in this video we're going to focus on significant figures the rules behind addition subtraction multiplication and division we're going to go over some examples including some combined operations problems where you have both addition subtraction and multiplication combined so let's begin the first thing that you need to be able to do is you need to tell how many sigfigs are in a certain number so for example how many significant figures are in the number 73 the first rule that you need to know is that nonzero numbers are always significant so seven and three are significant so there's two sig figs in this number 468 has three significant figures 7,600 and 51 has four significant figures now what about 703 how many significant figures are in this number so whenever you have a zero between sigfigs between significant numbers or non-zero numbers the zero is significant so therefore we have three significant numbers so what about 5,6 so here we have four sig figs now what about 7,30 how many sigfigs do we have trailing zeros that's a zero to the right of a nonzero number trailing zeros are nonsignificant if there's no decimal point so this zero is not significant so therefore we only have three significant figures now what if we have 7,030 with a decimal point so now this trailing zero is significant so we have a total of four significant figures 1 2 3 four try these examples 50 500 500 again 500.0 and 50,60 and 50, 60.0 Z so trailing zeros are non significant if there's no decimal point so this zero is not counted so here we only have one this is also one sigic now we have a decimal point so each of these are significant so therefore it's going to be three this zero is also significant due to the decimal point so this is four Sig fixs for this one the in between zeros are significant including the five and six but not this zero because there's no decimal point so this is four Sig fix now for this example because we have the decimal point every zero that is to the right of a nonzero number is significant so it's going to be one two three four five six seven so we have seven significant figures now what about 0.4 0.435 078 5060 and 304 leading zeros that's zer to the left of a nonzero number in this case the four the leading zeros are nonsignificant so anytime you have a zero to the left of a four a five or seven it's not counted so only this number is counted even though there's a decimal point so we have one sigic for the next number only the four three and five are counted so we have three Sig fics now here we have an in between zero it's in between two significant numbers so that zero is counted so we have three sigic a zero to the right of a nonzero number like a six this zero is significant because we have a decimal point and this zero which is between two significant numbers is also significant so therefore we have four significant figures and anytime you have a decimal point any zeros to the right of a nonzero number are significant so here we have five sigfigs so let's try a few more examples for the sake of practice try these 035 0306 07080 01 and 0 54 0760 so don't count the zeros on the left of a nonzero number so here we have two Sig fixs for this one it's three for this is going to be five for this one it's only one and for the last one it's 1 2 3 4 five six 7 so now let's go over some rules associated with addition let's say if we want to add 4231 + 3.51 how can we do it what you need to do is add the two numbers like this you want to make sure the decimal points are aligned so 1+ this imaginary zero is 1 3 + 1 is 4 2 + 5 is 7 4 + 3 is 7 so when you add these numbers you get 7741 that is the exact answer but we need to round it so notice that you have three digits to the right of the decimal point and for the other number it's two digits to the right of the decimal point you need to round it like this here the one is the digit that is uncertain you're not of that value should be a one a zero or two and this one is uncertain so you have to round it to the least digits after the decimal point so we're uncertain about the the hundred's place so we have to round it to the hundred's place here we're uncertain about the thousand's place but we need to round it here so all we need to do is draw the red line and stop after the four so the correct answer should be 7.74 if you wish to round it let's try a few more examples so let's add 2.36 and 5.4 so we're going to draw the line but we're going to add the numbers it's going to be six 7 and 7even 7.76 but we need to stop at 7.7 but we still need to use the six to decide if we should round it to 7.8 or 7.7 if this number is five or greater you need to round up so we're going to round it to 7.8 now here un certain about the hundred's place but here we're un certain about the 10th digit so we need to round it to the 10's digit so it's 7.8 try this one 7.23 61 and 8.42 so when we add it it's going to be 1 six 5 6 7 + 8 is 15 so we're uncertain about the 10,000th place and also the 100th place but we're going to round it to the 100 place so we need to round it to this number that's where we're going to stop at so we look at this number we see that six is greater than five so we need to round up to 1566 as opposed to 1565 so this is