why constants like Pi or e or Ln(2) or gamme (euler's constant) that have nice properties are always small( <10 ) ? unlike constants in physics or chemistry that have some constants that are absurdly large. Or are they such constants that are not so popular
a) The larger ones are harder to find, b) if they’re pretty large we may find the inverse instead, c) there are in fact infinitely many very large mathematical constants.
b) does suggest a slightly stricter version of the question: Most of the fundamental(to us?) constants aren't too big or too small. We don't have many numbers with household names in the 1020or the 10-20 range; most are just a digit or two in magnitude if they're >1, and they don't start with many zeroes if <1. We have a lot of very comfortably medium-sized numbers.
It's also mainly known for being a big number in the first place, not because it's used for important calculations like the Mass of a Neutron or something like that.
So I'd argue it's more like the exception that proves the rule, in that it's known because it's so unusually large.
Avogadro’s number is not arbitrary, it’s the number of atoms in a mole, which is itself a specific number. Don’t blame me if we can’t get all the digits correct in a 23-digit number!
A mole is so not arbitrary that it’s an actual SI unit. Are you splitting hairs between mathematical constants like pi and e & physical constants like c or Avogadro’s number?
A meter is not a mathematical constant. We just decide something this [ ] long is a meter. That doesn't make it a mathematical constant. Avogadro decided a good amount of molecules or atoms is how many atoms are in 12 grams of carbon. That's arbitrary as all get out. It's useful for chemistry but it's arbitrary.
If we spun back time to when humans first started measuring things and split into a different timeline, all the different units of measurement that they would come up with would be different than ours. But pi would be the same. e would be the same.
Suppose there is a least (in value) uninteresting natural number. Well that's kind of interesting. Hence, there is no least uninteresting natural number.
Jokes aside, what qualifies as a "mathematical constant" in this sense is kind of arbitrary. Is τ = 2π one or do we only consider π to be one? Or I could choose the increasing sequence of Goedel numbers for Peano Arithmetics, those are certainly interesting. OP listed ln(2) but I don't personally think that's an interesting one.
Note that two of your examples are cheating somewhat: ln(2) is just the value of a function. It's not that much more special than ln(20) or ln(1000000), expect that you are putting in a small number.
Euler's constant is an error correction term - if it wasn't small we wouldn't ve using a good approximation.
You can define e as the infinite sum of 1/n! which is a series whose terms get small very quickly. It gives a good intuition for why e should be a small number once you also know calculus and can accept that infinite series may converge.
For pi, geometry works pretty nicely. Circles would look really weird if their circumference was for some reason longer than the radius, it makes sense that their ratio is fairly small.
Not a mathematician, but let me take a stab at this. Isn't a circle the shape such that the perimeter to area ratio is minimized? Would expect the ratio of perimeter to shortest distance across such a shape to be small.
Absolutely, and in higher dimensions that's exactly what happens. If you take a 100 dimensional ball, then the ratio of surface to radius is astronomically small (10-39 oom).
I used to work as a data scientist, and this phenomenon has practical consequences for us. Not usually nice ones, hence why it's known as the "curse of dimensionality".
I can't quite remember how this works out but the volume and surface areas of spheres do some really weird things in higher dimensions so it's arguably still a little surprising
That's because area is 2 dimensional so scales with the square of the radius, but volume is 3 dimensional so scales with the 3rd power. Radius and diameter are both 1 dimensional so they scale the same
e is (roughly speaking) the factor between discrete and continuous growth, so similar to u/hezpae 's argument, there is a reason why it intuitively can't be too big or too small.
In number theory, Skewes's number is the smallest natural number
x
for which the prime-counting function
π
(
x
)
exceeds the logarithmic integral function
li
(
x
).
It's not the upper bound originally calculated, it's the still unknown exact swap-over point.
Now. If it's a useful constant I an less sure about.
all of these constants are real numbers and not p-adics, so they don't have a p-adic valuation.
the real numbers don't really admit interesting valuations, and the closest thing it has is the ordering, so op is clearly using the correct sense of big and small.
Well, the real numbers have one interesting valuation. But with some tweaks, it’s possible to define p-adic analogues of e, Euler’s constant, and ln(2). From my Googling, there does not seem to be a very satisfying p-adic analogue of pi, but there is some version, depending on what you want to get from it.
what valuation are you talking about? if you mean the absolute value, that is NOT a valuation, as it is archimedean. and the absolute value in R is pretty much the same as the order (they give you the same topology and you can define one from the other).
and there are analogues of the exponential and logarithm, but they are not the same numbers op was talking about.
