Transcript for:
Significant Figures Overview

in this video we're going to go over significant figures how to count them how to round them to the right digit we're going to talk about addition subtraction multiplication division scientific notation and we're going to go over some hard examples some combined operations and things like that so the first thing you need to know is that any nonzero number is counted as a significant figure so 100 26 has three significant digits one two three so it has three Sig fics 95 has two significant figures and 4,873 has four significant figures what about this one 404 how many significant figures does this number number has now in between zeros zeros that are in between nonzero numbers are always counted as significant so therefore all three of these numbers are significant so it has three Sig fics so what about 50006 and this has four significant figures now what about 70,80 how many sig figs are in this number so the zeros that are in between seven and eight are counted as significant however the trailing zeros which is the zero to the right of the eight trailing zeros are not significant if there's no decimal point but if there is a decimal point they become significant so therefore we don't have a decimal point so we only have four significant figures so what about the number 8,000 how many sigfigs are there so there's no decimal points so all the zeros to the right of the eight are trailing zeros so therefore this only has one SigFig 800 also has one significant figure now let's say if we put a decimal point how many sigfigs do we have so now the zeros to the right of the eight are counted as significant figures so we have three sigfigs in this case now what about 800.0 so we count all the zeros including the one to the right of the decimal point since all of the zeros are to the right of the eight so in that example it has four Sig fix now what about this one 50,60 with a decimal point how many sigfigs are in this number so since we have a decimal every zero to the right of a non-zero number will be counted as significant so we have five significant figures now what about this one so all of the in between zeros and the three trailing zeros to the right of the six they're all counted so in this case we have seven significant figures now let's say if we have a very small number 0153 how many sig figs do we have now the zeros to the left of a non-zero number are known as leading zeros and they're not counted as significant so these two leading zeros are not counted even if there's a decimal point so we only have three significant figures now what about point 0 0 1 0 08 in between zeros are always count as significant so that's four now what about this one Z 01 070 0 how many sigfigs are in that number so we have two nonzero numbers we have an in between zero and also we have two trailin zeros which are to the right of a nonzero number but but the leading zeros are not counted as significant the trail leading zeros are significant since we have a decimal point so there's a total of five sigfigs in this example now what about this one 05 07 0 080 so every zero to the right of the five is going to be counted as significant since we have a decimal point so we have a total of seven significant figures in this example now what about numbers in scientific notation so let's say if you have 2.53 * 10 4 how many significant figures are in this number so for a number expressed in scientific notation you could ignore um this part the multiplier so simply count the number of sigfigs that you see here in this case it's three now what about 3.06 * 105 so the in between Zer counted as significant so we have three sigfigs what about 1. * 10 8 so all of the trailing zeros are counted as significant so we have a total of four since there's a decimal point and finally what about 4.20 * 107 so all three of these numbers are counted as significant now the next thing that we need to be able to do is that we need to be able to round a certain number to the appropriate number of Sig fix for example let's say if we have 4,000 25716 how can we round this number to one significant figure how would you do it if you only have one significant figure that figure is going to be uncertain your decision is should you keep it as a four or should you round it up to five so what you need to do is look at the next number the next number is a two it's less than five so you have to round it down or you going to keep it at four since we only want one significant figure everything else is going to be a zero so we need something that's very close to 4,257 the closest answer that we can get is 4,000 so 4,000 has one significant figure and it's relatively close to 4,257 now let's say if we want to round it to two significant figures so the first digit is going to be very certain we're not uncertain of this um number the last digit in a number is the uncertain digit that's the last nonzero number so the two is uncertain should we keep it at two or should we round it to three so looking at the next number five since we have a five or if we had anything greater than five that means we got to round up so it's going to be 43 now that we have the two sigfigs that we want we're going to add two zeros so 4,300 has two significant figures and it's fairly close to 42576 notice that 4,300 is closer to the actual value of 4,257 than 4,000 4,000 is further away the more significant figures that you have the more accurate your answer will be now how can we round it to three significant figures and four significant figures so to round it to three sig figs the first two numbers are certain four and two we're uncertain about the five so should we keep it at five or should we round it up to six looking at the next number is greater than five so we need to round it up to six and then we're going to add a zero so 4260 is even closer to 42576 and it has three sigfigs now for the next one if we went around to four sigfigs the first three digits are certain so all you got to do is copy what you see here 425 the last digit is uncertain should we keep it at seven or should we round it up to eight so looking at the next number it's one it's less than five so we're going to keep it down at seven so we're going to say 4257 now what if we want to round it to five significant figures so the first four digits are C ER the last one is uncertain should we keep it at one or should we round it up to two 6 is greater than 5 so we're going to round it up to two so it's 42 57.2 try this one 1581 054 go ahead and round this number to one significant figure two sigfigs three four and five feel free to pause the video as you work out this example and then unpause it to see the solution so if we only have one significant figure that figure is going to be uncertain so should we keep it at one or should we round it up to two looking at the next digit we have a five so we got to round it up so we're going to say 200 so 200 has one SigFig but it's not to too far away from 158 now if we want two sigfigs the first digit is certain so we're going to write the first digit the second digit we're not too sure should we keep it at five or should we round it up to six looking