Where transcendental numbers hide in everyday math(quantamagazine.org)
quantamagazine.org
Where transcendental numbers hide in everyday math
https://www.quantamagazine.org/where-transcendental-numbers-hide-in-everyday-math-20211027/
5 comments
I like this explanation from Numberphile as well[1]
[1]https://www.youtube.com/watch?v=AuA2EAgAegE
[1]https://www.youtube.com/watch?v=AuA2EAgAegE
I mean it is an infinite polynomial. Sometimes it’s a lot more useful to think of it that way and not in terms of compounding.
Or you can think of the function exp() represented by the Taylor series and e just happens to be exp(1). But the particular number itself isn’t as important as the function.
Or you can think of the function exp() represented by the Taylor series and e just happens to be exp(1). But the particular number itself isn’t as important as the function.
<<< this is important
<<< this is not important
To be fair, neither is important. You don't have to reach for a transcendental if you want infinite polynomials, you can manufacture one for integers, rationals, irrationals...the familiar example -
f(x) = 1 + 1/x + 1/x^2 + 1/x^3 + 1/x^4 + ... f(1) = 2
To be fair, neither is important. You don't have to reach for a transcendental if you want infinite polynomials, you can manufacture one for integers, rationals, irrationals...the familiar example -
f(x) = 1 + 1/x + 1/x^2 + 1/x^3 + 1/x^4 + ... f(1) = 2
Oops, I deleted that part before you posted the comment; didn't think it was worded well. I guess my point was that the specific value of 2.718... isn't as important as a lot of pop-math sources make it out to be.
Hmm. Did you end up studying differential equations?
I feel like the most clear statement from that article is "it's the base of exponential growth", but that (for a math inclined audience) the best way to show that is via equations like dy/dt = \alpha y.
I feel like the most clear statement from that article is "it's the base of exponential growth", but that (for a math inclined audience) the best way to show that is via equations like dy/dt = \alpha y.
Another good way to see the limit definition is by solving the discrete case of that diff. equation (y' = y, y(0)=1).
Replace dy/dx with ∆y/∆x, so
∆y = y(x+∆x) - y(x)
Once you solve that (use some induction and numerical methods), see what happens when ∆x appraches to 0 and x=1.
For example try with ∆x=1 and then generalize that case to ∆x=n.
Replace dy/dx with ∆y/∆x, so
∆y = y(x+∆x) - y(x)
Once you solve that (use some induction and numerical methods), see what happens when ∆x appraches to 0 and x=1.
For example try with ∆x=1 and then generalize that case to ∆x=n.
You didn't justify why you picked y' = y, though, and I'm having a hard time connecting that differential equation with the one that follows "so" in your message.
Could you break it down further?
Could you break it down further?
My favourite use of e is determining how many (independent) trials it would take to encounter an event of some likelihood.
Eg. if an event has a 1/10 chance of occuring, how likely am I to encounter it at least once after 10 trials? What about a 1/1000 event after 1000 trials?
Well, if it has probability p, then there is 1-p of it NOT happening. So the chance of seeing a 1/p event after p trials is:
= 1 - (1-p)^p (looks a lot like e) = 1 - 1 / (1-p)^-p = 1 - 1/e (63.2%)
This comes up a lot in real-life, since it feels like if we do something with a 1% chance 100 times, it should occur, but there is really a more than 1/3 chance against it!
Eg. if an event has a 1/10 chance of occuring, how likely am I to encounter it at least once after 10 trials? What about a 1/1000 event after 1000 trials?
Well, if it has probability p, then there is 1-p of it NOT happening. So the chance of seeing a 1/p event after p trials is:
= 1 - (1-p)^p (looks a lot like e) = 1 - 1 / (1-p)^-p = 1 - 1/e (63.2%)
This comes up a lot in real-life, since it feels like if we do something with a 1% chance 100 times, it should occur, but there is really a more than 1/3 chance against it!
I can recommend "e: The Story of a Number". It has a good explanation of transcendentals along with an informative dive into the history and significance of e.
I was going to post this Mathologer video (https://www.youtube.com/watch?v=CaasbfdJdJg) based on e being "the most irrational number" -- but I misremembered. It's the golden ratio, phi.
But its still a really interesting video!
But its still a really interesting video!
