Significant Figures

Added: 2026-05-13

[00:00:04]all right so we are going to pick up with our math concepts and look at significa significant figures um this looks at kind of how exact a measurement is um so if we're using like a ruler or graduate cylinder looking at where we should include an estimated digit uh whether we're doing math this also is going to give us a clue of where we should round to so as significant figures these are the digits of a measurement including all known digits and an estimated digit at the end so this a lot of times we will use when we're measuring so say that we have a ruler and here is 51.

[00:00:59]and here is 52.

[00:01:02]say that our Mark is right here well it's we know it's at least 51 centimeters but it's more than 51 so we would need an add an estimated digit on there um so we could say that this is about 51.3 centimeters now you're kind of eyeballing it because you don't have tense places there you're just looking at how far between 51 and 52 is it so you might have it down as 51.2 51.4 something like that but you're always going to add one extra digit smaller than your smallest graduation or your smallest marking on that tool so if this had tense marks on it we would need to add a hundredths place that that estimated digit on there um we are going to be responsible for counting significant figures in writing numbers with the correct significant figures so whenever you're given a number being able to figure out how many significant figures or when you're doing a calculation where you should round two so let's explore that a little bit more so with significant figure rules we can count how many significant figures or how many digits are in a number based on the Atlantic Pacific rules and Apollo I apologize for my drawing skills but I draw a kindergartner so here's my map of the U.S we have the Atlantic Ocean is on the right the Pacific Ocean is on the left so the Atlantic Pacific rules kind of give you an idea of what side of the number you should start counting from um so if the number does not have a printed decimal point the decimal is absent we're going to start counting on the Atlantic side of the number so start from the right of the number and we're going to read this number from right to left and we're going to start counting digits once we reach a non-zero number and then once you start counting you count every digit after you have started Counting so for example here we have the number 400 400 this does not have a printed decimal point in it so we start from the right side and we go from right to left our first non-zero digit from the right is the four so this number only has one significant figure anytime you see SF that significant figure another way you might see this is if we have like 1070.

[00:04:13]as a number there's no printed decimal point so we would start from the right and read it from right to left our first non-zero digit from the right is seven and once we start counting we continue counting yet again we're reading from right to left so this number has three significant figures then there's what if there is a decimal point so if the number has a decimal point we say that the decimal point is present we start on the Pacific side of the number so we start at the left of the number and we read the number from left to right um and yet again we start counting once we reach a non-zero digit and we count every digit until the end of the number once we've started Counting so here I also have the number 400 but it is four zero zero point zero zero this has a printed decimal point so we start on the Pacific side we read from left to right our first non-zero digit from the left is four and reading from left to right we have to count every digit once we've started so this number has five significant figures and this is also going to come into play if we have a decimal number so say that it's 0.030 this has a printed decimal point so we would start on the left and read from left to right our first non-zero digit is three and once we start counting we continue counting every digit after that so this number has two significant figures and this will take a little bit of practice getting used to how you're determining which zeros count and which zeros do not count all right so let's look at a couple of these um it wants to know how many significant figures are in the following measurements so yet again you have to pay attention to is there a printed decimal point or is the decimal point absent so this first number 0.05730 there is a printed decimal point so remember we are reading it from left to right we're starting on that specific side our first non-zero digit from the left is five and reading from left to right we count every digit after we've started so this has four significant figures looking at the next one we have 8065 meters this does not have a printed decimal point so we're reading it from right to left our first non-zero digit from the right is five and we count every digit after we start it so this also has four significant figures let's look at the next one the 0.00073 grams this has a printed decimal point so we start on the Pacific side we're reading from left to right our first non-zero digit is seven and once we count the seven we continue counting from left to right so this has two significant figures I'm going to skip the next one I'm gonna go to the 1430 grams this the decimal point is absent so we're going to start on the Atlantic side so we're reading from right to left our first non-zero digit from the right is three and then reading from right to left we count every digit once we've started Counting so this number has three significant figures the last example I wanted to bring up because we do deal with numbers in scientific notation quite often and pardon me that should be to the negative two my superscript disappeared whenever you have a number like this you're only going to pay attention to that coefficient you're kind of ignoring the times 10.

