There’s several factors at play here. An athletic heart is not only more efficient down to the conductance of the cardiac tissue, but it also has a larger stroke volume. With those 120 bpm each beat could be pumping 40cc of blood while in the other heart each beat might only be able to pump 30cc. This is because an athletic heart is able to more fully contract to squeeze out all of the internal volume. Think of the conductance of the heart as a snowy hillside. The first time you take a sled down the hill the snow hasn’t been compacted to make a path. The more often you take your sled down that path, the more compacted it gets and the faster you and your sled will go down the hill the next time. Plus I haven’t even mentioned blood pressure decreasing in an athlete due angiogenesis and dilation of already present veins and arteries.
So to summarize, it’s not just bpm that need to be accounted for here. You also have to consider:
conductance lowering the cardiac impulse threshold
Indeterminate forms come from limits. It’s not the question you asked, and I think this answer was a little off the mark because of it. For the sake of shared knowledge, I will explain anyways:
When looking at a limit, it’s important to note that you aren’t working with zero (or infinity, or any number you are studying the limit of), what you are working with are numbers approaching the limit. For example, for (x+1)/(x), the expression has no equivalent value at x=0, as 1/0 does not exist. We can see why if we use the limit as x approaches zero. The numerator will approach 1, and the denominator approaches 0. The numerator has little impact on the value of the expression, but the denominator… dominates the value, for the pun. And, while we can’t evaluate at 0, we can put really small numbers in there and see what happens- and what happens is the expression becomes incredibly large. I’m sure that if you don’t see where this is going, you can go to Desmos or some other graphing calculator and try it for yourself.
As far as the indeterminate form- 0/0 is always undefined, at least in most mathematics. However, if you were to look at equations :
y = x/x
y= x^2^/x
y= x/x^2^
you’ll see the curves behaving differently around x=0. The first makes 0/0 look like 1, the second makes 0/0 look like 0, and the last will make 0/0 look like infinity*. Once again, note, however: 0/0 does not exist, and there is discontinuity on all of these curves at x=0.
*Edit: or negative infinity, I forgot that this limit doesn’t exist. Even though the limit doesn’t exist, it is still a useful example.
It’s a calculus thing. We can only give the expression a value if we know the functions giving us a zero value that are being devided. For example if we were dividing the function (X) by the function (X^2) at zero our we would get infinity (Wikipedia has a pretty good page on indeterminate forms).
You could also think of it like multiplying both the numerator and denominator of a fraction by 0. This should preserve the fractions value, but multiplying by 0 essentially erases both values so we can no longer know what the fraction equals unless we know how both values came to be 0.
There are several ways of approaching that particular question. And none are simple, actually.
First, just to frame why 0/0 is so weird, consider 1/0. Asking “what’s 1/0” is like asking “what number when multiplied by 0 equals 1?” There’s no answer because any number multiplied by zero is zero and no number multiplied by zero is one.
So now on to 0/0. “What’s 0/0” is like asking “what number when multiplied by zero gives zero?” And the answer is “all of them.” 1 times 0 equals 0, so 1 is an answer. But also 2 times 0 is 0. And so is pi. And 8,675,309.
So, you could say that 0/0 doesn’t have a single answer, but rather an infinite number of answers. That’s one way to deal with 0/0.
Another way is with “limits”. They’re a concept usually first introduced in calculus. Speaking a bit vaguely (though it’s definitely worth learning about if you’re curious, and it seems you are), limits are about dealing with “holes” in equasions.
Consider the equasion y=x/x. With only one exception, x/x is always 1, right? (5/5=1, 1,000,000,000/1,000,000,000=1, 0.00001/0.00001=1, etc.) But of course 0/0 is a weird situation for the reasons above.
So limits were invented (by Isaac Newton and a guy named Leibniz) to ask the question “if we got x really close to zero but not exactly zero and kept getting closer and closer to zero, what number would we approach?” And the answer is 1. (The way we say that is “the limit as x approaches zero of x divided by x is one.”)
