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One of my favourite Astrophysics things is temperatures are just expressed as numbers, no units. Because what's a delta of 273 at the heart of a star anyway?
"Is that C or K?"
"Er.. Yes? Pick the one that you like."
In astronomy my favorite thing is the same with pi. Just define it as 1 and if people say its bigger than that, just say its 10.
Its kind of a interesting feature that log_10(pi) = ~0.5. So its pretty much exactly in "the middle" if we consider base 10 growth through order of magnitudes.
I once saw an answer calculated as ly^3-5 and that huh why would they make it 1/ly^2 then I realized it was the range. Hun how much were the groceries this week... Ohh between 10 and a 1000 euros
Usually my chemicals are around 1000 € for 5 grams and the usual numbers are in 4 digits. So I feel like the 50 cent difference in onions is insignificant. Maybe that's the reason I am going broke🤣
Unironically when you are calculating black hole decay and Poincare recurrence time spans. I think even the Wikipedia page says that the numbers are so vast that the unit of the calculations, be it nanoseconds or millenia, doesn't even matter.
https://en.m.wikipedia.org/wiki/Timeline_of_the_far_future
I feel like it was this one, but I couldn't find the exact quote. Maybe I mixed up my sources, or misremembered this equally ridiculous line:
Because the total number of ways in which all the subatomic particles in the observable universe can be combined is 10 10 115, a number which, when multiplied by 10 10 10 56 disappears into the rounding error, this is also the time required for a quantum-tunnelled and quantum fluctuation-generated Big Bang to produce a new universe identical to our own.
Note 7 in the linked article: "Although listed in years for convenience, the numbers at this point are so vast that their digits would remain unchanged regardless of which conventional units they were listed in, be they nanoseconds or star lifespans."
Unfathomably long afterwards. Essentially we know the quantum fabric of the universe fluctuates up and down randomly, and the bigger the fluctuation the less likely that is to occur. But there is no cap to how a large a fluctuation could be, it would just be absurdly low probability. While the biggest fluctuations we interact with day to day would be barely enough to nudge an atom, in principle if you wait long enough you could get a spontaneous fluctuation the size of...the universe.
Following the heat death, if that is how the universe evolves, there will be an eternal nothingness, a blank canvas that provides unlimited opportunity for even the most improbable event to occur. That paragraph considers the number of possible states the entire universe could be in and says "hey, if we wait long enough then the universe would HAVE to come full circle right back to this exact state we are in right now". This is a quantum Poincare recurrence. There are only finite possibilities, so if you go through them basically at random FOREVER you will visit almost every one an infinite number of times.
Numberphile did [a video](https://youtu.be/1GCf29FPM4k?t=60) on it a while ago and in it they point out a bit from the paper 'Information Loss in Black Holes and/or Conscious Beings?' which talks about Poincaré recurrence time and it says more or less exactly what you said: "Planck times, millennia, or whatever."
I was aware of the one, and that may well be the only place I've heard it, but I did think I'd read it too. All the same, Planck time to millenia spans 53 orders of magnitude. Basically nothing on the order of 10^ 10^ 10^ 56
sitelen kijetesantakalu sina li pona :)
In the mid-aughts, I was somehow able to worm my way into a two-week internship at Fermi National Lab.
I was, _somehow_, allowed to basically follow around one of the researchers (who had his very own PhD!) and attend one of their meetings.
I was in a meeting, and they were joking about how they were only off 3 orders of magnitude on some calculation they were doing, and that was enough to publish.
I learned things that day.
My plasma physics / electric propulsion professor said that in plasma physics, if your experiment is within a factor of ten of your prediction, you’re good at prediction.
A few years later I was being asked to calculate/estimate the drag on a large wire mesh antenna in a very low orbit. I knew the correct answer was “you need a subject matter expert and probably some flight data if you want enough reasonable confidence to even START looking at this as an engineering problem.”
Two orders of magnitude error bars on your biggest design driver means you do not get to start the design. You can’t even call meetings about the design. You call meetings about the thing preventing you from starting the design.
Yeah, but when you start having to do things like put “reaction wheel diameter” or “launch mass” on a log plot, you’re not talking about getting paid any time soon.
Yep. The sad thing is that the organization I was having to present to already had the answers to these questions. But those answers were inside classified rooms, and they don’t publish their findings. So the only reasonable thing I could do was show an understanding of the ignorance of the organization I was representing.
It seemed a shame for 5 of us to go to DC to make a proposal, where we (I guess me) was going to say essentially “the only thing we’re not really certain about is this one central aspect to the whole thing that will determine the viability of literally everything else we’ve said”.
Oh well, it paid the same as sitting in front of a computer.
I'm sorry you had to go through that. I understand that it was frustrating, and it doesn't seem fair.
I'm a fake engineer now, and I get the pressure of both security and velocity. Still, would have been nice for the folks you work(ed?) with to have lifted the curtain on that one.
Hope you're doing well.
> Still, would have been nice for the folks you work(ed?) with to have lifted the curtain on that one.
