• (disco) in reply to David_C
    David_C:
    If you have the system x² = y², and you want to solve for x, there are two answers: y and -y. Any answer that mentions only one is wrong.

    The general rule is that if you're solving some simple polynomial involving xk as the highest-order term, you can expect there to be k roots, though they might be coincident and might be only on the complex number plane.

  • (disco) in reply to dkf
    dkf:
    might be only on the complex number plane.

    Yes, well, even the roots of x2 = 4 are on the complex plane. They happen to be on the real axis of the plane, but they're on the complex plane.

  • (disco) in reply to xaade
    xaade:
    Anything can be represented in finite space.

    Nope. There are such things as indescribable numbers. And there are a lot of them.

  • (disco) in reply to PleegWat
    PleegWat:
    That does not necessarily follow. Consider exponentiation (`0^b=0`, `a^0=1`, `0^0=?`)

    Not sure what you're getting at, but to answer the (probablyhopefully rhetorical) question, 0^0=1, according to Google's search calculator..

  • (disco) in reply to David_C
    David_C:
    For instance, if you have the equation y = 2x (plotting a straight line), then y/x will represent that line's slope (1/2 2).

    FTFY.

    Also, the slope is not y/x, it's m in the equation y = mx + b, or ∆y/∆x more generally. This is not the same as 0/0 having a defined value.

    David_C:
    If you take the derivative of y/|x|, you end up with 1/(x/|x|)

    I think you mean if you take the derivative of y = |x| with respect to x, and the result can more simply be written as x/|x| for x ≠ 0.

    David_C:
    If we try to take the derivative of that, we end up with dy/(1/(x/|x|))

    The second derivative of y = |x| with respect to x is 0, for x ≠ 0.

    Note: I'm assuming we're working in the reals. I don't know enough to extend to the complex numbers yet.
  • (disco) in reply to Dreikin

    At minimum it's a discontinuity. Google, android calculator, and windows calculator give 1. Linux calculator and wolfram alpha give undefined. I'd check my old TI-82, but that was paid by my parents and I never got to take it home.

  • (disco) in reply to Dreikin
    Dreikin:
    Note: I'm assuming we're working in the reals. I don't know enough to extend to the complex numbers yet.

    Basically the same thing. |x| on the complex plane is the distance from the point to the origin, or sqrt((Re x)^2 + (Im x) ^ 2).

  • (disco) in reply to PleegWat
    PleegWat:
    At minimum it's a discontinuity.

    Technically, the function $y=|x|$ is not differentiable at $x=0$. Since it isn't differentiable there, talking about its slope is TDEMSYR moment, but if you were to compute the limit from below and the limit from above, you'd get different values ($-1$ and $1$ respectively).

  • (disco) in reply to dkf

    Completely correct, but that post was about a^b.

  • (disco) in reply to PleegWat

    One of the representations for complex numbers uses $r\ (= |x|)$ as one of the coordinates (with the other coordinate, $\theta$, being the angle round from the positive real line axis). It's a useful one to know, as it's the one where multiplication (and division) is trivially easy to compute; cartesian-style coordinates are better for addition and subtraction.

  • (disco)

    Math still sucks.

  • (disco) in reply to PleegWat

    Hm. I just checked Wolfram's zero page. Aside from zero being a PITA as usual, it says

    Therefore, the choice of definition for 00 is usually defined to be indeterminate.

    However, defining 00=1 allows some formulas to be expressed simply (Knuth 1992; Knuth 1997, p. 57)

    I can't claim any special knowledge or thought behind that post though - I just thought it might be defined and went to google to check, when I really should have gone to Wolfram, of course. However..

  • (disco) in reply to PleegWat
    PleegWat:
    Basically the same thing. `|x|` on the complex plane is the distance from the point to the origin, or `sqrt((Re x)^2 + (Im x) ^ 2)`.

    I did go to Wolfram to check that, where it says:

    Note that the derivative (read: complex derivative) d|z|/dz does not exist because at every point in the complex plane, the value of the derivative of |z| depends on the direction in which the derivative is taken (so the Cauchy-Riemann equations cannot and do not hold).
    Which is why I put that note there. :confused:
  • (disco) in reply to Dreikin

    Only if they are uncountable.

    If you could come up with a mechanism to count and place them, then you can represent them.

    Even if the representation is

    Blarg(0), Blarg(1), and so on.