There's something ironic about calling the "most irrational number" the "golden ratio"
The defining characteristic of the exponential function is that d/dx e^x = e^x. The series construction makes this really obvious, because each element of the series is the first derivative of the next element, ad infinitum. And it wasn't as if this series was somehow luckily discovered, it was constructed from the defining characteristic.
Visual Complex Analysis, by Tristan Needham, covers this lucidly.
Visual Complex Analysis, by Tristan Needham, covers this lucidly.
I think I find viewing at the result of some series to be more satisfying. After seeing series solutions to differential equations, the number e kind of faded into the background of the numerical process that gives us e.
Where did you see e as a series before seeing it as the limit of some growth? I remember seeing the intuitive explanations in school, before anything else.
IMO the most elegant definition of pi and e comes from differential equations.
e is the base of the exponential function, the solution of
e is the base of the exponential function, the solution of
y = y'
y'' = -yNot arbitrary transcendental numbers, but the e specifically.
Yeah, this was disappointed. Information about e is widely available, but I don't recall ever really reading much about transcendental numbers in general, other than a brief introduction to e.
Definitely a missed opportunity.
Definitely a missed opportunity.
OK, here's a fun fact: most numbers are transcendental, which is to say, if you choose a real number uniformly at random in any finite interval the probability of choosing a transcendental number is 1. (The proof is easy and left as an exercise.) But actually describing a transcendental number is very hard. The first transcendental number other than pi and e was not described until the mid 19th century, and even today only a few dozen classes of transcendental numbers are known.
Another thing that tingles my mind: it's easy to describe numbers that are likely to be transendential (just define a turing machine that spews out digits by some more or less chaotic algorithm – most would agree that it seems extremely unlikely, that the output would correspond to some algebraic number by chance!), it's just hard to PROVE that they are.
If such a program ever terminates, then the result is rational. If it doesn't terminate, then pausing at any point will always describe an infinite number of irrationals.
Sorry, I caught your comment only a few days later. What do you mean by "infinite number of irrationals"? We are talking about a deterministic machine here, so surely it describes only one number?
But you have to let the Turing machine run forever for this to work! Every number with a terminating decimal representation in rational.
Maybe it's not so easy to describe a transcendental number, at least in our lifetimes...
Maybe it's not so easy to describe a transcendental number, at least in our lifetimes...
You don't have to let the Turing machine run forever anymore than you have to actually fully write out every digit of PI. It's possible to determine strictly from the description of the Turing Machine that the value it outputs will converge to a specific number equal to the output of some other process, such the process of writing out 4 - 4/3 + 4/5 - 4/7 + 4/9 + ...
The equivalence class of all such representations is the number we call PI.
The equivalence class of all such representations is the number we call PI.
So, I was thinking that the Turing machine itself is a description of the number, because a description of the machine uniquely determines the output. If you require the digits to be spelled out, then yeah, that's impossible.
Possible spoiler (I don't know if my proof is correct)--
Algebraic numbers are countable, since their description is a countable combination of rationals which are also countable. The measure of a countable set is zero, so the measure of its complement is 1, and thus the probability of choosing a transcendental is 1.
Algebraic numbers are countable, since their description is a countable combination of rationals which are also countable. The measure of a countable set is zero, so the measure of its complement is 1, and thus the probability of choosing a transcendental is 1.
Exactly right!
On this line of thought, the Numberphile YT video titled "All the Numbers" hits on this and other interesting truths. (I'd never learned the term 'normal number' until watching this.) Runtime 14m27s. Highly recommended.
Not just very hard, but actually impossible for almost all transcendentals. Reason being, any description must be finite in length; descriptions therefore form a countable set; but transcendentals are uncountable. The subset of reals which are describable are known as "definable real numbers" [1]. (Notably, all algebraic numbers -- the reals of which form the complement of transcendentals -- are definable by definition.)
[1] https://en.wikipedia.org/wiki/Definable_real_number
[1] https://en.wikipedia.org/wiki/Definable_real_number
Yes, that's true. But what I intended to convey was that it's very hard in the sense that we have only described a few dozen classes of transcendentals. So the actual limiting factor has to be something other than this counting argument. The hard part is finding descriptions of transcendentals within the space of describable numbers.
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I read in Hacker's Delight a proof that e is the theoretically optimal base for computation. But since we lack the technology to make a base-e computer system, we have to settle for either binary, or base-3.
So you're probably onto something.
So you're probably onto something.
To be more clear, e is the optimal base for representing approximate numbers, when it is desired to minimize the relative errors.