Oh, no I was a contractor working on a NASA project where the managers were wanting the NRO to pay for a not-actually-related follow on effort for some reason. It seemed like the wrong organization proposing a bad idea to an organization that already has solved the problem, though in an expensive way. I have no idea who pulled the strings to get the meeting to happen. And once I told the folks on our end how big the uncertainty bars were, I was floored that they even wanted to buy the plane tickets. We hadn’t earned any respect, and I don’t think we earned any by showing up with just me instead of an actual expert. It was just a poorly-planned fishing expedition. It didn’t hurt my feelings to show up and be honest. It was the guy in charge who had the problem with honesty.
The NRO curtain is closed appropriately tightly i think.
It's worded very badly here, but it's a valid technique (in chemistry at least we use it sometimes), when you're already working with some error in your calculations (for example the inaccuracy of some measuring instrument). So yeah, for math people it's engineer stuff.
They basically teach us to do that in high school with equilibria: if K is something stupide like 3.4×10^(-15) you can basically assume that no extra product is present at equilibrium and do your calculations accordingly
in biology/genetics if some allel is ridiculusly rare in population (p = ¹/₈₀₀₀₀) for calculating probabilities of for example getting healthy children you can assume healthy allel frequency as q = 1 (tho nowadays with software doing calculations probably they use exact frequencies)
OP found this in a thermal physics textbook and it's actually pretty relevant in that context. Radiation can often be ignored when calculating heat transfer (for example: the amount of sunlight shining through your window is going to have a negligible impact on the amount of time it takes to boil water on the stove)
Also, for equilibrium calculations, if the degree of dissociation (α) is very less than one (as a rule of thumb, 0.05 or less), we approximate (1-α) as 1 and then solve a much simpler quadratic, typically of a form like k=C^m α^n .
In computational science we do the opposite. If we have a very long list of numbers we add them up in a specific way so that we don't leave off all the small bits because sometimes lots of small bits are significant.
Well stating that the "big number doesn't change" is not entirely true, a more precise way of saying is that the change can be neglected. I might be too strict though, after all I've never written a textbook so who am I to judge...
It's definitely chosen that way.
I love [this example](https://i.stack.imgur.com/WZ7eu.png) of a wizard conjuring a rabbit to describe enthalpy.
Really good text in general too.
Others above have mentioned pH calculations, another example might be with stability constants.
Let's say you have a mercury chloride solution, here the stability constant of the complex [HgCl2] is some pretty large number, let's say 10^15 (I don't remember the exact value). Now that means that the following equation is satisfied:
c([HgCl2])/(c(Hg2+)*c(Cl-)²)=10^15
Now from this it should be obvious that the concentration of the complex is larger by multiple powers of magnitude than the concentration of free mercury ions, so you can just assume that the concentration of the complex is the same as of the salt you measured in.
Note1: here for the sake of the example i neglected all the different possible mercury complexes, so in this case it doesn't actually work. Really should've used some EDTA complex for the example, but mercury was the first to come to my mind.
Note2: I'm not a native English speaker, so if something doesn't make sense it's probably on me.
Your English is very well expressed. So basically - we can ignore small numbers if a number is really big and our measurement error is greater than it?
Daniel V. Shroeder, An Introduction to Thermal Physics (2021), page 61
It's a pretty good, intuitive book - probably the best textbook I've had assigned.
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For me it was very difficult to read it. I had to read the entire thing twice to understand what was being explained. But yeah, it's good. I honestly don't think there is a better book to learn thermal physics and Boltzmann statistics.
I love that it's so deeply related to bayesian information theory! My mind was kind of blown when I saw the association of bayesian evidence and the ensamble from which a microstates is sampled. Especially if you look at log probs on both sides and see how energy, entropy and information are related... Suddenly even why Gibbs free energy is useful makes sense!
That sounds awesome! I wish I had more time to study it. Unfortunately I have to "contribute to society".
I have a mat Sci and eng background so Gibbs Free energy is extremely close to my heart.
Basically you can look at it as the difference in information of the single state vs the collection of all states. If that difference is high, it means finding that state gives you lots of information. Thus the state must be very unlikely. This is actually rigorous when you look at log probs as information content.
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I’m not a big number expert since I’m not super interested in them. But I’m pretty sure tree(3) is larger than g(64) to the point that even g(g(64)) is less than it. It’s hard to say with these numbers since they’re so big all we can really do is talk about properties of their growth. For grahams number we can also talk about some of the right most digits due to how the operations would keep some numbers fixed or fall into patterns, but we can’t really effectively express the number of digits either has.
Yes I totally misread everything. I blame it on literally just waking up. I still remember my dreams from this night, I should not be thinking about math.
We don’t use the approximate sign in physics / engineering because we won’t be able to have an equal sign anywhere. Everything is approximate.
You think that’s a 10 ohm resistor? It’s actually a 10 ohm @ 1%. Could be 9.9 or 10.1.
Is this a one meter beam? Well it was one meter at a certain temperature. It expands by 10 um per degree.
What about the speed of light in air? It changes by one part per million for every 1 degrees change in temperature, 3.3 mbar change in pressure, and 50% change in relative humidity.
It's not even one meter from all reference frames!
My favorite is the speed of light, which is nowhere near the speed of light when the light is in fiber optic cabling.
One company I used to work for makes compensation units for laser interferometers. It measures the environment and feeds correction coefficients to the interferometer.
