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Miles per gallon being equivalent to length^-2 is so amusing! Dimensional analysis can also help catch mistakes when solving diff eq's; if you ever get sin, log, exp, etc. with an argument that isn't unitless, then you know you've made an error.
Conversion factors that need context to create, don't equal one. They instead represent whatever the context is. When you turn context into a conversion factor, you can multiply the situation by it to get the result. For example, 3dollar/10apple (I use incorrect grammar with no spaces when dealing with units as variables) is the context in the form of a conversion factor, and 5apple is the situation, then the equation written is "(3dollar/10apple)5apple" is equal to the cost of 5 apples if 10 apples cost 3 dollars. The "if 10 apples cost 3 dollars" is part of the equation, not context for the equation. And in the students, minutes, and problems problem, the conversion factor "problem/minute/student" (a "1" is not needed since we're treating them as variables) should've been said, and again, it doesn't equal one, it equals the context (which is the student's ability to solve these specific problems).
This video is legitimately SO GOOD. It is one of the best videos I've seen, and I've seen thousands of math videos. It seems few are talking about units despite how incredibly powerful and useful they are to any scientific or engineering field. Every high school student needs to see this, and I'd be willing to bet a great portion of university students would benefit from seeing this as well. Phenomenal job.
360 probably stuck because it is a practical number, i.e. every number from 1 to 360 can be made by adding its factors, and no factor is repeated. 360's prime factorization is 2³ • 3² • 5, so we have 1, 2, 3, 4, 5, 6, 8, 9, 10, 12, 15, 18, 20, 24, 30, 36, 40, 45, 60, 72, 90, 120, 180 and 360. Any number that is greater that 180 can made with 180 + leftover. We've split the size of numbers we have to prove can be made in half. We can do that again with 90 and 45, which is what you get when you remove the 2^n from the prime factorization. Here are all the numbers from 1 to 45: 1 = 1 2 = 2 3 = 3 4 = 4 5 = 5 6 = 6 7 = 6 + 1 8 = 8 9 = 9 10 = 10 11 = 10 + 1 12 = 12 13 = 12 + 1 14 = 12 + 2 15 = 15 16 = 15 + 1 17 = 15 + 2 18 = 18 19 = 18 + 1 20 = 20 21 = 20 + 1 = 9 + 10 22 = 20 + 2 23 = 20 + 3 24 = 24 25 = 24 + 1 26 = 24 + 2 27 = 24 + 3 28 = 24 + 4 29 = 24 + 5 30 = 30 31 = 30 + 1 32 = 30 + 2 33 = 30 + 3 34 = 30 + 4 35 = 30 + 5 36 = 36 37 = 36 + 1 38 = 36 + 2 39 = 36 + 3 40 = 40 41 = 40 + 1 42 = 40 + 2 43 = 40 + 3 44 = 40 + 4 45 = 45
Excellent video! This really fills in the gaps of knowledge I have from middle school/early high school also, If you change the timestamp of the intro from 0:05 to 0:00 you will have chapters instead of key moments.
But really, that's just a way to represent two different data points under one set with its own term. I know you know this because you used sort of, but I still don't believe you can say they're added.
- We can add commensurable units - We can multiply units by unitless value - We can multiply and divide any units - Ce can exponentiate units by unitless values But could we take a unit to the power of another? Like seconds^meters? And, how could we intuitively represent ourselves units to the power of unitless values? Sure, with units of lengh, it's just dimentions as we naturally think about it. But multiple dimensions of time? (sec²). Furthermore, how could we represent ourselves meters^½? These are the strange questions I've always wondered even before the video 😅. (Tho for meters^1.585 for example, we could think of it as units of "fractal aeras" using Sierpiński's triangles instead of squares.)
There are three problems I see with this kind of approach (in general): - All of the 'commensurate' units are actually the same as the contextually convertible units in that we do have hidden assumptions underlying them based on the model we use (that model is the context). - Many formulas stated in a mathematical context will have been simplified, already cancelling out units that may have been important, you need to understand where the formula comes from in order to insist that any term in it has any kind of particular dimension. - Some units are not linearly composable when converting to other units: take decibels for example which exist on an exponential scale / logarithmic scale depending on how you want to convert them to/from SI units, or farenheit vs celsius which need an affine transformation (not a linear transformation) to convert them.
Really great points here, especially the last one! I wanted to keep this introduction as accessible as possible to a general public, so some omissions were intentional, but I do regret that I didn't cover that last point more.
15:25 I don't get why everyone agrees that radians are the only natural choice for angle measurement. If I turn around such that I face the same way, I wouldn't say that I turned 2π radians or 360°, I would say that I made 1 full rotation. Why isn't there a unit that is 2π larger than a radian? Revolution is 1, half revolution is 1/2, third revolution is 1/3. Seems much more logical. And as for formulas, we can just accept that these formulas require arc length and not angle, and a conversion unit between them is 2π rev^-1 because arc length of 1rev is 2π (aka circumference of a unit circle). And radians are totally not unitless, the same way Joule and newton-meter are different units. Joule is a scalar and newton-meter is a bivector, because torque needs a plane to rotate in. The bivectorness of newton-meter comes from the radian, if you divide it by the angle that torque caused (in radians) you'll get work in Joules. Rotation of 1rad clockwise around z axis is a different thing than rotation of 1rad clockwise around x axis.
As I mentioned in the video, dimensional analysis can be very arbitrary. It just depends on the context you're working in, as long as it is well-defined, consistent, and meaningful. It would be unrealistic to go over every possible interpretation of angle measurement and their consequences in a video, nor would it be very productive. But what you're doing by being flexible and creative in how we think about units, challenging accepted norms, is, in my opinion, a healthy mindset to have.
I think it's better to think about the unit of permutation being choices^n rather than the object^n. Permitting 3 people isn't 3 people x 2 people x 1 person, it's 3 choices of people x 2 choices of people x 1 choice of people. Choices^3 clearly shows that the abstract object being quantified is the result of 3 success choices. That way if you chose a permutation of 3 people and then chose a card, the unit would be choices^4 rather than some bizarre people^3•cards. It may also be useful to specify the number of options, eg. Rolling 3 dice and then flipping a coin would be (6-choice)^3(2-choice), and the measurement could be equated with other phenomena like colouring a random vertex of a hexagon either black or white. This approach does mean that permutations wouldnt reduce though, (3-choice-of person)! Would be 6 (3-choice)(2-choice)(1-choice). Maybe we can just factorialise the unit so the result is 6 (3-choice)!
The "product of choices" you speak of is generally considered the purview of combinatorics. Combinatorics is concerned with the cardinality/"size" of sets, and as such tend to treat the values as dimensionless. That said, there is still a sensible way to interpret the units "people^3" or "people^3•cards". Notice what a set of "permutations of three people" actually contains; each member of the set is actually a list, or tuple, of three people. (In fact the "handshake" / people^2 mentioned in this video can be considered a tuple of two people, a.k.a a pair of people.) Tuples can also contain unlike items, so you can have a tuple of three people and a card, hence "people^3•card" Coming back to combinatorics however, the reason cardinalities are generally treated as dimensionless is because you *can* add units together that would normally be incommensurable, as sets could be heterogenous (containing unlike items), you *can* put people, apples, and "(dog, tree) tuples" in the same set if you wanted or needed, for example.
awesome video, just a small add no 360 degress is not quite that useless, the reason why it was used is because 360 has the most primefactors which means it can be divided into different parts very easily
An observation: Despite "degree" and "radian" both being dimensionless, it is not okay to add these. Then it is better to keep the unit-name, as being a higher level of (more nuanced?) "truth" than just the dimension .Then again, "degree per radian" as a conversion factor makes complete sense - I have pi radian - multiply by {360 deree per 2*pi radian --> boom --> 180 degree. I know that this is completely obvious to _you_ - but spelling it out literally may be good for some of your viewers :) How about this other story: A certain forest is visited by people taking their dogs for walks. Each dog, compulsively, _must_ pee on every tree. As there are twelve dogs visiting on a certain day, and twenty trees, there is 240 dog.tree worth of pee deposited in the forest that day. As one dog.tree is equivalent to 1e-4 m3, let's say that's about 2.4e-2 m^3 of pee that day. 24 liter. Nice.