the answer if we add the numbers now what if you get a problem that looks like this 420 plus 3 51 and let's say there's a decimal point how would you do it so if we add these numbers it's going to be 42351 we surround it here so the answer is simply 424 now what if you have 500 and 1365 if you add it you get 5013651949 which is here so this is going to be approximately 500 be careful with that question because that could be tricky now what about this one 4,300 plus 17621 so let's add the numbers first 1 + 3 is 4 the other numbers are simply going to fall down so it's 45,47 6.21 that's the exact answer but where should we round it so the four five and three are significant figures the zeros are non-significant so therefore we need to draw the line at this point right after the last significant figure so what is the answer that means this has to be the last significant figure that we're going to write the rest should be zeros so what we have is 45,47 6.21 and to the right of the line we see that it's a 7 is greater than five so we got to round up instead of down so it's going to be be 455 and then after that we're going to add zeros so 4,476 is close to 45,500 so this is going to be the answer now we're not going to add a decimal point and make these zeros because if we do that we're going to have too many significant figures we're only supposed to have three sig figs and not seven so we won't add the decimal point if we write it like this only these numbers will be significant but if we put the decimal point these zeros will become sigfigs so we don't want that so we're going to leave it as 45,500 without a decimal point now let's go over some examples associated with subtraction what's 3542 minus 2.1 so even though we don't have zeros here you can treat it as if it's Z because nothing is zero so 2 - 0 is 2 4 - 0 is 4 5 - 1 is 4 3 - 2 is 1 so the answer is 1. 1442 but now where should we round it so here we only have two SigFig and for the other number we have four so we're uncertain about this digit so we're going to draw the line right here and since this number is less than five we're going to leave it as 1.4 we're not going to round it up to 1.5 So This Is The Answer try this one 6357 minus 2.47 so 7 minus a zero is 7 5 - 7 that's going to give us a negative number so what we need to do is borrow a one so the three is going to become two and the five will be 15 15 - 7 is 8 and you know what let's keep the two here 2 - 4 is negative so we need to borrow a one this is going to turn to five and that's going to be a 12 12 - 4 is 8 5 - 2 is 3 so we can confirm that with the calculator if you type it in 6357 minus 2.47 that's 3887 now where should we round it we need to draw the line here and we can see the seven is greater than five so we're going to round it to 3.89 9 so we have three Sig fix so what is 300 minus 47465 so let's use the calculator if you subtract these two numbers you should get 25253 now notice that the zeros are non-significant so we're not certain about those digits this is the most accurate number for 300 that's a significant number so we're going to draw a line right here so what should we round it to so we're going to have a two but should we run it down to two or up to three notice that we have a five so if it's five or more you got to round up so this is going to be three and then these two numbers we're going to make them zeros so it's going to be 300 we're not going to make these numbers zeros because if we do that we're going to have to write the decimal point and these zeros will become significant so we're just going to leave it as 300 so that's what you got to do for a problem like that now what is 8.4 4 * 5 8.4 * 5 is 42 now what are the rules for multiplication and division so for multiplication and division you don't need to line it up as we did for addition and subtraction you simply need to round the final answer according to the least number of significant figures so here we have two sigfigs and for this number we have one so therefore our final answer has to be or has to contain one signific so we need a number that's close to 42 but that has one significant figure the best number that we can write is 40 40 is close to 42 and it has one sigic so keep in mind you want to keep the first number but you need to decide if you should round it down to four or up to five five we need to keep the first number because we need one significant figure because the two is under five we need to round down to four not up to five if we had 46 for example and we wish to round it to one significant figure because the six is greater than five we would round it up to 50 but if we had like 43 this would round down to 40 so what if we have this number 279 * 83 so let's find the exact answer first the exact answer is 23,5 so in this number we have three sigfigs here we have two sigfigs so we need to round the final answer to two sigfigs so what number is close to 23,000 157 and has two sigfigs so at two fig sigfigs excuse me we need to stop here so should we round it up to 24 or should we keep it down at 23 so the one is less than five so we're going to keep it down at 23 so these numbers after two sigfigs we're just going to make it zero so it's going to be 23,000 