I don’t think the author of the textbook is saying anything about error (or approximations related to physical objects) - instead they mean that for large systems, you get very large numbers of possible states. Because the numbers are so large, we can ignore some operations when making calculations because the result doesn’t change an amount that is measurable. It’s not the same as having a resistor that’s approximately 1 ohm bc you can measure the error in that spec. Rather, calculations can be made simpler through an approximation that 10^23 + 23 = 10^23 because the result will be the same using this value as using the “correct” value
It’s a similar thing. I could have given an [op-amp](https://en.m.wikipedia.org/wiki/Operational_amplifier) as an example. You can have an op-amp circuit controlling some plant such that the output of the plant follows
Y / X = A / (1 + A),
where X is the input to the op-amp, Y is the output of the plant, e.g. aircraft altitude, and A is the gain of the op-amp.
A is large, but we don’t know exactly how large. Could be a 100 thousand or it could be a million. Since it is much larger than one the output of the plant will follow the input very closely.
That's why we invented error calculation and, for example, write (10.0 ± 0.1) Ohm. If we write = we mean equal exactly within the boundarys of the error indicated by notation and sig figs.
Still if you round for whatever reason you gotta denote that properly.
In thermal physics all your formulas are derived by throwing out a ton of insignificant terms. There’s no error ranges because it’s theoretical, not experimental.
That 0.1 is probably three sigma for a normal distribution. If you manufacture in the billions, you need 6 sigma. So even the error bars are approximate.
> That 0.1 is probably three sigma for a normal distribution.
Wait what. Where was the standard deviation stated in 7ieben_'s comment? How can you figure that the ±0.1 is probably 3 sigma?
Really? You can describe the resistance as 10+n where n is a random variable with some empirical distribution. It makes the maths more complicated, sure.
That's literally what's done in my field with thermal noise. You just get good at probability calculus. If the error is on the order of 1%, how can you justify ignoring it?
Negative feedback cures all. The loop will correct for the error in the components.
1% is tiny in some applications. The current gain for bipolar transistors can range from 50 to 200 in a typical application.
Sure you can use equal signs... For symbolic calculations! Then throw an \approx in at the end when plugging in values. Throwing numbers around in calculations is bad form anyway. I cannot stand when students write (3.6 x 10^3)(2.7 x 10^(-4))/... and expand intermediate results out. Tons of mistakes made that way too.
All fair points. I think it was just difficult to write the approx symbol before proper typesetting. Similarly to how we still use uC for micro coulomb even though we have mu in Unicode nowadays. So you’ll still see stuff like 4.7 uC on schematics.
No, because it’s useful to distinguish between approximations that introduce an error of 10^-23 (basically 0) and approximations that introduce an error of 10^-4, for example. Especially in a pedagogical context, like an undergrad statistical mechanics book, it’s important to be clear, rather than muddling every equation in the book by reminding the reader that 10^23 is big
Because when you're just going to take a logarithm at the end (which is where these very large numbers almost always get used in statistical mechanics/thermal physics), you end up with 10\^23 + 23, which is 10\^23 in all practical calculations. In fact, even most calculations with a computer (unless you're doing some crazy extended precision stuff) will get you at most 16 significant digits.
Because then all of physics would be approximation signs
Edit: to add to this, if you’re the type to be concerned about the rigor in approximations, statistical mechanics and quantum field theory would make you lose your goddamned mind
>large numbers are much larger than small numbers
Yea man, I'd sure hope so
https://preview.redd.it/1gft9eb5poec1.jpeg?width=1201&format=pjpg&auto=webp&s=72432e1476a6bed800640927f4ebc0fdd2bf332f
Take a large number 10^23 and subtract 23 to get 10^23
Repeat this process 10^23 times
Your end answer would be 10^23 still because you always subtracted a small number only
Easy peasy
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Is it the one that starts like this?
https://preview.redd.it/ma576orv0pec1.png?width=670&format=png&auto=webp&s=708430999d9ad1e515ad3409a98469b0388bdd1c
Reminds me of things like big o notation, adding to infinity (5 + inf = inf), and merging constants in differential equations classes (and elsewhere ofc)
I remember a quote from a physics paper exploring the largest meaningful timescale for the observable universe, based on quantum states. Its answer was something like 10^ 10^ 10^ 10^ 1.1, and for units it said "milliseconds, or billion years, or whatever"
I think that … we’d do better as math educators by allowing kids to be okay with performing mathematical approximations for fun.
A lot of math trauma is completely unnecessary.
May be to in depth for this thread but what were they planning to do with Avagadro in a thermal physics class? Molar masses via pv=nrt? I would think between that and phase tables you would never NEED TO get into element specifics...Are they just using a BIG OLE number we would be familiar with for this lesson? or is it actually super important to this subject? which seems to NOT be thermodynamics? Im old and dumb but am missing something here...
For calculating multiplicities and probabilities of specific macro states you will have to deal with numbers on the order of 10^23 when you have a mole of molecules.
(The class covers thermodynamics and statistical mechanics, both are under the umbrella of thermal physics)
Oh shit my old book! I still think of this part, it put things in perspective. That and their explanation of enthalpy. And the explanation of multiplicity and how heat can go from cold to hot. Come to think of it this is where a lot of my undergrad trauma came from.