at the next number which is eight tells us we got to round it up to six once you get your two sigfigs add a zero so 160 is close to 158 now if we want three sigfigs the first two digits which are the one and five are certain so let's write that the last digit is uncertain should we keep it at eight or round it up to nine looking at the next number is less than five so we're going to keep it at eight now what about the next one how can we round it to four significant figures so if you want four significant figures the first three digits are certain so we're going to write 1 5 8 the last digit is uncertain so should we keep it at one or round it up to two looking at the next number zero Z is less than five so we're going to keep it down at one so it's so it's going to be11 now let's try one more example like this but dealing with a very small number so let's say if we have the number 0.35 047 so go ahead and round this number to one Sig fig two significant figures three and four so looking at the this number the first digit the first significant figure is uncertain that's the three should we keep it at three or should we round it up to four to looking at the five we need to round it up to four so it's going to be4 now let's move on to the next one if we want two six fig the first significant figure is certain so we're going to keep the three now let's focus on the five should we keep it at five or round it up to six the next number is zero Which is less than five so we have to keep it down at five now what about three Sig fixs what is it going to be so the first two digits will be significant so we're going to write 035 now the next digit is the uncertain one should we keep it down at zero or round it up to one looking at the four it's less than five so we're going to keep it down at zero so we have three Sig fixs now if we want four we need to keep the first three digits or the first three significant numbers now the four is uncertain but looking at the seven that tells us that it we need to round up since seven is greater than five so we're going to round the last number to a five so by now I'm pretty sure you have a good handle on these types of problems it's easy to round when you have a lot of significant figures but what if you don't have a lot of significant figures what if you need to round your answer to a number that has more significant figures than the number that you currently have for example take a 100 100 has only one significant figure the zeros to the right are not significant how can you round this answer to two Sig fics if you only have one Sig fic now think about it feel free to pause the video and see if you can come up with the solution I really want you to think deeply about this one because these are the hard questions that you might see on the test and they could be quite tricky so take your time see what you think the answer will be what number can you you write that has two sig figs and that has a value that's 100 or very close to it so using the technique that we've been using we know the first digit is certain so we're going to keep the one the second digit is uncertain now typically we would look at this number since it's less than five we won't round it up to one we're going to keep it at zero but the next number is going to be zero and we only have one six fix so that technique doesn't work here so we got to do something else in this type of situation you want to bring in scientific notation it's going to save you from a problem like this so in scientific notation we can write 1.0 * 10 2 if we want to but notice that this has three sigfigs 1 2 3 we can also write it as 1.0 * 10 2 now this has two Sig fix 1. * 10 2 and 1.0 * 10 2 they have the same value they're both equal to 100 10 s means 10 * 10 which is 100 so this number is 1 time 100 that's 100 this is also 1 time 100 but the purpose of writing it in scientific notation is that you can control how many significant figures there are if you want three sig figs use this answer but if you want two use this which is what we want now what about the number 50,000 how would you convert that into a number that has let's say three significant figures actually let's go through the list let's say one Sig fig two sigfigs three and four right now it's already one SigFig but is there another way we can write 50,000 where it's still one significant figure and the answer is using scientific notation so if we move the decimal point one 2 3 4 units to the left this is going to be 5 * 10 4 10 4 that's 10 * 10 * 10 * 10 four * that's 10 000 10 4 is basically 1 with four zeros since we have an exponent four 5 * 10,000 is 50,000 and don't worry if scientific notation confuses you I'm going to go through some examples in which you can get some practice converting scientific notation into decimal numbers and numbers in standard notation back into scientific notation so we'll go over that shortly so this is how you can put it in scientific notation um with one significant figure now if you want two sigfigs it's going to be 5.0 * 10 4 make sure you add the decimal point and then add a zero after that 5.0 has the same value as five so you don't have to worry about rounding here now if you want three sigfigs simply write 5.00 * 10 4 and if you want four sigfigs 5.00 * 10 4 now let's try an example that has a lot of significant figures and nonsignificant figures so let's say if we have this number 36 5 z0 z0 go ahead and round it to one significant figure two 3 4 and five so the first number is uncertain should we keep it at three or four so we're going to round it up to four since the second number is larger than five after that we need to add some zeros so this is going to be four and after the three there's six numbers so we need to add six zeros this is 3.6 65 million which is not too far away from 4 million if we miss a zero this would be 400,000 which is very far away from 3.6 million so you don't want to miss a zero since there's six numbers here make sure you add six zeros afterward now let's say if we want to round to two sigfigs so the first digit is certain the second one is uncertain should we keep it at six or should we round it up to seven looking at the five it tells us to round it up to seven so after six we have five digits or five numbers after that so we're going to replace those five numbers with zeros now what about rounded it to three Sig fix the answer that we already have has three Sig fix so it's going to stay the same now the next problem that's where it changes because this number has three sigfigs and if you want it to round it to a number that has more s figs that is present than this number that's when you need to switch to scientific notation mode so what we want to do at this point is move the decimal somewhere between the first two numbers in scientific notation you want a number that is between 1 and 10 so we need 3.65 so one two three four five six we need to move it six units to the left so it's going to be 3.65 that's three Sig fix and we're going to add a zero to get four * 10 6 since we move the decimal six places to the left now if we want five signic it's going to be 3.65 but with two extra zeros and it's going to still be time 10 6 so we have five sigfigs now so now you know how to round a number to any number of sigfigs now let's move on to our next topic addition let's say if we want to add 2314 to 5.23 how can we add these numbers and round it to the correct number of Sig fix the first thing you want to do is you want to align the two numbers you want to rewrite it like this and you want to get the exact answer first so there's an invisible zero at this um position 4 + 0 is 4 1 + 3 is 4 3 + 2 is 5 and 2 + 5 is 7 so 7544 is the exact answer now we need to round it so for the first number the 2. 