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# Introduction to scientific notation

## Video transcript

I don't think it's any secret that if one were to do any kind of science we're going to be dealing with a lot of numbers it doesn't matter whether you do biology or chemistry or physics numbers are involved and in many cases the numbers are very large there are very very large numbers very large numbers or or they're very small very small numbers very small numbers you could imagine some very large numbers if I were to ask you how many atoms are there in the human body or how many even how many cells are in the human body or the mass of the earth in kilograms those are very large numbers if I were to ask you if I were to ask you the mass of an electron that would be a very very small number so any kind of science you're going to be dealing with these and just as an example let me show you one of the most common numbers you're going to see and especially chemistry is called Avogadro's number Avogadro's number and if I were to write it in just the standard way of writing a number it would literally be written as do it a new color it would be six zero two two and then another 2001 two three four five six seven eight nine 10 11 12 13 14 15 16 17 18 19 20 and even if I were to throw some commas in here it's not going to really help the situation to make it more readable let me still throw some commas in here it still this is still a huge number and I don't even you know if I have to write this on a piece of paper if I were to publish some paper on you know using Avogadro's number take me forever to write this and even more it's hard to tell if I forgot to write a 0 or if I maybe wrote too many zeros so there's a problem here is there a better way to write this so is there a better way better way to write this then to write it all out like this to write literally the six followed by the 23 digits or the 6 0 to 2 followed by the 2000 and to answer that question and in case you're curious Avogadro's number if you had 12 grams of carbon especially 12 grams of carbon-12 this is how many atoms you would have in that and just so you know 12 grams is like a fiftieth of a pound so that just gives you an idea of how many atoms are are laying around at any point in time this is a huge number that's the point of here isn't to teach you some chemistry the point of here is to talk about an easier way to write this and the easier way to write this we call scientific notation scientific scientific notation and take my word for it although it might be a little unnatural for you with this video it really is an easier way to write things like things like that so let me before I show you how to do it let me show you the underlying theory behind scientific notation if I were to tell you what what is 10 to the 0 power we know that's equal to 1 what is 10 to the 1 power less equal to 10 what's 10 squared that's 10 times 10 that's 100 what is 10 to the third 10 to the third is 10 times 10 times 10 which is equal to 1,000 I think you see a general pattern here 10 to the 0 has no zeros no zeros in it right 10 to the 1 has 1 0 10 to the 2 2 10 to the second power let's see the tooth power 10 to the second power has 2 zeroes finally 10 to the third has 3 zeros so want to beat a dead horse here but I think you get the idea 3 zeroes if I were to do if I were to do 10 to the hundredth power 10 to the hundredth power what would that look like I don't feel like writing it all out here but it would be 1 followed by you could guess it 100 zeros so it'd just be a bunch of zeros and if we were to count up all of those zeros you would have 100 zeros right there and actually this might be interesting just as an aside do you you may or may not know what this number is called this is called this is called a Google Google in the early 90s if someone said hey that's a you know that's a Google you wouldn't have thought of a search engine you would have thought of the number 10 to the 100th power which is a huge a huge number if you actually it's it's more than the number of atoms or the estimated number of atoms in the known universe in the known universe I mean you know this is question of what else is there out there but you know I was reading up on this not too long ago and if I remember correctly the known universe has the order of 10 to the 79th - 10 to the 81 atoms and this is of course rough no one can really count this people are just kind of estimating it or even better guesstimating it but there's a huge number but maybe even more interesting to you is this number was the motivation behind the naming a very popular search engine Google Google Google is essentially just a misspelling of the word Google with the Oh L and I don't know why they called it Google maybe they got the domain name maybe maybe they want to hold this much information maybe that many bytes of information or it's just a cool word whatever it is maybe it was you know the founders favorite number but it's a cool thing to know but anyway I'm digressing this is a Google it's just one with 100 zeros but I could equivalently have just written that as 10 to 100 which is you know clearly an easier way this is an easier way to write this this is easier in fact this is so hard to write that didn't even take the trouble to write it who have taken me forever this was just 2008 here 100 zeros I would have filled up and I would I would have filled up this green and you'd have found it boring so I didn't even write it so clearly this is easier to write and so you might well how can we write this is just good for powers of 10 right but how can we write something that isn't a direct power of 10 how can we use the power of the simplicity how can we use the power of the simplicity somehow and to do that you just need to make the realization this number this number we could write it as so this has how many digits in it has 1 2 3 and then 2003 digits after the 6 right 23 digits after the 6 so what happens if I have if I use this if I try to get close to it with the power of 10 so what if I were to say 10 to the 23 dude in this magenta 10 to the 23rd power that's equal to what that equals 1 with 23 zeros the 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 you get the idea that's 10 to the 23rd now can we somehow write this guy is some multiple of this guy what we can