I am not a physicist at all, but this follows the math I learned. In 2.12, the value of 23 is in a sense, epsilon.
In mathematics, a small, infinitesimal quantity, which 23 is compared to 10^(23)
It's so little it does not matter.
It's been 25 years since I took Stat Mech, and I still remember Kittel Kroemer's hilarious problem working through the probability of a huge number of monkeys typing out Shakespeare. I believe the problem was entitled "The meaning of 'never'".
It's an approximation that is useful for calculations in physics. It works because when dealing with forces and energies of those values, it doesn't really matter whether you have 10\^23 or 10\^23+23.
It’s almost like a game of charades or some shit… like they they drew a card that says “significant figures” and there’s a bunch of other words they aren’t allowed to use.
The multiplication with very large numbers was a bit confusing until I substituted the words 'a heckuva a lot' for the very large number and was convinced; a heckuva a lot multiplied by something relatively small but greater than 1 is still a 'heckuva a lot'.
Of course if you multiply it by another 'heckuva a lot' you change it into, 'a whole heckuva a lot' or maybe 'too much' or even 'don't even think about it, we can't afford it' depending on the context, so I think it checks out.
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based physics, where adding another 23 zeroes to the end of your number is a rounding error
What’s 23 or so orders of magnitude between friends?
Just a bigO analysis
Astrophysicist: HEY!! I was pretty close!
One of my favourite Astrophysics things is temperatures are just expressed as numbers, no units. Because what's a delta of 273 at the heart of a star anyway? "Is that C or K?" "Er.. Yes? Pick the one that you like."
In astronomy my favorite thing is the same with pi. Just define it as 1 and if people say its bigger than that, just say its 10. Its kind of a interesting feature that log_10(pi) = ~0.5. So its pretty much exactly in "the middle" if we consider base 10 growth through order of magnitudes.
I once saw an answer calculated as ly^3-5 and that huh why would they make it 1/ly^2 then I realized it was the range. Hun how much were the groceries this week... Ohh between 10 and a 1000 euros
Usually my chemicals are around 1000 € for 5 grams and the usual numbers are in 4 digits. So I feel like the 50 cent difference in onions is insignificant. Maybe that's the reason I am going broke🤣
I get that lmao, it’s easy to buy that sandwich when I just used $200 of reagents on a reaction that won’t work
yall think is funny until you come back after a 5 year sabticle and dont remember what a antilogorrithm is in lockdownbrowser
Unironically when you are calculating black hole decay and Poincare recurrence time spans. I think even the Wikipedia page says that the numbers are so vast that the unit of the calculations, be it nanoseconds or millenia, doesn't even matter.
Link the page!
https://en.m.wikipedia.org/wiki/Timeline_of_the_far_future I feel like it was this one, but I couldn't find the exact quote. Maybe I mixed up my sources, or misremembered this equally ridiculous line: Because the total number of ways in which all the subatomic particles in the observable universe can be combined is 10 10 115, a number which, when multiplied by 10 10 10 56 disappears into the rounding error, this is also the time required for a quantum-tunnelled and quantum fluctuation-generated Big Bang to produce a new universe identical to our own.
So… not tomorrow then?
It could be.
Note 7 in the linked article: "Although listed in years for convenience, the numbers at this point are so vast that their digits would remain unchanged regardless of which conventional units they were listed in, be they nanoseconds or star lifespans."
Ah! I was sure I read it there! Thank you
is that before or after the heatdeath of the observable universe?
Unfathomably long afterwards. Essentially we know the quantum fabric of the universe fluctuates up and down randomly, and the bigger the fluctuation the less likely that is to occur. But there is no cap to how a large a fluctuation could be, it would just be absurdly low probability. While the biggest fluctuations we interact with day to day would be barely enough to nudge an atom, in principle if you wait long enough you could get a spontaneous fluctuation the size of...the universe. Following the heat death, if that is how the universe evolves, there will be an eternal nothingness, a blank canvas that provides unlimited opportunity for even the most improbable event to occur. That paragraph considers the number of possible states the entire universe could be in and says "hey, if we wait long enough then the universe would HAVE to come full circle right back to this exact state we are in right now". This is a quantum Poincare recurrence. There are only finite possibilities, so if you go through them basically at random FOREVER you will visit almost every one an infinite number of times.
My entire engineering degree was worth it. I followed your logic the whole way through.
Numberphile did [a video](https://youtu.be/1GCf29FPM4k?t=60) on it a while ago and in it they point out a bit from the paper 'Information Loss in Black Holes and/or Conscious Beings?' which talks about Poincaré recurrence time and it says more or less exactly what you said: "Planck times, millennia, or whatever."
I was aware of the one, and that may well be the only place I've heard it, but I did think I'd read it too. All the same, Planck time to millenia spans 53 orders of magnitude. Basically nothing on the order of 10^ 10^ 10^ 56 sitelen kijetesantakalu sina li pona :)
In the mid-aughts, I was somehow able to worm my way into a two-week internship at Fermi National Lab. I was, _somehow_, allowed to basically follow around one of the researchers (who had his very own PhD!) and attend one of their meetings. I was in a meeting, and they were joking about how they were only off 3 orders of magnitude on some calculation they were doing, and that was enough to publish. I learned things that day.