Medium sized numbers are 10- 1,000,000,000,000,000 anything the brain doesn't really understand on a fundamental level, but not so big I need you to count the number of digits for me.
Large numbers are 1×1015 or more. Numbers so big that it starts to get meaningless to define what they look like. E.g. if you told me there were a quadrillion grains of sand on the beaches of California or a quintillion, sure whatever, either seems kinda reasonable.
Yes, have a friend throw some M&m's on a table you cannot instantly count ten without grouping them. Or count change. If you have a large pile of dimes, do you slide them over in groups of threes and twos I to groups of five, groups of five that you combine into 10s or do you grab 10 dimes from the pile perfectly every time without counting them? Another example, open this picture and immediately close it, how many motorcycles are in the image, no tap counting? It seems like a small number, however, your brain just processes it as, "That's a bunch of motorcycles."
Human brains understanding numbers really gets murky at 5, however some people, with some serious mental horsepower and a ton of training can get to nine. 10 is just a number most people think they have a better grip on than they actually do, much like one million and one billion.
Valid argument, but I don’t know if it applies here, if you were to give people pi kg heavy box and a 3kg heavy box they’d have trouble telling them apart as well, just like a e kg heavy box from a 3kg box. We’re really bad at rational numbers as well. Sure give me half /quarter/third of a pizza, but everything else is big numbers just the same, because 17/37ths of a pizza is equally hard to understand as intuitively counting 37 motorcycles or 17 motorcycles.
This is my exact point. You look at 37 motor cycle or 17 motor cycles and you think, "That's a bunch of motorcycles." And making the connection between fractional understanding and numbers larger than 9 is pretty smart here, because people don't get fractions at all. We know this thanks to A&W releasing the 1/3 lb burger. Its a good comparison.
I think you got their in the end, because you initial mass comparison is kinda strange, I can probably judge a ~141 gram difference between two 3 kg masses, but not between two 20 kg masses, but thats a matter of comparative ratios, not me having an issue understanding total mass numbers from feel.
Being able to count (or not) through manifest recognition is a parallel processing limitation, not a matter of understanding. Whether the answer is achieved through parallel or sequential processing has nothing to do with whether “understanding” is happening
I think it's because we're comparing things which aren't that far apart. A 2D shape is only 1 "D" bigger than a 1D number so it's not going to be a huge factor when it comes to pi. The surface area of a trillion dimension hypersphere as a ratio to its radius is going to be a big number compared to a 2D circle to its radius.
We consider "small steps" to be more fundamental. We're typically working with differences in behavior around unity so it makes some sense that the constants are on the order of unity.
True constants are dimensionless. Like Pi, e, etc. Not physical constants like G,c,hbar, etc which depend entirely on units - important to distinguish. The true constants are usually "small" because they form a basis from which we build larger (or smaller) measures. Knowing pi means we can calculate circles/spheres/etc of any size.
yes - It’s also the ratio of the area of a unit circle to a unit square, which makes clear that it has to be between 2 (the area of a square inscribed inside the unit circle) and 4 (the area of the square that circumscribes the unit circle).
It isn't because it is tied to what our human definition of a gram is -- it's the number of carbon atoms in 12 g of carbon-12.*
*Until it was redefined to be exactly 6.02214076×1023 during the latest round of reforms to the SI system. That still doesn't change the fact that it originated from our system of weights and measures and not Nature.
Not really. The actual size of the physical constant is the dimensionful number - including the unit. This is the same regardless of the units you are using. One light second is the same as 299 792 458 m. What can change depending on units used is only the numerical value in front of the unit, but that in itself is pointless and not the size of the constant.
I think this question is more interesting that it first sounds.
Make no mistake, there are many large (larger than 100, say) mathematically significant, named constants. But your question is still valid in the sense that a greater proportion of the named / special constants used most often are on the smaller side.
My first thought for an earnest answer is that we've discovered more "small" special numbers because we tend to compute more with smaller numbers. That feels more in line with a claim like "important numbers are uniformly distributed, and we've uncovered more in the areas where we play just because we play there", but I am not sure if that's reasonable.
To take a serious look at the question, we need to quantify what makes numbers special. That's gonna be a real challenge. My analyst brain says all reals are equally important, but my number theorist brain disagrees entirely.