314 the last number is the uncertain digit which is in the thousands place for the second number the 5.23 the three is the uncertain digit and that's in the 100th place so if you have an uncertain digit in the H hundredth place and the uncertain digit in the thousandth place your answer is going to have an uncertain digit in the 100 place a technique that can help you with this is to draw a line so notice that this number only has two numbers to the right of the decimal point whereas this one has three so you want to draw a line where your final answer has the least number of digits to the right to of the decimal point so here this has three digits to the right of the decimal point this one has two so your final answer should have two digits to the right of the decimal point it's the smaller of these two numbers so this is the number that is uncertain in our final answer so that's the number we got to focus on should we keep it at four or should we round it up to five so look at at the number to the right it's less than five so we have to round it down so our final answer answer is going to be 7.54 rounded to the correct number of significant figures so let's go ahead and try some more examples try this one 3.45 + 5.3 so let's write it let's line it up like this so 5 + 0 is 5 4 4 + 3 is 7 3 + 5 is 8 so the five is the uncertain digit it's the last significant figure in 3.45 so for that number it's uncertain in the hundreds place now the three is the last significant figure in 5.3 so it's the uncertain digit so therefore we're going to be uncertain about the seven since we're uncertain about the three so the uncertainty is going to be the 10's place this time so we're going to draw a line right here the line is going to be to the right of the last uncertain digit not this one but the one that's close to the left so therefore we're uncertain about the seven should we leave it at seven or should we round it up to eight looking at the five that tells us we have to round it up to eight so the answer is going to be approximately 8.8 try this one 82354 plus 9.35 so first let's get the exact answer the four and five is going to drop down 3 + 5 is 8 and 2 + 3 is 5 8 + 9 is 17 so we're uncertain about this number and this number so therefore this number will be uncertain if you're uncertain about the 10,000 place and the 100th place then your final answer if you're adding or subtracting you're going to be uncertain about the hundred's place so let's draw a line right here so we need to round this number we're un certain about the eight should we keep it that eight or round it up to 9 looking at the five we need to round it up to 9 so the answer is approximately 1759 rounded to the correct number of Sig fix now sometimes you might be unsure of where to draw the line if you're unsure about that this example might help you let's say if we're adding 3.65 14.1 8 .36 now where should we put the line should we put it here should we draw it here here here or here where would you put the line which line is the correct line to draw so you want to draw the line in such a way that on the left side of the line you you only have significant figures there's no empty space on the left side on the right side you need to have at least one empty space so draw the line in such a way that there's no empty space on the left but you have at least one empty space on the right so let's go through each one first let's look at the blue line if we put it here it's not going to work because we have empty space to the left so let's not do that now let's say if we put the line here it's not good because we have this empty space to the left now if we put the line here this could work we have no empty space on the left but we have at least one empty space on the right and that's where you want to put the line you do not want to put the line here because even though there's no empty space on the left there's no empty space on the right of it so you went too far in that case so you want to put the line where there's no empty space on the left but there's at least one empty space just to the right of it so let's add these numbers to get the exact number 0 + 6 is 0 5 + 3 is 8 6 + 1 + 1 is 8 3 + 4 is 7 7 + 8 is 15 we need to write the five carry over the one 1 + 1 is two so we have 25.88 and we know to put the line right here so we're uncertain about the 10's digit so should we keep it at eight or should we round it to nine looking at the next number we have an eight which is greater than five so we got to round this eight to a nine so then the answer is going to be 25.9 now let's go over some unusual examples with addition let's say if you want to add 530 with 4.63 try this example and round it to the appropriate number of significant figures so first let's add everything 0o and four is four the five and three is going to fall down so if you add these two numbers you're going to get 5346 3 that's the exact answer but now where should we put the line should we put it here should we put it here where should we put it now as you mentioned before you want to put a line next to you want to draw the line where you have no empty space to the left but you want to have an empty space to the right so you want to put a line here by the way when drawing the line you want to put it to the right of the significant numbers the zero is significant and the four is significant if the zero wasn't significant then that could change everything so we'll go over an example relating to that but in this case the zero and the four are significant so we want to put it to the right of those numbers and we have an empty space here so now we just got around it so looking at the four which is the uncertain digit because the last Sig fig 530 is uncertain we need to round a four should we keep it at four or should we turn it up to five since six is higher than five we need to round it up to five so the answer is going to be 535 now let's see if you have an example that looks like this 36,500 plus 14356 where would you draw the line where would you round it to so the least significant figure in 36,500 or I should say the last significant figure is the five the last significant figure of 143.5 6 is a six so the better way to describe where to put the line is you want to put the line to the right of the the last significant figure that is further to the left so we want to put the line right here because five is the last significant figure of 36,500 and if you compare these two last significant figures you want to choose the one that's on the left side so you can treat these zeros as being um non-existent like empty