that's close to 23157 and it only has two sigic if if we had 23768 we would round it up to 24,000 if we want two sigfigs so let's go over some rounding examples so let's say if we have 34,000 78 now let's say if you want to round the number to one signic what would it be so we need to keep the first number three that's the number that we're focused on now that number should we keep it at three or should we round it up to four because this number is less than five we're going to round it down to three so we have our first significant figure every other number is going to be zero so the decimal point is really here so we need to replace each of these numbers to the left of the decimal point with a zero so it's going to be 30,000 now what if we want to round the same number but to two significant figures so we need to keep the three now the second number which is going to be the second sigic we need to decide if we're going to round it down to four or round it up to five so we look at this number the seven is greater than five so we need to round it up so we're going to round it up to 35,000 now that we have our two sigfigs everything else is going to be a zero so now what if we want to round it to three significant figures so we're going to need the three and the four now the last significant figure which is a seven we got to find out if we should keep it at seven or round it up to eight so let's look at the next number which is a one one is less than five so we're going to keep it down at seven after that we're just going to add zeros until we reach the decimal point but we're not going to put the decimal point because that's going to change our significant numbers now what if we need to round it to four Sig fix so we're going to keep the three we're going to keep the four and the seven so now let's focus on the one should we keep it at one or should we round up to two so looking at the next number eight it's above five so we got to round it to two and then the last number is going to be a zero so that's how you can round to the appropriate number of significant figures but now let's say if we have some numbers to the right of a decimal point so let's say if we have 5,7 26 302 so what I want you to do is round this number to one 2 3 four and five significant figures so let's start with the first one we only need one number or one significant figure should we keep it at five or should we round it to six looking at the next number 7 is greater than 5 so we're going to round it up to six so we're going to have a six and then we need to add three zeros because at this point there's going to be a decimal point but we can't write it 6,000 is close to 5,726 so we don't want 600 and we don't want 60,000 that's too far away from 5,726 now how can we round it to two sigfigs so we're going to keep the first number five and the second number is the one that we need to round which is this number so looking at the two we got to keep it at seven we're not going to round it to eight since two is less than five after that we're going to add two zeros because we have a total of four numbers to the left of the decimal point so in both cases we're going to have those four numbers now what about rounding it to three sig figs so the first two numbers is going to be the same the third significant figure that's the one we got to worry about and that's a two because the number to the right of it is greater than five we're going to round a two up to a three and so we only need to add one zero because we just need four numbers since there's four numbers to the left of the decimal point now what if we want to round it to four Sig fix so we're going to keep the five the seven and the two we're going to FOC focus on a six the next number is still less than five so we're going to keep it at six and we don't need to add anything because we have four Sig fixs now if we want five Sig fixs we're going to keep the first four numbers the same but we're going to look at the fifth number since the number to the right of that is less than five we're going to keep the three and we need the decimal point at this point and so here we have five Sig fix so now you know how to round a number to the appropriate number of Sig fixs so now let's do a few more examples with multiplication try these two multiply 3.46 by 2.1 and 9365 * 14. 315 so the first thing we need to do is just type it in 3.46 * 2.1 that's 7266 so here we have three sigfigs and here we have two significant figures so rules for multiplication and division need to round to the least number of Sig fixs so we want two Sig fix so we're going to write the seven the last Sig fig is the one that we have to worry about the two so should we round up or down looking at the six it's more than five so we got to round up so we're going to round it up to 7.3 so that's the answer for that one now what about multiplying the other two numbers 9365 * 14 this is a one by the way 14. 