9 ms·

What Is Dimensional Analysis?

104 points by cgadski 3y ago

sixo 3y ago
Dimensions behave somewhat like a "type system" for math. These dimensional-analysis tricks act like the trick you see in Haskell sometimes, where you can easily guess an implementation of an expression once you know it's type (or e.g. search by type signature https://hoogle.haskell.org/ https://hoogle.haskell.org/ )
nonrandomstring 3y ago
Brings back memories. We had a particularly strict physics teacher who only awarded full marks if our answers were numerically correct, with correct units, and we proved (with a little marginal aside simplifying all mass, length and time) that the dimensions were also correct.
- zwieback 3y ago
  Same here, our high school physics teacher drilled that into our heads and 35 years later I still carry through units or do a separate calculation every time.
- jeffwass 3y ago
  My chemistry teacher was good like that. Sadly my HS physics teacher often gave us unreasonable values on homeworks and exams so you couldn’t sanity check easily. Eg - a kangaroo jumps 245 meters high, or the barrel of a gun is 1.5 kilometres long, etc.
  - HarryHirsch 3y ago
    Why do people do such silly things? Part of working assignments is to grow an intuition, and no kangaroo jumps hundreds of meters. Such crap just reinforces the notion that what you do in school is completely meaningless and divorced from reality.
    - hirundo 3y ago
      Reality includes jet packs and moon walks where such numbers can be relevant. It also includes virtual realities like Fortnite in which such jumps are mundane. And the oddity can add interest to the problem while helping to train the intuition with examples outside of the normal box.
      - barkingcat 3y ago
        then make them relevant. ... don't say kangaroo jumps 500 metres, say an alien from mars jumps x martian units and ask the student to derive a conversion factor between the martian and earthling units. or some problems with time dilation. BUT don't call it "kangaroo jumps 50000 km and we need it cause reality says so"
        NeoTar 3y ago
        It can also help you understand what physics you should be using / taking account of. A metal ball travelling 1m at 3 m/s and I’m pretty confident I can get a good approximation using the basic Newtonian equations. An irregular shaped object like a Kangaroo travelling 50.000 km at presumably many hundred of meters of per second and I need to think about a lot more! Not least that this kangaroo os travelling about one and a quarter times around the world so the spherical nature of the world needs to be taken into account!
- wtallis 3y ago
  I used the HP48 family of calculators for most high school and college science classes, so it was easy to keep track of units throughout the entire chain of calculations with automatic conversions and a degree of type checking (not quite fully automatic dimensional analysis). I developed a very strong habit of always being explicit about units for any quantity, and was appalled at how lenient many students and instructors were. Worst were the textbooks that defined obscure domain-specific constants without explicitly stating the units, often forcing the reader to reference several pages to be sure of the units in use for that context.
  - nxobject 3y ago
    Similarly, I would have never survived college physics without a cheapo CAS calculator that could tag and symbolically manipulate units. I simply made far too many algebra errors without it.
- Sharlin 3y ago
  Hmm, that's very standard where I'm from. You're never supposed to operate on bare numbers, but carry the units along in your equations, which automatically proves that the answer has the correct units if you didn't make mistakes. To get full marks you're definitely not allowed to just assume the units in your answer.
  - pacman2 3y ago