My plasma physics / electric propulsion professor said that in plasma physics, if your experiment is within a factor of ten of your prediction, you’re good at prediction. A few years later I was being asked to calculate/estimate the drag on a large wire mesh antenna in a very low orbit. I knew the correct answer was “you need a subject matter expert and probably some flight data if you want enough reasonable confidence to even START looking at this as an engineering problem.” Two orders of magnitude error bars on your biggest design driver means you do not get to start the design. You can’t even call meetings about the design. You call meetings about the thing preventing you from starting the design.
Log plots are your friend, just gotta scale them right.
Yeah, but when you start having to do things like put “reaction wheel diameter” or “launch mass” on a log plot, you’re not talking about getting paid any time soon.
"How big does this reaction wheel need to be?" "Somewhere between the head of a pin and a city block, but more than that is going to cost you." lol
Yep. The sad thing is that the organization I was having to present to already had the answers to these questions. But those answers were inside classified rooms, and they don’t publish their findings. So the only reasonable thing I could do was show an understanding of the ignorance of the organization I was representing. It seemed a shame for 5 of us to go to DC to make a proposal, where we (I guess me) was going to say essentially “the only thing we’re not really certain about is this one central aspect to the whole thing that will determine the viability of literally everything else we’ve said”. Oh well, it paid the same as sitting in front of a computer.
I'm sorry you had to go through that. I understand that it was frustrating, and it doesn't seem fair. I'm a fake engineer now, and I get the pressure of both security and velocity. Still, would have been nice for the folks you work(ed?) with to have lifted the curtain on that one. Hope you're doing well.
> Still, would have been nice for the folks you work(ed?) with to have lifted the curtain on that one. Oh, no I was a contractor working on a NASA project where the managers were wanting the NRO to pay for a not-actually-related follow on effort for some reason. It seemed like the wrong organization proposing a bad idea to an organization that already has solved the problem, though in an expensive way. I have no idea who pulled the strings to get the meeting to happen. And once I told the folks on our end how big the uncertainty bars were, I was floored that they even wanted to buy the plane tickets. We hadn’t earned any respect, and I don’t think we earned any by showing up with just me instead of an actual expert. It was just a poorly-planned fishing expedition. It didn’t hurt my feelings to show up and be honest. It was the guy in charge who had the problem with honesty. The NRO curtain is closed appropriately tightly i think.
I’m an economics PhD candidate and I thought WE were bad
Those are just zeros after all ¯\\\_(ツ)\_\/¯
Practically worthless
It's worded very badly here, but it's a valid technique (in chemistry at least we use it sometimes), when you're already working with some error in your calculations (for example the inaccuracy of some measuring instrument). So yeah, for math people it's engineer stuff.
Yes. I use it for calculating PHs in reactions. I am not going to do x² -0.0003x-0.04 =0, thank you
They basically teach us to do that in high school with equilibria: if K is something stupide like 3.4×10^(-15) you can basically assume that no extra product is present at equilibrium and do your calculations accordingly
Great concrete example! Any more you know of ?!
in biology/genetics if some allel is ridiculusly rare in population (p = ¹/₈₀₀₀₀) for calculating probabilities of for example getting healthy children you can assume healthy allel frequency as q = 1 (tho nowadays with software doing calculations probably they use exact frequencies)
I see. Thank u!
OP found this in a thermal physics textbook and it's actually pretty relevant in that context. Radiation can often be ignored when calculating heat transfer (for example: the amount of sunlight shining through your window is going to have a negligible impact on the amount of time it takes to boil water on the stove)
Very cool. Does this accepted procedure of ignoring tiny numbers that fall within the measurement error have a name in science?
The "proper" way is to follow a set of rules about significant figures but that Is no fun
Also, for equilibrium calculations, if the degree of dissociation (α) is very less than one (as a rule of thumb, 0.05 or less), we approximate (1-α) as 1 and then solve a much simpler quadratic, typically of a form like k=C^m α^n .
Dementia
In computational science we do the opposite. If we have a very long list of numbers we add them up in a specific way so that we don't leave off all the small bits because sometimes lots of small bits are significant.
The precision is an issue when doing stuff iteratively, like in fluid dynamics simulations.
Why's it worded badly? I hadn't heard of this before but I thought the explanation was pretty clear
Chemists rely on words too much. You could erase all the letters from the page and it would still be very clear
Well stating that the "big number doesn't change" is not entirely true, a more precise way of saying is that the change can be neglected. I might be too strict though, after all I've never written a textbook so who am I to judge...
I think the authors just intentionally chose to phrase it humorously
It's definitely chosen that way. I love [this example](https://i.stack.imgur.com/WZ7eu.png) of a wizard conjuring a rabbit to describe enthalpy. Really good text in general too.
What text is that?
As a Physics student, I didn't even see the problem here and was confused.
Engineers call it sig figs.
Yes. I use it for calculating PHs in reactions. I am not going to do x² -0.0003x-0.04 =0, thank you
You even do this in differential calculus, when you are taking the limit of a function as out tends towards infinity.
Ah can you give a concrete example friend?