because if we multiply this guy by 6 so what is 6 if we multiply 6 times 10 to the 23rd what do we get well we're just going to have a 6 with 23 zeroes we're going to have a 6 and then you're going to have 2300 that you're gonna have 2306 times this you don't have to multiply it down to 6 times this one you get a 6 and all the six times the zeros will all be 0 so we'll have 6 followed by 23 zeros so that's pretty useful but still we're not getting quite to this number I mean this had some this had some twos in there so how can you do it a little bit better well what if we wrote it as a decimal this number this number right here is identical to this number as if these tools were zeros but if we have we want to put those twos there what can we do we could put some decimals here we could say that this is the same thing as 6.022 times 10 to the 23rd and now this number is identical to this number but it's a much easier way to write it and you could verify it if you like it'll take you a long time maybe we should do it with a smaller number first if you multiply 6.022 times 10 to the 23rd and you write it all out you will get this number right there you will get Avogadro's number of Oh God rose Avogadro x' number and although this is complicated or it looks a little bit unintuitive to you at first you know this was just a number written out this has a multiplication and then a 10 to a power you might say hey that's not so simple but it really is because you immediately know how many zeros there are and it's obviously a much shorter way to write this number let's do a couple of more this was I started with Avogadro's number because it really shows you the need for a scientific notation so you don't have to write things like that over and over again so let's do a couple of other numbers we'll just write them in scientific notation so let's say I have the number let's say I have the number seven thousand three hundred and forty five and I wanted to write it a scientific notation so the I guess the best way to think about it is it's seven thousand three hundred and forty five so how can I represent kind of a thousand well I wrote it over here ten to the third is a thousand so we know that 10 to the third is equal to one thousand so that's essentially the largest power of 10 that I can fit into this this is seven thousand so if this is seven thousands and then it's point three thousands then it's point O four thousands I don't know if that helps you we can write this as seven point three four five times 10 to the third because it's going to be seven thousand plus point three thousands what's point three times a thousand point three times 1000 is three hundred what's 0.04 times a thousand that's 40 what's 0.005 times a thousand that's a five so seven point three four five times a thousand is equal to seven thousand three hundred and forty-five let me multiply it out just to make it clear so if I took seven point three four five times one thousand the way I do it is I ignore the zeros I says essentially multiply one times that guy up there so I get seven three four five then I had three zeros here so I put those on the end and then I have three decimal places one two three so one two three put the decimal right there and there you have it seven point three four five times 1000 is indeed seven thousand three hundred and forty five let's do a couple of them let's say we wanted to write the number six in scientific notation obviously there's no need to write in scientific notation but how would you do it well what's the largest power of ten that fits into six well the largest power of ten that fits into six is just one so we could write it as something times 10 to the zero this is just 1 right that's just one so six is what times one what's just 6 so 6 is equal to 6 times 10 to the zero you wouldn't actually have to write it this way this is much simpler but it shows you that you really can express any number in scientific notation now what if we wanted to represent something like this I started off the video saying in science you deal with very large and very small numbers so let's say you had the number I'll do it in this color let's see the number one and you have one two three four and then let's say five zeroes and then you have followed by a seven once again this is not an easy number to deal with but how can we deal with it as a power of 10 as a power of 10 so what's the largest power of 10 that fits into this number that this number is divisible by so let's think about it ever all the powers of 10 we did before we're going to positive or going to well yeah positive powers of 10 we could also do negative powers of 10 we know the 10 to the 0 is 1 let's start there 10 to the minus 1 is equal to 1 over 10 which is equal to 0.1 we switch colors I'll do pink 10 to the minus 2 is equal to 1 over 10 squared which is equal to 1 over 100 which is equal to 0.01 and you I think you get the idea that they well let me just do one more so that you can get the idea 10 to the minus 3 10 to the minus 3 is equal to 1 over 10 to the third which is equal to 1 over 1000 which is equal to point zero zero one so the general pattern here is 10 to the whatever negative power is however many places you're going to have behind the decimal point so here it's not the number of zeros in here is 10 to the minus 3 you only have two zeros we have three places behind the decimal point so what is the largest power of 10 that it goes into this well how many decimal places behind the decimal point do I have I have one two three four five six so ten to the minus six is going to be equal to point and we're going to have six places behind the decimal point and the last point is going to be or the last place is going to be one so you're gonna have five zeros and a-one that's ten to the minus six now this number right here is seven times this number right if we multiply this times 7 times 7 we get 7 times 1 and then we have one two three four five six numbers behind the decimal point so one two three four five six so this number times seven is clearly equal to the number that we started off with so we can rewrite this number we can rewrite we can instead of writing this number every time we can write it as being equal to this number or we could write it as seven this is equal to seven times this number but this number is no better than that number but