    Agreed. A plain number as an answer could give you zero points. It is always UNITS UNITS UNITS. 10 what? Kg? Apples? kW? I remember actually a chemistry test where I did not use units in an important calculation but only in the answer. If I had used units I would have realized that I totally fucked up. I went belly up. :-( Dimensional analysis was part of my degree but honestly, I remember little.
    - emidln 3y ago
      I had a high school physics teacher who would have the instructions "Answer in SI units unless specified" at the top of his exams, but there was always a last bonus problem that referenced the previous answer and promised extra credit equal to a full problem for the answer in "most creative or interesting units (with work shown)". It was a competition in the class for the extra credit and he'd typically award 1-3 of the answers full extra credit.
  - MichaelZuo 3y ago
    I agree, this was a basic sophmore year concept at my mechanical engineering department.
m0llusk 3y ago
Kind of odd that this doesn't really go into metrics directly nor does it mention one of the more interesting cases which is infinite dimensionality.
- deleted 3y ago
  [deleted]
- code_biologist 3y ago
  Is this the infinite dimensionality you refer to? I asked ChatGPT4: In the context of dimensional analysis, what is "infinite dimensionality"? Keep your answer brief, only one or two short paragraphs. In the context of dimensional analysis, the term "infinite dimensionality" typically doesn't refer to the physical dimensions (like length, time, and mass) that one might immediately think of. Instead, in mathematical and theoretical physics contexts, it often pertains to spaces, particularly function spaces, that have an infinite number of dimensions. For instance, the space of all possible functions of a particular type can be thought of as having infinite dimensionality, as each function can be uniquely described by an infinite series of coefficients. One of the most commonly encountered infinite-dimensional spaces in physics is the Hilbert space, used in quantum mechanics. Here, the state of a quantum system can be represented by a vector in an infinite-dimensional space. While the idea of "infinite dimensionality" might sound abstract and counterintuitive, it provides the necessary mathematical framework to accurately describe and predict a variety of physical phenomena.
  - hiAndrewQuinn 3y ago
    That looks off the mark to me - ChatGPT is jumping to the idea of R^∞ spaces and the like. I have no idea what the concept of a "meter^∞/second^∞" dimension would mean.
    - code_biologist 3y ago
      I suspected so and I hope m0llusk will clarify. I found it frustrating to hear about an interesting missed case without enough context to google, unless you already know about infinite dimensionality in the first place.
chowells 3y ago
Dimensional analysis crimes are the best crimes. Express velocity in meter-hertz. It's wrong, but the units are right!
- jeffwass 3y ago
  Ooh, you could have gone one further and done hertz-per-diopter.
- zwieback 3y ago
  Good point. Hertz is weird because it automatically makes one think of frequency but it's really just "per-second". I just read up about Hertz - weird he lived not that long ago. What did they use for frequency before Hertz?
  - chowells 3y ago
    Hertz are really only appropriate for describing cyclical events. But by convention the units are just 1/s, so have fun abusing it.
    - diracs_stache 3y ago
      A period of infinity is cyclical /s
    - nyanpasu64 3y ago
      One interesting property of Hertz is that it's partially appropriate for describing sampling rates. I've found use in annotating separate units for "cycles/sec" and "samples/sec" so I avoid mixing them up (for example, the Nyquist rate can be expressed as 0.5 cyc/smp).
  - ericbarrett 3y ago
    I've seen older literature that just uses "cycles," "kilocycles," and so forth. It's never abbreviated, so I always assumed the use of a named unit came from the desire to be able to write shorthand like kHz.
smitty1e 3y ago