Units of angle are dimensionless, but not scaleless; same dimensional coefficients, different units. It's comparable to nautical miles versus meters (both of which are based on navigation angles originally: 60×360 nautical miles ≈ 40 megameters ≈ length of 1 equator). For math reasons we like to use radians as the base unit, but it's also pretty intuitive to use turns.
@@kamikeserpentail3778 Depends on how you think about it. From a notational perspective, yes, but from a quantity perspective, no. a = b => a + c = b + c, 45° = π/4 => 45° + π = π/4 + π (= 5π/4).
If there are budding game devs in here, here's why we need delta time, what's meant by it, and why you shouldn't haphazardly multiply everything with it. The game's visual state is refreshed in succession of independent still image "frames", and the rate of this update is denoted with a unit of FPS or "frames per second". For example a value of 60 FPS means there are 60 frames per second which equals the update frequency of 60 Hz. Yes, the derived units of FPS and Hz are essentially equivalent because 1 second is the base unit used in both of them. From there, to gather a duration of a single frame, you take a reciprocal, which is 1 second / 60 Hz = 16.66.. milliseconds, a result that represents the ideal delta time between two frames. However because we don't live in an ideal world, the hardware varies as well as the computation, so the delta time is measured and may oscillate around the ideal value. This measured delta time value is supposed to be used somehow if we want, for example, to produce smooth motion over some arbitrary time duration. Typically we define the speed in standard derived units, such as 'meters per second', however video games live in a virtual space so it makes more sense to define the speed in 'game space units per second' or 'world units per second'. Let's call this unit of length "w", and let's denote 'frames' with "F". FPS = 60 F/s Tdi = 1/FPS = 0.0166.. s (an ideal delta time) Speed = 10 w/s (a desired speed of our object) Td = 0.0168 s (a measured delta time at some point in time) It is fairly easy to intuitively find how much the object should move over just one frame, just by multiplying the speed with the delta time. Speed * Td = 10 w/s * 0.0168 s = 0.168 w But why this works? Observe that when we derived the two delta times we've lost some units in the conversion. The FPS reciprocal is in fact X [s] / Y [F/s] = X/Y [s/F] but people usually ignore the divisor because it is always 1 by definition, and erroneously dismiss the 'frames' unit as well. This is why many people miss to appreciate the actual units involved in this conversion. This is what truly happens when we multiply the two Speed * Td = 10 w/s * 0.0168 s/F = 0.168 w */F* So when you multiply the two you basically convert the Speed expressed in w/s to a more meaningful w/F because the goal is to apply it in each singular frame of the animation. This means that after approximately 60 frames (which should last for exactly 1 second), your object should move by 10 world units, if it would move 0.166.. w/F *on average*. Now from the classical V = s/t we can check V out as s * t = V, and also switch the continuous measurement of time with discrete frames. 0.166.. w/F * 60 F/s = 10 w/s You can see that this hold up even when delta time varies, making the parameters change on the fly. 0.168 w/F * 59.52.. F/s = 10 w/s So this is why this technique is used to maintain the original speed set in w/s, otherwise the speeds would be scaled differently on different systems and under different performance profiles, which is not how real time video games are supposed to work. Also this is why you shouldn't use this approach when computing absolute positions with other math tools such as i.e. linear interpolation. It simply doesn't work the same way and doesn't involve incremental steps that scale with measured delta times. Another takeaway from this example is that the measured delta time always lags by exactly 1 frame, so keep in mind that this kind of math is never perfect, but usually good enough when the actual FPS is high enough or stable enough!
THE SEARCH IS FINALLY OVER I've been looking for this exact thing for the past 3 years! This...Work Of Art has answered so many of my questions, and provided a pathway to answer so many more. Eternally grateful.
I just got goosebumps! I was thinking about *_units_* as *_variables_* this evening and found out it makes sense! But I didn't search for any vids about this online but exactly 3hrs later this vid popped up on my feed! Now I understand it better 😊
I love this video. It is not a topic I have seen covered like this before and it is something many people do not understand. You definitely did it justice here, I've added it to my playlist of best maths videos.
I came here, knowing about the Tree Function. (Like Tree(3) which is an absurdly large number). Thinking, that Dog is smth Similar. Like Dog(5) would be a huge Number aswell for a different reason. I waited trough the whole video for the big reveal, just to realize that you were talking about a literal Tree and a literal dog... Still a great video and kinda shifted my mind on how I think about units
Unit and dimension simply shouldn't be interchangeable terms, which is the origin of the complication you mentioned, and doing so missed an opportunity to make these ideas far clearer. Radians and degrees are dimensionless units, but both refer to the same quantity of "angle". They are not "unitless" precisely because they are different units themselves, references for measuring angles: one is 1/360 of a turn, the other 1/2pi of a turn. A unit is simply a reference value for a quantity. A quantity may or may not have dimensions, but regardless, it is still a conceptual property associated to a plain number, like "angle".
Quantity and quality are interchangeable. Dimension(quality:dog,tree ...) and curvature(quantity: 1 2 3 ..) may interchange with each other. take some linear(dimensionles) operator, for example, derivative: D(f*g)=1*D(f)*g + 1*f*D(g). It is represented by 2-convolution of binom '1 1" row_0= 1 row_1= 1 1 row_2= 1 2 1 row_3= 1 3 3 1 row_4= 1 4 6 4 1 row_5= 1 5 10 10 5 1 ... Each row is face-vector of simplex. If to replace binom "1 1" by "1 2" then we can represent derivative by cube: row_0= 1 row_1= 1 2 row_2= 1 4 4 row_3= 1 6 12 8 row_4= 1 8 24 32 16 row_5= 1 10 40 80 80 32 ... Each row is face-vector of cube. operator look like this: D(f*g)=1*D(f)*g + 2*f*D(g). Second derivative is DD(f*g)=1*(DD(f)*g+2*D(f)*D(g))+ 2*(D(f)*D(g)+2*f*DD(g))= 1*DD(f)*g+4*(D(f)*D(g)+4*f*DD(g)) wich is row_2 of above Pascal Matrix. If to replace binom "1 1" by 3-nom "1 1 1" ( sequence "1 1 1 1 1 . . ." , of course ,is a face-vector of sphere) and 2-convolution by 3-convolutoin then we can represent derivative by simplex3 wich appears only in odd dimensions: row_0= 1 =1 row_1= 1 1 1 =3 row_2= 1 2 3 2 1 =9 row_3= 1 3 6 7 6 3 1 =27 row_4= 1 4 10 16 19 16 10 4 1 =81 row_5= 1 5 15 30 45 51 45 30 15 5 1 =243 ... Each row is face-vector of simplex3. D(f*g*h)=1*f*D(g*h) + 1*g*D(f*h)+ 1*h*D(f*g) . Of course, D(m*n)=a*D(m)*n + b*m*D(n). a and b depends on wich nested shape we choose (see above). Unlike previose, this derivative operator is nested(dimensional). Why?, because of quantity-quality inversion happens in even dimensions. Instead of Simplex3 to be curved into shape, dimension itself is curved into operator wich becames nested(dimensional).
Good video, have been teaching this to my Kids for ages and wish schools would do that since it makes understanding and dealing with units so much easier. Dealing with percent as a unit also makes that that much easier to understand. A former customer still insisted it was a bug to display 700(%^2) instead of 7% ;) Also fun fact 0m = 0s = 0
I've always loved dimensional analysis. My favorite application of it is in the use of values in (non-polynomial) analytic functions like exp or sin or cos or ln. Whenever you have one of these kinds of functions in an expression, the argument must be dimensionless. I don't know if there's an ultimate answer as to "why," but I prefer the argument that, since you can't add apples to apples^2 and have it make sense, the power series demands the arguments be dimensionless, as you briefly addressed at 15:49. Of course there are some "generalized functions" that don't follow this rule, like the Dirac delta, but even those functions show where their dimension comes from in their limit definitions
You mention assigning units to traditionally dimensionless quantities like radians, but I think it’s more common in physics to do the reverse. Physicists often treat the speed of light as exactly 1, dimensionless; length and time are the same thing in this analysis. 1 second + 1 meter is 299,792,459 m ≈ 1.000000003 s. (Honestly I do a bit of both; I feel like the mol should in most cases be dimensionless but radians should be a separate unit)
Dog x tree is simply a unit of square energy: convert dog and tree by dividing by average mass of dog and tree respectively then convert mass into energy by multiplying by c^4
The best explanation I have heard for degrees is that it stems from the compass construction of an equilateral triangle, and 60 being the base of the Babylonians made it a natural choice to denote it.