  • (disco) in reply to Dreikin
    Dreikin:
    Therefore, the choice of definition for 00 is usually defined to be indeterminate. However, defining 00=1 allows some formulas to be expressed simply (Knuth 1992; Knuth 1997, p. 57)

    That makes math subjective.

    And therefore, not real.

  • (disco) in reply to xaade

    That's by far not the only definition that isn't consistent throughout mathematical literature.

    And what does mathematics have to do with reality?

  • (disco) in reply to xaade
    xaade:
    Only if they are uncountable.
    They are, of course.

    Consider the fact that the set of all representations consisting of finite strings over a countable alphabet is countable. Thus, no matter what grammar you invent, it is impossible to represent more than a countable subset of numbers in a finite space.

  • (disco) in reply to Dreikin
    Dreikin:
    Not sure what you're getting at, but to answer the (probablyhopefully rhetorical) question,

    I'll wait while you plot

    0x lim x->0+

    and

    y0 lim y->0+

    And tell me what happens as both x and y approach zero.

    [spoiler]0x = 0 as x approaches 0 ­ y0 = 1 as y approaches 0. ­ Discontinuity @ x=0, y=0.[/spoiler]

    Unless you're a mathematician in which case you can ignore all that and simply define it.

  • (disco) in reply to xaade
    xaade:
    That makes math subjective.

    Your decision of which type of math to think about is subjective at least.

    PWolff:
    And what does mathematics have to do with reality?

    Math is more real.

    Filed Under: Let's go shopping!

  • (disco) in reply to PWolff
    PWolff:
    And what does mathematics have to do with reality?

    It is the language of the universe!

    [spoiler]INB4 :whoosh:[/spoiler]

  • (disco) in reply to RaceProUK
    RaceProUK:
    It is the language of the universe!

    TIL universes talk to each other. And they use mathematics as their common language. According to your belief system.

    But: Don't start another religious controversy!

  • (disco) in reply to PJH
    PJH:
    I'll wait while you plot

    0x lim x->0+

    and

    y0 lim y->0+

    And tell me what happens as both x and y approach zero.

    [spoiler]0x = 0 as x approaches 0 ­ y0 = 1 as y approaches 0. ­ Discontinuity @ x=0, y=0.[/spoiler]

    Unless you're a mathematician in which case you can ignore all that and simply define it.

    You were :hanzo:d by the discussion starting at:

    PleegWat:
    At minimum it's a discontinuity. Google, android calculator, and windows calculator give 1. Linux calculator and wolfram alpha give undefined. I'd check my old TI-82, but that was paid by my parents and I never got to take it home.

    However, in response to your last sentence, I am working on that so :stuck_out_tongue: (and thanks for that link, it was interesting).

  • (disco) in reply to anotherusername
    anotherusername:
    How about 2 * infinity? Is that larger than infinity?
    In cardinal numbers, infinity * 2 = infinity. (However 2 ^ infinity > infinity, although we're not sure by how much more.) In ordinal numbers, 1 + infinity = infinity, but infinity + 1 > infinity, and infinity * 2 > infinity + N for all finite N. You can then define infinity * 3 and so on up to infinity ^ 2 which is greater than infinity * N for all finite N. In surreal numbers, you can also divide infinity by 2, but I forget how that works.
  • (disco) in reply to Dreikin
    Dreikin:
    There are such things as indescribable numbers. And there are a lot of them.
    Has anyone actually discovered one yet? If so, which was the first to be discovered?

    Edit: I just realised that @xaade probably pointed this out already, but I didn't notice because he hadn't quoted @Dreikin.

  • (disco) in reply to urkerab
    urkerab:
    In surreal numbers, you can also divide infinity by 2

    In surreal numbers, you can divide infinity by marshmallow and get fried eggs.

  • (disco) in reply to dkf
    dkf:
    In surreal numbers, you can divide infinity by marshmallow and get fried eggs.

    Breakfast is always surreal until coffee is applied.

  • (disco) in reply to dcon
    dcon:
    Breakfast is always surreal until coffee is applied.

    ah.... is Tea an acceptable substitute?

    /me hates the flavour of coffee

  • (disco) in reply to accalia
    accalia:
    ah.... is Tea an acceptable substitute?

    As a 100% certified British person, I can confidently claim that tea is always an appropriate drink, no matter the occasion ;)

  • (disco) in reply to accalia
    accalia:
    ah.... is Tea an acceptable substitute?

    Of course!

  • (disco) in reply to accalia
    accalia:
    dcon:
    Breakfast is always surreal until coffee is applied.

    ah.... is Tea an acceptable substitute?