[00:08:33]so here there is a printed decimal point so we start on the Pacific side so we're reading from left to right our first non-zero digit is the eight and once we start counting we count every digit after that so this number has four significant figures so practice that and really it's going to be those zeros either at the front or end of a number that we're going to be questioning whether they count foreign so let's look at what we do when we're doing math and trying to figure out where we round to um there are two different rules regarding rounding with scientific notation or significant figures excuse me there's one rule for if we are multiplying and dividing then there's another rule if we are adding and subtracting so let's look at at the multiplication division rule in general the calculated number cannot be more significant than the number of digits and the measurements used in the calculation in other words your answer cannot have more digits than your original numbers but you use to get that answer so for example if we need to find the area of a floor with a length of 7.7 meters and a width of 5.4 meters when you do 7.7 times 5.4 your calculator gives you an answer of 41.58 meters squared notice that our two original numbers the 7.7 and the 5.4 both have two significant figures 41.58 has four significant figures our answer cannot have more digits than our original numbers so since both original numbers only have two significant figures our answer cannot have more than two significant figures so have we round it to two places where rounding to the ones place so that would Round Up to 42 meters squared so whichever one of your original numbers has the least number of significant figures that's how many significant figures your answer should have so if the two measurements have two different numbers of significant figures it should have the least number so if one number has two significant figures and the other has three your answer can only have two it's whatever is the least number of digits or the least number of significant figures so let's look at what happens when we add and subtract and we're going to do a couple examples with these so the rule for adding and subtracting is that the calculated number should be rounded to the same number of decimal places as the measurement with the least number of decimal places so we said a multiplication division you're rounding to the least number of digits addition subtraction you're rounding to the least number of decimal places so let's do some examples and kind of put this to use and I would suggest putting this in the calculator and figuring out your calculator answer first before trying to figure out where to round two if you try to do too much at once you tend to miss things so let's do a couple from the top section so it says solve the file solve each of the following and give the answer using the correct number of significant figures so our first problem we have is 61.2 meters plus 9.35 meters plus 8.6 meters I would punch that in the calculator first and then figure out your rounding so 61.2 plus 9.35 plus 8.6 your calculator answer is this is 79.15 meters now we're adding so remember the rule for addition subtraction is you round to the least number of decimal places so 61.2 has one decimal point 9.35 has two decimal points and 8.6 has one decimal point so our least number of decimal points is one decimal point that tenths place so our answer we want to round to the tenths place so that is going to change to 79.2 meters that would be our correct answer to the correct number of significant figures let's get the next one and do the more word problem on where it says find the total mass of three diamonds that weigh 14.2 grams 8.73 grams and 0.912 grams if you add those three in the calculator it gives you a total mass of 23.842 grams now to figure out where to round two yet again we added so we're looking to the least number of decimal points 14.2 has one decimal point 8.73 has two and.912 has three so our least number of decimal points is one decimal point so we're rounding to that tenths place so this would be 23.8 grams that would be our answer in the correct number of save bags so now let's look at multiplication division so solve the following putting the answer in scientific notation with the correct number of significant figures don't forget units so this kind of puts together sig figs and scientific notation that we looked at yesterday so and this tends to be what we come across more in chemistry because we tend to multiply and divide more than anything else so the first one we have 8.3 meters times 2.22 meters when you put that on the calculator you get 18.426 and that's meters times meters is meters squared so remember multiplication division you are looking at the least number of digits the least number of significant figures 8.3 has two significant figures 2.22 has three significant figures two sig figs is less than three so our answer needs to be rounded to two sig figs which rounds to that ones place so this would end up being 18 meters squared would be our correct number in sig figs to put this in scientific notation because sometimes especially on multiple choice problems the answer might be in scientific notation even though that's not what your calculator answer is so remember scientific notation there has to be one non-zero digit in front of the decimal point so the decimal point would go after the one so it'd be 1.8 then that would be times 10 and we move the decimal one place so that is a possible answer so don't be alarmed if you're not seeing what is on your calculator as an answer check to see if it is rounded or if it is in scientific notation that's kind of a common thing that has messed up people in the past all right let's look at the next one we have 8432 divided by 12.5 our calculator answer is 674.56 um that would end up being meters based off of the units that we have looking at because we're dividing we're looking for the least number of digits 8432 has four digits 12.5 has three digits so our answer we need to round to three digits which would round to the ones place which would end up being 675 meters looking at this last one it says calculate the volume of a warehouse that has inside dimensions of 20.4 meters by 11.3 meters by 5.2 meters so volume is length times width times height so you multiply the three together and this gives us an answer of 1 316.224 and that's meters cubed um Sig Pig wise 22.4 has three digits 11.3 has three digits 5.2 has two digits so our answer needs to be rounded to two digits this is something that messes people up a lot so two digits rounds to the hundreds place you still need to show that this is close to 1300 not 13. so put placeholders in if needed so that would just stay the same so it would be thirteen hundred meters cubed um scientific notation I forgot to do on the last one so we'll go back and do that as well um scientific notation we would have one non-zero digit in front of the decimal so it would be 1.3 times 10 and we move the decimal three places and don't forget to include your units on there um back to the previous one scientific notation the non-zero digit would be six so it would be 6.75 times 10 and we move the decimal two places that would be to the second so just be conscious of figuring out where you need a round two but I said if you need to figure out that calculator answer first and then figure out the rounding that may help you a little bit more you can continue working on your unit one math homework the very next section is all significant figures so counting significant figures and then it has some math problems where you're figuring out where to round two