Tree - Carbon Sequestration: Trees absorb and store carbon dioxide from the atmosphere, a process known as carbon sequestration. This can be quantified as a rate, such as kilograms of CO2 sequestered per year. Dog - Carbon Footprint: The environmental impact of keeping a dog can be partly quantified by its carbon footprint, which includes the resources used for its food, care, etc. This can also be expressed as a rate, such as kilograms of CO2 equivalent produced per year. Multiplying the Attributes: Combining the Rates: By multiplying the carbon sequestration rate of a tree by the carbon footprint of a dog, we can create a unit that represents the balance between carbon sequestration and carbon production. Example Calculation: If one tree sequesters 20 kg of CO2 per year, and one dog produces a carbon footprint equivalent to 100 kg of CO2 per year, multiplying these gives 20 kg CO2/year/tree×100 kg CO2/year/dog20kg CO2/year/tree×100kg CO2/year/dog. The result is 2000 kg2CO2/year2/tree-dog2000kg2CO2/year2/tree-dog With that we can finally answer the question that we all want to know, How many trees need to be planted to offset the carbon footprint of owning a dog?
Excellent, you've found a meaningful conversion and interpretation for dog and tree units that I have not yet seen in the comments yet! However, after you convert both units to the amount of carbon dioxide, this is not actually dog * tree but dog + tree. Multiplying units means that every pair of a dog and a tree needs to describe something, whereas here, the dog and the tree each represent a quantity that is just combined to a total. That being said, dog + tree is equally amusing, and you've given me an example to highlight and distinguish!
Dogs are well know for what they do when they visit trees. So if you have 3 dogs and 5 trees your going to get 15 pees. (Not to be considered with 15p, a monetary amount in British currency)
This video gave me an idea about programming(I don't know if this can be implemented IRL). Basically in your code you create separate numerical type for each base unit of measure (like one for combinations, another apples, third for yens, etc). And this is done in such way that you are forbidden from adding or subtracting values between different numerical types, yet you can divide and multiply them (you can't add apples to oranges, but you can multiply apples by oranges)
This can in fact be done! Create an abstract class for the unit, then use an object/dictionary property to track each unique unit factor. The name of the unit factor is the key, the exponent is the assigned value. You can then manually code the logic of addition and multiplication, where addition applies a check beforehand for validity, and multiplication simply sums the object/dictionary by key to obtain the new derived unit.
15:58 hot take, turns are the least arbitrary unit of rotation. 1 as a full rotation amount is just objectively less arbitrary than 2pi. and in computer math libraries, often a factor of pi or 2pi is divided out first, introducing extra floating point error.
Yes, turns are also a fairly natural unit for angles/rotation. As mentioned in the video, the unit definitions can be pretty arbitrary. You can justify whatever definitions you want as long as it is coherent, consistent, and meaningful!
I don’t think I agree, on account of: (1): when you differentiate trig functions, having the trig function take a number of turns rather than a number of radians, you end up getting a factor of 2pi, instead of 1. (2): Using turns, complicated the relationship with the e^x function. e^{i x} = cos(x) + i sine(x) when cosine and sine are considered to take their inputs as in radians, but if they take their inputs in turns, then you would have to say e^{i x} = cos(x/2pi) + i sine(x/2pi) .. or, I suppose instead you could write e^{2 pi i x} = cos(x) + i sine(x) ? Still, I don’t think this is natural. If we are solving some differential equation, it is more likely that we will get stuff relating to e^{i k x} than to e^{i 2pi k x} . ... “natural” shouldn’t mean the same thing as “most intuitive seeming when considering for the first time”. It is more like, “the convention that one would eventually find that the math ‘wants’ one to set”. I feel like using cosine(x) = (1/2) * ( exp(2pi i x) + exp(- 2pi i x) ) as the standard , instead of the version without the 2pi, is kind of like, ignoring what the math is telling us about what is natural? ... unless, perhaps, if it might be more natural to use (2pi i) as the base thing to use instead of i? But I don’t think that’s right. The complex numbers being the algebraic completion of the real numbers is important, and this connects much more closely to them being the real numbers adjoin a square root of -1, then the real numbers adjoin a square root of - (2pi)^2 . Like... I don’t think adjoining 2pi i onto the integers, or rationals, gives nearly as natural of an object than adjoining i does. So, I think i is more fundamental than 2pi i. (Though perhaps 2pi i is more fundamental than 2pi ?)
@@drdca8263 yea maybe in pure math it makes more sense. i’m thinking of it in part from the perspective of using it in practice, math libraries that take in radians afaik typically remove a factor of pi or 2 pi first, and in that case, using turns makes it easier for the user and a little faster since it doesn’t need to remove pi or 2pi from the input first
When you said "turns" I thought you meant 'right-angle turns' because when I 'perform a turn' I am generally turning through a right angle (e.g. navigating in a grid), I would use the term 'revolution' or 'rev' for short to refer to the full circumference. BTW right-angle turns (sometimes called eta, like pi is a half the circumference and tau is the full circumference) are also a nice unit.
I wouldn't say you are directly multiplying students with minutes to get the number of problems solved. I'd say you are getting the number of student-minutes, and then using a meaningful conversion factor: one student can solve 1 problem per minute, which converts to a ratio of 1 problem per minute per student. From there you can get x student minutes * (1 problem / 1 minute) / 1 student = xsm * (p/m) * (1/s) = xp or x problems.
I guess we could look at digits (or natural numbers, if you want) themselves as commensurable units. Like, "one `pair` is two `units`", e.g. the dimensionless one is all we need: it equals to `(one plus one) divided by (two)`. Thus, when we use any number notation (numerals, I guess?) we effectively do _measure numbers_ using this unit system.
At 13:10 I have a problem with this part about using dimensional analysis as a sanity check: The proposed (misremembered) formula could indeed be correct, even if the dimensions *by naive substitution of the terms* don't align. Let's suppose we're operating in a different space to the usual euclidean plane, let's say such that the area of a rectangle is not a*b but something arbitrary (we could insist this still forms a related measure space and metric space etc so we still have a notion of lengths that subtend an area). Can you tell me that it's impossible for the incorrect formula to hold in such a space? (not necessarily this particular formula, but any formula that's 'wrong' by this kind of argument). If it's not impossible, then how can this method be obviously applicable? You effectively have to do as much work figuring out if you can apply the sanity check as actually solving the problem. In general you can't just substitute units into the terms without knowing the whole process behind the derivation. For example, if I have a problem "the volume of a 1 unit long cylinder", I'll solve it by giving the answer "the area of a circular face" but clearly these do not have the same dimensions and yet *the formula holds* mathematically.
I would have always taken it as the average eggs per chicken, which is slightly different than assuming each chicken lays the same amount of eggs every day.
really good video, however at 16:00 i think you're wrong because sin(x)'s taylor series would need to be rewriten to accomodate for degrees or grade if they were assumed as the default angle unit.
`Can you do Division? Divide a loaf by a knife--what's the answer to that?' `I suppose--' Alice was beginning, but the Red Queen answered for her. `Bread-and-butter, of course.
What about logarithmic units like decibels? How do conversions and stuff get affected by something like that. Theres also temperature degrees and that fun trick about 0C + 0C = 84F or something, but i saw a video about that, and am realizing you may have made
As you kind of imply, I think anyone actually caught by the hook of making sense of dog * tree might not be satisfied, but I'm pretty happy with getting a formal statement of this system. I'm thrown off by 15:57 because this doesn't seem like something that should be inherent to radians. Would the derivative and thus the Taylor expansion be different in degrees?
The derivative would not be characteristically different. The only things that changes is you would need to divide the input x by degrees to convert it back into the more natural radian unit.
