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Science

Greatest Equations Ever 1017

Posted by timothy
from the comic-shop-guy's-revenge dept.
sgant writes "What is your favorite equation? This was the question asked by Physics World in a recent poll. This is also covered in a New York Times article about the same poll. Some of the equations mentioned were the simplistic 1+1=2 and Euler's equation, ei + 1 = 0. What are some of your favorite equations?"
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Greatest Equations Ever

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  • correction (Score:5, Insightful)

    by schematix (533634) * on Monday October 25, 2004 @04:23AM (#10618970) Homepage
    Euler's equation is actually Exp[i*Pi] + 1 = 0 not Exp[i*n] +1 = 0 (unless they say n = Pi, which they don't). I'd have to say this is the most elegant equation of all time. It combines the 5 most important numbers in all of mathematics into a single formula. This formula also has tremendous applications in many fields of engineering and other areas of applied mathematics. If it wasn't for this equation, your cell phone wouldn't work.
    • Re:correction (Score:5, Insightful)

      by niks42 (768188) on Monday October 25, 2004 @04:36AM (#10619025)
      Actually, isn't Euler's formula Exp[i*theta] = cos[theta] + i*sin[theta] ? and then substitute in the value of pi into theta, and the more famous result appears.
    • Re:correction (Score:5, Informative)

      by Ford Prefect (8777) on Monday October 25, 2004 @04:43AM (#10619058) Homepage
      It combines the 5 most important numbers in all of mathematics into a single formula.

      It's also got the other important mathematical concepts - exponentiation (i.e. raising something to the power of something else), multiplication, addition and equals. Essentially, it's a huge nugget of maths in a tidy little wrapper.

      I've got an old Sharp graphics calculator, which has both proper notation layout and a complex numbers mode. I still like keying in the 'e^(pi*i)+1', pressing 'Enter', then getting the zero, all perfectly laid out on a little LCD display...
      • by Anonymous Coward on Monday October 25, 2004 @04:54AM (#10619089)
        Too bad my mathematical abilities don't reach beyond spelling rude words on calculators held upside-down.

        Oh well. 5318008.
        • Joke Time (Score:5, Funny)

          by TrentL (761772) on Monday October 25, 2004 @07:27AM (#10619653) Homepage
          George Bush still doesn't know if Bin Laden is alive! After numerous rounds of "We don't even know if Osama is still alive", Osama himself decided to send George Bush a message in his own handwriting to let him know that he was still in the game.
          Bush opened the letter and it appeared to contain a coded message:

          370HSSV-0773H

          Bush was baffled, so he typed it out and e-mailed it to Colin Powell. Colin and his aides had no clue either so they sent it to the CIA. No one could solve it, so it went to the NSA and then to MIT and NASA and the Secret Service.

          Eventually they asked Britain's M I6 for help. They cabled the White House: "Tell the President he is looking at the message upside down."
        • by UnknowingFool (672806) on Monday October 25, 2004 @08:48AM (#10620119)
          Too bad my mathematical abilities don't reach beyond spelling rude words on calculators held upside-down.

          Oh well. 5318008.

          Wouldn't it more appropriate to be: 55378008

    • Actually... (Score:4, Insightful)

      by yoshi_mon (172895) on Monday October 25, 2004 @04:46AM (#10619066)
      If it wasn't for this equation, your cell phone wouldn't work.

      If it wasn't for the laws of nature things wouldn't work. The mathematical formulas are our way of expressing them.
      • Re:Actually... (Score:3, Informative)

        by beaverfever (584714)
        it wasn't for this equation, your cell phone wouldn't work.

        If it wasn't for the laws of nature things wouldn't work. The mathematical formulas are our way of expressing them.


        Mathematical formulas indicate an understanding of such laws, so without that understanding, your cell phone wouldn't work.
      • Re:Actually... (Score:5, Insightful)

        by Rich0 (548339) on Monday October 25, 2004 @06:34AM (#10619454) Homepage
        e^i*pi=-1 isn't a law of nature.

        It is a mathematical relationship which is completely abstract - none of those values are physical quantities, although all of them are used in other physical equations.

        In theory an alien in a completely different universe could come up with the same formula.

        Think about it - e is related to the integral of 1/x on a flat plane - which doesn't exist in real life. i is the square root of -1, which is about as abstract a concept as you'll ever come up with - it certainly doesn't correspond to any physical quantity (unless you define a physical system using complex coordinates for the sake of convenience). Pi is a number which is very useful in practical measurements, but which can be described completely in the abstract.

