Reinventing the magic log wheel: How was this missed for 400 years?

[Music] welcome to another mythology video last october a very curious mathematical wheel popped up on reddit this wheel gives a previously unknown incredibly simple way to visualize and explain the magic behind an ingenious gadget that was invented 400 years ago until quite recently this gadget was the engineering counterpart to the doctor's stethoscope and during the 20th century alone about 40 million of these gadgets were produced what's this ingenious gadget can you guess and what's this mysterious reddit wheel well that's what today's video is about buckle up and prepare for some real magic okay here's the simple setup we start with a wheel and a rubber band that's labeled with the numbers from 0 to 10. the zero end of the rubber band is fixed to a wall and the 10 end is fixed to the top of the wheel now start spinning the wheel in the clockwise direction watch what happens okay so the rubber band wraps around the wheel and it does so without slipping as a consequence the numbers spread out more and more along the wheel or clear great let's keep on spinning there there okay so far we've just dealt with the integers from 1 to 10 but of course there are other numbers in between those integers in particular let's highlight the tens decimal fractions there now the inventor of the wheel chose the wheel to be of just the right size so that after one revolution the one will end up in exactly the same spot as the 10. watch okay well that was pretty cool can you see what's happening on the rubber band there's now a 0.1 where originally there was a 1 there's a 0.2 where originally there was a 2 and so on now can you see where all those zero point numbers go when we rotate the wheel a second time well obviously in the same spots as one two three four up to ten give the wheel another spin so more stretching and you end up here and so on what all this shows is that all numbers that just differ in the positions of their decimal points end up in exactly the same spot on the wheel for example seven 0.7 0.07 0.007 and so on all these numbers just differ in the position of their decimal points and all of them occupy the same spot on the wheel of course if instead of 10 our initial rubber band went up to 100 or 1000 or whatever we'd get exactly the same scale and so it's clear that this diagram also extends to all larger numbers maybe a bit unusual but definitely not hard to understand so far right again all numbers that just differ in the position of the decimal point sit on one particular point of the wheel there and there okay maybe what we've seen so far is a tiny little bit magical but where is that really potent magic that i promised you well just around the corner to begin make a copy of our strange scale let's spin the outer scale a bit have a close look at how the numbers on the two scales line up can you see anything remarkable no okay there is the two over one right at the top look again four over 2 can you see it now the ratio of the outer number to inner number is always the same 2 over 1 is 2 4 over 2 is 2 6 over 3 is 2 8 over 4 yep 1 over 5 that seems strange until you remember that the one here also stands for 10 and 10 over 5 is 2. great next nothing over 6 well it's not really nothing let's just refine our scale a little bit point two over six or in other words twelve over six that's two fourteen over seven yep yep yep back to two over one which of course also stands for twenty over ten and two hundred over one hundred and so on curious huh but is it maybe just a coincidence that all these ratios are the same what do we spin the outer disk further there three over one yeah is the same as six over two is the same as nine over three pretty cool isn't it in fact it's possible to prove that this always works spin the outer disk anywhere you like and all the matched up outer and inner numbers will be in the same ratio always amazing and not only amazing but also super useful can you see why no okay as i said green over black is the same as red over blue three over one well that's just three now solve for the red number so green times blue equals red always this means that we can use our strange wheel gadget to multiply numbers we'll do this in a second but before we do this let's refine our scale a little bit more great now let's say we want to multiply 1.3 times times what well let's multiply 1.3 by itself there both the green and the blue are 1.3 so what is 1.3 times 1.3 well the red 1.7 well actually not quite we have to move the red arrow one step back to align it with the blue 1.3 arrow at the bottom and that takes us to 1.69 so 1.3 times 1.3 equals 1.69 and of course on our strange wheel 1.3 also stands for 13 and so we also find that 13 times 13 is equal to 169 okay but 13 times 13 i can do in my head so is this actually useful yes it is let's do another more complicated example let's multiply pi by something pi that's roughly 3.14 there three point one four times what well let's go for the other mathematical superconstant e that's roughly 2.72 for this we just have to spin both wheels locked together until we see 2.72 at the bottom okay so let's navigate to 2.72 at the bottom first we spin to 2.5 okay now here's 2.6 2.7 and finally 2.72 that's 2.72 at the bottom and we can read of the product at the top about 8.54 the exact result is 8.5408 well 8.54 is good enough for most practical purposes in fact the precision of this gadget is close to three significant figures not better especially if your life depends on quickly figuring out a reasonable estimate and you forgot your computer but that's not all our gadget is also really good at dividing just quickly and easy example 2 divided by 5. okay find the two on the outside and the five on the inside now just spin the two wheels locked together until you see the one at the bottom the result is four at the top two divided by five is four what well of course not but with all these calculations you always have to remember that the true answer will be the number in front of you with the position of the decimal point shifted appropriately this mental shifting of the decimal point takes a bit of practice but is really not that hard anyway in this case the answer is clearly not 4 but 0.4 can you see what we have here a mechanical device that can easily multiply and divide hmm yep as a lot of you will have guessed by now our machine is a circular slide rule just by winding a rubber band around a wheel we've reinvented the incredible analog computer that ruled the world of engineering for 400 years fantastic the rubberband also supplies a spectacular one-glance explanation for why this gadget works an explanation that primary school kids can understand i'll keep that explanation for the killer finale of this video something to look forward to but first it's time for your history lesson [Music] are you taking notes there will be a test the circular slide rule was invented in 1622 by the mathematician william outright of cambridge slide rules are pretty rare nowadays but most of you would have seen a slide rule before in thrift shops or op shops as we australians call them or somewhere similar but of course