This was interesting, but I was expecting something more along the lines of base units => https://en.wikipedia.org/wiki/International_System_of_Units https://en.wikipedia.org/wiki/International_System_of_Units
- HPsquared 3y ago
  Dimensions are a level below units.
  - smitty1e 3y ago
    Ah, so: "dimensions are the abstract theory of units", then?
    - cgadski 3y ago
      In the formalism of both this article and https://news.ycombinator.com/item?id=37517118 https://news.ycombinator.com/item?id=37517118, "base units" are specially designated physical quantities, while "dimensions" describe how physical quantities scale under changes to the scale of measurement. Customarily, capital letters like M, T, L, etc., refer to dimensions, and you would something like [1 kg] = M to say that 1 kg is a quantity that scales like a mass. A PLT analogy analogy would be to compare physical quantities to values and dimensions to types. In practice we can refer to dimensions by base units, but it has some mathematical benefits to view them as distinct. Hope that clarifies things!
      - smitty1e 3y ago
        It does, thank you.
thsksbd 3y ago
This is criminally under taught and under used
- maxbond 3y ago
  Once I casually mentioned dimensional analysis/quantity calculus to a programmer who worked at CERN, doing scientific computing. They had never heard of it. They had already worked out the hard way that you could do this stuff, but they thought of it as a hack. They were pumped to find out there was a theory behind it and it wasn't just ad-hoc duct taping. That really surprised me. I'm not that smart or educated. I think they either had a masters degree or were a master's student (can't recall precisely).
  - thsksbd 3y ago
    Myself i saw it once in a one semester fluid class for two lectures. A class on the Buckingham pi theorem and another on using units to hack simple physical models. To be fair, fluid engineers and the like see this a lot though (because of All their dim-less numbers). But ECE, physics, CS, chemistry etc majors? Almost nada.
- devnullbrain 3y ago
  That makes it an advantage if you learn it early enough. Some physics questions all the way up to college can be trivialised by understanding the units of the inputs and answer.
  - thsksbd 3y ago
    You can trivialize a lot past grad school with this.
- hirundo 3y ago
  I oppose making a failure to teach dimensional analysis with the physical sciences a crime, but it fits nicely as a tort of professional negligence, aka malpractice. It might be tricky to get that by the teachers unions though.
10000truths 3y ago
Dimensional analysis is useful when you want to convert between compositions of units. A good example is comparing the price of petrol where different units are being used. If you know the conversion rate between litres and gallons, and the conversion rate between USD and GBP, then you can figure out the conversion rate between USD/gal and GBP/L: USD/gal * GBP/USD * gal/L = GBP/L 1.24 USD = 1 GBP 1 gal = 3.78541 L GBP/USD = 1.24 gal/L = 3.78541 USD/gal * 1.24 * 3.78541 = GBP/L USD/gal * 4.6939084 = GBP/L $4.6939084/gal = £1/L
- HPsquared 3y ago
  Agreed, it's super useful to write out all the units and check that they cancel out properly. I usually write out the Witt I want to convert, complete with units, and successively multiply by dimensionless conversion factors which equal 1. That is to convert 45 MPG to "km/l" I'd write out: 45 (mile / UK gallon) * (1 UK gallon / 4.55 l) * (1.6 km / 1 mile) The expression (1.6 km / 1 mile) is dimensionless and equal to 1 (it's a length divided by an equal length). Cancelling the units gives (45 / 4.55 * 1.6) km/l
  - sokoloff 3y ago
    I taught an hour-long class at work where I'd prepped an example to warm the group up, which involved converting the sales material for a US car that got 26 mpg to a Canadian market using liters per 100 km. 26 mpg is carefully chosen because between 1/3 and 2/3 (usually on the lower end) would get it right (~9 liters per 100km) and the others would often come up with around 11 (km per liter). I could then ask "who got a figure around 9?" and "who got a figure around 11?" and both groups would raise their hand feeling pretty confident they'd gotten it right. I'd then introduce and walk them through dimensional analysis and finish with an example that was more complex and error-prone, but was pretty straightforward if you just applied DA. (The main point of the class was "use correct units" rather than "learn dimensional analysis".)