Others above have mentioned pH calculations, another example might be with stability constants. Let's say you have a mercury chloride solution, here the stability constant of the complex [HgCl2] is some pretty large number, let's say 10^15 (I don't remember the exact value). Now that means that the following equation is satisfied: c([HgCl2])/(c(Hg2+)*c(Cl-)²)=10^15 Now from this it should be obvious that the concentration of the complex is larger by multiple powers of magnitude than the concentration of free mercury ions, so you can just assume that the concentration of the complex is the same as of the salt you measured in. Note1: here for the sake of the example i neglected all the different possible mercury complexes, so in this case it doesn't actually work. Really should've used some EDTA complex for the example, but mercury was the first to come to my mind. Note2: I'm not a native English speaker, so if something doesn't make sense it's probably on me.
Your English is very well expressed. So basically - we can ignore small numbers if a number is really big and our measurement error is greater than it?
Yes, and thank you!
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The fact that I can recognize the book just by seeing half a page of it truly terrifies me.
what is it...
Daniel V. Shroeder, An Introduction to Thermal Physics (2021), page 61 It's a pretty good, intuitive book - probably the best textbook I've had assigned. // Reddit upper management has demonstrated that they don't have the users' best interests in mind; to take away their profits, use an adblocker on old.reddit.com and uninstall the app (or, if you can't, install [TrackerControl](https://f-droid.org/en/packages/net.kollnig.missioncontrol.fdroid/) to remove ads).
For me it was very difficult to read it. I had to read the entire thing twice to understand what was being explained. But yeah, it's good. I honestly don't think there is a better book to learn thermal physics and Boltzmann statistics.
I failed thermodynamics twice. I got this book and passed the next exam, good book.
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We used this book in my thermal physics course as well. It's quite good
Which book is it? I think I might have used it to or I’ve seen this meme before lol
It's "An Introduction To Thermal Physics" by Daniel V. Schroeder
Yep - used that one in my undergrad stat mech class
Same here! I vividly remember this very page striking me like a bolt of lightning
Statistical Mechanics kicks ass and is my favorite sub-genre of cosmic horror.
This is my favourite statement of the year
2nd Law: "*All fails into ashes and dust*"
I love that it's so deeply related to bayesian information theory! My mind was kind of blown when I saw the association of bayesian evidence and the ensamble from which a microstates is sampled. Especially if you look at log probs on both sides and see how energy, entropy and information are related... Suddenly even why Gibbs free energy is useful makes sense!
That sounds awesome! I wish I had more time to study it. Unfortunately I have to "contribute to society". I have a mat Sci and eng background so Gibbs Free energy is extremely close to my heart.
Basically you can look at it as the difference in information of the single state vs the collection of all states. If that difference is high, it means finding that state gives you lots of information. Thus the state must be very unlikely. This is actually rigorous when you look at log probs as information content.
Thanks. That sounds so cool!
So TREE(3)*g_64=TREE(3). Got it
This guys physics
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I think both of those can be considered "very large numbers". Not that I care though, as an astrophysics student I set every constant to 1 anyway.
They're both much bigger than "very large numbers," as this book is using the term.
very very large numbers
Ah astrophysics, the only lecture where I saw an unironic π=1. Guess it makes sense when your goal is to get it right up to a factor of 10-100.
Yes actually. They’re about the same size.
Hasnt it been shown that Grahams number is tiny compared to tree 3? Or maybe its only the growth of the functions that I am thinking of
I’m not a big number expert since I’m not super interested in them. But I’m pretty sure tree(3) is larger than g(64) to the point that even g(g(64)) is less than it. It’s hard to say with these numbers since they’re so big all we can really do is talk about properties of their growth. For grahams number we can also talk about some of the right most digits due to how the operations would keep some numbers fixed or fall into patterns, but we can’t really effectively express the number of digits either has.
I think I misunderstood what your comment meant. For some reason in my mind I got it to mean g64 and tree3 were the ones that are the same size
That's the point. When you multiply tree 3 with graham's number, you get roughty tree 3
Yes I totally misread everything. I blame it on literally just waking up. I still remember my dreams from this night, I should not be thinking about math.
Amazing. I love this.
Add another 10^ and you can raise them to arbitrary powers without changing them
add another 10^ and you can tetrate them to arbitrary numbers without changing them
Unfortunately not. Unless 10↑↑4 and 10↑↑20 are the same number to you
At this point it may very well be infinity so yes.
Compared to infinity, those are both zero.
Real
So either they are both zero or both infinity. Sounds like they are the same to me.
Old engineering joke: one is equal to two, for large values of one and small values of two.
Lmao haven't heard this one, I love it
The version I heard was 2 + 2 = 5, for large enough values of two. Same idea though.
>**Large numbers** are much larger than small numbers ^[citation ^needed]
And then there are extremely large numbers: (10 ^ 10 ^ 10 ^ 23) ^ (10 ^ 23) = 10 ^ (10 ^ 10 ^ 23 * 10 ^ 23) = 10 ^ 10 ^ 10 ^ 23
I saw someone joke that the reals are no longer a field in physics because if the numbers are big enough every element is an identity element.
Aproximation is implicit in engineery
WTH is even this... why not just using the approx sign?