those slide rules would have been straight something like this well maybe not quite like this more like this a slide rule that fits into your hand such a slide rule features a sliding part in the middle there slide side side okay the main attraction of this linear slide rule is the highlighted green part which on closer inspection is really just a straightened out version of our circular scale have a look [Music] and so the scale used for both devices is really exactly the same also in either device the two copies of the scale are used to multiply and divide numbers in essentially the same way well in the linear slide rule we'll slide the middle part instead of rotating one scale against the other and some pac-man wrap-around action is sometimes required when we run out of slide rule when sliding in theory the circular slide rule is the better device because it does not require any pacman action however in practice is a lot easier to make linear slide rules that are super precise and user friendly and so most slide rules that were produced were linear and hence also the name slide rule in any case from the time they were invented in 1622 these diversely ingenious analog computers were produced in the millions until the advent of the electronic computing machines but even after modern computers had taken over slide rules were and are still often used as emergency backup computers for example all apollo missions carried slide rules and these were often used during flight one slide rule even made it to the moon and even today slide rules are still produced for emergency use by aircraft pilots in fact this particular slide rule built into the e6b flight computer will be used for many hundreds of years to come as evidenced by this snapshot from the distant future if you want to play with a slide rule right now just download one of the many really nice slide rule apps i've included links to some of my favorites in the comments if you're interested in investing in a bit of a conversation piece consider buying a fancy pilot watch with built-in circular slide rule you never know this might come in handy when your fancy cast calculator fails you in the middle of a high stakes exam if you know of any good slide rule tidbits and stories of the beaten track please share them with the rest of us in the comments of course no mathologer video is complete without addressing the why okay so what's the source of the magic of the slide rule one new super simple answer also uses nothing but the new reddit rubber band setup ready to be amazed here we go what we need to explain is why no matter how we rotate the scales matching numbers always give the same ratio well to start with remove the scales except for the two and the one on top of the scales attach rubber bands now obviously the rubber band at the bottom is stretched twice as much as the one at the top this means that any of the numbers at the top is always exactly double the number just below easy right but now remember we can recreate the scales on the two wheels by simply rotating balls wheels once and since the rubber bands don't slip corresponding numbers on both rubber bands will keep their doubling alignment throughout the synchronized rotation consequently the ratio of matching numbers on the outside and the inside is always the same well up to shifting decimal points of course and obviously we can do the exactly the same thing for three ratio or pi ratio or whatever this always works and so what an amazingly simple proof absolutely made my day when i first discovered it how about you okay okay i'm sure the chloe among you are going log log log log on yes yes pre rubber band any proper discussion of slide rules would have been built around the properties of logarithms so to finish off let me also quickly show you where logarithms are hiding in our magic scales let's say the original rubber band is one unit long then a little bit of calculus shows that the wheel has a circumference equal to the natural logarithm of 10. well maybe one of you matt's demons can supply a proof in the comments all of you non-demons who don't breathe logarithms just relax and go with the flow we're almost there anyway in general the clockwise distance along the wheel between the one at the top and one of the numbers from 1 to 10 is the natural logarithm of that number for example the distance between 1 and 2 is equal to the natural logarithm of 2 and the distance between 1 and 3 is equal to the natural logarithm of 3. let's highlight this distance on the second scale okay there that blue distance is equal to the natural logarithm of three now we want to multiply two times three remember how that works just rotate the green two to the top from earlier we also know that 2 times 3 equals 6 is opposite the blue 3. yep there it is now the distance from the outer 1 to the 6 is the natural logarithm of 6 but that red distance is also green plus blue and so log 2 plus log 3 is equal to log 6 or log 2 plus log 3 is equal to log of 2 times 3. there you have it the magic of the slide rule illuminated from a different angle right the sum of the logarithms of two numbers is the logarithm of the product of the two numbers and so from the logarithmic point of view what the slide rule does is to translate a complicated operation of multiplying two numbers into the much simpler operation of adding two numbers historically the invention of the slide rule by william outright in 1622 was a direct consequence of the invention of logarithms by john napier in 1614 just a couple of years earlier there that's napier is one of his famous log tables fantastic stuff don't you agree and so powerful and at the same time so incredibly easy to understand when you look at it in just the right way okay today's coding challenge can you code a circular slide rule with infinite precision so what i'm after is a virtual slide rule in which you can zoom in on a dynamically refining scale this has not been done before i think should be fun as usual i'll put my wishlist for this app in a comment pinned to the top of the comment section below and as usual anybody who comes up with an app like this will automatically enter into a draw for one of my and marty's books ok now to finish off the wheel of logarithms that inspired this video was invented by dmitry zakolowski who is a software engineer and entrepreneur based in new york city and who in his spare time is also one of the directors of the new york math circle here then is dimitri's original post on reddit as well as the wheel the way he presented it and that's all for today [Music] [Applause] [Music] uh [Music] [Applause] [Music] [Music] [Music] um [Applause] [Music]

Channel: Mathologer

Views: 200,372

Keywords:

Id: ZIQQvxSXLhI

Length: 19min 5sec (1145 seconds)

Published: Sat Apr 02 2022

Reddit Comments

[u/final_cut]

My boss has one she uses sometimes if I’m thinking of the right thing. Little cardboard wheel chart.

[u/rcreveli]

Not much anymore but, when I started out we had one at each analog copier, the press camera & in the prepress area.

[u/scottdave]

We have a few in the shop, still. I use it at least once a week. It's usually quicker than typing into a calculator.