315 so when you multiply these two numbers you should get 134.5 997 you always want to get the exact answer at first so now in this number we have four four sigfigs here we have five so we need to round the final answer to four sigfigs so we're going to keep the first three numbers the fourth Sig fig we got to focus on it which is the zero so looking at the number to the right of the zero it's a five that means we got to round up to one and not down to zero so it's going to be 1 34.1 now the rules for division is the same as that for multiplication so let's try these two examples 32.6 over 2.8 that's a form of division sometimes you might get numbers that are presented this way so go ahead and try these two examples feel free to pause the video so let's divide 32.6 by 2 .8 so the exact answer is about 11. 64286 so here we have three Sig fix and here we have two so we got to round it to two sigic so we're going to keep the first number the second number we need to round it so since the six is higher than five we're going to round the one up to two so we're just going to leave it as 12 since it has two six figs and 12 is fairly close to 11.6 so now for the next example let's type the numbers in the calculator 464 6895 / 12145 so you should get 38. 26179 498 so in this number we have five sigfigs here we have seven so we got around to five so we're going to keep the first four sigfigs the same 3 8 and 26 the last Sig fig is the one that we may have to round so we're either going to keep it at one or round it up to two seven is greater than five so we're going to round it up to two so it's going to be 38262 so now let's try some examples where we have combined operations let's say if we have multiplication and division so if you have 3.6 * 4.6 39 plus 5.83 1 what do you need to do here now what you want to do first is find the exact answer you do not want to round in the middle you don't want to round intermediate answers so let's multiply 3.6 by 4. 639 that's 1674 by the way you don't want to add 4.6 39 and 5.83 because you need to follow the order of operations so if you remember pem do please excuse my dear unsal you need to do a multiplication before you perform addition whenever you're evaluating numbers so now let's add [Music] 167004 + 5.83 one so the final answer or the exact answer is 22.5 314 so now we got to figure out where we should round this answer so looking at multiplication which was the first step that we did here we have have two SigFig and here we have four so the result for that number we need to round it to two SigFig if you round this number it should be 17 but you don't want to add 17 and 5.83 1 if you do that's going to change your answer but you want to keep track of it so therefore this number should contain two sigfigs but we're going to use the exact answer to get this value now this number has four sigfigs but we're doing addition and for addition we need to line up the numbers so let me see if I can fit it in here 5.83 1 plus 16.7 004 so the seven is the last significant number that we need from 1674 and it's in the 1 digit or the ones unit place so here's the ones unit that's where we have to round so we need to put the line right here now when we add these numbers we do get 22.5 314 so we require only two Sig fix we got to stop here so looking at this number it's a five so instead of rounding down to 22 we need to round up to 23 so this is going to be the answer so let's try another example like that 8.9 * 3564 plus 2. 316 so 8.9 * 3564 that's 31 7196 now let's add 2316 to that and the answer is 34.3 56 so this is the exact answer so here we have two sigfigs and in this number we have four so this answer should be rounded to two Sig fixs which would be about 32 now if we add these two numbers 31.7 one96 and 2. 316 we get 34.0 356 but notice that this number here is the last significant number for 31.7 7196 after multiplication so since this is the last sigic we need to draw the line right here therefore we only need two Sig fixs since we have two numbers to the left of that line looking at this number we need to round down to four and not up to five so our final answer is about 34 try this one 9.24 minus 1342 ID 2.3 so let's find the exact answer first 9.24 minus 1342 that is equal to 7. 898 and if we divide that by 2.3 the exact answer is 3.4 3391 so since we perform subtraction first we need to line it up so if we draw the line we could see that the hundreds digit is the least significant number so that's what we have to round so if we were to round this value notice that we have to round this number up because 8 is greater than five so to round 7.89 up it's going to be 7.90 which means after subtraction you get a number that has three Sig fix but you don't use this number in your calculation you simply use it subract the number of Sig fixs so after subtraction we have three sigfigs and so for division we're going to divide it by 2.3 which has two fig so for division you need to use the number of least significant figures that's two so our final answer should have two sigfigs so therefore it should be 3.4 since this number is less than five we're going to keep the four the way it is we're not going to round it to five and so that's the answer for that problem try this one 7.85 5 6 + 8.23 + 11.4 / 8.9 now we need to add first so let's line up the numbers 7.85 6 + 8.23 + 11.4 that's 27486 so the least significant number is the 10th digit so let's draw a line here so after addition we should get a number that has three significant figures that