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Where transcendental numbers hide in everyday math
- nine_k 5y agoNot arbitrary transcendental numbers, but the e specifically.
- mywittyname 5y agoYeah, 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.
- lisper 5y agoOK, 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.
- JackFr 5y agoPossible 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.
- lisper 5y agoExactly right!
- GolDDranks 5y agoAnother 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.
- ladams 5y agoBut 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...
- Kranar 5y agoYou 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.
- GolDDranks 5y agoSo, 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.
- 613style 5y agoIf 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.
- GolDDranks 5y agoSorry, 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?
- AceyMan 5y agoOn 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.
- colanderman 5y agoNot 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 https://en.wikipedia.org/wiki/Definable_real_number
- lisper 5y agoYes, 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.
- bsedlm 5y ago[deleted by me, the poster]
- mywittyname 5y agoI 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.
- adrian_b 5y agoTo 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.
- anamax 5y agoThe 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.)
- adrian_b 5y agoI 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).
- vishnugupta 5y agoUntil very recently I had no clue what "e" really stood for, though I was good at maths in college. I took it as the result of some complicated infinite series. Until that is I came across this[1] wonderful article. That site is a treasure trove of very good insightful articles. Can't recommend enough. [1] https://betterexplained.com/articles/an-intuitive-guide-to-exponential-functions-e/ https://betterexplained.com/articles/an-intuitive-guide-to-e...
- rrauenza 5y agoI was going to post this Mathologer video (https://www.youtube.com/watch?v=CaasbfdJdJg 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!
- Smithalicious 5y agoThere's something ironic about calling the "most irrational number" the "golden ratio"
- gnatman 5y agoI like this explanation from Numberphile as well[1] [1]https://www.youtube.com/watch?v=AuA2EAgAegE https://www.youtube.com/watch?v=AuA2EAgAegE
- jstx1 5y agoI 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.
- dxbydt 5y ago<<< 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
- unanswered 5y agoPersonally I find "transcendental" to be very problematic language. It definitely feels like it is referring to historical contexts steeped in white supremacy.
- QuercusMax 5y agoCan you expand on this?
- Jtsummers 5y agoPresumably they're trolling because that comment is completely nonsensical. Best to ignore them, maybe downvote.
- alejohausner 5y agoAn 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.