Old IBM mainframes used base 16 for floating-point numbers, which (like also the base 10) is more distant from e than 2, and that caused larger computation errors than in modern computers with binary floating-point numbers.
For exact numbers (e.g. integers), the base does not matter, except on how it influences the cost of the hardware needed to implement the arithmetic operations. Base 2 normally results in minimal cost.
Old IBM mainframes used base 16 for floating-point numbers, which (like also the base 10) is more distant from e than 2, and that caused larger computation errors than in modern computers with binary floating-point numbers.
For exact numbers (e.g. integers), the base does not matter, except on how it influences the cost of the hardware needed to implement the arithmetic operations. Base 2 normally results in minimal cost.
The argument for base e is based on the physics of sensors and storage; what's the right number of signal levels.
The problem with IBM's floating point scheme was that the multi-digit exponent granularity threw away mantissa bits.
Reasonable precision with reasonable range is hard in 32 bits and IBM effectively had less. (36 bit FP can be much better.)
The problem with IBM's floating point scheme was that the multi-digit exponent granularity threw away mantissa bits.
Reasonable precision with reasonable range is hard in 32 bits and IBM effectively had less. (36 bit FP can be much better.)
I do not believe that there is any reason related to the physics of sensors and storage for which e can be an optimal base, unlike the case for minimizing errors in arithmetic operations, where e is optimal.
In order to determine an optimal value for anything, you need some quantity that must be minimized or maximized.
For storage, I do not see what quantity might have anything to do with the representation base except the probability of errors when retrieving a previously stored digit.
However that probability decreases monotonically with the decrease in the number of values per digit, so the error probability is smallest for the smallest base, i.e. for 2, not for e.
The same is for a communication channel affected by noise, the optimal base is 2.
On the other hand, one may add additional constraints that can define an optimization problem, e.g. for a communication channel, if the bandwidth and the acceptable error probability are given, to maximize the quantity of information transmitted per time, or for a storage device, if the volume used for storage is given and the acceptable error probability is given, to maximize the quantity of information stored in that volume.
However, for these 2 problems determined by physics the solutions cannot be achieved by using a certain numeric base. The optimal solutions for both problems is to use an appropriate error-correcting code, the right code depending on the parameters of the problem (but almost equivalent error-correcting codes can be defined for stored or transmitted symbols belonging to different symbol sets, e.g. for digits in numbers represented with different bases).
In order to determine an optimal value for anything, you need some quantity that must be minimized or maximized.
For storage, I do not see what quantity might have anything to do with the representation base except the probability of errors when retrieving a previously stored digit.
However that probability decreases monotonically with the decrease in the number of values per digit, so the error probability is smallest for the smallest base, i.e. for 2, not for e.
The same is for a communication channel affected by noise, the optimal base is 2.
On the other hand, one may add additional constraints that can define an optimization problem, e.g. for a communication channel, if the bandwidth and the acceptable error probability are given, to maximize the quantity of information transmitted per time, or for a storage device, if the volume used for storage is given and the acceptable error probability is given, to maximize the quantity of information stored in that volume.
However, for these 2 problems determined by physics the solutions cannot be achieved by using a certain numeric base. The optimal solutions for both problems is to use an appropriate error-correcting code, the right code depending on the parameters of the problem (but almost equivalent error-correcting codes can be defined for stored or transmitted symbols belonging to different symbol sets, e.g. for digits in numbers represented with different bases).
While 2 values per storage element is optimal for the sensor, it is the worst case for "data per storage element".
The quantity optimized was "cost to store data". The three parameters are sensor characteristics, storage element characteristics, and the amount of data per storage element. Increasing the last makes the other two worse, but that doesn't tell us that minimizing the last is the optimum solution.
Yes, ECC across multiple storage elements changes things.
The quantity optimized was "cost to store data". The three parameters are sensor characteristics, storage element characteristics, and the amount of data per storage element. Increasing the last makes the other two worse, but that doesn't tell us that minimizing the last is the optimum solution.
Yes, ECC across multiple storage elements changes things.
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An k-ary search tree has k children per node, and k-1 keys per node. To find a value in a k-ary tree with n nodes, the maximum number of comparisons will be (k-1)log_k(n).
This is minimized when k = e.
This is minimized when k = e.
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[1] https://betterexplained.com/articles/an-intuitive-guide-to-e...