Per the chain rule, the inner scaling factor comes out if you derive: diff(sin(x degrees)) = (π × cos((πx) / 180)) / 180 ≈ 0.01745329252 × cos(0.01745329252x) diff(sin(x radians)) = cos(x) That happens for every step of derivation, and matches up with the input in the Taylor series: Taylor (sin x) = x - x³/6 + x⁵/120 ... Taylor (sin (fx)) = fx - (fx)³/6 + (fx)⁵/120 ... And thus it's more convenient to apply the scaling factor and calculate by radians rather than some other unit of angle, specifically when taking derivatives or relating radius with circumference. Turns are more intuitive and degrees provide more convenient integer factors, however.
As we know, during a walk, every dog marks the area by peeing on, among other things, trees. The unit "dog" x "tree" tells us how many pissings on trees will be made by a specific group of dogs... :)
I thought it was about finding the value of (100xD + 10xO +1xG) x (1000xT + 100xR + 1xE + 1xE) where each letter represented an integer between 0 and 9.
I have a dog who I take to walk everyday. And I can say 1 dog tree = 15 seconds. It is the average time a dog spends either sniffing or doing its business on a tree.
In fact there is a metric form of angles called gradians that defines a right angle as 100! (So a full circle is 400 gradians.) One of those things like metric time that never took on, although surveyors in France used (and sometime still use) them.
@@algotkristoffersson15Either name works. A square meter is a meter squared. The same unit having multiple names is nothing new, just look at pound-feet and foot-pounds, or newton-meters and joules.
@@algotkristoffersson15 The way you’re using “meters squared” is not as a unit at all. You’re using it as a unit (meters) and an operation (squaring). Granted, you can do that, but that’s definitely not the only or even the most common way of using “meters squared.”
There has been a modern trend of viewing 2pi as the proper circle constant rather than just pi. We define tau=2pi in this case. The argument you give here is the reason why tau is preferred by many over pi.
It is a really good video to explain a concept, but I might simply be weird, but I intuitively understood this as a kid as soon as I saw square and cubic volumetric values, and km/h. Is this something that is genuinely hard for people to understand? Am I just a freak? (I am not trying to humble brag here, but the concept is so simple to me that I am really confused...)
It may not necessarily be difficult to understand, it may just be some people haven't considered it before. It is my experience working as an educator that led me wanting to work on this video. I have encountered many students who did not have an understanding of these ideas, and even when they did, they may not realize they could be applied in a wide range of applications. I think humility here serves a very functional purpose. Once I understood that this was a concept I took for granted, I examined ways to explain and teach it, and that exploration led me to learn new ideas. No matter how simple the concept, by recognizing it and not taking it for granted, you could end up learning new things about it. I think you'll really appreciate my next video to be released this month. :)
@@zhulimath I don't know, I never had to even really think about it, I felt it was super intuitive. meter x meter = meter^2 meter x meter x meter = meter^3, super intuitive to me. I never really took it for "granted", it just seemed... intuitive. Because I had some "alternative schooling", I was never really taught it.
Here's a use case for dogxtree unit: on a park there are 5 dogs each of which will pee to 3 trees. At the end of day how many trees contain dog pee on them? Answer 15 dog (peed) trees.
dog * tree =/= dog + tree ! The first i would saw Wolf, and the second a Dog-tree creature =/= tree-dog creature edit : ah i belived it was in langage models >
This is neither a fourier-analysis not neural-networks approach to this question ;) "tree/bush" and "fish" are broad categories, that have absolutely no basis in biology or anchestry, because most trees/bushes are closer related to specific grasses as closest common ancestor, than to other trees/bushes, and this also concerns their core biological functions. Trees are just tall brothers of some grasses. The same is true for all "fish". Such broad classes barely manage to categorize by shapes+sizes, but they utterly fail to describe biological function or ancestry.
16:36 I believe you have made an error. When we have a fuel efficiency in miles per gallon, that specifically means miles of distance per gallon of gasoline. After your manipulations, we are not left with any mention of gasoline, and the original meaning was lost. To see it more easily, let G represent a unit of one gallon of gasoline. Can you do anything with the ratio 30mi/G? You cannot, unless you want to re-express G as something like "gallons*gasoline", and I don't see how this is useful.
@@zhulimath I don't see what information the cylinder adds. What you have done is taken the reciprocal of the fuel efficiency and multiplied it by some constant, namely 2.35239984775268 of some unit. Yes, mathematically you can do this, but what information does it add? How does knowing a fuel efficiency in square millimeters tell me something that knowing fuel efficiency in miles per gallon doesn't? If we have two positive numbers, x and y, and if x < y, then 1/x > 1/y, and c/x > c/y for any positive constant c. If you know that a fuel mileage of x is less than a fuel mileage of y, then you will gain no information by converting to mm^2. All I care about when I look at fuel mileage is which number is bigger.
It explains why area is a coherent quantity for measuring fuel efficiency. We can visualize an area that represents fuel efficiency. When you are flexible in conceptualizing dimensional analysis like this, a lot of more abstract units become easier to understand and comprehend, and could have practical applications. For instance, units that measure impedance or frequency or power, or even weirder more niche applications.
I have a strong suspicion that this is one of those deceptively straightforward ideas that seem easy until you start to dive into them. Like, your brain takes information in from your senses, and then performs some type of clustering analysis which classifies two distinct objects into "dog" and "Tree". Obviously, observations from your senses have a unit, so there exists some relationship between those three dimensions.
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Dog times tree. Start by factoring out the common factor. Obviously the result will contain the unit bark^2.
hahahahaha
underrated comment, thats great lol
This made me laugh more than it should.
Actually a good pun
That is gold lmao
i appreciate your use of τ at 15:28
Miles per gallon being equivalent to length^-2 is so amusing!
Dimensional analysis can also help catch mistakes when solving diff eq's; if you ever get sin, log, exp, etc. with an argument that isn't unitless, then you know you've made an error.
Conversion factors that need context to create, don't equal one. They instead represent whatever the context is. When you turn context into a conversion factor, you can multiply the situation by it to get the result. For example, 3dollar/10apple (I use incorrect grammar with no spaces when dealing with units as variables) is the context in the form of a conversion factor, and 5apple is the situation, then the equation written is "(3dollar/10apple)5apple" is equal to the cost of 5 apples if 10 apples cost 3 dollars. The "if 10 apples cost 3 dollars" is part of the equation, not context for the equation.
And in the students, minutes, and problems problem, the conversion factor "problem/minute/student" (a "1" is not needed since we're treating them as variables) should've been said, and again, it doesn't equal one, it equals the context (which is the student's ability to solve these specific problems).
This video is legitimately SO GOOD. It is one of the best videos I've seen, and I've seen thousands of math videos. It seems few are talking about units despite how incredibly powerful and useful they are to any scientific or engineering field. Every high school student needs to see this, and I'd be willing to bet a great portion of university students would benefit from seeing this as well. Phenomenal job.
70th person who likes, hates these stuff
360 probably stuck because it is a practical number, i.e. every number from 1 to 360 can be made by adding its factors, and no factor is repeated.
360's prime factorization is 2³ • 3² • 5, so we have 1, 2, 3, 4, 5, 6, 8, 9, 10, 12, 15, 18, 20, 24, 30, 36, 40, 45, 60, 72, 90, 120, 180 and 360. Any number that is greater that 180 can made with 180 + leftover. We've split the size of numbers we have to prove can be made in half. We can do that again with 90 and 45, which is what you get when you remove the 2^n from the prime factorization. Here are all the numbers from 1 to 45:
1 = 1
2 = 2
3 = 3
4 = 4
5 = 5
6 = 6
7 = 6 + 1
8 = 8
9 = 9
10 = 10
11 = 10 + 1
12 = 12
13 = 12 + 1
14 = 12 + 2
15 = 15
16 = 15 + 1
17 = 15 + 2
18 = 18
19 = 18 + 1
20 = 20
21 = 20 + 1 = 9 + 10
22 = 20 + 2
23 = 20 + 3
24 = 24
25 = 24 + 1
26 = 24 + 2
27 = 24 + 3
28 = 24 + 4
29 = 24 + 5
30 = 30
31 = 30 + 1
32 = 30 + 2
33 = 30 + 3
34 = 30 + 4
35 = 30 + 5
36 = 36
37 = 36 + 1
38 = 36 + 2
39 = 36 + 3
40 = 40
41 = 40 + 1
42 = 40 + 2
43 = 40 + 3
44 = 40 + 4
45 = 45
I love how you said when $10 =3 apples, that you would show a practical example later, even though this example is quite practical
your feelings are irrational
@@Fire_AxusAlright, Spock
Excellent video! This really fills in the gaps of knowledge I have from middle school/early high school
also, If you change the timestamp of the intro from 0:05 to 0:00 you will have chapters instead of key moments.