    - stkdump 3y ago
      The great thing about that conversion is that due to the reciprocal relationship the same conversion formula works both ways, so you don't have the risk of converting the wrong way.
maxbond 3y ago
Check out this NIST technical note on implementing quantity calculus (the broader topic in which dimensional analysis resides - it has nothing to do with integration or differentiation, don't let the 'calculus' part turn you off) in software. https://www.nist.gov/publications/architecture-software-assisted-quantity-calculus https://www.nist.gov/publications/architecture-software-assi...
unchocked 3y ago
Best one-lesson favor I ever got was dimensional analysis in high school. Do it, and most basic physics is trivial.
ttoinou 3y ago
Great. Now let's talk about the addition operation not being available for some units / dimensions like Temperature, Density, Speed etc.
- sheepshear 3y ago
  How would you convert between Kelvin and Celsius if temperatures couldn't be added?
  - defrost 3y ago
    What does it mean to add temperatures though (serious question)? Eg: Suppose (one of many subtly different scenarios) you have 1 kg of iron at a constant temperature of 20 C and 5 kg of Aluminium at a constant temperature of 15 C. What does it mean to add those temperatures?
    - sheepshear 3y ago
      You tell me. Units help describe what you modeled, they can't stop you from modeling nonsense. I heated up your iron by 4 degrees and now it's 20C+4C=24C. I modeled the physical process of heating and the units provide some additional information about that.
      - defrost 3y ago
        > I heated up your iron by 4 degrees and what happened when you applied the same amount of heat to the greater mass of Aluminium? What's the difference between intensive properties (tempreture, pressure, density) and extensive properties, such as mass or volume?
        sheepshear 3y ago
        Again, you tell me. Units have no meaning without the context of what you're modeling. I can model the average temperature of your pieces of metal as (20C+15C)/2=17.5C.
Remnant44 3y ago
Shout out to https://instacalc.com/ https://instacalc.com/ . It's like a simple spreadsheet or calculator but with awesome support for dimensional analysis -- you can attach any units you want to a quantity and it will track it throughout the calculations, and do a best-faith effort to figure out conversions / "what you meant". For example, here's a quick scratchpatch I used for calculating memory bandwidth / CPU transfer rates: https://instacalc.com/56842 https://instacalc.com/56842
kingkongjaffa 3y ago
I learned this in fluid dynamics class for learning about non dimensional value that are used like Reynolds numbers, but more broadly it was a bit like a cheat code because often we had an exam question with units of inputs and then the final answer in another set of units, and even without knowing the correct equation you could with a little intuition, derive the correct solution by combining the input units in fun ways, Understanding the MLT system in this way made most of mechanical engineering much more straightforward, and I used this almost every day up to my masters degree in computational fluid dynamics.
touisteur 3y ago
I always go back to the reference https://gmpreussner.com/research/dimensional-analysis-in-programming-languages https://gmpreussner.com/research/dimensional-analysis-in-pro... for programming language support. It's updated from time to time, and incorporated the work done on Ada in the 2010s. Good read.
amai 3y ago
Why do we have square meters but not square seconds?