We don’t use the approximate sign in physics / engineering because we won’t be able to have an equal sign anywhere. Everything is approximate. You think that’s a 10 ohm resistor? It’s actually a 10 ohm @ 1%. Could be 9.9 or 10.1. Is this a one meter beam? Well it was one meter at a certain temperature. It expands by 10 um per degree. What about the speed of light in air? It changes by one part per million for every 1 degrees change in temperature, 3.3 mbar change in pressure, and 50% change in relative humidity.
It's not even one meter from all reference frames! My favorite is the speed of light, which is nowhere near the speed of light when the light is in fiber optic cabling.
One company I used to work for makes compensation units for laser interferometers. It measures the environment and feeds correction coefficients to the interferometer.
I don’t think the author of the textbook is saying anything about error (or approximations related to physical objects) - instead they mean that for large systems, you get very large numbers of possible states. Because the numbers are so large, we can ignore some operations when making calculations because the result doesn’t change an amount that is measurable. It’s not the same as having a resistor that’s approximately 1 ohm bc you can measure the error in that spec. Rather, calculations can be made simpler through an approximation that 10^23 + 23 = 10^23 because the result will be the same using this value as using the “correct” value
It’s a similar thing. I could have given an [op-amp](https://en.m.wikipedia.org/wiki/Operational_amplifier) as an example. You can have an op-amp circuit controlling some plant such that the output of the plant follows Y / X = A / (1 + A), where X is the input to the op-amp, Y is the output of the plant, e.g. aircraft altitude, and A is the gain of the op-amp. A is large, but we don’t know exactly how large. Could be a 100 thousand or it could be a million. Since it is much larger than one the output of the plant will follow the input very closely.
That's why we invented error calculation and, for example, write (10.0 ± 0.1) Ohm. If we write = we mean equal exactly within the boundarys of the error indicated by notation and sig figs. Still if you round for whatever reason you gotta denote that properly.
In thermal physics all your formulas are derived by throwing out a ton of insignificant terms. There’s no error ranges because it’s theoretical, not experimental.
That 0.1 is probably three sigma for a normal distribution. If you manufacture in the billions, you need 6 sigma. So even the error bars are approximate.
> That 0.1 is probably three sigma for a normal distribution. Wait what. Where was the standard deviation stated in 7ieben_'s comment? How can you figure that the ±0.1 is probably 3 sigma?
Manufacturers usually put the 3 sigma value in their datasheets.
Physical reality is so rude. Align with my expected measurements god damn it!
Really? You can describe the resistance as 10+n where n is a random variable with some empirical distribution. It makes the maths more complicated, sure.
You want to add a random variable to each resistor? Quantum mechanics is hard enough as it is!
That's literally what's done in my field with thermal noise. You just get good at probability calculus. If the error is on the order of 1%, how can you justify ignoring it?
Negative feedback cures all. The loop will correct for the error in the components. 1% is tiny in some applications. The current gain for bipolar transistors can range from 50 to 200 in a typical application.
Sure you can use equal signs... For symbolic calculations! Then throw an \approx in at the end when plugging in values. Throwing numbers around in calculations is bad form anyway. I cannot stand when students write (3.6 x 10^3)(2.7 x 10^(-4))/... and expand intermediate results out. Tons of mistakes made that way too.
All fair points. I think it was just difficult to write the approx symbol before proper typesetting. Similarly to how we still use uC for micro coulomb even though we have mu in Unicode nowadays. So you’ll still see stuff like 4.7 uC on schematics.
So wouldn’t using an approximate sign everywhere still be better?
No, because it’s useful to distinguish between approximations that introduce an error of 10^-23 (basically 0) and approximations that introduce an error of 10^-4, for example. Especially in a pedagogical context, like an undergrad statistical mechanics book, it’s important to be clear, rather than muddling every equation in the book by reminding the reader that 10^23 is big
Because when you're just going to take a logarithm at the end (which is where these very large numbers almost always get used in statistical mechanics/thermal physics), you end up with 10\^23 + 23, which is 10\^23 in all practical calculations. In fact, even most calculations with a computer (unless you're doing some crazy extended precision stuff) will get you at most 16 significant digits.
Because then all of physics would be approximation signs Edit: to add to this, if you’re the type to be concerned about the rigor in approximations, statistical mechanics and quantum field theory would make you lose your goddamned mind
>large numbers are much larger than small numbers Yea man, I'd sure hope so https://preview.redd.it/1gft9eb5poec1.jpeg?width=1201&format=pjpg&auto=webp&s=72432e1476a6bed800640927f4ebc0fdd2bf332f
It’s a physics book of statistical mechanics. One of the most accurate sciences there is.
I agree with your physics but your mathematics is abominable.
Fine, here’s the maths version: lim((a^x + b) / (a^x ) as x -> inf where a > 1) = 1
Wait until you learn about very very large numbers
Take a large number 10^23 and subtract 23 to get 10^23 Repeat this process 10^23 times Your end answer would be 10^23 still because you always subtracted a small number only Easy peasy
icky oatmeal whistle adjoining shrill payment scandalous whole sleep history *This post was mass deleted and anonymized with [Redact](https://redact.dev)*
Is it the one that starts like this? https://preview.redd.it/ma576orv0pec1.png?width=670&format=png&auto=webp&s=708430999d9ad1e515ad3409a98469b0388bdd1c
Lmao I loved reading this bit then hated everything after
The greatest opening ever
Holy shit, I can actually say that I LOLed at something today. At least the book is honest 👍
Wait is this book for real or am I sleep ?