        In any case, an equation like Euler's formula reflects our understanding of mathematics in general more than it reflects our knowledge of any particular physical process.
        • Re:Actually... (Score:5, Interesting)

          by rsidd (6328) on Monday October 25, 2004 @08:53AM (#10620156)
          i is the square root of -1, which is about as abstract a concept as you'll ever come up with - it certainly doesn't correspond to any physical quantity (unless you define a physical system using complex coordinates for the sake of convenience).

          Quantum mechanical wavefunctions are complex. You could define them as two real wavefunctions and work out the appropriate algebra, but it's exactly complex algebra. So i could correspond to the phase difference of two wavefunctions, which would be observable via interference effects.

          Not disagreeing with what you're saying though -- the equation is fundamental mathematics, independent of the physical universe, it doesn't make sense to imagine an "alternative universe" where it doesn't apply.

      • by dwbassett42 (752317) on Monday October 25, 2004 @06:43AM (#10619485) Homepage
        This difference in views is similar to a fundamental difference between engineers and physicists: Engineers feel their equations are a reasonable approximation of reality, and physicists feel that reality is a reasonable approximation of their equations. And mathematicians? They see no relation between the two. ;)
      • by Smidge204 (605297) on Monday October 25, 2004 @07:28AM (#10619654) Journal
        Nonono, any researcher will tell you that. They just want to cover up the TRUTH. Those complex mathematical formulas are actually mystical runes that describe ancient spells. The formulas themselves DO make your phone work!

        =Smidge=
        • It's what I always said, maths is in fact magic.

          It would be cool with a game like final fantasy where spells were named after mathemathical concepts.

          Quick! Do a Laplace transformation and invoke the Jacobi symbol!
    • Re:correction (Score:4, Informative)

      by Enigma_Man (756516) on Monday October 25, 2004 @08:20AM (#10619926) Homepage
      that is Pi up there, not n. It's a very small font, so it may look like an n, but it's actually a PI symbol...

      -Jesse
    • Richard Feynman once famously remarked that Euler's Identity was the most remarkable equation in mathematics, since it combined all the really important numbers into one formula. Recently while attempting to formulate a technically-oriented conlang, I was considering what numbers really were important and concluded that there was one number of massive significance that was left out, and another was formulated somewhat arbitrarily.

      Firstly, 2 is a very important number. 0 is null and the origin, 1 is unity
  • sum of cubes (Score:5, Interesting)

    by themusicgod1 (241799) <themusicgod1NO@SPAMzworg.com> on Monday October 25, 2004 @04:23AM (#10618971) Homepage Journal
    a^3+b^3 = (a+b)(a^2-ab+b^2)
    first proof, that i'd seen at least, of the existance of negative numbers.
    • by Kippesoep (712796) on Monday October 25, 2004 @08:13AM (#10619876) Homepage
      I wish I could deny the existence of negative numbers. My bank, on the other hand, insists that is how much money I have...
  • V=IR (Score:5, Interesting)

    by oddbudman (599695) on Monday October 25, 2004 @04:23AM (#10618973) Journal
    Gotta Love V=IR. Works pretty well, I use it daily, well that and P=VI.
    • Re:V=IR (Score:5, Interesting)

      by djdead (135363) <seth@wenc[ ].com ['hel' in gap]> on Monday October 25, 2004 @06:12AM (#10619382)
      When I was taking physics for the first time in high school, a EE from MIT taught me the following:

      twinkle twinkle little star
      power equals I squared R

      I remembered it.
      • Re:V=IR (Score:5, Funny)

        by Quantum Jim (610382) <{jfcst24} {at} {yahoo.com}> on Monday October 25, 2004 @11:33AM (#10621744) Homepage Journal

        There was once a football player who was teetering on the edge of academic eligibility. To help the poor guy with his physics test, the coach told him:

        Remember this ryme, to get the power in a circuit:
        Twinkle twinkle little star,
        Power equals I squared R.

        Well the school day before the exam, the football player also had a big game. He tackled alot of people and had a really good day. However, the next day he failed his test! The coach couldn't understand, so he asked the player if he remembered the ryme. The football player said:

        Of course, coach:
        Twinkle twinkle star in the sky,
        Power equals R squared I!

        There's a moral in there somewhere. :-)

    • Re:V=IR (Score:3, Interesting)

      by Hatta (162192)
      I happen to like the Gibbs free energy equation:

      delta G = delta H - T(delta S)

      This equation balances the contributions of entropy (S) and enthalpy (H) and tells you if a given reaction is energetically favorable. delta H is the total energy in a reaction, while T(delta S) is the energy unavailable for work. A quick rearrangment shows that delta G is the energy available for doing work.

