.00 times 10 squared that also has 3 significant figures because these zeros are important. So chances are you're going to have to actually do significant figures in some calculations. So let's discuss how it applies to multiplication and division and addition and subtraction. So for multiplication and division the numbers of sig figs and the result is the same number as that in the measurement with the smallest number of sig figs. So we call the number with the smallest number of sig figs basically your limiting term.
Okay so you're limited by how many, by how accurate your smallest measurement is. So for instance here I've written 5.16 times 1.3 so you're going to be limited by the 2 sig figs here in 1.3 so the answer as the result to this multiplication is 6.708. So something to note here too is that you want to write out all the digits first before you round otherwise you're not going to be as accurate in your final measurement. So if we're dividing let's say we have 5.168 divided by 1.43 so 1.43 is going to be our limiting term meaning we can only have 3 significant figures in our final answer. So the result of this division is 3.61398601 all the numbers that showed up on my calculator and then I will round that down to 3 sig figs which should be 3.61 again I'm limited by limiting term which had 3 sig figs. So let's lastly do addition and subtraction so it's a little bit different as the limiting term here is the one with the smallest number of decimal places. So if I had 16.15, 14.1 and 1.06 and I'm adding them together, I'm going to be limited by 14.1 because it only has one decimal place. So then my answer on my calculator would become 31.266 and I would round that to 31.3 because again I'm limited in my answer by the one decimal place.
Same thing for subtraction, if I had 16.15 subtracted by, and I'm subtracting 0.4 my limiting term is going to be point 4 again I'm limited to one decimal place in my answer, so my answer on the calculator would be 15.75 which I would round to 15.8 having one decimal place. And that's significant figures.