Sometimes there’s still weirdness, though. If we look at y=x/|x| (where “|x|” means “the absolute value of x” which basically means to remove any negative sign – so if x is -3, |x| is positive 3) when x is positive, x/|x| is positive 1. When x is negative, x/|x| is negative 1. When x is 0, x/|x| still simplifies to 0/0, so it’s still helpful to our original problem. But when we approach x=0 from the negative side, we get “the limit as x approaches 0 from the negative side of x/|x| is -1” and “the limit as x approaches 0 from the positive side is (positive) 1”. So what gives?
Well, the way mathematicians deal with that is just to acknowledge that math is complex and always keep in mind that limits can differ depending which direction you approach them from. They’ll generally consider for their particular application whether approaching from the left or right is more useful. (Or maybe it’s beneficial to keep track of how the equasion works out for both answers.)
I’m sure there are other ways of dealing with 0/0 that I’m not directly aware of and haven’t mentioned here.
So, to wrap up, there are some questions in mathematics (like “what’s 0/0?”) that don’t have a single simple answer. Mathematicians have come up with lots of clever ways to deal with a lot of these cases and which one helps you solve one particular problem may be different than which one helps you solve a different problem. And sometimes “there’s no right answer” is more helpful than using clever tricks. Sometimes the problem can also be restated or the solution worked out in a different way specifically to avoid running into a 0/0.
It’s definitely unfortunate that they don’t teach some of the weirdness of mathematics in school. But something I haven’t even mentioned yet is that all of what I’ve said above assumes a particular “formal system.” And the rules can be quite vastly different if you just tweak a rule here or there. There’s not technically a reason why you couldn’t work in a system which was just like Peano Arithmetic (conventional integer arithmetic) except that 0/0 was by definition (“axiomatically” – kindof “because I said so”) 1. (Or 42, or -10,000, or whatever.) That could have some weird implications for your formal system as a whole (and those implications might render that whole formal system in practice useless, maybe), or maybe not. Who knows! (Probably someone does, but I don’t.) (Edit: looks like howrar knows and it does indeed kindof fuck up the whole formal system. Good to know!)
One spot where mathematicians have just invented new axioms to deal with weirdness is for square roots of negative numbers. The square root of 1 is 1 (or -1), but there’s no number you can multiply by itself to get -1.
…right?
Well, mathematicians just invented something and called it “i” (which stands for “imagionary”) and said “this ‘i’ thing is a thing that exists in our formal system and it’s the answer to ‘what’s the square root of negative one’ just because we say so and let’s see if this lets us solve problems we couldn’t solve before.” And it totally did. The invention(/discovery?) of imagionary numbers was a huge step forwards in mathematics with applications in lots of practical fields. Physics comes to mind in particular.
Zero divided by zero is undefined. In that it literally does not meet the definition of division (from a mathematical perspective.)
This is a bit tricky because the reason that 0/0 is undefined is separate from why any other number divided by zero is undefined.
If I divide 6 by 0, there’s no number I can multiply by zero to get back to 6. Since I can’t get back to the 6, this is undefined.
If I divide 6 by 2, I get 3. And I can multiply 2 by 3 to get 6. Now it’s genuinely important that there is no other number I can multiply 2 by to get 6. There has to be a single unique result for both the division and the going back via multiplication.
Now, if we assume 0/0 = 1, that is fine. And I can multiply 1 times 0 to get back to 0. Checks out so far. However, 1 isn’t unique in getting back to 0. If I try 5 x 0, I get 0. Which, by the rules of division should mean that 0/0 = 5. Which clearly it wouldn’t.
So zero divided by zero is undefined because there is an infinite amount of numbers that would get me back to zero.
It does not. If you enforce 0/0=1, then you end up in a situation where you can prove any two numbers are equal to each other and you end up with a useless system, so we do not allow for that.
e.g. 0=0*2 -> 0/0 = (0/0)2 -> 1=12 -> 1=2
If you get into calculus though, you’ll have ways to deal with this to some extent using limits.
There's no such thing as a single nothing. All of the nothings are all the same nothing, there are infinite amounts of nothing but no nothing at all.
Of course there is a limit to nothing because there is something, in a more real and universal sense, but mathematically speaking there is no nothing.
So whenever you divide or multiply anything by nothing all that you get is nothing. If you multiply nothing by itself you end up with nothing. If you divide nothing by itself you get nothing.
When I was a kid I got really fascinated with Zen Koans.