space because they're not significant so now let's go ahead and add the numbers so it's going to be 3664 3.56 so the last significant figure or the uncertain digit is in the the hundreds place so therefore this digit will be uncertain in the hundred's place so we got a round the six looking at the four we need to keep it down at six and not round it to seven so this is going to be 366 and then everything else will be a zero so you don't want to stop at 366 because 366 is very far away from the exact answer of 36,6 43.5 6 so you want to add two more zeros 36,600 is close to 36643 now you do not want to add a decimal if you add the decimal these zeros become significant and you don't want that because that would mean that this zero is the last Sig fig or the the uncertain digit whereas six has to be the uncertain digit so therefore you do not want to put the decimal point our final answer should have three significant figures the six has to be the last uncertain digit so we don't want to put the decimal point if we do it's going to be 56 figs and that's not going to be right but without it it's three sigfigs and since the zeros are not significant the six is the the last SigFig which is the uncertain digit now let's say for add in three numbers and let's say there's plenty of zeros as well where would you draw the line so let's identify the last significant figure or the uncertain digit in each number for 350,000 the last s fig is the five for 57,000 the last Sig fig is the one these zeros which are in between the two nonzero numbers the seven and the one they're significant so the one is the uncertain digit and the eight is the UN certain digit so when you have a lot of zeros particularly for these harder examples you want to draw the line to the right of the first uncertain digit that you see so here's the first one which is on the left side you want to draw the line to the right of that number because the answer is going to be uncertain in the 10,000 place so that's where we have to round it now let's get the exact answer so if we add everything this is going to be 8 6 + 1 is 7 2 plus a bunch of zeros is 2 this is four Z 7 5 and five is 10 carry over the one 3 and 1 is four so if we add these numbers we should get this number as the exact answer so this zero should be the last significant figure should we keep it at zero or should we round it up looking at the seven it's bigger than five so so we have to round it up to one so it's going to be 4 one after the one everything is going to be insignificant so these four remaining digits to the left of the decimal point we're just going to replace it with zero and we're not going to put the decimal point because we only want two Sig fix we want the one to be the last significant digit so 410,000 is very close to 47,000 and 42 make sure your final answer is as close as possible to the exact answer and at the same time you want to make sure it has the appropriate number of Sig fixs here's the last example for addition go ahead and add 72,000 4,300 and 160,000 so the question is where to draw the line so the last significant figure in 72,000 is the two for the 4300 it's the three and for the 160,000 is a six so make sure you draw the line to the right of the first uncertain digit or last sigic that you see so make sure you draw the line to the right of the first Circle so right here now let's add everything to get the exact answer so this is going to be 0 0 3 4 and two is 6 7 and 6 is 13 carry over the 1 1 + 1 is 2 so 3 is going to be the last Sig fic but should we keep it at three or should we round it up to four looking at the six we need to round it to four so it's going to be 2 4 and then every other number we're just going to put zero so 240,000 is very close to the exact answer of 2363 but it has the appropriate number of Sig fix two Sig fixs so looking at the first answer we can see that the uncertain digit is in the thousand's place here it's in the 100 place and here it's in the 10,000 place our final answer should be should have an uncertain digit in the 10,000 place which is which corresponds to the six so that is it for the addition part of Sig fix now let's move on to subtraction the rules for subtraction is the same as the rules for addition so let's say if we want to subtract 4671 by 2.1 so first let's get the exact answer 1 minus the invisible Z is 1 we can bring down to seven 6 - 1 is 5 4 - 2 is 2 so the exact answer is 2.5 5 71 now the last significant digit in 4671 is the one the last significant digit or the uncertain digit in 2.1 is one so we're going to draw the line to the right of the circle that is on the left so we have two circles draw the line to the right of this one as you can see we have significant figures to the left and we have our first empty space or nonsignificant figure to the right so we got round it around 2.5 so the five is the uncertain digit should we keep it at five or round it up to six looking at the seven it's bigger than five so we're going to round it up to six so we're going to say the answer is about 2.6 so as you can see the rules are the same so let's try some more practice problems let's subtract 7.46 3 by 3.58 so 3 minus the invisible Z is 3 6 - 8 is a negative number so we need to borrow a one so let's take away a one from four so the four becomes a three and the six will now be 16 so 16 - 8 is 8 now 3 minus 5 is a negative number so we got to borrow a one so this is going to become a six and the three is now 13 13 - 5 is 8 and 6 - 3 is 3 so we have 3.88 3 and let's confirm that with the calculator if you type in 7.46 Dre - 3.58 you indeed get that answer so now where should we draw the line the last Sig fig for the first number is the three and for the second number is the eight so draw the line to the right of the first Circle which is going to be right there so eight is the uncertain digit three is less than five so let's keep eight at eight let's not round it to n so the answer is going to be 3 88 try this one what is 200 minus 28.4 so this is one of those unusual examples now let's use the calculator to find the difference between the two numbers 200us 2814 is 171.8cm now where should we draw the line for the second number the uncertain digit is the four for the first number the uncertain digit is a two that's the only significant figure we have so we need to draw it to the right of the first Circle that we see on the left so let's put it right there so our final answer should only contain one sigic should we keep it at one or should we round it up to two looking at the seven it's greater than five so we got to round it up to two so it's going to be two and then the numbers to the left of the decimal we need to replace it with zero so 200 is the only number that we can write that has one Sig fig but that is very close to 171.8cm another way you could write it but this is going to be the answer we need to round up instead of down 100 is further away from 171 than 200 200's closer and that's all we can do that's the only way we can round it to one significant figure let's try one more like that so let's say if we have 5,000 minus 62.4 13 so first let's get the ex exact answer 5,000 minus that number is 4,9 37587 