You can sort of add incommensurable units, you just have to keep the parts separate as a vector
But really, that's just a way to represent two different data points under one set with its own term. I know you know this because you used sort of, but I still don't believe you can say they're added.
I suppose this becomes useful if you later convert both units to the same units, by (for example) converting both to dollars using their price.
@@Tzizenorec Or if you work with complex numbers
- We can add commensurable units
- We can multiply units by unitless value
- We can multiply and divide any units
- Ce can exponentiate units by unitless values
But could we take a unit to the power of another? Like seconds^meters?
And, how could we intuitively represent ourselves units to the power of unitless values?
Sure, with units of lengh, it's just dimentions as we naturally think about it.
But multiple dimensions of time? (sec²). Furthermore, how could we represent ourselves meters^½?
These are the strange questions I've always wondered even before the video 😅.
(Tho for meters^1.585 for example, we could think of it as units of "fractal aeras" using Sierpiński's triangles instead of squares.)
we need a part 2
Taking units to fixed fractional powers is fine.
I don’t think taking a unit to the power of another unit is sensible.
There are three problems I see with this kind of approach (in general):
- All of the 'commensurate' units are actually the same as the contextually convertible units in that we do have hidden assumptions underlying them based on the model we use (that model is the context).
- Many formulas stated in a mathematical context will have been simplified, already cancelling out units that may have been important, you need to understand where the formula comes from in order to insist that any term in it has any kind of particular dimension.
- Some units are not linearly composable when converting to other units: take decibels for example which exist on an exponential scale / logarithmic scale depending on how you want to convert them to/from SI units, or farenheit vs celsius which need an affine transformation (not a linear transformation) to convert them.
Really great points here, especially the last one! I wanted to keep this introduction as accessible as possible to a general public, so some omissions were intentional, but I do regret that I didn't cover that last point more.
15:25 I don't get why everyone agrees that radians are the only natural choice for angle measurement. If I turn around such that I face the same way, I wouldn't say that I turned 2π radians or 360°, I would say that I made 1 full rotation. Why isn't there a unit that is 2π larger than a radian? Revolution is 1, half revolution is 1/2, third revolution is 1/3. Seems much more logical.
And as for formulas, we can just accept that these formulas require arc length and not angle, and a conversion unit between them is 2π rev^-1 because arc length of 1rev is 2π (aka circumference of a unit circle).
And radians are totally not unitless, the same way Joule and newton-meter are different units. Joule is a scalar and newton-meter is a bivector, because torque needs a plane to rotate in. The bivectorness of newton-meter comes from the radian, if you divide it by the angle that torque caused (in radians) you'll get work in Joules.
Rotation of 1rad clockwise around z axis is a different thing than rotation of 1rad clockwise around x axis.
As I mentioned in the video, dimensional analysis can be very arbitrary. It just depends on the context you're working in, as long as it is well-defined, consistent, and meaningful. It would be unrealistic to go over every possible interpretation of angle measurement and their consequences in a video, nor would it be very productive. But what you're doing by being flexible and creative in how we think about units, challenging accepted norms, is, in my opinion, a healthy mindset to have.
I think it's better to think about the unit of permutation being choices^n rather than the object^n. Permitting 3 people isn't 3 people x 2 people x 1 person, it's 3 choices of people x 2 choices of people x 1 choice of people.
Choices^3 clearly shows that the abstract object being quantified is the result of 3 success choices. That way if you chose a permutation of 3 people and then chose a card, the unit would be choices^4 rather than some bizarre people^3•cards. It may also be useful to specify the number of options, eg. Rolling 3 dice and then flipping a coin would be (6-choice)^3(2-choice), and the measurement could be equated with other phenomena like colouring a random vertex of a hexagon either black or white. This approach does mean that permutations wouldnt reduce though, (3-choice-of person)! Would be 6 (3-choice)(2-choice)(1-choice). Maybe we can just factorialise the unit so the result is 6 (3-choice)!
The "product of choices" you speak of is generally considered the purview of combinatorics. Combinatorics is concerned with the cardinality/"size" of sets, and as such tend to treat the values as dimensionless.
That said, there is still a sensible way to interpret the units "people^3" or "people^3•cards". Notice what a set of "permutations of three people" actually contains; each member of the set is actually a list, or tuple, of three people. (In fact the "handshake" / people^2 mentioned in this video can be considered a tuple of two people, a.k.a a pair of people.) Tuples can also contain unlike items, so you can have a tuple of three people and a card, hence "people^3•card"
Coming back to combinatorics however, the reason cardinalities are generally treated as dimensionless is because you *can* add units together that would normally be incommensurable, as sets could be heterogenous (containing unlike items), you *can* put people, apples, and "(dog, tree) tuples" in the same set if you wanted or needed, for example.
awesome video, just a small add no 360 degress is not quite that useless, the reason why it was used is because 360 has the most primefactors which means it can be divided into different parts very easily
Yes. The French grad system is a romantic idea but it isn't very practical outside of civil engineering.
Isn't that related to why we have 60 seconds per minute and 60 minutes per hour.
superior highly composite numbers, my favorite!
An observation:
Despite "degree" and "radian" both being dimensionless, it is not okay to add these. Then it is better to keep the unit-name, as being a higher level of (more nuanced?) "truth" than just the dimension .Then again, "degree per radian" as a conversion factor makes complete sense - I have pi radian - multiply by {360 deree per 2*pi radian --> boom --> 180 degree.
I know that this is completely obvious to _you_ - but spelling it out literally may be good for some of your viewers :)
How about this other story:
A certain forest is visited by people taking their dogs for walks. Each dog, compulsively, _must_ pee on every tree. As there are twelve dogs visiting on a certain day, and twenty trees, there is 240 dog.tree worth of pee deposited in the forest that day. As one dog.tree is equivalent to 1e-4 m3, let's say that's about 2.4e-2 m^3 of pee that day. 24 liter. Nice.
Units of angle are dimensionless, but not scaleless; same dimensional coefficients, different units. It's comparable to nautical miles versus meters (both of which are based on navigation angles originally: 60×360 nautical miles ≈ 40 megameters ≈ length of 1 equator). For math reasons we like to use radians as the base unit, but it's also pretty intuitive to use turns.
It *is* ok to add them, if you don't do it wrong. 1k + 3 = 1.003k, and 45° + π = 5π/4
@@MCLooyverse Clearly there's some conversion going on there though.
@@kamikeserpentail3778 Depends on how you think about it. From a notational perspective, yes, but from a quantity perspective, no. a = b => a + c = b + c, 45° = π/4 => 45° + π = π/4 + π (= 5π/4).
If there are budding game devs in here, here's why we need delta time, what's meant by it, and why you shouldn't haphazardly multiply everything with it.
The game's visual state is refreshed in succession of independent still image "frames", and the rate of this update is denoted with a unit of FPS or "frames per second". For example a value of 60 FPS means there are 60 frames per second which equals the update frequency of 60 Hz. Yes, the derived units of FPS and Hz are essentially equivalent because 1 second is the base unit used in both of them. From there, to gather a duration of a single frame, you take a reciprocal, which is 1 second / 60 Hz = 16.66.. milliseconds, a result that represents the ideal delta time between two frames.
However because we don't live in an ideal world, the hardware varies as well as the computation, so the delta time is measured and may oscillate around the ideal value. This measured delta time value is supposed to be used somehow if we want, for example, to produce smooth motion over some arbitrary time duration.