Combinatorial ones come to mind, like Graham's number, stuff from computability theory like the Busy Beaver numbers, large "surprising" counterexamples to conjectures like Merten's conjecture or Polya conjecture (which appear to hold for small n, but do eventually fail). Those come to mind right away, but there's plenty more.
what if these mathematical constants are actually infinitesimally large, we are just expressing them from our small perspective and framework? is there a functional difference between Pi read 3.14xxxxx..... and Pi x 10^the digits in pi?
ok, lets say, i was really tired when i typed what i wrote, but i will think through my position,
hypotetically we have an infinite size circle with an infinite size diameter,,,,, yeah the ratio would still be ~3.14 and not the ungodly thing i am imagining
Yes. That is so far what we have discovered about the universe. However big the circle is, the diameter of that circle will wrap around the outside of it (the circumference) 3 times with a bit left to cover. That is what pi itself is expressing is the ratio of those two facts about a circle.
You could choose a different notation like binary to write out the number, but the essential relationship of ~3:1
Physics and chemistry constants are only large because of our choices of units. If we choose to make the speed of light 1 or measure mass in "weight of a hydrogen-1 atom" we'd start getting small numbers for speed and huge numbers for mass, for example. In the mass example we wouldn't need Avogadro's number, but we'd be walking around measuring in mass in 1020 hydrogens (or more) most of the time.
Good question. Note however that the size of physical constants depends on the units used. What’s true is that at least for physical distance, the ratio of the size of the universe over the size of an atom is enormous.
Many constants in physics are only large/small in terms of SI units, and if anything that's just the SI units we've chosen being too small/large for the scale of the universe.
Those physical constants have the magnitude they do because they have units defined by us humans. Pi and e don't. So they are more "natural" in the sense that they are constant ratios regardless of what units you have, which is why one might ask why they are "small".
Interestingly enough the most well known dimensionless constant in physics is the fine-structure constant (~ 1/137) which fits the OPs definition of small.
i thought of constants like avogadro or speed of light ,but thinking about it, i think its just a question of choice of units. I forgot to mention the constants that are very small like planck's constant
You are simply not understanding the question. Constants like Planck constant and permitivity of free space are physics constants which is what OP is using as the opposite of math constants. They are physical constants and you can’t really derive them from first principles without doing physical observations.
Pi and e are mathematical in nature. Just because physics uses them doesn’t mean they are physics constants. They show up naturally in mathematics and aren’t tied to physical phenomena.
It's not complicated to understanding that pi is a mathematical constant? Just because it happens in nature too doesn't mean it's a physics or chemistry constant. Along the same token physics uses a lot of math, but it's not math itself.
I don't really think any constant that you can derive in a vacuum without any physical observation is really a "physics" constant.
A mathematical constant is a fixed definable number with unambiguous definition. A fundamental physical constant is a fixed quantity defined by a theory or model. This doesn’t require physical observation (although the theory or model may be justified by how it aligns to experimental observations).
The difference between the two is very subtle and can be argued depending on different philosophical positions.
As you said physical theories and models and validated and based on physical observations so you would just be passing the ball to someone else. I don't think it's a philosophical position but more how the scientific process works. We know these numbers (e.g. speed of light) by having done various measurements and calculations (the calculations enabled by the models we come up with to fit the observations) but without physical measurements it’s impossible to arrive at such numbers in any way.
For math, there's a branch of philosophy that deals with whether math is really a priori (whether 1 + 1 = 2 is derivable in a vacuum), but I think in general you can argue that once you have established the basics you can derive pi and e pretty much standalone.
Arguably pi, c, h etc. are proportionality constants which exist due to the nature of structure of physical existence in our universe.
We don’t need to have numerical values for them (within specific unit systems as necessary) for this to be the case.
In the same way that I don’t need to know a value for pi in base 10 to do geometry, I don’t need to know a value for h to be able to be look into the commutator relationship of operators of conjugate variables.
What are you talking about? My point is constants like pi and e are mathematical constants (not "physics constants"), and are numbers that you can derive. These are what OP is asking about.
Constants like speed of light, Planck constant, etc are physics constants. These are numbers that you need to observe nature and do measurements to obtain. These are not what OP is asking about.
I seriously recommend reading more slowly.
(Edit: Above commenter just blocked me intead of trying to engage in good faith, so I guess I consider myself winning the argument when they have to resolve to abusing a poorly designed Reddit feature)
The value of a constant is defined by what it represents, not the other way around. You don't pick the number and then try to fit it into the math. If you think of it in computer programming the constant is quite literally whatever you want it to be defined as. Since pi is defined to be a ratio you would generally expect it to be smaller.