104 points by cgadski 3y ago

sixo 3y ago

Dimensions behave somewhat like a "type system" for math. These dimensional-analysis tricks act like the trick you see in Haskell sometimes, where you can easily guess an implementation of an expression once you know it's type (or e.g. search by type signature https://hoogle.haskell.org/ https://hoogle.haskell.org/ )

nonrandomstring 3y ago

Brings back memories. We had a particularly strict physics teacher who only awarded full marks if our answers were numerically correct, with correct units, and we proved (with a little marginal aside simplifying all mass, length and time) that the dimensions were also correct.

zwieback 3y ago
Same here, our high school physics teacher drilled that into our heads and 35 years later I still carry through units or do a separate calculation every time.
jeffwass 3y ago
My chemistry teacher was good like that. Sadly my HS physics teacher often gave us unreasonable values on homeworks and exams so you couldn’t sanity check easily. Eg - a kangaroo jumps 245 meters high, or the barrel of a gun is 1.5 kilometres long, etc.
- HarryHirsch 3y ago
  Why do people do such silly things? Part of working assignments is to grow an intuition, and no kangaroo jumps hundreds of meters. Such crap just reinforces the notion that what you do in school is completely meaningless and divorced from reality.
  - hirundo 3y ago
    Reality includes jet packs and moon walks where such numbers can be relevant. It also includes virtual realities like Fortnite in which such jumps are mundane. And the oddity can add interest to the problem while helping to train the intuition with examples outside of the normal box.
    - barkingcat 3y ago
      then make them relevant. ... don't say kangaroo jumps 500 metres, say an alien from mars jumps x martian units and ask the student to derive a conversion factor between the martian and earthling units. or some problems with time dilation. BUT don't call it "kangaroo jumps 50000 km and we need it cause reality says so"
      - NeoTar 3y ago
        It can also help you understand what physics you should be using / taking account of. A metal ball travelling 1m at 3 m/s and I’m pretty confident I can get a good approximation using the basic Newtonian equations. An irregular shaped object like a Kangaroo travelling 50.000 km at presumably many hundred of meters of per second and I need to think about a lot more! Not least that this kangaroo os travelling about one and a quarter times around the world so the spherical nature of the world needs to be taken into account!
wtallis 3y ago
I used the HP48 family of calculators for most high school and college science classes, so it was easy to keep track of units throughout the entire chain of calculations with automatic conversions and a degree of type checking (not quite fully automatic dimensional analysis). I developed a very strong habit of always being explicit about units for any quantity, and was appalled at how lenient many students and instructors were. Worst were the textbooks that defined obscure domain-specific constants without explicitly stating the units, often forcing the reader to reference several pages to be sure of the units in use for that context.
- nxobject 3y ago
  Similarly, I would have never survived college physics without a cheapo CAS calculator that could tag and symbolically manipulate units. I simply made far too many algebra errors without it.
Sharlin 3y ago
Hmm, that's very standard where I'm from. You're never supposed to operate on bare numbers, but carry the units along in your equations, which automatically proves that the answer has the correct units if you didn't make mistakes. To get full marks you're definitely not allowed to just assume the units in your answer.
- pacman2 3y ago
  Agreed. A plain number as an answer could give you zero points. It is always UNITS UNITS UNITS. 10 what? Kg? Apples? kW? I remember actually a chemistry test where I did not use units in an important calculation but only in the answer. If I had used units I would have realized that I totally fucked up. I went belly up. :-( Dimensional analysis was part of my degree but honestly, I remember little.
  - emidln 3y ago
    I had a high school physics teacher who would have the instructions "Answer in SI units unless specified" at the top of his exams, but there was always a last bonus problem that referenced the previous answer and promised extra credit equal to a full problem for the answer in "most creative or interesting units (with work shown)". It was a competition in the class for the extra credit and he'd typically award 1-3 of the answers full extra credit.
- MichaelZuo 3y ago
  I agree, this was a basic sophmore year concept at my mechanical engineering department.

m0llusk 3y ago

Kind of odd that this doesn't really go into metrics directly nor does it mention one of the more interesting cases which is infinite dimensionality.

deleted 3y ago
[deleted]
code_biologist 3y ago
Is this the infinite dimensionality you refer to? I asked ChatGPT4: In the context of dimensional analysis, what is "infinite dimensionality"? Keep your answer brief, only one or two short paragraphs. In the context of dimensional analysis, the term "infinite dimensionality" typically doesn't refer to the physical dimensions (like length, time, and mass) that one might immediately think of. Instead, in mathematical and theoretical physics contexts, it often pertains to spaces, particularly function spaces, that have an infinite number of dimensions. For instance, the space of all possible functions of a particular type can be thought of as having infinite dimensionality, as each function can be uniquely described by an infinite series of coefficients. One of the most commonly encountered infinite-dimensional spaces in physics is the Hilbert space, used in quantum mechanics. Here, the state of a quantum system can be represented by a vector in an infinite-dimensional space. While the idea of "infinite dimensionality" might sound abstract and counterintuitive, it provides the necessary mathematical framework to accurately describe and predict a variety of physical phenomena.
- hiAndrewQuinn 3y ago
  That looks off the mark to me - ChatGPT is jumping to the idea of R^∞ spaces and the like. I have no idea what the concept of a "meter^∞/second^∞" dimension would mean.
  - code_biologist 3y ago
    I suspected so and I hope m0llusk will clarify. I found it frustrating to hear about an interesting missed case without enough context to google, unless you already know about infinite dimensionality in the first place.

chowells 3y ago

Dimensional analysis crimes are the best crimes. Express velocity in meter-hertz. It's wrong, but the units are right!