Typically we define the speed in standard derived units, such as 'meters per second', however video games live in a virtual space so it makes more sense to define the speed in 'game space units per second' or 'world units per second'. Let's call this unit of length "w", and let's denote 'frames' with "F".
FPS = 60 F/s
Tdi = 1/FPS = 0.0166.. s (an ideal delta time)
Speed = 10 w/s (a desired speed of our object)
Td = 0.0168 s (a measured delta time at some point in time)
It is fairly easy to intuitively find how much the object should move over just one frame, just by multiplying the speed with the delta time.
Speed * Td = 10 w/s * 0.0168 s = 0.168 w
But why this works?
Observe that when we derived the two delta times we've lost some units in the conversion. The FPS reciprocal is in fact X [s] / Y [F/s] = X/Y [s/F] but people usually ignore the divisor because it is always 1 by definition, and erroneously dismiss the 'frames' unit as well. This is why many people miss to appreciate the actual units involved in this conversion.
This is what truly happens when we multiply the two
Speed * Td = 10 w/s * 0.0168 s/F = 0.168 w */F*
So when you multiply the two you basically convert the Speed expressed in w/s to a more meaningful w/F because the goal is to apply it in each singular frame of the animation. This means that after approximately 60 frames (which should last for exactly 1 second), your object should move by 10 world units, if it would move 0.166.. w/F *on average*.
Now from the classical V = s/t we can check V out as s * t = V, and also switch the continuous measurement of time with discrete frames.
0.166.. w/F * 60 F/s = 10 w/s
You can see that this hold up even when delta time varies, making the parameters change on the fly.
0.168 w/F * 59.52.. F/s = 10 w/s
So this is why this technique is used to maintain the original speed set in w/s, otherwise the speeds would be scaled differently on different systems and under different performance profiles, which is not how real time video games are supposed to work.
Also this is why you shouldn't use this approach when computing absolute positions with other math tools such as i.e. linear interpolation. It simply doesn't work the same way and doesn't involve incremental steps that scale with measured delta times. Another takeaway from this example is that the measured delta time always lags by exactly 1 frame, so keep in mind that this kind of math is never perfect, but usually good enough when the actual FPS is high enough or stable enough!
THE SEARCH IS FINALLY OVER
I've been looking for this exact thing for the past 3 years!
This...Work Of Art has answered so many of my questions, and provided a pathway to answer so many more.
Eternally grateful.
I just got goosebumps! I was thinking about *_units_* as *_variables_* this evening and found out it makes sense! But I didn't search for any vids about this online but exactly 3hrs later this vid popped up on my feed! Now I understand it better 😊
This just helped me figure out optimization in calculus one.
I love this video. It is not a topic I have seen covered like this before and it is something many people do not understand. You definitely did it justice here, I've added it to my playlist of best maths videos.
I came here, knowing about the Tree Function. (Like Tree(3) which is an absurdly large number). Thinking, that Dog is smth Similar. Like Dog(5) would be a huge Number aswell for a different reason. I waited trough the whole video for the big reveal, just to realize that you were talking about a literal Tree and a literal dog... Still a great video and kinda shifted my mind on how I think about units
Unit and dimension simply shouldn't be interchangeable terms, which is the origin of the complication you mentioned, and doing so missed an opportunity to make these ideas far clearer. Radians and degrees are dimensionless units, but both refer to the same quantity of "angle". They are not "unitless" precisely because they are different units themselves, references for measuring angles: one is 1/360 of a turn, the other 1/2pi of a turn. A unit is simply a reference value for a quantity. A quantity may or may not have dimensions, but regardless, it is still a conceptual property associated to a plain number, like "angle".
Quantity and quality are interchangeable. Dimension(quality:dog,tree ...) and curvature(quantity: 1 2 3 ..) may interchange
with each other.
take some linear(dimensionles) operator, for example, derivative:
D(f*g)=1*D(f)*g + 1*f*D(g).
It is represented by 2-convolution of binom '1 1"
row_0= 1
row_1= 1 1
row_2= 1 2 1
row_3= 1 3 3 1
row_4= 1 4 6 4 1
row_5= 1 5 10 10 5 1
...
Each row is face-vector of simplex.
If to replace binom "1 1" by "1 2" then we can represent derivative by cube:
row_0= 1
row_1= 1 2
row_2= 1 4 4
row_3= 1 6 12 8
row_4= 1 8 24 32 16
row_5= 1 10 40 80 80 32
...
Each row is face-vector of cube.
operator look like this:
D(f*g)=1*D(f)*g + 2*f*D(g).
Second derivative is
DD(f*g)=1*(DD(f)*g+2*D(f)*D(g))+
2*(D(f)*D(g)+2*f*DD(g))=
1*DD(f)*g+4*(D(f)*D(g)+4*f*DD(g))
wich is row_2 of above Pascal Matrix.
If to replace binom "1 1" by 3-nom "1 1 1" ( sequence "1 1 1 1 1 . . ." , of course ,is a face-vector of sphere) and 2-convolution by 3-convolutoin
then we can represent derivative by simplex3 wich appears only in odd dimensions:
row_0= 1 =1
row_1= 1 1 1 =3
row_2= 1 2 3 2 1 =9
row_3= 1 3 6 7 6 3 1 =27
row_4= 1 4 10 16 19 16 10 4 1 =81
row_5= 1 5 15 30 45 51 45 30 15 5 1 =243
...
Each row is face-vector of simplex3.
D(f*g*h)=1*f*D(g*h) + 1*g*D(f*h)+ 1*h*D(f*g) .
Of course, D(m*n)=a*D(m)*n + b*m*D(n). a and b depends on
wich nested shape we choose (see above).
Unlike previose, this derivative operator is nested(dimensional). Why?, because of
quantity-quality inversion happens in even dimensions.
Instead of Simplex3 to be curved into shape, dimension itself is curved into operator wich becames nested(dimensional).
Good video, have been teaching this to my Kids for ages and wish schools would do that since it makes understanding and dealing with units so much easier.
Dealing with percent as a unit also makes that that much easier to understand. A former customer still insisted it was a bug to display 700(%^2) instead of 7% ;)
Also fun fact 0m = 0s = 0
I've always loved dimensional analysis. My favorite application of it is in the use of values in (non-polynomial) analytic functions like exp or sin or cos or ln. Whenever you have one of these kinds of functions in an expression, the argument must be dimensionless. I don't know if there's an ultimate answer as to "why," but I prefer the argument that, since you can't add apples to apples^2 and have it make sense, the power series demands the arguments be dimensionless, as you briefly addressed at 15:49.
Of course there are some "generalized functions" that don't follow this rule, like the Dirac delta, but even those functions show where their dimension comes from in their limit definitions
You mention assigning units to traditionally dimensionless quantities like radians, but I think it’s more common in physics to do the reverse. Physicists often treat the speed of light as exactly 1, dimensionless; length and time are the same thing in this analysis. 1 second + 1 meter is 299,792,459 m ≈ 1.000000003 s.
(Honestly I do a bit of both; I feel like the mol should in most cases be dimensionless but radians should be a separate unit)
Learning natural units in my special relativity course
Dog x tree is simply a unit of square energy: convert dog and tree by dividing by average mass of dog and tree respectively then convert mass into energy by multiplying by c^4
The best explanation I have heard for degrees is that it stems from the compass construction of an equilateral triangle, and 60 being the base of the Babylonians made it a natural choice to denote it.
Tree - Carbon Sequestration: Trees absorb and store carbon dioxide from the atmosphere, a process known as carbon sequestration. This can be quantified as a rate, such as kilograms of CO2 sequestered per year.
Dog - Carbon Footprint: The environmental impact of keeping a dog can be partly quantified by its carbon footprint, which includes the resources used for its food, care, etc. This can also be expressed as a rate, such as kilograms of CO2 equivalent produced per year.
Multiplying the Attributes:
Combining the Rates: By multiplying the carbon sequestration rate of a tree by the carbon footprint of a dog, we can create a unit that represents the balance between carbon sequestration and carbon production.
Example Calculation:
If one tree sequesters 20 kg of CO2 per year, and one dog produces a carbon footprint equivalent to 100 kg of CO2 per year, multiplying these gives 20 kg CO2/year/tree×100 kg CO2/year/dog20kg CO2/year/tree×100kg CO2/year/dog.