number would be rounded to 27.5 even though we're not going to use 2 7.5 in the calculation to get the exact answer we're going to take 27.4 A6 and not 27.5 so we're going to divide 27.4 A6 by 8.9 and so the exact answer is about 3.88 3146 so after addition we get a number that has three Sig fix and we're dividing it by a number of two sigfigs so according to the rules of division we need to round to the least number of significant figures which is two two is less than three so we get around this answer to two Sig fix so we need the three and we need one more number which is a zero so looking at the next number we see that it's an eight which is greater than five so we're going to round up to 3.1 instead of down to 3.0 so this is the answer for that problem now sometimes you may get a problem that has scientific notation with Sig fix so let's say if you get a question that looks like this 3.61 * 10 3 plus 5.1 * 10 3 whenever you add the numbers in scientific notation to add 3.61 and 5.1 the multiplier has to be the same since it is we could just add 3.61 + 5.1 but let's line it up we still have to follow the rules of addition so when we add it we get 8.71 so if we draw the line we can see that the uh 10th digit is the uh least significant it's the last significant figure in 5.1 so we're going to round 8.71 to two sigs to 8.7 so the final answer is going to be 8.7 * 10^ the 3 you don't have to use this number um for Sig fix you only need to focus on these two numbers now what if we need to add two numbers that contain different multipliers what should we do we need to change one number into the other now it's going to require less work if you start with the smaller number and change it to the large number you can work with the four and change to a three but eventually once you get your answer you're going to have to change it back into a four so it's easier if you change the smaller number into the large number so how can we do that this how can we make the 10^ the 3 10^ 4th 5.4 is basically 0.54 * 10 do you agree if you multiply 0.54 * 10 you get 5.4 so we can replace 5.4 with 0.54 * 10 and 10 is the same as 10 to the first power so therefore 5.4 * 10 3 is .54 * 10 4 whenever you move the decimal point one unit to the left the exponent is going to increase by one if you move the decimal point one unit to the right it's going to decrease by one so we need to add 0.54 * 10 4 with 2.1 * 10 4 since the multipliers are the same we can now write it like this 2.1 + 54 so we need to place the line right here since the one is the last significant figure for the first number so when we add it it's 2.64 but we only need two sigic since four is less than five we're going to round it down to 2.6 so the final answer is 2.6 * 10 10 4 now what if we need to multiply two numbers in scientific notation let's say if you have 2.1 * 10 3 and we're going to multiply that by 4 * 10 5 so first multiply 2.1 and 4 and separately multiply 10 3r by 10 5 2.1 * 4 is 8.4 now whenever you multiply common bases you need to add the exponents 3 + 5 is 8 so it's 8.4 * 10 to the 8 so now we need to round notice that the 2.1 has two sigfigs but the four has one for multiplication you need to round to the least number of sigfigs so our final answer should have one sigic which is just eight since four is less than five we're going to round it down to 8 instead of up to 9 so it's going to be 8 * 10 8 try this one 1 235 * 10 5 multiplied by 2.3 * 10 6 so first let's multiply 1.23 by 2.3 so that's 2.84 five and if we add five and six that's going to give us 11 so notice that we have four sigfigs here and two sigfigs here so our final answer should only contain two Sig fix that's 2.8 since the four is less than five we're going to leave it as 2.8 and not as 2.9 so the final answer is 2.8 * 10 to 11 consider this example 7.86 * 10 3 multiplied 5.4 * 10 2 so 7.86 * 5.4 that's 42444 and if you add 3 and two you're going to get five now notice that this number is not in proper scientific notation so we need to move the decimal point one unit to the left whenever you move it to the left the exponent is going to increase to one so it's going to increase by 1 to six the proof for that is you can rewrite 42.4 44 as 42444 * 10 4.2 * 10 is 42 10 1 * 10 5 you need to add 1 + 5 is 6 so it's 4.2 444 * 10 6 so that's the exact answer that's the answer you should get if you type the original problem exactly the way you see it in the calculator now where should we round it to so notice that we have three sigfigs for this number and here we have two sigfigs so the final answer should contain only two sigfigs since the four is less than five we're going to leave it as 4.2 * 10 6 so now let's go over an example that contains division let's say if we want to divide 9.37 * 10 6 by 2.4 * 10 2 so divide 9.37 by 2.4 first and you should get 3 3941 67 when you divide common bases you need to subtract the exponent 6 - 2 is 4 so now we can round so here we have two sigfigs here we have three so we got it around to two so the final answer is 3.9 * 10 4 so that's all I got for you today hopefully this video will help you in your next chemistry or physics exams on significant figures so thanks for watching this video and I hope you have a great day