jeffwass 3y ago
Ooh, you could have gone one further and done hertz-per-diopter.
zwieback 3y ago
Good point. Hertz is weird because it automatically makes one think of frequency but it's really just "per-second". I just read up about Hertz - weird he lived not that long ago. What did they use for frequency before Hertz?
- chowells 3y ago
  Hertz are really only appropriate for describing cyclical events. But by convention the units are just 1/s, so have fun abusing it.
  - diracs_stache 3y ago
    A period of infinity is cyclical /s
  - nyanpasu64 3y ago
    One interesting property of Hertz is that it's partially appropriate for describing sampling rates. I've found use in annotating separate units for "cycles/sec" and "samples/sec" so I avoid mixing them up (for example, the Nyquist rate can be expressed as 0.5 cyc/smp).
- ericbarrett 3y ago
  I've seen older literature that just uses "cycles," "kilocycles," and so forth. It's never abbreviated, so I always assumed the use of a named unit came from the desire to be able to write shorthand like kHz.

smitty1e 3y ago

This was interesting, but I was expecting something more along the lines of base units => https://en.wikipedia.org/wiki/International_System_of_Units https://en.wikipedia.org/wiki/International_System_of_Units

HPsquared 3y ago
Dimensions are a level below units.
- smitty1e 3y ago
  Ah, so: "dimensions are the abstract theory of units", then?
  - cgadski 3y ago
    In the formalism of both this article and https://news.ycombinator.com/item?id=37517118 https://news.ycombinator.com/item?id=37517118, "base units" are specially designated physical quantities, while "dimensions" describe how physical quantities scale under changes to the scale of measurement. Customarily, capital letters like M, T, L, etc., refer to dimensions, and you would something like [1 kg] = M to say that 1 kg is a quantity that scales like a mass. A PLT analogy analogy would be to compare physical quantities to values and dimensions to types. In practice we can refer to dimensions by base units, but it has some mathematical benefits to view them as distinct. Hope that clarifies things!
    - smitty1e 3y ago
      It does, thank you.

thsksbd 3y ago

This is criminally under taught and under used

maxbond 3y ago
Once I casually mentioned dimensional analysis/quantity calculus to a programmer who worked at CERN, doing scientific computing. They had never heard of it. They had already worked out the hard way that you could do this stuff, but they thought of it as a hack. They were pumped to find out there was a theory behind it and it wasn't just ad-hoc duct taping. That really surprised me. I'm not that smart or educated. I think they either had a masters degree or were a master's student (can't recall precisely).
- thsksbd 3y ago
  Myself i saw it once in a one semester fluid class for two lectures. A class on the Buckingham pi theorem and another on using units to hack simple physical models. To be fair, fluid engineers and the like see this a lot though (because of All their dim-less numbers). But ECE, physics, CS, chemistry etc majors? Almost nada.
devnullbrain 3y ago
That makes it an advantage if you learn it early enough. Some physics questions all the way up to college can be trivialised by understanding the units of the inputs and answer.
- thsksbd 3y ago
  You can trivialize a lot past grad school with this.
hirundo 3y ago
I oppose making a failure to teach dimensional analysis with the physical sciences a crime, but it fits nicely as a tort of professional negligence, aka malpractice. It might be tricky to get that by the teachers unions though.

10000truths 3y ago

Dimensional analysis is useful when you want to convert between compositions of units. A good example is comparing the price of petrol where different units are being used. If you know the conversion rate between litres and gallons, and the conversion rate between USD and GBP, then you can figure out the conversion rate between USD/gal and GBP/L: USD/gal * GBP/USD * gal/L = GBP/L 1.24 USD = 1 GBP 1 gal = 3.78541 L GBP/USD = 1.24 gal/L = 3.78541 USD/gal * 1.24 * 3.78541 = GBP/L USD/gal * 4.6939084 = GBP/L $4.6939084/gal = £1/L