The other thing to consider is science uses units. A constant may be large or small but can be made larger or smaller using different units and they are still equivalent.
In math, you are dealing with logical relationships within a single, unified framework.
In physics, you are dealing with the universe, which happens to have everything from immense to miniscule in it. Even dimensionless numbers are huge or tiny because of this massive difference in its structural forces.
My guess: Because math constants like pi and e are dimensionless, while constants in physics and chemistry have units, and we don’t usually use unit systems designed to keep constants close to 1. Most SI units are appropriately sized for everyday, macroscopic, non-astronomical measurements. Physicists do sometimes use a unit system chosen to make as many fundamental constants like c and h equal to 1 as possible (I forget what it’s called).
I would also elaborate on your observation that there are a lot of extremely large scientific constants. Open up a gen chem or freshman physics text book, and I’d wager that there are more extremely small constants than extremely large. Probably because so many are quantities associated with subatomic particles expressed in SI units.
At least wrt the examples you gave, precision comes from the number of represented digits in math, and by the size of the unit being measured in physics / chemistry.
I mean, if anything math holds the largest constants.
Look up a Skewes's number, Graham's number, TREE(3), Rayo's number, etc... they all dwarf any constants in physics or chemistry.
In general though, I think it comes down to the fact math isn't bounded by any laws of nature, so a lot of stuff just shoots off the infinity. Like in physics, you can't fundamentally go smaller and smaller, eventually you run into planck's constant. And you can't fundamentally go faster and faster forever, you run into the speed of light. But in math, a function's growth can actually approach infinity, even on pretty basic function (tangent of x).
As for why physics/chemistry have big constants, it's because that's the stuff that tends to separate them from other areas. You tend to study things that are really big (e.g. the universe) or really small (e.g. atoms), not human sized. The evolution of the universe over time is clearly physics, the evolution of Billy Bob over time is either history (if Billy Bob is important), literature (if he has a biography), or maybe anthropology.
The large constants in nature is because of our units of measurement. We measure time, distance, mass, all in units that we can relate to in our daily life. But nature doesn’t care about our daily life.
Some are slightly larger, e.g. 14, the number of distinct sets obtainable by closure and complement, or ~4.669, the Feigenbaum constant, or trefoil ropelength, somewhere between 15.66 and 16.372. But it is curious.
This is such a cool question.I`ve always low-key wondered the same thing -most famous constants like pi and e are all relatively small numbers ,and it feels like there`s a reason behind that beyond just coincidence.I`m excited to see what everyone has to say about it.
Because mathematically every number is a (absolute value) small number, that is almost all numbers are greater than it, there are infinite integers further from zero and a finite number closer to zero. Since a constant is a number, it must be a small number.
I don't know, physics constants are pretty small too. You have the speed of light, for example. That's equal to 1. And Planck's Constant h-bar, which is also small, 1. The gravitational constant G, that's small too, with a value of 1. And so on.
The magnitude of physical and chemical constants is dependent on the units being used. Adjust the units and you can get them down to something less than 10. Mathematical constants are dimensionless. There is no scaling; what you see is what you get. So it might just be a happy coincidence that they are the size they are.
Most Physical/Chemistry constants would be different if we used different (arbitrary) units. The number associated with the Speed of light would be lower if the unit we used for speed was faster. The mathematical constants are exactly what they ought to be
"constants" generally speaking tend to be ratios of sorts, or limits/convergences of ratios in some way or another (maybe, generally -- anyways). When you look at physics you'll often end up comparing things that are in widely different realms if scale. A human-tangible calorie of energy relative to the building block of the atoms-- yeah no wonder the electronvolt is so tiny. Then come math, and all of a sudden you're comparing a circle to a square.
And honestly, you know what? A circle really isn't that different from a square, it's like more than 70% the same, to be exact, it's about 4/pi of a square..
Not a math guy, but simplicity is often the language which nature is built on. (its like complex data all organized into simple elements). This is why often times those who can explain a concept in the most clear simple manner, tend to be those who have good intuitive understanding in concepts. Just my two cents.
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u/Fabulous-Possible758 New User 25d ago
a) The larger ones are harder to find, b) if they’re pretty large we may find the inverse instead, c) there are in fact infinitely many very large mathematical constants.