so the three is the last SigFig for the second number but the five is the only SigFig or the uncertain digit for the first number so we're going to draw the line to the right of the first Circle so looking at the four which is the only SigFig that we have for our answer should we leave it at four or should we round it up to five so looking at the next digit which is a nine that tells us we got round the four up to five so it's going to be five and then there's three numbers to the left of the decimal point so we're going to replace those with zeros so it's going to be 5,000 we can't say 4,000 because even though 4,000 has one sigic it's F it's two too far away from 4,900 4,937 is closest to 5,000 than 4,000 so 5,000 is the best answer that has one sigic now we could write it as 5 * 10 3 which is five with three zeros now let's move on to multiplication let's say if we wish to multiply 9.6 by 7 where 7 is not an exact number but it's a measurement value how can we round this to the appropriate number Sig fix we'll talk about exact numbers later in this video so for multiplication and division you want to round your final answer to the least number of sig figs that you see in these individual numbers so 9.6 has how many significant figures this has two Sig fixs and seven has one so the final answer should be rounded to the least number of sig figs which is one so first let's get the exact answer 9.6 * 7 is exactly 67.2 so how can we round this number to one sigic so if we only want one sigic the first number is going to be the uncertain number should we keep this number at six or should we round this a seven looking at the next number which is a seven it's greater than five so we got to round a first number up to seven so it's going to be a seven and then we only have one number to the left of the decimal after the seven so we're just going to replace that with zero so 70 is very close to 67.2 and 70 has one SigFig so we can write the answer as 70 or we could say 7 * 10 1 in science of notation 7 * 10 is 70 so both answers are acceptable what about this one 325 * 75 feel free to pause the video and work on that example so first let's get the exact answer 325 * 75 is 24375 now the first number has three Sig fixs the second number has two we need to round the final answer to the least number of sigfigs which is two so if we want two sigfigs the first number is going to be the same the second number the last Sig fig is the uncertain digit should we keep it at four or should we round to five so looking at this number three is less than five so we're going to keep it at four now the decimal is somewhere over here so after this four our last SigFig the remaining three numbers we're going to replace it with zeros so the answer is 24,000 or we could say 2.4 * 10 to the fourth power if we move the decimal 1 2 3 four units then it's going to be to the fourth power so both of these numbers have two Sig fixs and 24,000 is fairly close to 24375 so what's the answer for this one so if we get the exact answer first it's going to be 10.74 now where should we round it to 4.38 has three sigfigs 2.3 has two so our final answer should contain only two so the first number is going to stay the same the second number is zero we're going to keep it at zero because this number is zero now we have 10 10 only contains one sigic but if we put the decimal point now it has two Sig fix so that's going to be our answer we could say 10 or we could say 1.0 * 10 the 1st power both of these two numbers is equal to 10 and they both contain two significant figures which is close to 10.74 try this one the first thing you should always do is you should always get the exact answer first if you multiply these two numbers you should get 14441 492 so the first number has four sigfigs the second one has five so for multiplication we need to round it to the one with the least number of Sig fix so we want four Sig fixs so we're going to keep the first three digits 144 the last one the fourth SigFig is the uncertain one so looking at the next number it's one it's less than five so we're going to keep the four so we're going to say it's 1444 now here's the last example for multiplication try this one 4,000 * 8.17 so let's get the exact answer first so the exact answer is 32680 so 4,000 only has one SigFig 8.17 has three so our final answer should have only one sigic so what should that number be so the three is the UN digit should we keep it at three or round it up to four two is less than five so we're going to keep it at three the other four numbers which is to the left of the decimal point we're going to replace with zero so we're going to say it's 30,000 or 3 * 10 4 power so 30,000 is not too far away from 32680 but it does have one sigic so we can write the answer in these two ways now let's move on to division the rules for division is the same as that for multiplication so let's say if we want to divide 34.7 by 3.1 so just like before the first thing you want to do is get the exact answer if you divide these two numbers you should get 11.1 1935 48 and so forth now the first number has 36 the second number has two so you want to round your final answer to the least number of sigfigs so that's two sigfigs so the first number is going to be the same the last SigFig is based on this number which one is less than five so we're going to keep it at one and not round it to two so the final answer is simply going to be 11 how about this one 536 3172 / 13.2 so let's see what that's going to be equal to so the answer is 40. 6300 9091 so this number has three Sig fix and this one has seven so we're going to round it to the small smaller one the three sigfigs so the first two sigfigs is going to be the same these are certain digits the last SigFig is the uncertain digit so this three is less than five so we're going to keep the six and not round it up to seven so it's going to be 40.6 that's our final answer for this example now let's try one unusual example so let's try 250,000 / 3176 so this is going to be 78,79 65 so this number has four Sig fixs this one has two so we need to round our final answer to two Sig fixs so we're going to keep to seven eight is the last SigFig or the uncertain digit and looking at the next number is bigger than five so we got to round it up so instead of keeping that eight we're going to round it to nine and the three remaining digits to the left of the decimal we're going to replace it with zeros so it's going to be 79,000 which is the same as 7.9 * 10 to the 4th power so both numbers have two significant figures and 79,000 is very close to 7875 so 79,000 is the answer for this problem now let's work on some examples where we have multiple operations so let's say if we wish to multiply 4.3 * 5231 + 6814 now the first thing we want to do is get the exact answer so according to the rules of order of operations perhaps you heard of pemos parentheses exponents multiplication division addition and subtraction you want to multiply before you add 4.3 * 5231 the exact value is 22. 