HPsquared 3y ago
Agreed, it's super useful to write out all the units and check that they cancel out properly. I usually write out the Witt I want to convert, complete with units, and successively multiply by dimensionless conversion factors which equal 1. That is to convert 45 MPG to "km/l" I'd write out: 45 (mile / UK gallon) * (1 UK gallon / 4.55 l) * (1.6 km / 1 mile) The expression (1.6 km / 1 mile) is dimensionless and equal to 1 (it's a length divided by an equal length). Cancelling the units gives (45 / 4.55 * 1.6) km/l
- sokoloff 3y ago
  I taught an hour-long class at work where I'd prepped an example to warm the group up, which involved converting the sales material for a US car that got 26 mpg to a Canadian market using liters per 100 km. 26 mpg is carefully chosen because between 1/3 and 2/3 (usually on the lower end) would get it right (~9 liters per 100km) and the others would often come up with around 11 (km per liter). I could then ask "who got a figure around 9?" and "who got a figure around 11?" and both groups would raise their hand feeling pretty confident they'd gotten it right. I'd then introduce and walk them through dimensional analysis and finish with an example that was more complex and error-prone, but was pretty straightforward if you just applied DA. (The main point of the class was "use correct units" rather than "learn dimensional analysis".)
  - stkdump 3y ago
    The great thing about that conversion is that due to the reciprocal relationship the same conversion formula works both ways, so you don't have the risk of converting the wrong way.

maxbond 3y ago

Check out this NIST technical note on implementing quantity calculus (the broader topic in which dimensional analysis resides - it has nothing to do with integration or differentiation, don't let the 'calculus' part turn you off) in software. https://www.nist.gov/publications/architecture-software-assisted-quantity-calculus https://www.nist.gov/publications/architecture-software-assi...

unchocked 3y ago

Best one-lesson favor I ever got was dimensional analysis in high school. Do it, and most basic physics is trivial.

ttoinou 3y ago

Great. Now let's talk about the addition operation not being available for some units / dimensions like Temperature, Density, Speed etc.

sheepshear 3y ago
How would you convert between Kelvin and Celsius if temperatures couldn't be added?
- defrost 3y ago
  What does it mean to add temperatures though (serious question)? Eg: Suppose (one of many subtly different scenarios) you have 1 kg of iron at a constant temperature of 20 C and 5 kg of Aluminium at a constant temperature of 15 C. What does it mean to add those temperatures?
  - sheepshear 3y ago
    You tell me. Units help describe what you modeled, they can't stop you from modeling nonsense. I heated up your iron by 4 degrees and now it's 20C+4C=24C. I modeled the physical process of heating and the units provide some additional information about that.
    - defrost 3y ago
      > I heated up your iron by 4 degrees and what happened when you applied the same amount of heat to the greater mass of Aluminium? What's the difference between intensive properties (tempreture, pressure, density) and extensive properties, such as mass or volume?
      - sheepshear 3y ago
        Again, you tell me. Units have no meaning without the context of what you're modeling. I can model the average temperature of your pieces of metal as (20C+15C)/2=17.5C.

Remnant44 3y ago

Shout out to https://instacalc.com/ https://instacalc.com/ . It's like a simple spreadsheet or calculator but with awesome support for dimensional analysis -- you can attach any units you want to a quantity and it will track it throughout the calculations, and do a best-faith effort to figure out conversions / "what you meant". For example, here's a quick scratchpatch I used for calculating memory bandwidth / CPU transfer rates: https://instacalc.com/56842 https://instacalc.com/56842

kingkongjaffa 3y ago

I learned this in fluid dynamics class for learning about non dimensional value that are used like Reynolds numbers, but more broadly it was a bit like a cheat code because often we had an exam question with units of inputs and then the final answer in another set of units, and even without knowing the correct equation you could with a little intuition, derive the correct solution by combining the input units in fun ways, Understanding the MLT system in this way made most of mechanical engineering much more straightforward, and I used this almost every day up to my masters degree in computational fluid dynamics.

touisteur 3y ago

I always go back to the reference https://gmpreussner.com/research/dimensional-analysis-in-programming-languages https://gmpreussner.com/research/dimensional-analysis-in-pro... for programming language support. It's updated from time to time, and incorporated the work done on Ada in the 2010s. Good read.

amai 3y ago

Why do we have square meters but not square seconds?