The result is 2000 kg2CO2/year2/tree-dog2000kg2CO2/year2/tree-dog
With that we can finally answer the question that we all want to know,
How many trees need to be planted to offset the carbon footprint of owning a dog?
Excellent, you've found a meaningful conversion and interpretation for dog and tree units that I have not yet seen in the comments yet!
However, after you convert both units to the amount of carbon dioxide, this is not actually dog * tree but dog + tree. Multiplying units means that every pair of a dog and a tree needs to describe something, whereas here, the dog and the tree each represent a quantity that is just combined to a total.
That being said, dog + tree is equally amusing, and you've given me an example to highlight and distinguish!
This just solved all my problems in physics
for context, if a dog spends x seconds when visiting a tree, then "seconds" is just another name for "dog-trees"
Dogs are well know for what they do when they visit trees. So if you have 3 dogs and 5 trees your going to get 15 pees.
(Not to be considered with 15p, a monetary amount in British currency)
This video gave me an idea about programming(I don't know if this can be implemented IRL). Basically in your code you create separate numerical type for each base unit of measure (like one for combinations, another apples, third for yens, etc). And this is done in such way that you are forbidden from adding or subtracting values between different numerical types, yet you can divide and multiply them (you can't add apples to oranges, but you can multiply apples by oranges)
This can in fact be done! Create an abstract class for the unit, then use an object/dictionary property to track each unique unit factor. The name of the unit factor is the key, the exponent is the assigned value. You can then manually code the logic of addition and multiplication, where addition applies a check beforehand for validity, and multiplication simply sums the object/dictionary by key to obtain the new derived unit.
15:58 hot take, turns are the least arbitrary unit of rotation. 1 as a full rotation amount is just objectively less arbitrary than 2pi. and in computer math libraries, often a factor of pi or 2pi is divided out first, introducing extra floating point error.
Yes, turns are also a fairly natural unit for angles/rotation. As mentioned in the video, the unit definitions can be pretty arbitrary. You can justify whatever definitions you want as long as it is coherent, consistent, and meaningful!
I don’t think I agree, on account of:
(1): when you differentiate trig functions, having the trig function take a number of turns rather than a number of radians, you end up getting a factor of 2pi, instead of 1.
(2): Using turns, complicated the relationship with the e^x function.
e^{i x} = cos(x) + i sine(x) when cosine and sine are considered to take their inputs as in radians, but if they take their inputs in turns, then you would have to say e^{i x} = cos(x/2pi) + i sine(x/2pi)
.. or, I suppose instead you could write e^{2 pi i x} = cos(x) + i sine(x) ?
Still, I don’t think this is natural.
If we are solving some differential equation, it is more likely that we will get stuff relating to e^{i k x} than to e^{i 2pi k x} .
...
“natural” shouldn’t mean the same thing as “most intuitive seeming when considering for the first time”. It is more like, “the convention that one would eventually find that the math ‘wants’ one to set”.
I feel like using cosine(x) = (1/2) * ( exp(2pi i x) + exp(- 2pi i x) ) as the standard , instead of the version without the 2pi, is kind of like, ignoring what the math is telling us about what is natural?
... unless, perhaps, if it might be more natural to use (2pi i) as the base thing to use instead of i?
But I don’t think that’s right. The complex numbers being the algebraic completion of the real numbers is important, and this connects much more closely to them being the real numbers adjoin a square root of -1, then the real numbers adjoin a square root of - (2pi)^2 .
Like...
I don’t think adjoining 2pi i onto the integers, or rationals, gives nearly as natural of an object than adjoining i does.
So, I think i is more fundamental than 2pi i. (Though perhaps 2pi i is more fundamental than 2pi ?)
@@drdca8263 yea maybe in pure math it makes more sense. i’m thinking of it in part from the perspective of using it in practice, math libraries that take in radians afaik typically remove a factor of pi or 2 pi first, and in that case, using turns makes it easier for the user and a little faster since it doesn’t need to remove pi or 2pi from the input first
@@morgan0 for that purpose I can agree, yeah
Thanks for clarifying
When you said "turns" I thought you meant 'right-angle turns' because when I 'perform a turn' I am generally turning through a right angle (e.g. navigating in a grid), I would use the term 'revolution' or 'rev' for short to refer to the full circumference. BTW right-angle turns (sometimes called eta, like pi is a half the circumference and tau is the full circumference) are also a nice unit.
I wouldn't say you are directly multiplying students with minutes to get the number of problems solved. I'd say you are getting the number of student-minutes, and then using a meaningful conversion factor: one student can solve 1 problem per minute, which converts to a ratio of 1 problem per minute per student.
From there you can get
x student minutes * (1 problem / 1 minute) / 1 student
= xsm * (p/m) * (1/s)
= xp or x problems.
I guess we could look at digits (or natural numbers, if you want) themselves as commensurable units. Like, "one `pair` is two `units`", e.g. the dimensionless one is all we need: it equals to `(one plus one) divided by (two)`. Thus, when we use any number notation (numerals, I guess?) we effectively do _measure numbers_ using this unit system.
Thank you so much I’ve always wanted a video talking about this
Your videos are always top quality. I appreciate you!
At 13:10 I have a problem with this part about using dimensional analysis as a sanity check:
The proposed (misremembered) formula could indeed be correct, even if the dimensions *by naive substitution of the terms* don't align.
Let's suppose we're operating in a different space to the usual euclidean plane, let's say such that the area of a rectangle is not a*b but something arbitrary (we could insist this still forms a related measure space and metric space etc so we still have a notion of lengths that subtend an area).
Can you tell me that it's impossible for the incorrect formula to hold in such a space? (not necessarily this particular formula, but any formula that's 'wrong' by this kind of argument). If it's not impossible, then how can this method be obviously applicable? You effectively have to do as much work figuring out if you can apply the sanity check as actually solving the problem.
In general you can't just substitute units into the terms without knowing the whole process behind the derivation. For example, if I have a problem "the volume of a 1 unit long cylinder", I'll solve it by giving the answer "the area of a circular face" but clearly these do not have the same dimensions and yet *the formula holds* mathematically.
This isn't 3Brown1Blue but it's still Goodly Zhuli
My dogs and trees want to know!!!
See, this is why everyone should switch to measuring liters per hundred kilometers. The dimensional analysis makes so much more sense!
Zhuli, do the thing!
i love the kapustin intro
Everybody gangsta until the pH and pKa start walking in
I would have always taken it as the average eggs per chicken, which is slightly different than assuming each chicken lays the same amount of eggs every day.
I never stick to your videos, I threw the stick the dog fetched it.
really good video, however at 16:00 i think you're wrong because sin(x)'s taylor series would need to be rewriten to accomodate for degrees or grade if they were assumed as the default angle unit.
Yes, the point is that it is wrong, and this is why radians is the most "natural" choice for angle measure units in this context.
Really nice video.
Another video recommendation about this topic is "a joke about measurement" by Jan Misali
assumed this was going to point to last common ancestor
and broadcast the array of coherent lifeforms with a lesser genetic distance.
`Can you do Division? Divide a loaf by a knife--what's the answer to that?'
`I suppose--' Alice was beginning, but the Red Queen answered for her. `Bread-and-butter, of course.
Haha, my/our first video centers on dimensional analysis and also includes the example of gallons/mile being the area of the string left behind a car
Frankly, from the premise of the video I expected a talk on exterior algebra. But what you've done is fine too =)
Nice good video. I invented my own unit of sqrt(ha) (square-root hectare), which is 100m.
What about logarithmic units like decibels? How do conversions and stuff get affected by something like that. Theres also temperature degrees and that fun trick about 0C + 0C = 84F or something, but i saw a video about that, and am realizing you may have made
As you kind of imply, I think anyone actually caught by the hook of making sense of dog * tree might not be satisfied, but I'm pretty happy with getting a formal statement of this system.
I'm thrown off by 15:57 because this doesn't seem like something that should be inherent to radians. Would the derivative and thus the Taylor expansion be different in degrees?
The derivative would not be characteristically different. The only things that changes is you would need to divide the input x by degrees to convert it back into the more natural radian unit.