4933 now if we add the numbers our final exact answer is 2930 73 so now we need to look at the Sig fix now looking at the first operation which is multiplication 4.3 has two sigfigs 5.2 31 has four sigic so we need to round this answer to two Sig fixs so that's going to be about 22 since the four is less than five now you don't want to add 22 to 6814 rather you want to keep track of the Sig fix you want to use the exact answer to get this value you don't want to say 22 + 6814 which is 28.8 one4 if you do that you're going to get the wrong answer don't do it always get the exact answer first but use the intermediate values to keep track of the Sig fix so therefore since the 22.49% three now for 6. 814 the uncertain digit is the last sigic it's the four for the 22.49% now looking at this number is less than five so we're going to keep the last s fig at 9 and we're not going to round it up to zero or 30 so our final answer is 29 for this example so let's try this one 7.35 * 4265 plus 7.34 7 1.35 * 4265 the exact value for that is 3134 775 and let's add it to 7.34 so the final exact answer is 38. 68 775 now this number has three sigic and this one has four so for the first step which is multiplication we need to round it to the least number of sigfigs so that's three sigfigs so the uncertain digit is at the three this one has three sigfigs but since we're dealing with addition we need to line it up so let's line up the last two numbers so for 7.34 the 4 is uncertain and for 31. 34775 the three is uncertain so the line is going to be to the right of the first uncertain digit or the first Circle and so we know the final answer is 38.6 8775 so we need to round it at this point so looking at the eight that's greater than five so we need to round the six up to seven so the final answer is going to be 38.7 the uncertain digit is in the 10's place which should be the same for The Final Answer let's try this example 8.46 minus 5312 let's divide it by 2.8 so first we have to work with the numerator let's perform subtraction and let's line it up so what is 8.4 6 - 5312 so the exact answer is 3.48 so the six is the uncertain digit and the two is the uncertain digit so we're going to draw the line to the right of the first uncertain digit or the first Circle that we see so even though 3.14 is the exact answer for the first part if we we had to round it we would need to round it to 3.15 which has 3 Sig fix but remember don't use the rounded answer to get the exact Final Answer use this value so 3148 / 2.8 is going to be 1.12 4 286 now we need to use this number to keep track of the Sig fix so this value should have three Sig fix which means that the uncertain digit is the four 2.8 has two Sig fics since we're dealing with division we need to round the final answer to two sigic so the final answer is just going to be 1.1 try this one 8431 plus 9.25 + 12.6 / 4.7 so let's add the three numbers first so if we add the three numbers the exact value is going to be 30. 281 so for the first number the 8431 one is the uncertain digit and here it's five and here it's six so we're going to draw the line to the right of the first Circle so the rounded answer should be 30.3 so the answer has three Sig fics but don't use the r answer make sure you use the exact answer so 30281 divid 4.7 this is going to be 6.44 277 so you want to get enough digits there's more numbers after the seven but we don't need that much so 30. 281 it should be rounded to three Sig fix based on what we see here and 4.7 has two Sig fixs since we're dealing with division our final answer should have two Sig fixs so this is less than five so we're going to keep this at four so the final answer is going to be 6.4 let's try this one 5312 * 2. 86 / 19.3 - 17.2 so let's multiply the two numbers on top so that's going to be 1519 232 and if we subtract the two numbers on the bottom 19.3 minus 17.2 that's 2.1 so both of these numbers are uncertain so we got to draw the line to the right of the uncertain number so this is going to stay 2.1 so this should have two Sig fixs and this number should have three Sig fix this number has four this one has three so for multiplication we need to round this value to three Sig fix so let's divide 15.1 19232 by 2.1 so that's going to be 7.23 4438 so we're dividing two numbers one with three sigfigs the other with uh two sigfigs so we got around the final answer to two sigfigs so it's just going to be 7.2 so you have to keep track of the sig figs for every step whenever you have these complicated combined operations here's another one for you 4.83 1 iD 2.1 plus 76. 381 / 364 plus 9830 / by. 215 so let's divide each number first 4831 / 2.1 that's about 2.300 476 I'm not going to write all the numbers just enough of them though 76. 381 divid 364 that's going to be about 209 8379 and then for the last one 98 8 3 id. 215 that's going to be 4.57 209 so now let's keep track of the Sig fix so for the first part we're dividing the first number has four Sig fics the second one has two so this answer should have two Sig fics the rounded value should be like 2.3 now for the next one we have five sigfigs and three sigfigs so this one should be rounded to like 210 for the last one here we have four sigfigs and three SigFig so it should be rounded to three SigFig now we need to perform addition so let's line it up so we have 2.3 0476 and then 209 8379 and 4.57 209 okay let's add these numbers so you should get 26 71 04 66 now for this number we should have two sigfigs so the three is the uncertain digit for this number should be three sigfigs so the nine is uncertain and for this number is three Sig fix so seven is uncertain so we need to draw the line to the right of the first Circle the first uncertain digit which is here so we got around our final answer to three Sig fix so looking at the seven we need need to round the six to a seven so our final answer is going to be 217 now let's say if you want to find the average of two numbers let's say you want to average 8.4 3.21 and 5436 you would have to add them up and divide by three exactly 3 now how would you round this operation to the appropriate number of Sig fix now this is one exception to the significant figures rule if you're dealing with an average and if you're dividing by an exact number then you don't count this number as one sigic exact numbers have an infinite number of Sig fics so that's one exception to the significant figures rule so when we divide we're not going to base our answer on one SigFig we're going to base it on what we get in the numerator so let's add the three numbers first you should get 17046 in this example the six is uncertain the four is uncertain and the one is uncertain so we need to draw the line right here so our intermediate answer should have three sigic but let's divide the exact value of 17.04 6 by 3 so the exact average is 5682 but we're not going to round it to one sigic we're going to round it to three Sig fix since exact numbers have an infinite number of Sig fix so if we round a final answer to three Sig fix it's going to be 5.68 now we're going to spend some time using numbers in scientific notation but before we do let's review how to convert a number in scientific notation into decimal form and vice versa so let's say if you have 3.4 * 10 the 3 what is this value in standard notation whenever you have a positive exponent it's going