    /me hates the flavour of coffee

    I sure hope so. IMNSHO, bread and coffee don't work together.

  • (disco) in reply to PleegWat
    PleegWat:
    IMNSHO, bread and coffee don't work together.

    Coffee goes better with toast.

  • (disco) in reply to dkf
    dkf:
    Coffee goes better with toastponcy 'Continental' breakfasts hipsters eat, thinking they're cool, when really, they're just being posh douches.

    <I've got a lovely bunch of coconuts...>

  • (disco) in reply to RaceProUK

    TIL I'm a posh douche! :) :) :)

  • (disco) in reply to urkerab
    urkerab:
    Has anyone actually discovered one yet? If so, which was the first to be discovered?

    Such a number would be describable, wouldn't it?

    (The reasoning is basically:

    • The cardinality of the set of all possible representations of numbers is ℵ0.
    • The cardinality of the set of real numbers is ℵ1.
    • ℵ0 < ℵ1.
    • Therefore, some real numbers can not be represented (i.e. described).
    • There are more real numbers that can not be represented than can be represented.)
  • (disco) in reply to Dreikin

    So, what's an example of a number that can't be represented? :trolleybus:

  • (disco) in reply to accalia
    accalia:
    ah.... is Tea an acceptable substitute?
    RaceProUK:
    As a 100% certified British person, I can confidently claim that tea is *always* an appropriate drink, no matter the occasion ;)

    As a person of Irish descent, I must concur. I actually have a tea bowlmug that resembles that, which I use regularly.

  • (disco) in reply to Dreikin

    i should clean the pens out of my Mug Of TEA and use it for tea again...

    [image]

    yes those are full sized pens in there

  • (disco) in reply to accalia
    accalia:
    yes those are full sized pens in there
    :open_mouth: ***WANT!!!***

    If it was any bigger, it'd be a keg!

  • (disco) in reply to RaceProUK

    fits a full pint of tea it does. that's rather why i stopped using it actually. eitehr i could drink it fast enough that it stayed hot by the time i finished it, but i wouldn't enjoy it, or i drank slow enough that i enjoyed it and it was cold at the end. better to have multiple small cuppas.

  • (disco) in reply to accalia

    Get something like this: http://www.amazon.co.uk/Mustard-Hot-Cookie-USB-Warmer/dp/B00A3A02SC/ref=pd_cp_147_2?ie=UTF8&refRID=1YJ0MYV606TYWQJ0054M ;)

  • (disco) in reply to accalia
    accalia:
    yes those are full sized pens in there
    [image]

    Here I go getting into another size contest with @accalia..

  • (disco) in reply to accalia
    accalia:
    i should clean the pens out of my **Mug Of TEA** and use it for tea again... [image]

    Had this one for about 12 years:

    [image]

    Usually full of coffee rather than tea however...

  • (disco) in reply to PJH
    PJH:
    Usually full of coffee

    HISSS! the evil brew!

  • (disco) in reply to Dreikin
    Dreikin:
    The cardinality of the set of real numbers is ℵ1.

    Only if the continuum hypothesis is true.

    Nevertheless, ℵ0 < ℵ1 ≤ 2ℵ0 = | ℝ | , so the logical chain still holds.

    Edit: To my delightdistressconfusion HTML code and no-double-blank-lines-in-between nested quotations seem to work the last few days.

  • (disco) in reply to PWolff
    PWolff:
    Only if the [continuum hypothesis][1] is true.

    Nevertheless, ℵ0 < ℵ1 ≤ 2ℵ0 = | ℝ | , so the logical chain still holds.

    True. I should perhaps have used ℶ0 and ℶ1.

  • (disco) in reply to Dreikin
    Dreikin:
    True. I should perhaps have used &beth;0 and &beth;1.
    why are you using beth?

    what does she have to do with this?

  • (disco) in reply to accalia
    accalia:
    why are you using beth?

    what does she have to do with this?

    Huh. That works in preview..

    [image]

    @discoursebot?

  • (disco) in reply to Dreikin

    i noticed. it works in quites too. i had to mess with your raw to make the bug appear in the quote

  • (disco) in reply to accalia
    accalia:
    i noticed. it works in quites too. i had to mess with your raw to make the bug appear in the quote

    That is...that is quite the wtf. It's so belgiumed up it wraps back around to working.

  • (disco) in reply to Dreikin

    indeed!

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