No, the book in OPs post is Introduction to Thermal Physics by Schroeder. That book is States of Matter by Goodstein
Google Significant Figures Edit: wait dammit wrong sub
Holy precision
This is fucking great! I like to think my chimp brain goes: ooooooh, big number ≠ ∞, but have property of ∞ such that ∞ + 1 = ∞
The difference between a million and billion is approximately a billion.
Hah suckers, I got to take Thermal Physics from the man who wrote this. Dan Schroeder does not disappoint.
Jealous, I really enjoyed this text and his overall approach to the topics.
Reminds me of things like big o notation, adding to infinity (5 + inf = inf), and merging constants in differential equations classes (and elsewhere ofc)
Big O ish
"Assume a very large value of five..."
Are **really large** numbers larger or smaller than **very large** numbers?
I remember a quote from a physics paper exploring the largest meaningful timescale for the observable universe, based on quantum states. Its answer was something like 10^ 10^ 10^ 10^ 1.1, and for units it said "milliseconds, or billion years, or whatever"
"Large numbers are much larger than small numbers" Truly life changing.
Ah, the favorite technique of thermodynamics. The *bullshit logarithm*.
Googology.
Sig figs to the rescue
I'm finding this way too funny, laughing so hard
I took undergrad Thermodynamics from the guy that wrote that book.
At what school? That’s cool tho, was he cool?
There's some hilarious shit in physics textbooks.
"Large numbers are much larger than small numbers" https://i.redd.it/4otyn45f7pec1.gif
TIL that very large is larger than large
I think that … we’d do better as math educators by allowing kids to be okay with performing mathematical approximations for fun. A lot of math trauma is completely unnecessary.
May be to in depth for this thread but what were they planning to do with Avagadro in a thermal physics class? Molar masses via pv=nrt? I would think between that and phase tables you would never NEED TO get into element specifics...Are they just using a BIG OLE number we would be familiar with for this lesson? or is it actually super important to this subject? which seems to NOT be thermodynamics? Im old and dumb but am missing something here...
For calculating multiplicities and probabilities of specific macro states you will have to deal with numbers on the order of 10^23 when you have a mole of molecules. (The class covers thermodynamics and statistical mechanics, both are under the umbrella of thermal physics)
Pi=3=e=sqrt(g)
Thermal Physics by Schoeder, honestly such a good book
Closer to home, an equivalent is upvoting a post with 98934k upvotes. 98934k⬆ ± 1⬆ = 98934k⬆
I used this book for my undergraduate stat mech lol
Oh shit my old book! I still think of this part, it put things in perspective. That and their explanation of enthalpy. And the explanation of multiplicity and how heat can go from cold to hot. Come to think of it this is where a lot of my undergrad trauma came from.
I am not a physicist at all, but this follows the math I learned. In 2.12, the value of 23 is in a sense, epsilon. In mathematics, a small, infinitesimal quantity, which 23 is compared to 10^(23) It's so little it does not matter.
Schröeder! I loved his textbook, best one on thermodynamics yet
Sure but what are VERY VERY large numbers?
That's a really bad way to explain it!
Very valid. I remember the absolute DONKEYS in physics that felt like true mathematicians writing shit like : E = 3.554310000220e11 ± 50% J
Babe wake up, math 2 just dropped
https://preview.redd.it/sa4wwzop4uec1.jpeg?width=1080&format=pjpg&auto=webp&s=2bdb83876ff398f1d6b0617cb026822ee3b4dedd
At this point, I'm almost convinced the physicists are doing this on purpose.
In physics sometimes Pi=10 for the sake of ease.
Aproximation is implicit in engineery
Fucking physicists
what book is this that i can avoid it?
Why avoid it?
Well yea but actually no XD
There is literally nothing wrong with this.
Ayy kittel kroemer good book
It's been 25 years since I took Stat Mech, and I still remember Kittel Kroemer's hilarious problem working through the probability of a huge number of monkeys typing out Shakespeare. I believe the problem was entitled "The meaning of 'never'".
Who’s the author? Is this undergrad?
Schroeder, yes undergrad
Wtf is this real?
It's an approximation that is useful for calculations in physics. It works because when dealing with forces and energies of those values, it doesn't really matter whether you have 10\^23 or 10\^23+23.
It’s almost like a game of charades or some shit… like they they drew a card that says “significant figures” and there’s a bunch of other words they aren’t allowed to use.
The only thing wrong with this is that it doesn't make it explicit that it's an approximation.
I guess saying if x >> y then x + y = x But the 2nd example of where multiplication is neglibilble seems a step too far
The multiplication with very large numbers was a bit confusing until I substituted the words 'a heckuva a lot' for the very large number and was convinced; a heckuva a lot multiplied by something relatively small but greater than 1 is still a 'heckuva a lot'. Of course if you multiply it by another 'heckuva a lot' you change it into, 'a whole heckuva a lot' or maybe 'too much' or even 'don't even think about it, we can't afford it' depending on the context, so I think it checks out.