      I'm also fond of Shannons juggling theorem. [bc.edu]
  • by oneandoneis2 (777721) * on Monday October 25, 2004 @04:23AM (#10618975) Homepage
    Some of the equations mentioned were the simplistic 1+1=2 and Euler's equation, e^in + 1 = 0. What are some of your favorite equations?"

    Take a look at the username, and take a guess at mine :o)

  • Geometry and Algebra (Score:5, Interesting)

    by metlin (258108) * on Monday October 25, 2004 @04:24AM (#10618979) Journal
    In my opinion, the most important equations are those that brought together Algebric representation of Geometry -- that has been the single most fundamental basis for today's advancement in mathematics and physics.
  • Well... (Score:5, Funny)

    by Anonymous Coward on Monday October 25, 2004 @04:24AM (#10618981)
    I'm quite fond of this one...

    B*u*pi * integral of e^x

    Hint: Try writing it in mathematical notation.

  • H = F ^ 3 (Score:4, Insightful)

    by Rob_Warwick (789939) <(moc.rettirfelppa) (ta) (kciwraw)> on Monday October 25, 2004 @04:28AM (#10618996) Homepage Journal
    Happiness = Food x Friends x Fun
    From Woz.

    It's the most important and beautiful equation I've ever seen.
  • by Paster Of Muppets (787158) on Monday October 25, 2004 @04:29AM (#10618999)
    x = x0 + V * cos (theta)* t, y = y0 + V * sin (theta) * t - (1/2) * g * t^2.

    Projectile equations of motion [anl.gov], very useful in FPS games.

  • by peculiarmethod (301094) on Monday October 25, 2004 @04:29AM (#10619003) Journal
    First we state that women require time and money:
    Women = Time X Money

    And as we all know "time is money"

    Time = Money

    Therefore by substituting Money for Time we get:

    Women = Money X Money

    Women = (Money)2

    And because "money is the root of all evil" we therefore can state:

    Money = (Evil)1/2

    And Since

    (Money)2 = Women

    Then (Money)2 = Evil

    And we are forced to conclude by substituting "women" for "(money)2" from above that:

    Women = Evil
    • you've made an error in your initial assumption, which gives a wrong answer

      First we state that women require time and money:
      Women = Time X Money
      error--^
      this should be
      Women = Time + Money

      and from there onwards ..

      And as we all know "time is money"

      Time = Money

      Therefore by substituting Money for Time we get:

      Women = Money + Money

      Women = 2(Money)

      And because "money is the root of all evil" we therefore can state:

      Money = (Evil)^1/2

      And Since

      2(Money) = Women

      and

      (Mone
  • by bokmann (323771) on Monday October 25, 2004 @04:30AM (#10619010) Homepage
    My favorite is the thinkgeek tshirt that says "2+2=5 for extremely large values of 2".

    It is not just funny... if you consider the numbers not as integers, but as any float value with that integer as the first number, it is true.
  • by Jesrad (716567) on Monday October 25, 2004 @04:31AM (#10619012) Journal
    ih/2Pi dPhi/dt = hc/2iPi (A1 dPhi/dx1 + A2 dPhi/dx2 + A3 dPhi/dx3) + A4 mc(squared)Phi

    Said by Hotson to be the Equation of Everything. First part [zeitlin.net], second part [zeitlin.net]. Worth a read IMO.
  • 0 = 0 (Score:5, Funny)

    by RAMMS+EIN (578166) on Monday October 25, 2004 @04:33AM (#10619018) Homepage Journal
    My favorite is 0 = 0, because it's the one that most often indicates you're done with the math exercise. :-)
    • Re:0 = 0 (Score:3, Funny)

      by _xeno_ (155264)
      Conversely, my "favorite" is 1 = 0, because it means that you get to spend another 5 minutes figuring out what you did wrong.
  • dupe of old poll (Score:5, Informative)

    by Gathers (78832) on Monday October 25, 2004 @04:38AM (#10619031) Homepage
    "What is your favorite equation? ..."
    Shashdot has already covered this in a poll! We all already know that E=mc^2 is the overall favorite, closely followed by F=ma.
    http://slashdot.org/pollBooth.pl?qid=804 [slashdot.org]
  • by physicsphairy (720718) on Monday October 25, 2004 @04:39AM (#10619043) Homepage
    The equation everyone knows offhand is E=mc^2 (even if they don't know what it means), but few people know that the full equations is E=m^2c^4 + p^2c^2. 'p' is momentum, so when you're talking about just the rest mass of the particle you have E=mc^2.

    Anyway, just thought I'd share that because E=m^2c^4 + p^2c^2 is my favorite equation and most people think it looks a little familiar but wouldn't know what it was without a little additional explanation.