to be a large number 10 3r is 10 * 10 * 10 that's a th000 so what you're really saying is 3.4 * 1,000 it's a one with three zeros 3.4 * 1,000 is 3400 so what you could do is move the decimal point three units to the right and you can get 3,400 that way so go ahead and convert these numbers into standard notation so let's start with the first one 4.5 * 10^ 2 10 squ is 100 that's 10 * 10 4.5 * 100 is 450 or you can move the decimal point two units to the right and then add a zero here so it's 450 now 3.78 * 10 4th let's move the decimal four units to the right 1 2 3 4 so it should now be here and we need to add two zeros so this is the same as 37,800 now what about the last one we need to move it five units to the right so if we start with 1.70 this is going to be 1 2 3 four five so we need to add three zeros so it's going to be 70,000 now let's say if we have a negative exponent try these 1.36 * 10 to Theus 2 4.2 * 10 -3 5.61 * 105 and 6.8 * 10us 1 go ahead and convert it into standard notation so let's start with the first one 1.36 * 10 -2 we need to move the decimal two units to the left so this is going to be 0136 now 4.2 * 10us 3 we need to move the decimal three units to left so 1 2 3 so we're going to add two zeros and it should be here now so the answer is going to be 042 now for this one we need to move the decimal five units to left 1 2 3 4 five so we need to fill these with zeros and so it's going to be [Music] 0.005 61 so the decimal it was here we move it 1 2 3 4 5 now the last one 6.8 * 10us one we just got to move it one unit to the left so it's simply 68 so now let's work backwards let's convert a number in standard notation into scientific notation so let's say if you have 400 5,700 365,000 and 15800 Z go ahead and convert these numbers into scientific notation so we need to move the decimal two unit to the left and because we're dealing with large numbers the exponent is going to be positive so we know this is going to be 4 * 10 pos2 now for this one we need to move the decimal three units to the left we want the decimal to be between the five and the seven in scientific notation this number has to be between 1 and 10 so this is going to be 5.7 * 10 to the 3 power now for the next one we need to move the decimal 1 2 3 4 five units to the left so it's 3.65 * 10 5 what about the last one what's the answer so this is three units six 7 8 so this is going to be 1.58 * 10 to the 8th power now let's try a smaller numbers so let's say if we have 014 036 1784 and 0.000000 52 so since we're dealing with small numbers the exponent is going to have to be negative so we need the decimal to be between the one and the four so we need to move it two units to the right so this is going to be 1.4 * 10 to minus 2 This original number is the same as 014 * 10 0 whenever you move the decimal point to the right the exponent decreases it went down by two from 0 to -2 in the case of a large number like 1500 if you move the decimal to the left in this case it the exponent is going to go up from 0 to positive3 now we're going to use that fact when solving questions in the future so make sure you're aware of that if you move the decimal to the left the exponent is going to go up if you move it to the right it's going to go down so in this case we're moving the decimal point three units to the right so it's going to go down by three units from 0 to-3 now for the the next example we just need to move it one unit to the right so it's going to be 1784 * 101 for the last example we need to move it three six units to the right so this is going to be 5.2 * 10 to the -6 now let's say if you want to add two numbers in scientific notation how would you do it and how would you write to the appropriate number of significant figures if the multiplier is the same you can simply add the numbers in front you can add 4.23 and 5.1 because they're like terms it's like adding 3x + 4x you can say 3x + 4x is 7 x since 3 + 4 is 7 if the multiplies were different you can't add them you can't add 3x + 4 Y and say it's 7x y it just doesn't work to add the coefficients the terms must be like terms so we have like terms here so we can add 4.23 and 5.1 so let's line it up so three is the least significant figure in 4.23 or it's the uncertain digit one is the uncertain digit in 5.1 so we need to draw the line to the right of the first Circle and if we add these numbers it's 9.33 so this is less than five so we're going to keep this number at three and not round it up to four so it's going to be 9.3 * 10 3 power we need to round it to two significant figures now what would you do if you had to add two numbers in scientific notation that contain different multipliers so we have two options we need to change either the four into a three or change the three into a four then we can add the coefficients so to speak right now we can't add them we can't add 3x and 4 y so we can't add 3.6 and 2.3 so we have to adjust it now if you change the four into a three you're going to get an answer where you're going to have to change the three back to a four so you want to change the smaller exponent into a larger exponent if you do that you won't have to change back it's going to require less work so let's convert the three into a four so if we want to increase the three to a four should we move this decimal to the left or to the right whenever you move the decimal point to the left the exponent will increase if you move the decimal point to the right the exponent will decrease now let's understand why let's move the decimal points to left 2.3 is the same as23 * 10 do you agree if you type in 23 * 10 in your calculator you're going to get 2.3 so this is equivalent to this so we can replace two .3 with 23 * 10 now we still have the 10 3r so let's rewrite that now 10 is the same as 10 to the first Power whenever you multiply two common bases you can add the exponents 3 + 1 is 4 so this is the same as 23 * 10 4 so as you can see when we move the decimal point one unit to the left the exponent is going to go up by one from three to four so now we can add the two numbers so we can add 3.6 * 10 4 with 23 * 10 4 now that the multipliers are the same so let's line it up 3.6 + 23 this is going to be 3.83 so the six is the the uncertain digit and three is the uncertain digit in point 23 so let's draw the line to the right of the first Circle so we're going to round it to 3.8 so our final answer is 3.8 * 10 4th power so that's how you can add two numbers in scientific notation if the multipliers are different and now you know how to round it to the appropriate number of significant figures consider this example 4231 * 10 - 5 + 7.6 * 10 -6 + 2.73 * or 2.73 1 * 104 go ahead and add the three numbers so let's start with the smallest number and convert it to the largest number we need the exponents to be the same so which of these is the smallest number or which one has the lowest value it's not4 it's -6 on the number line -6 is to the left of4 so on the left side the numbers have a lower value so with respect to a number line we're going to change -6 into -4 and5 to4 so to change to4 we need to move the decimal one unit to the left remember if you move the decimal to the left the