[HN Gopher] Where transcendental numbers hide in everyday math
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Where transcendental numbers hide in everyday math
Author : theafh
Score : 76 points
Date : 2021-10-27 13:40 UTC (9 hours ago)
(HTM) web link (www.quantamagazine.org)
(TXT) w3m dump (www.quantamagazine.org)
| bsedlm wrote:
| [deleted by me, the poster]
| [deleted]
| mywittyname wrote:
| 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.
| adrian_b wrote:
| 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.
| anamax wrote:
| 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.)
| [deleted]
| unanswered wrote:
| Personally I find "transcendental" to be very problematic
| language. It definitely feels like it is referring to historical
| contexts steeped in white supremacy.
| QuercusMax wrote:
| Can you expand on this?
| Jtsummers wrote:
| Presumably they're trolling because that comment is
| completely nonsensical. Best to ignore them, maybe downvote.
| vishnugupta wrote:
| Until 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-e...
| rrauenza wrote:
| 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!
| Smithalicious wrote:
| There's something ironic about calling the "most irrational
| number" the "golden ratio"
| boulos wrote:
| 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.
| gnatman wrote:
| I like this explanation from Numberphile as well[1]
|
| [1]https://www.youtube.com/watch?v=AuA2EAgAegE
| huachimingo wrote:
| 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.
| hota_mazi wrote:
| 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?
| User23 wrote:
| 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.
| kevin_thibedeau wrote:
| 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.
| jstx1 wrote:
| 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.
| dxbydt wrote:
| <<< 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
| jstx1 wrote:
| 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.
| Gunax wrote:
| 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!
| chobytes wrote:
| 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.
| alejohausner wrote:
| 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.
| nine_k wrote:
| Not arbitrary transcendental numbers, but the _e_ specifically.
| mywittyname wrote:
| 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.
| lisper wrote:
| 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.
| GolDDranks wrote:
| 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.
| ladams wrote:
| 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...
| Kranar wrote:
| 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.
| 613style wrote:
| 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.
| AceyMan wrote:
| 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.
| colanderman wrote:
| 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
| lisper wrote:
| 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.
| JackFr wrote:
| 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.
| lisper wrote:
| Exactly right!
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