  • by Ibag (101144) on Monday October 25, 2004 @04:41AM (#10619049)
    The integral of a differential form on the boundry of a manifold is equal to the integral of the exterior derivative on the manifold itself.

    S_{dM)w=S_(M)dw

    An important special case is the fundamental theorem of calculus. Not only is this a beautiful looking theorem, but important too.

    Other special cases are the classical forms of green's theorem, stoke's theorem, and the divergence theorem.

    I dunno if its my favorite equation, but its up there.
    • That's very similar to Farfegnugen's Law of Inverse Transients relating the comb structure of the polymassive decay groupings to the unthorped resident pressures:

      S_{pD0^(42e)}pi=23^ln(volume)

      Oh, crap, why don't you admit that we both just made this stuff up to sound intelligent on Slashdot?

  • by marcovje (205102) on Monday October 25, 2004 @04:43AM (#10619057)

    Everything = 42 :-)
    • Re:Everything = 42 (Score:3, Interesting)

      by martinde (137088)
      Reminds me of a fun trick with google. Google's calculator knows all kinds of constants - "c", "pi", "e", etc. (Just put those in the standard search box and hit search and you'll see what I mean. Now you can use them in equations - "2*pi+7" or whatever.)

      Anyways, it knows this constant too:
      "the answer to life the universe and everything"

      Made me chuckle the first time I saw it...
  • by lewger (648379) on Monday October 25, 2004 @04:51AM (#10619078) Journal
    I always liked this one that my calc teacher says he saw once on a students paper
    Sin x / n = 6
    The logic of this was that the n on the bottom cancelled out the n on the top so the result was Six. Oh well I laughed when I was shown it.
  • Paper on this (Score:3, Interesting)

    by mukund (163654) on Monday October 25, 2004 @04:56AM (#10619097) Homepage
  • khinchin's constant (Score:3, Interesting)

    by harlows_monkeys (106428) on Monday October 25, 2004 @05:02AM (#10619122) Homepage
    For almost all real numbers r, let {Pn/Qn} be the sequence of convergents of the continued fraction expansion of r. Then limit as n goes to infinity of Qn^(1/n) exists and is equal to exp(pi^2/(12 ln 2)).

    That's my favorite.

    I used to even use "exp(pi^2/12ln2)" as my name in Quakeworld.

  • by Knx (743893) on Monday October 25, 2004 @05:05AM (#10619133) Homepage
    If you don't have a math tool handy, you can use the Google calculator to check some of the equations mentioned here. For instance:

    sqr(-1) [google.com]
    e^(i.pi)+1 [google.com]
    1+1 [google.com]

    That is, if you're suspicious...
  • 1+1=10 (Score:5, Funny)

    by notany (528696) on Monday October 25, 2004 @05:13AM (#10619161) Journal

    There are 10 kinds of people: those who understand binary and those who don't.
    • Re:1+1=10 (Score:3, Funny)

      by 216pi (461752)
      and what are the other eight?
  • by SamSim (630795) on Monday October 25, 2004 @05:14AM (#10619162) Homepage Journal
    Point nine recurring equals one.
  • by jolyonr (560227) on Monday October 25, 2004 @05:36AM (#10619254) Homepage
    Of special importance to slashdot:

    garbage in = garbage out
    Jolyon
  • by syntap (242090) on Monday October 25, 2004 @05:47AM (#10619288)
    I use this in day-to-day life probably more than anything else. Helpful for calculating my home theater projector screen sizes when I need to one-up friedns that get new televisions.
  • by tootlemonde (579170) on Monday October 25, 2004 @05:58AM (#10619332)
    Over at the Historia-Matematica [mathforum.org] discussion list, the members debated [mathforum.org] a similar question:

    As you know, notation has helped the progress of mathematics. Consider, for example, the limitations of the Roman number system, the importance of the invention of a symbol for zero, etc.

    Which were, in your opinion, the notations that have permitted the greatest advances in mathematics?

    Apropos to the current discussion was this response [mathforum.org]:

    the interest of the question:

    > Which were, in your opinion, the notations that have permitted the
    > greatest advances in mathematics?

    (which is very different from any question concerning the history of math. notations) is very close to the interest of the question: who has been the greatest mathematician in the history, e.g. near zero.

  • by tod_miller (792541) on Monday October 25, 2004 @06:29AM (#10619430) Journal
    us {all,your,base}

    Of course if sets aren't your thing...
  • by dysprosia (661648) on Monday October 25, 2004 @06:29AM (#10619431)
    I would have to say, at the moment, my favorite equation would have to be the one giving the coefficients of the generalized Fourier series [wikipedia.org] involving a set of eigenfunctions {p_n}, ie., c_n = <f, p_n>/||p_n||^2.