Koans being the kind of self-evident riddles that are most famously popularized by the question, "if a tree falls in the woods, and no one is around to hear it, does it make a sound?"
And ultimately I ended up creating my own Zen Koan by accident.
That Koan being, "What does nothing look like?"
I did everything in my power to visualize what nothing would look like. I imagined infinite black spaces with nothing around and nothing in it, a void of eternal darkness. Vast mental landscapes devoid of heat or cold for light or dark or sound or wind or air, but it just wasn't "nothing", and I didn't know why.
I tried and I tried and I tried but I couldn't help but feel like I had failed to understand.
Then one day I was standing on a hill, the wind was blowing through my hair, the Sun was shining on me, it was spring outside and the birds were singing and I was trying to visualize what nothingness would look like when I realized the reason why I couldn't visualize nothingness.
I couldn't visualize nothingness because I was looking at it.
If there was nothingness I wouldn't be there to see it.
It was at that moment a space in my brain opened up for the concept of nothing and I understood.
You can't divide nothing by nothing because there's nothing to divide with. You can multiply something by nothing because there is something to multiply with, yes, but you can't divide something by nothing because there's nothing to divide with.
Nothing is really definite. The right word to use would be consistent. We dont know the larger or the smaller picture, just that the small pocket of all physics we know is related in a certain way in a comprehensible manner.
I’ve read that all math breaks down as you approach the big bang. I’m not educated enough in math to understand how, or why, but apparently they cannot mathematically understand the origin of the universe.
But if you condense it all into something infinitely dense, then is it suddenly finite in size? Does it still have infinite size and simultaneously infinite density? Why didn’t the immense density cause it to form a black hole?
I don’t think current understanding of things is that the universe is infinite. We can estimate the size of the universe we know, because we know how fast it is spreading and for how long. Wiki says: "Some disputed estimates for the total size of the universe, if finite, reach as high as 10 10 10 122 10^{10^{10^{122}}} megaparsecs. We don’t know whether that’s all there is though. We don’t even know whether the universe has the same properties everywhere, which complicates things.
My understanding is that it has a 14 billion light-year radius from any given point. We can only see 14 billion light years away, since the universe is only 14 billion years old (actually 13.8). Light can only travel at a given speed, so we can’t see beyond the distance light has traveled during the existence of the universe. But since the universe expanded in all directions, from everywhere all at once, it’s truly infinite. If you were to teleport 14 billion light-years in any direction, you would still see 14 billion years away, since the universe expanded from that point too during the big bang. It’s mindfuck level stuff.
That understanding is intuitive but very wrong. We can see parts of the universe that are up to 46 billion light years away because of the expansion of space. The actual physical universe extends beyond that, further than we can observe.
The light didn’t travel 46 billion lightyears, but the objects whose light we are seeing are 46 billion lightyears away by the time we collect that light due to expansion. So the agreed on “radius of the observable universe” is 46.something GLY
youtu.be/XBr4GkRnY04 this old video from Veritasium explains the concept of the hubble sphere and the particle horizon, both of which are further than 13.8.Billion lightyears away
youtu.be/eVoh27gJgME this newer video from PBS Spacetime goes into much further detail about how they’re calculated
They use the Lamda-CDM model which outputs the rate of expansion of the universe at every moment in past present and future. You measure the amount of light+matter+dark matter+dark energy that your universe has, plug those values into the Friedmann equation, and it spits out the rate.
You can try out an online calculator yourself! It already has those values filled in, all you need to do is enter the z value - the “redshift” - and click generate. So for example when you hear in the news something like “astronomers took a photo of a galaxy at redshift 3”, you put in 3 for “z”, and you see that the galaxy is 21.1 Gly (billions light years) away! That’s the “comoving distance”, a convenient way to define distance on cosmic scales that is independent of expansion rate or speed of light. It’s the same definition of distance that gives you that “46 Gly” value for the size of observable universe. But the light from that galaxy only took 11.5 Gyr to reach us. The universe was 2.2 Gyr old when the light started. So the light itself only traveled 11.5 Gly distance, but that distance is 21.1 Gly long right now because it kept expanding behind the photon.
Crucially, we are able to determine the distance by redshift via the observations of objects with known distance (like standard candles) and their redshifts. The ΛCDM model only becomes necessary for extrapolating to redshifts for which we otherwise don’t know the distance, but this extrapolation cannot be made without the data of redshifts of known distances.