exponent is going to increase by one so we're going to add 1 to5 so it's going to be 4231 * 10 to the5 + 1 is4 which is what we want now for the next one we're going to take this number and convert it to a large number we need to convert it to4 so we're going to have to move two spaces to the left so it's going to be 076 times 104 now the last one we're going to keep it the same way now that we have the same multiplier we can add the numbers so let's line it up we have 4 231 plus .76 plus 2.7 31 so if we add the numbers the one is going to drop down 1 + 6 + 3 is 10 so let's carry over the one 3 and 7 is 10 + 2 and 1 1 is 13 let's carry over another one 4 and 7 is 11 + 1 is 12 so let's carry this one over and so it's going to be 32301 let me confirm with the calculator to make sure my math is correct because mistakes do happen and it is so now we need to know where should we round it so one is the uncertain digit in 423 six is uncertain and this one is uncertain so we need to draw the line here which incorporates the zero so then it's going to be 3.23 * 10 to Theus 4 so this is the final answer now what about multiplying two numbers in scientific notation what's 1.5 * 10 2 * 2.13 * 10 3 now first let's calculate the answer let's multiply 1.5 and 2.13 and then we're going to multiply the bases 10^ s and 10 3 power so 1.5 * 2.13 that's going to be 3.1 95 and to multiply 10^ 2 by 10 Cub we need to add the exponents 2 + 3 is 5 so it's 10 5th power now how should we round it for multip ication and division we need to round to the least number of Sig fixs here we have two Sig fix and here we have three significant figures so our final answer should contain three Sig fix I mean not three but two Sig fixs the small of the two values so we're going to round the final answer to 3.2 this is the uncertain digit and because we have a nine we need to round up to two instead of keeping it at one so it's going to be 3.2 * 10 to 5th power try this one 1316 * 10 3r power multiplied 3.4 * 10 the 6 power so first let's multiply 1316 * 3.4 so that's going to be 4.47 45 4 * 10 let's add 3 + 6 which is 9 so in this number we have four Sig fix and here we have two so we got to find our final answer to the least number of Sig fix which is two so it's going to be 4 point now this is the uncertain digit should we keep it at four should we round it up to five so looking at the next digit seven is greater than five so we're going to round it up to five so it's going to be 4.5 * 10 9th power so that's the answer for this example try this one 8.46 * 10 the 4th power multiplied by 6.7 * 10 the 3 power so just like before let's multiply 8.46 by 6.7 so that's going to be 56. 682 10 4 * 10 3r is 10 7 4 + 3 is 7 now in this case what number should we round it to but before we do that before we round it to a certain number we need to put this in proper scientific notation because we have a number that's greater than 10 and we want it between one and 10 so let's move the decimal one unit to the left so it's gonna be 5.66 82 * 10 to the 8th power whenever you move the decimal one unit to the left the exponent is going to increase by one so now let's see where we should round it so for this one we have two sigf fix and in 8.46 we have three Sig fix so we got around the answer to two Sig fix so it's going to be 5.7 * 10 8 now let's try an example that is associated with division so go ahead and try this example so the first thing I would do is divide 9836 by 2.3 and so that's going to be 4 27652 time 10 to the whenever you divide the exponents you have to subtract them so if you divide 10 8 by 10 3r it's going to be 8 minus 3 which is 5 so it's 10 to the 5ifth power now we need to round it so in this number we have two Sig fix here we have four Sig fixs so the final answer has to contain the smaller of the two whenever you're dealing with addition I mean not addition but whenever you're dealing with multiplication or division so we need two sigfigs so the first one is a certain digit the last SigFig is uncertain so looking at the seven we need to round it up to three instead of keeping it at two so it's 4.3 * 10 to the 5th power so this is going to be the last problem for today so this is going to be a problem with combined operations all in scientific notation so let's add 4.6 * 10 to8 with 3.16 * 107 before we can add those two numbers we need to make sure the multiplies are the same so which number is smaller 8 OR7 on a number line8 is to the left of7 so let's convert 8 into A7 we can do that by moving the decimal point one unit to the left if we do do that it's going to be 46 * 108 I mean not8 it's going to be7 now since we need to add one to the exponent 8 + 1 is7 and this is going to be added to 3.16 * 10^ - 7 so now that the multipliers are the same we can add 3.16 and 46 6 + 6 is 12 carry over the 1 1 + 1 + 4 is 6 so we have 3.62 so it's going to be 3.62 * 10- 7 now this number is the uncertain digit and so is this one and the line has to be to the right of the uncertain digit which it is in both cases so we can leave it as 3.62 now let's focus on the two numbers in the bottom so let's convert the four into a five so we need to move the decimal point one unit to the left so it's going to be0 521 * 10 5 so we have 3.6 minus. 521 if we subtract these numbers let's use a calculator 3.6 minus 0.521 is 3.79 so the six is the least significant number or the uh uncertain digit in 3.6 one is the uncertain digit so we're going to put the line to the right of the first Circle that we see so we need to round it to 3.1 our answer should have two Sig fixs so it's going to be 3. 1 * 10 5 so remember this is 3.6 * 10 5 and this number is 521 * 10 5 so this is going to carry over anytime you move the decimal to the left add one to the exponent so now at this point we could divide the two numbers by the way I almost made a mistake we need to use the exact value not the rounded value the exact value is 3.79 * 10 5th power but the rounded value is 3.1 * 10 5th power so make sure you use the exact answer but keep track of the Sig fix so the red value should be 3.1 which means that this number should have two Sig fix fix and this one it was an exact answer we didn't have to round it so it's three Sig fix now we can divide the two numbers so 3.62 / 3.79 that's equal to 1.1 757 and now for the exponent it's going to be -7 - 5 7 - 5 is equal to -12 now we need to round it so this number is accurate to two Sig fix so therefore our final answer should contain only two sigic so one is the uncertain digit that's the number we have to round so because it is seven we need to round up so it's going to be 1.2 * 10^ -12 so so remember when you're adding or subtracting make sure um to line up the numbers and draw the line to the right of the first Circle or the first uncertain digit that you see now for multiplication or division make sure you round your final answer to the least number of significant figures so that is it for this video I hope you found it to be educational was a very long video but I hope these examples help you to understand how to round to the correct number of sig figs whenever you're doing any type of combined operation so thanks for watching and have a great day