    Simple stuff, but incredibly cool, considering that Fourier series don't always have to involve just sines and cosines, and you get similar sorts of behaviour.
  • Schrödinger! (Score:4, Informative)

    by k98sven (324383) on Monday October 25, 2004 @06:29AM (#10619432) Journal
    Come on, folks? The Schrödinger equation!

    H*Psi = E*Psi
    (note: H is an operator folks, not a number)

    Perhaps not as famous as E=mc^2.. or as exact as the Dirac equation (relativistic version of the S.E.),
    but.. in terms of practical benefit to mankind, I think this one has done more than any other equation during the last century.

    Atoms. Molecules. Semiconductors. Lasers.

    The number of things explained and modelled by the Schrödinger equation are just uncountable. You can explain almost* all of chemistry with that thing.

    Relativity is nice, but it hasn't had the technical uses quantum physics has.

    (*Relativistic effects are important in heavy elements. For instance the yellow color of gold is a relativistic effect.)
  • by ortholattice (175065) on Monday October 25, 2004 @06:35AM (#10619457)
    The answer is simple. The most beautiful equations, hands down, are those from which all of mathematics can be derived. These are the axioms of ZFC set theory. What could possibly be more beautiful or more important than that? And it's a shame so few people know about them. See Zermelo-Fraenkel Axioms [wolfram.com] and Metamath Proof Explorer [metamath.org].
  • Another.. (Score:5, Interesting)

    by deego (587575) on Monday October 25, 2004 @06:44AM (#10619489)

    Z = z^2 + c

    ^^ This simple equation is the generator of Mandelbrot (and Julia) set, arguably the richest fractal known to us..
  • by amightywind (691887) on Monday October 25, 2004 @07:24AM (#10619639) Journal

    Without a doubt the most important equation of all time is the theorem of Pythagoras:

    a^2 + b^2 = c^2

    It is found virtually everywhere in mathematics and physics: from grade school geometry, trigonometry, calculus, non-Euclidian geometry..., to the Tanyama Shimura Conjecture.

  • by radtea (464814) on Monday October 25, 2004 @08:05AM (#10619829)
    I was amused to see the ideal gas law amongst the contenders, written as PV = nRT where n is in some weird units and R is some weird constant.

    A much nicer form is:

    P = nkT

    where n is the number density of particles and k is Boltzmann's constant.

    For some reason chemists persist in using 12 divided by the mass of the proton in grams as the basis for all measurement, and this choice leads to a proliferation of strange constants and units. I know there are historical reasons for this, but one only has to look at the way physics has re-invented its notation and concepts repeatedly over the years to realize that historical reasons are no excuse.

    Written in a sensible form, the idea gas law is a very beautiful equation, though not so beautiful as the Dirac equation, which is the only differential equation in physics that I'm aware of that describes reality and only reality.

    All the other equations we use have non-physical as well as physical solutions, and we quietly throw out the non-physical solutions. We sometimes even try to maintain that mathematics is "unreasonably successful" as a means of describing reality, when we know perfectly well that half of what our equations describe has no physical counter-part, but is just an ugly artefact of an imperfect description.
  • by static0verdrive (776495) on Monday October 25, 2004 @08:24AM (#10619957) Homepage Journal
    Here's my favorite:

    First we state that women require time and money:
    Women = Time x Money

    And as we all know "time is money"
    Time = Money

    Therefore by substituting Money for Time we get:
    Women = Money x Money
    Women = (Money)^2


    And because "money is the root of all evil" we therefore can state:
    Money = (Evil)^1/2
    And Since
    (Money)^2 = Women
    Then (Money)^2 = Evil

    And we are forced to conclude by substituting "women" for "(money)^2" from above that:
    Women = Evil

    Can't argue with mathematical proof!
    • Re:Women = Evil (Score:3, Interesting)

      by rpillala (583965)

      It's such an old joke and I'm such a math teacher that I'm forced to point out that:

      let x = -3then x^2 = 9
      if you take the square root of both sides you get x = 3.

      Technically you should instead write |x| = 3 which covers the actuality that x is in fact -3. I had to find a way to explain the + or - part of the quadratic formula to my Algebra 2's and that's what I did.

      What you've really proved is that women are either evil or the opposite of evil.

  • A^2+B^2=C^2 (Score:4, Insightful)

    by Shotgun (30919) on Monday October 25, 2004 @09:19AM (#10620359)
    A^2+B^2=C^2

    This is the only equation that will give you the quickest way from here to there in an airplane.

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