Per the chain rule, the inner scaling factor comes out if you derive:
diff(sin(x degrees)) = (π × cos((πx) / 180)) / 180 ≈ 0.01745329252 × cos(0.01745329252x)
diff(sin(x radians)) = cos(x)
That happens for every step of derivation, and matches up with the input in the Taylor series:
Taylor (sin x) = x - x³/6 + x⁵/120 ...
Taylor (sin (fx)) = fx - (fx)³/6 + (fx)⁵/120 ...
And thus it's more convenient to apply the scaling factor and calculate by radians rather than some other unit of angle, specifically when taking derivatives or relating radius with circumference.
Turns are more intuitive and degrees provide more convenient integer factors, however.
As we know, during a walk, every dog marks the area by peeing on, among other things, trees. The unit "dog" x "tree" tells us how many pissings on trees will be made by a specific group of dogs... :)
What happens if you sqare root a unit?
I thought it was about finding the value of (100xD + 10xO +1xG) x (1000xT + 100xR + 1xE + 1xE) where each letter represented an integer between 0 and 9.
Anyone who read IEEE Standard for Floating Point Arithmetic knows that it's NaN
I have a dog who I take to walk everyday. And I can say 1 dog tree = 15 seconds. It is the average time a dog spends either sniffing or doing its business on a tree.
In fact there is a metric form of angles called gradians that defines a right angle as 100! (So a full circle is 400 gradians.) One of those things like metric time that never took on, although surveyors in France used (and sometime still use) them.
"dog times tree equals dog times tree"
Every 60 seconds in Africa, a minute passes.
Ah yes, the floor here is made of floor.
Open - closed is a double negative so its OpenOpen that is Closed
Heron's formula is A=√(S(S-a)(S-b)(S-c)). A=√(m*m*m*m)=√(m⁴)=m². Isn't incorrect unit
As I mentioned in the video, the unit would be incorrect if the first "s" factor was missing. With the correct formula, the unit is correct!
@@zhulimath Sorry, i didn't hear that
I'm base 36, that equals BB3OTQ8.
Adding space and time? Less impossible than you'd think. Thanks, Einstein...
In this episode, we compute the labour-value of chickens.
Open x 360 degrees equals inside out 😮
By the way, for the chicken problem, I got 16 days plus 16 hours
8:58 Polska gurom!
but what about trog?
Dog ÷ Tree = Bark
If each pair of dog and tree corresponds to a piss, then:
dog x tree = total number of pisses
9:02 Poland? or random variables
dog x tree = dog trees = volume of pee pee-ed
Hmmmm. I thought this was going to be a Buckingham Pi video...
4:27 multiply by 1 min/(problem*student) then
17:29 it's square milimeters, not milimeters squared
why
@@Fire_Axus Because that's what the unit is called
@@algotkristoffersson15Either name works. A square meter is a meter squared. The same unit having multiple names is nothing new, just look at pound-feet and foot-pounds, or newton-meters and joules.
@@Tom-jw7ii Jes a square meter is a meter squared, but two square meters isn't two meters squared, instead it is four meters squared
@@algotkristoffersson15 The way you’re using “meters squared” is not as a unit at all. You’re using it as a unit (meters) and an operation (squaring). Granted, you can do that, but that’s definitely not the only or even the most common way of using “meters squared.”
Doesn’t contain any arbitrary values: 2π contains a 2
There has been a modern trend of viewing 2pi as the proper circle constant rather than just pi. We define tau=2pi in this case. The argument you give here is the reason why tau is preferred by many over pi.
The dog is a "scalar" but not a very good one and a tree can be represented as a matrix, so clearly you will get another tree.
It is a really good video to explain a concept, but I might simply be weird, but I intuitively understood this as a kid as soon as I saw square and cubic volumetric values, and km/h.
Is this something that is genuinely hard for people to understand? Am I just a freak?
(I am not trying to humble brag here, but the concept is so simple to me that I am really confused...)
It may not necessarily be difficult to understand, it may just be some people haven't considered it before.
It is my experience working as an educator that led me wanting to work on this video. I have encountered many students who did not have an understanding of these ideas, and even when they did, they may not realize they could be applied in a wide range of applications.
I think humility here serves a very functional purpose. Once I understood that this was a concept I took for granted, I examined ways to explain and teach it, and that exploration led me to learn new ideas. No matter how simple the concept, by recognizing it and not taking it for granted, you could end up learning new things about it.
I think you'll really appreciate my next video to be released this month. :)
@@zhulimath I don't know, I never had to even really think about it, I felt it was super intuitive.
meter x meter = meter^2
meter x meter x meter = meter^3, super intuitive to me. I never really took it for "granted", it just seemed... intuitive.
Because I had some "alternative schooling", I was never really taught it.
Here's a use case for dogxtree unit: on a park there are 5 dogs each of which will pee to 3 trees. At the end of day how many trees contain dog pee on them? Answer 15 dog (peed) trees.
Nope, IMHO.. the number of trees is measured in Trees. A Dogtree is a measure of urine? (My mind was taking me in similarly unpleasant directions)
Dog trees is a measure of urine volume
I thought of a game
You start with only base units
Than you make new units
10^18 is a trillion in a consistent, clear and meaningful way.
dog×tree is obviously a measure of volume. It's used to express the average amount of piss a dog will leave on a tree.
5 apples*5 meters=25apple-minuites
dog * tree =/= dog + tree !
The first i would saw Wolf, and the second a Dog-tree creature =/= tree-dog creature
edit : ah i belived it was in langage models >
This is neither a fourier-analysis not neural-networks approach to this question ;)
"tree/bush" and "fish" are broad categories, that have absolutely no basis in biology or anchestry, because most trees/bushes are closer related to specific grasses as closest common ancestor, than to other trees/bushes, and this also concerns their core biological functions. Trees are just tall brothers of some grasses.
The same is true for all "fish". Such broad classes barely manage to categorize by shapes+sizes, but they utterly fail to describe biological function or ancestry.
anyone saw it too he put 1 złoty = 81 forints he just put 2 brother counties poland and hungary coincidence i think not
before watching, my kneejerk thought: It's pee
Seriously, are you telling me there are people who don't know this?
Open x 5 is OpenOpenOpenOpenOpen that equals Open
Dog times Tree is Cmjfich
I think the answer is a dog wood tree
It's dore, change my mind
16:36 I believe you have made an error. When we have a fuel efficiency in miles per gallon, that specifically means miles of distance per gallon of gasoline. After your manipulations, we are not left with any mention of gasoline, and the original meaning was lost.
To see it more easily, let G represent a unit of one gallon of gasoline. Can you do anything with the ratio 30mi/G? You cannot, unless you want to re-express G as something like "gallons*gasoline", and I don't see how this is useful.
If you continue watching, I explain in the video why this representation still makes sense!
@@zhulimath I don't see what information the cylinder adds. What you have done is taken the reciprocal of the fuel efficiency and multiplied it by some constant, namely 2.35239984775268 of some unit. Yes, mathematically you can do this, but what information does it add? How does knowing a fuel efficiency in square millimeters tell me something that knowing fuel efficiency in miles per gallon doesn't? If we have two positive numbers, x and y, and if x < y, then 1/x > 1/y, and c/x > c/y for any positive constant c. If you know that a fuel mileage of x is less than a fuel mileage of y, then you will gain no information by converting to mm^2. All I care about when I look at fuel mileage is which number is bigger.
It explains why area is a coherent quantity for measuring fuel efficiency. We can visualize an area that represents fuel efficiency. When you are flexible in conceptualizing dimensional analysis like this, a lot of more abstract units become easier to understand and comprehend, and could have practical applications. For instance, units that measure impedance or frequency or power, or even weirder more niche applications.
Dog × Tree = DogDogDog
I have a strong suspicion that this is one of those deceptively straightforward ideas that seem easy until you start to dive into them.
Like, your brain takes information in from your senses, and then performs some type of clustering analysis which classifies two distinct objects into "dog" and "Tree". Obviously, observations from your senses have a unit, so there exists some relationship between those three dimensions.
Dog x Tree = Spider + Spider + Spider, of course.