Seven Dimensions

[Music] this is the si system of units it contains seven base units each defined using a physical constant and any other unit you care about can be expressed as a product of powers of these base units in exactly one way the basic quantities of the si are time length mass current temperature luminous intensity and amount of substance this makes sense because those quantities are all reasonably fundamental and we measure them a lot however it's not the only approach you can take for instance there was this joke proposal made by jan measley a couple of years ago which i highly recommend you watch where everything is derived not from time length and mass but from frequency velocity and energy the joke being that all the base units can be represented by some variation on the letter c slightly more seriously there are systems of natural units such as the planck units the base quantities of the planck units are quite complicated to express in terms of si and it's not immediately obvious how you would untangle these to derive units of length mass and time there is a way to solve this not just for the planck units but in general for systems built on any base quantities you like but to get there we have to think about units in a completely different way very often in mathematics when we have situations involving several different changing values we find it helpful to think of these values as the coordinates of some kind of abstract space for example you've probably seen this puzzle before you have two jars that can hold three and five liters of water and your job is to get exactly four liters of water into the larger jar but both containers are unmarked so there are constraints on how exactly you're allowed to move water if you treat the amounts of water in the two jars as coordinates on a grid each square represents one possible state that the system can be in and solving the puzzle involves thinking about how you can move around this diagram within the constraints of the problem in terms of what action each movement corresponds to or we could think about a pendulum if we want to describe the state of this pendulum the only things that matter are the angle it makes of the vertical and the speed at which that angle is changing this means that we can represent every possible state of this system in a coordinate plane and then consider how a point moves around this plane for instance we might recognize that if there's no air resistance our point will move around an ellipse centered on the origin this isn't the real ellipse of course the pendulum itself is moving in a circular arc it's just an abstract construction that we're using to help make sense of what's going on and with this we come onto units specifically i care about coherent systems of units which refers to any system where we start with a handful of base units and just multiply and divide them together to derive everything else for example let's take a quantity that's complicated to express in terms of si base units like capacitance the si unit for capacitance is the farad which is a shorthand for this which really is a shorthand for this we don't normally write out the units that have powers of zero but it's still helpful to treat them as though they're there in fact to be precise about things let's define this as a fixed format anytime we want to write anything in s i base units we write them in this exact order seconds meters kilograms amps kelvin candela's moles and we include the ones that have powers of zero if we always do this then the units themselves don't really convey that much information they're just labels that always stay the same they're defined as part of the format so really this is just an ordered list of seven numbers and then maybe we make the arbitrary decision to instead of writing these numbers left to right put them top to bottom and would you look at that we have a column vector in other words what we've done is define a seven-dimensional abstract vector space that contains a vector representing every possible quantity you could ever derive within the s i mass has a vector thermal conductivity has a vector the 99th derivative of position has a vector and each one is completely unique so why why would you want to do this i mean with the examples i gave earlier those abstract spaces were useful because we were only dealing with two numbers so we could easily visualize things on a plane but you can't visualize a seven-dimensional vector space so what exactly is this representation good for the answer is that a lot of the mathematics behind using systems based on different units and converting between them becomes much more elegant when we can start doing linear algebra to it so in order for linear algebra to make any sense at all in this context we have to start with a few basic definitions about our system namely we have to define what each of the basis vectors mean in this case those basis vectors represent the seven base quantities that are used in the s i in this specific order what this means is that for instance the vector with a one in its first entry and zeros everywhere else means time the vector with a one only in its second entry means length and so on the next thing we need is to make sure that the basic operations of linear algebra vector addition and scalar multiplication are meaningful well since our numbers represent exponents vector addition looks like multiplying quantities for instance the mass vector plus the acceleration vector produces the force vector scalar multiplication meanwhile looks like raising something to a power like how the area vector is 2 times the length vector because area is length squared with those basic definitions out of the way we can check that the axioms of linear algebra apply these axioms are a set of 8 rules that our system must follow if we want anything we do to make any sense i won't go through them all but they're on the screen if you want to verify that they do apply to the situation we're working with so we now have everything we need to start working with this system using all the tools that linear algebra has to offer of course the most important of these tools is matrices so a natural question to ask would be what do matrices mean here what operation do they correspond to before we get to that though i should point something out the basis vectors we're working with represent quantities not units our first basis vector means time not one second this means that a derived quantity like velocity always looks like length divided by time regardless of whether we're expressing it in meters per second or miles per hour or furlongs per fortnight the vector for velocity is always the same in these cases because we always derive velocity from s i base units as length divided by time and those base units define the basis vectors of our system so what happens if we try using a different set of basis vectors what happens if we put in say the ridiculous seven seas units from earlier instead of time length mass and current we derive things from frequency velocity energy and charge this is a perfectly valid system of measurement and we can use it to write anything that can also be written in si it'll look different from what we're used to of course but this for instance is a completely acceptable way to write mass provided we've defined our basis vectors in a way that makes it acceptable what we'd like is a good way of converting between this system and the si system that we're familiar with in the case of the seven seats based quantities are straightforward enough that you could just work out the algebra by hand without too much difficulty but that's no fun and we'd like a method that works for any crazy system you might want to try this is where the matrices come in converting between two systems with different basis vectors is called appropriately enough a change of basis and it's an operation that we can describe with a square matrix each row of this matrix represents one of our si based quantities and each column represents one of our new base quantities written in terms of si the first column is the si vector for frequency the second is the si vector for velocity and so on this matrix will allow us to convert any 7cs vector into an si vector for instance if we put in the mass