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# Logarithmic scale (with Vi Hart)

## Video transcript

all right so I'm I'm by heart and I'm here with Sal Khan and hello yeah we're talking about just how we think about numbers and what what is the most natural way to think about them in our everyday lives and vie said that she was way to test me right now yeah all right can I can i yes yes I get to use the official Penance po8 no that's all scream yes I need it you need training by training looks like a pizza it's it's a triangle okay where's where's your testify okay you're diverging I'm sorry all right so here's the number line uh regular old numbered line that way I want to start at one all right we're interested in one I'm gonna go all the way to a million and I'm gonna give you the pen now and I'm gonna ask where is 1,000 where is 1,000 where's 1,000 I see I see what you're doing so you can see yes yes so I'll tell you what went through my mint through my brain my first knee-jerk reaction was to put 1000 was to put 1,000 like right over here that's that's what I was tempted to do and then my brain kicked in my highly analytical mind yeah she's you know there there is a correct answer to this problem we can think right because oh yeah yeah where where is 1000 on the number line related to a million well a million divided by a thousand is a thousand thousands right so it's not there I was gonna draw it like a tenth of the way no a thousand a thousand is like like a dare like alt like you know you would you barely notice a different I didn't even see the difference yes I wasn't let me let me so so this this is fascinating what what is what is this about like what why did I why did I do that yeah why why do we think of a thousands as being much closer to a million than it is and we do this actually all the time we're not so used to having to think about the difference between a thousand and a million but when we're thinking about the difference between 1 and 2 or the difference between 2 and 3 or 1 and 10 um we we think you know 1 & 2 there's a big difference there two is like twice what yes it's twice 1 right and the difference between 9 and 10 is the same distance when you're looking at it at the usual scale it's 1 right right but when we're thinking about real-life things well the difference between 9 and 10 isn't so big in any real-life situation no but but the different 1 & 2 is huge I realize it's double yes right so now we have to think on a logarithmic scale is what it is oh yes the old logarithmic scale so what you're saying is that we we as humans even though everything we're taught is these linear scales where we want to say you know this is 1 and that maybe this is 10 and that this is 20 even though that's where we're taught and that's where most of our mathematics we plot lines and that's how we draw it out on paper usually yes um but that doesn't make sense usually for for how we think about things because the difference between 5 gazillion and 5 gazillion and 10 is is nothing nothing nothing between 1 and 10 is huge right right right so that's why almost the multiple matters more than the the kind of apps of the distance between the numbers yeah absolutely and that's what the logarithmic scale captures it is and that's why we see the logarithmic scale and so many things in real life as a mathematician on the piano well we see it on it's actually the logarithmic scale so let's get our piano picture oh look at that there's a piano yeah okay okay there you go yes right let's see if I can figure this out right so here we have this C let's call it middle C and here we have this D and you know there's a certain distance between these and then here's this C in this day and right when we're listening to these notes we think all right they're OneNote apart this is like the same distance here between here and here and here and there but if you look at the actual frequencies the distances are not the same it's it's there's maybe a bad example because I don't know the frequency of day but no well we could well I'll give you an example I can give numbers too which is maybe the difference between this octave and the difference between this octave right if this is C called X where the frequency is right yes like a 440 killer I know for a is for four days well we call I like say 300 yes all right so this is three hundred or 300 X or just X then this frequency would be 600 600 right it doubles it doubles when you when you go up an octave and this would be 1,200 up here this C um alright we're we're in a weird scale here but alright so the difference between here is 300 and the difference between here it's 600 but you know when we're listening to octaves we feel like the difference between this octave shouldn't be half as much as a difference between these two notes right the distance from an octave should be an octave right right so our sense of pitch is fundamentally the way we perceive pitch is logarithmic it's fundamentally logarithmic if you want all these what if you want to have all your notes on the piano be right next to each other instead of having a piano where you have like one key for C here and one key for C here and the next C is going to be like twice as far yeah and the next and the piano would have to be like so if piano manufacturers made it so they day and they innately made it based on a logarithmic scale whether they knew it or not they could have made it on a linear scale and then the keys would just get fatter and fatter as we went to the right oh yeah fatter keys someone should make that a linear scale piano yes that would be awesome and that's but that's not how we do it it might be hard to play to play that listen and it's not it's not and it's just not pitch it would also even be how we perceive a kind of magnitude of the frequencies because we have the decibel scale which is a logarithmic scale yeah yeah there's a lot of natural intuitive logarithmic scale so when we're looking at how loud something is as also alright the difference between you know how I'm talking now and how I'm talking if I'm a little louder and that's going to be we feel like distances between loudness also are right we perceive it a lot it is harder to leave it there I don't know and we don't wanna bother people like it's getting ice cream yes yes very great yeah right know that that's fascinated us little game here that I mean this this is I'm gonna start doing this at the next party go to you it's good and it makes sense when we're looking at things a difference between how much a million dollars and ten million dollars yes just the world follows these kind of rules right right right very cool yeah awesome