That’s true! There is a kind of incestuous relationship between the cosmic distance measurements and the cosmic model. Astronomers are able to measure parallax only out to 1000 parsecs, and standard candles of type Ia supernovae to a hundred megaparsecs. But the universe is much bigger than that. So as I understand it they end up climbing a kind of cosmic ladder, where they plug the measured distances up to 100 Mpc into the the ΛCDM model to calculate the best fit values for the amounts of matter/dark matter and dark energy. Then they plug in those values along with the redshift into the model to calculate the distances to ever more distant objects like quasars, the Cosmic Microwave Background, or the age of the universe itself. Then they use observations of those distant objects to plug right back into the model and refine it. So those values - 28.6% matter 71.4% dark energy, 69.6 km/s/Mpc Hubble constant, 13.7 billion years age of the universe - are not the result of any single observation, but the combination of all observations taken to date. These values have been fluctuating slightly in my lifetime as ever more detailed and innovative observations have been flowing in.
Are you an astronomer? Maybe you can help me, I’ve been thinking - how do you even measure the redshift of the CMB? Say we know that CMB is at redshift 1100z and the surface of last scattering is 45.5 GLy comoving distance away. There is no actual way to measure that distance directly, right? Plugging in the redshift into the model calculator is the only way? And how do we know it’s 1100? Is there some radioastronomy spectroscopy way to detect elemental spectral lines in the CMB, or is that too difficult?
If we match the CMB to the blackbody radiation spectrum, we can say that its temperature is 2.726K. Then if we assume the temperature of interstellar gas at the moment of recombination was 3000K, we get the 1100z figure. Is that the only way to do it? By using external knowledge of plasma physics to guess at the 3000K value?
So if you take 2 things that started say ~3 billion light years apart (which would be ~1000x a megaparsec), that means every single second the universe has existed those 2 points have gotten 70,000km further apart. And now that they’re further apart, they separate even faster the next second.
For reference:
31.5 million seconds in a year. ( 3.15 x 10^7 )
universe is 13.8 billion years old ( 1.38 x 10^10 )
So we talking about this 70,000km getting added between the 2 points ~4 x 10^17 times.
Then you gotta bring calculus into it to factor in the changing distance over time.
It … adds up. Which is why you’ll see the estimates for the observable universe’s radius being ~46.5 billion light years (93 billion light year diameter), even though the universe had only existed for ~14 billion years.
And now that they’re further apart, they separate even faster the next second.
That’s a common misconception! Barring effects of matter and dark energy, the two points do NOT separate faster as they get farther apart, the speed stays the same! The Hubble constant H0 is defined for the present. If you are talking about one second in the future, you have to use the Hubble parameter H, which is the Hubble constant scaled with time. So instead of 70 km/s/Mpc, in your one-second-in-the-future example the Hubble parameter will be 70 * age of the universe / (age of the universe + 1 second) = 69.999…9 and your two test particles will still be moving apart at 70000km/s exactly.
The inclusion of dark energy does mean that the Hubble “constant” itself is increasing with time, so the recession velocity of distant galaxies does increase with time, but that’s not what you meant. Moreover, the Hubble constant hasn’t always been increasing! It has actually been decreasing for most of the age of the universe! The trend only reversed 5 billion years ago when the effects of matter became less dominant than effects of dark energy. This is why cosmologists were worried about the idea of a Big Crunch for a while - if there had been a bit more matter, the expansion could have slowed down to zero and reversed entirely!
Oh wow thanks. You learn something new every day! I’m definitely an “armchair physicist”, and still find it hard to think about things in a nonstacically geometric way.
Sounds like the Hubble Constant ain’t so constant :)
It’s impossible to measure precisely enough to know for sure that it is completely flat, or even saddle shaped (both being infinite in size). The generally accepted understanding by cosmologists is that it is infinite. But just due to the nature of measurement and tools we can’t completely rule out a finite universe. However we do know based on the measurements that it is really really… really really really big if it’s not infinite.