vector from earlier we get the vector 0 0 1 0 0 0 0 which does indeed represent mass and s i or put another way this is stating that energy divided by velocity squared equals mass if it's not clear why this should work you can think of this matrix vector multiplication as writing mass in terms of the seven c's substituting each base quantity with its s i equivalent and then simplifying the result and of course there's nothing special about the seven cs here this works for any system built on abnormal base units this is all lovely but converting out of a different system is the easy part the hard part is converting into it so we'd like to know if we can go the other way as a matter of fact we can we just invert the matrix this would be a horrible experience if we had to do it by hand but fortunately we have computers that will happily do it for us so let's try it out we can plug in some si vector say capacitance multiply it by our inverted conversion matrix and we find the capacitance equals charge squared divided by energy and you can verify that this is indeed a valid representation of capacitance so brilliant we now have a tool that can take in any coherent system of units and provide a way to convert to and from si so far this might seem sort of useless since the 7c system is openly meant as a joke so let's look at a slightly more practical use case the planck units are one of many examples of so-called natural units they're not meant for measuring things at a human scale rather they're designed to make the maths easier in certain areas of physics you start by picking some number of important physical constants and you define them all to have a value of one when expressed in the system then you derive everything else from those this turns out to be quite useful if for instance you want to describe some object moving at one-fifth the speed of light it's a lot easier if you can just refer to its speed as point two one thing you might notice is that there are seven si base units but only five planck base units there are no planck units involving luminous intensity or amount of substance no equivalent to the candela or the mole so we can't derive them this isn't really a problem because they don't come up much in situations where the planck units are helpful but it does mean we're essentially working in a five-dimensional space instead of a seven-dimensional one in other words we can only derive quantities whose s.i vectors would end zero zero which means we might as well not bother writing the last two rows anyway if i told you to look at this table and tell me how to derive a unit of say length well it's not clear where you would even start most of the units on these base quantities are not simple and actually running through all the algebra just seems like it will be a nightmare it's not even clear how you could get a computer to go through the process of untangling these if we use our conversion matrix however everything becomes much neater each column of this is the si vector representing one of our base quantities inverting this matrix which is a well-defined procedure so we can get a computer to do it produces a tool for expressing si quantities in terms of blank units and it looks like this okay so it's not pretty and most of the entries are no longer integers but all that means is that lots of our units are going to have square roots in them this might feel like an odd concept since it never happens using the si but there's nothing mathematically wrong with it if you imagine inventing a system which included area as a base quantity for example then your unit of length would obviously be derived as the square root of area this is just a more complicated version of that anyway if we want to use this to find for instance the planck unit of time which is referred to imaginatively as the planck time we just multiply this matrix by our time vector actually we don't technically need to do any calculations for this multiplying by any basis vector just pulls out one of the columns so the planck time is by definition represented by the first column of this matrix the planck length is represented by the second column and so on now if we wanted to find something more complicated like our old friend capacitance we can again multiply it by this conversion matrix and we find that the units of the plant capacitance are surprisingly nice i don't know if there's a situation where you'd ever need to use the plant capacitance but you can now here's the thing this is a very effective tool and to me at least watching it in action feels a bit like magic but actually putting together all these matrices and inverting them and doing the multiplication is a bit time consuming so i've put together a spreadsheet that does it automatically you can make a system with any base quantities you like with units of any size you like and the spreadsheet will do all the matrix calculation for you it will tell you the dimensions of all of your derived units and their size in terms of si there's a link to this spreadsheet in the description so you can make a copy and play around if you want there is one more thing i want to talk about regarding unit systems as vector spaces and it involves something i mentioned earlier called coherence earlier i defined a coherent system as one where you start with some base units and just multiply and divide them to derive everything else this definition works okay but i'm going to change it to something slightly more specific you might have heard people use coherence to refer to a system where you don't need to do any unit conversions so for instance a system that measures length in meters and time in seconds and speed in miles per hour is not coherent because dividing your unit of length by your unit of time does not produce your unit of speed you have to do extra maths to get the unit you want this is a better definition but it's still not perfect consider a system that looks like this where we're defining base units for time length mass and energy this is a problem because energy can already be derived from time length and mass or if you want to be symmetric about it you can say that time can be derived from length mass and energy or something like that the point is that if you have three of these base units you can always derive the other one in the language of linear algebra we would say that these basis vectors are not linearly independent this means that energy or indeed any unit in the system can be represented by infinitely many different vectors all of them equally valid so if you're converting into this system how do you know which one to use the answer is you don't if you write out the matrix to convert this into s i its determinant is zero this means that you can't take an inverse so there is no matrix to convert from s i into the time length mass energy system in other words we can define a coherent system as one where every possible quantity either can't be expressed or can be expressed in exactly one way by multiplying base units together and a way to test whether a system is coherent is to take the determinant of the matrix that converts from it into s i and check whether it's equal to zero i think this is an interesting exercise not because it's exceptionally useful in itself i mean no one is ever going to use say the 7c system for anything practical but because it demonstrates the power of thinking of one construct in terms of a different one when we stripped away all the labels that come with the quantity in the si we saw it for what it was an ordered list of numbers and that led naturally into talking about vectors and vector spaces which opened up the whole world of linear algebra and all the valuable tools that come with it if you've been conditioned to think of vectors as nothing more than arrows in space it might seem like a strange step to take but vectors like a lot of things in maths are not fixed with one meaning they are whatever they need to be to be useful for the problem at hand so yes out of context saying that mass is a vector is nonsensical nothing is a vector but vectors can represent anything we like so long as we find it helpful

Channel: Kieran Borovac

Views: 775,393

Keywords: fish chair

Id: bI-FS7aZJpY

Length: 14min 41sec (881 seconds)

Published: Mon Aug 15 2022