this number is the same thing is ten to the minus six seven times ten to the minus six so now you can imagine no numbers like imagine the number I what if what if we had a seven let me think of it this way what if we had a seven say we had a seven three over there so what would we do well we'd want to go to the first digit right here because this is kind of the the largest power of 10 that can be they could go into this thing right here so if we wanted to represent that thing let me write let me do another decimal that's like that one so let's say I did one point zero zero zero zero five one six and I wanted to represent this in scientific notation I'd go to the first non digit zero the first non-zero digit not nun not non digit zero which is there I'm like okay what's the largest power of 10 that'll fit into that so I'll go one two three four five so it's going to be equal to five point one six so I take five there and then everything else is going to be the behind the decimal point times 10 so this is going to be the largest power of 10 that fits into this first nonzero number so it's one two three four five so 10 to the minus 5 power let me do another example so I have the point I wanted to make is you just go to the first you go to the first if you're starting at the left the first nonzero number that's what you get your power from that's where I got my 10 to the minus 5 because I kind of 1 2 3 4 5 you got to count that number just like we did over here and then everything else will be bhai the decimal let me do another example let's say at Point and my wife always points out that I have to write a zero in front of my decimal points because she's a doctor and if people don't see the decimal point someone might overdose on some medication so let's write it her way zero point zero zero zero zero zero zero zero zero zero eight one nine two clearly this is a super cumbersome number right and you know you might forget about as your add too many zeros which could be costly if you're doing some important scientific research or maybe doing well you would prescribe medicine at this small dose or maybe you would I know I don't want to get into that but how would I write this in scientific notation so I start off with the first nonzero number if I'm starting from the left so it's going to be eight point one nine two I just put a decimal and write point one nine two x times 10 to the what I just count times ten to the one two three four five six seven eight nine ten I have to include that number ten to the minus ten and I think you'll find it reasonably satisfactory that this number is easier to write than that number over there now and this is another powerful thing about scientific notation let's say I have these two numbers and I want to multiply them let's say I want to multiply the number point zero zero five times the number times the number point 0:08 this is actually a fairly straightforward one to to do but sometimes it can get quite cumbersome especially if you're dealing with twenty or thirty zeros on either sides of the decimal point zero here to make my wife happy but when you're doing in scientific notation will actually simplify it this guy can be rewritten as five times ten to the what I have one two three spaces behind the decimal 10 to the third and then this is eight so this is times 8 times 10 to the sorry this is five times 10 to the minus 3 it's very important 5 times 10 to the three would have been five thousand be very careful lying about that now what does this guy equal to this is one two three four places behind the decimal so it's 8 times 10 to the minus or so for multiplying numbers so this is if we're multiplying these two things this is the same thing as 5 times 10 to the minus 3 times 8 times 10 to the minus 4 there's nothing special about the scientific notation it literally means what it's saying so for multiplying you could write it out like this and multiplication order doesn't matter so I can rewrite this as 5 times 8 times 10 to the minus 3 times 10 to the minus 4 and then what is 5 times 8 5 times 8 we know is 40 so it's 40 times 10 to the minus 3 times 10 to the minus 4 and if you know your exponent rules you know that when you multiply two numbers that have the same base you can just add their exponents so you just add the minus 3 and the minus 4 so it becomes it's equal to 40 times 10 to the minus 7 let's do another example let's say we were to multiply Avogadro's number so we know that 6.022 times 10 to the 23rd let's say we were multiply that times some really small number so times say let's say it's 7 point 2 3 times 10 to the minus 22 so this is some really small number you're going to have a decimal and then you're gonna have 21 zeros then you're gonna have a 7 and then 2 and a 3 so this is some really small number but the multiplication when you do it in scientific notation is actually fairly straightforward this is going to be equal to six point oh let me write it properly 6.022 times 10 to the 23rd times 7 point 2 3 times 10 to the minus 22 we can change the order so it's equal to six point oh two two times seven point two three that's that part so you can view it as these first parts of our scientific notation x times 10 to the 23rd times 10 to the minus 22 and now this this is you know you're gonna have to do some little decimal multiplication right here it's going to be some number forty something I think but I don't I can't do this one in my head but this part is pretty easy to calculate I'll just leave this as the way it is but this part right here is easy this will be times 10 to the 23rd times 10 to the minus 23 you just add exponents you get times 10 to the first power or times 10 and then this number whatever it's going to equal I'll just leave it out here since I don't have a calculator 7.23 let's see what it'll be 7.2 let's see 0.2 times it's like 1/5 it'll be like 41 something so this is approximately approximately 41 times 10 to the 1 or another way it's approximately it's a little bits going to be 410 something and to get the to get it right you just have to actually perform this multiplication so hopefully you see that scientific notation is one really useful for super large and super small numbers and that not only is it's more useful to kind of understand the numbers and to write the numbers but it also simplifies operating on the numbers