Another problem lies within the mathematical framework of the Standard Model itself—the Standard Model is inconsistent with that of general relativity, to the point that one or both theories break down under certain conditions (for example within known spacetime singularities like the Big Bang and the centres of black holes beyond the event horizon).[4]
My ELI5: Both theories work great, supported by vast amounts of evidence and excellent theoretical models. It seems they are two tools with distinct purposes. One for big and heavy stuff, the other for small and energetic stuff. The problem arises when big and heavy stuff is compressed into tiny spaces. This case is relevant for both theories, but here they don’t match, and we don’t know which to apply. It’s a strong hint we lack understanding, one of the biggest unsolved problems in physics.
So math itself is probably fine, we’re just at a loss how to use it in these extreme cases.
One of the first things you learn in college-level science is accuracy and precision. A measurement can have a degree of correctness and a degree of exactness about the value. For example a sensor may get the wrong reading 3% of the time. When you have a big pile of readings, you don’t always have the time to validate them all. So, there is some uncertainty that you accept. The same sensor may only be able to give you an accurate reading down to a specific decimal point, which is expressed as precision. Anything less is given as a range in which error exists. These ideas are important, because when you do calculations based on those readings, you have to take the error with you. There may be a point where the value you reach is overshadowed by the magnitude of the error.
Yes, there are non-deterministic parts in physics. For example atomic decay. While we can measure and work with half-life times for large amounts of radioactive atoms, the decay of a single, individual atom is unpredictable. So in a way, you can get your desired dose of vagueness by controlling how many atoms you monitor. The less, the more.
Or another example from the same field: There are atoms for which we believe they are stable, although they theoretically could decay. But we never observed it. So maybe they are in fact stable, or maybe they decay just slower than we have time. Or only when we don’t look. Examples would be Helium-4 or Lead-208.
I also like the idea, inspired by Douglas Adams, that the universe itself could be a weird and random fluctuation, which just happens to behave as if it was a predictable, rationally conceivable thing. That actually, it’s all a random chain of junk events, and we’re fooled into spottings some patterns. This apparence could last forever or vanish the very next moment, who knows. Maybe it’s all just correlation and there is zero causation. As far as I know, we’ll never be able to tell. So fundamentally, all of it is a vague guess, supported by mountains of lucky evidence.
Good answer! Thanks for that. Also, good use of ‘apparence’ - not a word i see often.
Apparently Bismuth-209 has what is considered an “alpha decay” with a half life longer than the lifetime of the universe - whatever that means. So yeah, entered into some fuzzy physics there.
There are atoms for which we believe they are stable, although they theoretically could decay. But we never observed it.
Bismuth-209 was for a long time considered to be the heaviest stable primordial isotope, it had been theorized for a while that it might technically decay, but no one proved that until 2003, it has a half-life of over a billion times the current age of the universe, and so for all practical purposes can be treated as if it is stable.
I’m no physicist, so I very well be way out of my element, but I would personally not be the least bit surprised if it turned out every atom was technically unstable, but since the decay is so incredibly slow we may never be able to accurately detect it. Using the lead-209 example you gave, if it ever is proven to be unstable, the half life should be at least 10^25^ (10,000,000,000,000,000,000,000,000/ten septillion) times longer than the age of the universe. Smarter people than myself probably have some ideas, but I couldn’t imagine how you could possibly attempt to measure something like that.
Oh wow, thanks for the details! 10^25^ years … no, times … yeah, crazy. I mean, that’s beyond homeopathic. Since I learned about this topic as an interested layman, I somehow assumed everything can decay, and we simply call the things “stable” which do so very slowly. Which can mean as many atoms decay over the course of a billion years as there are medically effective molecules in homeopathic “medicine”; none.
The standard model predicts that hydrogen-1 is the only stable nuclide because electroweak instantons allow three baryons (such as nucleons: protons and neutrons) to decay into three antileptons (antineutrinos, positrons, antimuons, and antitauons), which imply the instability of any nuclide with a mass number of at least three; or for two baryons to decay into an antibaryon and three antileptons, which would imply that deuterium could decay into an antiproton and 3 antileptons.
This is very rarely discussed because the nuclides that can only decay through baryon anomalies would be predicted by the standard model to have ludicrously long half lives (to my memory, something roughly around 10^150 years, but I might be wrong).
Hydrogen-1 is stable in the standard model, as it lacks a mechanism for (single) proton decay.
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