# Knowledge without Proof **URL:** \ **Category:** Great Debates\ **Created:** [October 13, 2005, 8:15am UTC](https://boards.straightdope.com/t/knowledge-without-proof/326040 "2005-10-13T08:15:46Z")\ **Posts on this page:** 13\ **Page:** 10
**Author:** ![Kozmik](https://avatars.discourse-cdn.com/v4/letter/k/8dc957/32.png) [@Kozmik](https://boards.straightdope.com/u/Kozmik)\ **Post date:** [October 22, 2005, 12:29am UTC](https://boards.straightdope.com/t/knowledge-without-proof/326040/181 "2005-10-22T00:29:12Z")
> [@Bytegeist](#): > > the cube root of 12 is an irrational number, like e and pi, and cannot be expressed in decimal with a finite number of digits. The same goes for the cube root of 18. But the product of these two numbers is the cube root of 216, which is _exactly 6_. You can compute this result, and know it to be true, without making any approximations of either number First I find the cube root of 12 and 18 Cube root of 12 = 2.28942849 Cube root of 18 = 2.62074139 Then I add the two cube roots Cube root of 12 \* Cube root of 18 = 6 Therefore, Cube root of 12 \* Cube root of 18 = Cube root of 216 Can you please explain this: 2.28942849 \* 2.28942849 \* 2.62074139 \* 2.62074139 \* 2.62074139 = 216 ---
**Author:** ![Mangetout](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/mangetout/32/19_2.png) [@Mangetout](https://boards.straightdope.com/u/Mangetout)\ **Post date:** [October 22, 2005, 12:30am UTC](https://boards.straightdope.com/t/knowledge-without-proof/326040/182 "2005-10-22T00:30:51Z")
I’m sure we’ve done this already, but: x=x – I think we all agreed that one, didn’t we? - (whatever something is, that’s what it is) from that: x-x=0 – (whatever something is, removing it results in its removal) and x/x=1 – whatever something is, that’s what it is - exactly one instance of itself So… x-x = 0 = (x/x)-1 or π-π = 0 = (π/π)-1 Whatever π is, removing it results in its removal; this is the same as removing one from the number of instances π (whatever it is) is of itself. What am I missing? ---
**Author:** ![Ms2001](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/ms2001/32/18463_2.png) [@Ms2001](https://boards.straightdope.com/u/Ms2001)\ **Post date:** [October 22, 2005, 12:45am UTC](https://boards.straightdope.com/t/knowledge-without-proof/326040/183 "2005-10-22T00:45:30Z")
> [@Mangetout](#): > > I think that’s the point that \*_kozmik_ is labouring to make; that the process you describe, of subtracting the digits one from the other, cannot be completed. Sure, but it doesn’t need to be. We know the answer right from the start: it will be 0 followed by an infinite series of 0s. Actually going through with the calculation would be pointless, because there’s no conceivable way we could ever get anything but 0 by subtracting a digit from itself. ---
**Author:** ![ultrafilter](https://avatars.discourse-cdn.com/v4/letter/u/3d9bf3/32.png) [@ultrafilter](https://boards.straightdope.com/u/ultrafilter)\ **Post date:** [October 22, 2005, 1:08am UTC](https://boards.straightdope.com/t/knowledge-without-proof/326040/184 "2005-10-22T01:08:37Z")
> [@Kozmik](#): > > Cube root of 12 = 2.28942849 > Cube root of 18 = 2.62074139 Neither of these statements is true. ---
**Author:** ![Mangetout](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/mangetout/32/19_2.png) [@Mangetout](https://boards.straightdope.com/u/Mangetout)\ **Post date:** [October 22, 2005, 1:18am UTC](https://boards.straightdope.com/t/knowledge-without-proof/326040/185 "2005-10-22T01:18:16Z")
> [@Mr2001](#): > > Sure, but it doesn’t need to be. We know the answer right from the start: it will be 0 followed by an infinite series of 0s. Actually going through with the calculation would be pointless, because there’s no conceivable way we could ever get anything but 0 by subtracting a digit from itself. Sure, but what I think he’s driving at, after all this, is just that the one way you can’t prove it is _longhand_; you have to infer it logically, rather than calculate it mechanically. But, as others have said, this is true for integers, which still have all those decimals after them, they just happen to be filled with zeroes. ---
**Author:** ![alterego](https://avatars.discourse-cdn.com/v4/letter/a/6bbea6/32.png) [@alterego](https://boards.straightdope.com/u/alterego)\ **Post date:** [October 22, 2005, 1:26am UTC](https://boards.straightdope.com/t/knowledge-without-proof/326040/186 "2005-10-22T01:26:40Z")
> [@Kozmik](#): > > First I find the cube root of 12 and 18 > Cube root of 12 = 2.28942849 > Cube root of 18 = 2.62074139 > > Then I add the two cube roots > Cube root of 12 \* Cube root of 18 = 6 > > Therefore, > Cube root of 12 \* Cube root of 18 = Cube root of 216 > Can you please explain this: > 2.28942849 \* 2.28942849 \* 2.62074139 \* 2.62074139 \* 2.62074139 = 216 your math is so terrible it’s bolstering my confidence =) here: [(the cube root of 12)(the cube root of 12)(the cube root of 12)(the cube root of 18)(the cube root of 18)(the cube root of 18)](http://www.google.com/search?hs=xnm&hl=en&lr=&safe=off&client=firefox-a&rls=org.mozilla%3Aen-US%3Aofficial&q=%28the+cube+root+of+12%29%28the+cube+root+of+12%29%28the+cube+root+of+12%29%28the+cube+root+of+18%29%28the+cube+root+of+18%29%28the+cube+root+of+18%29&btnG=Search) ---
**Author:** ![Bytegeist](https://avatars.discourse-cdn.com/v4/letter/b/9dc877/32.png) [@Bytegeist](https://boards.straightdope.com/u/Bytegeist)\ **Post date:** [October 22, 2005, 1:35am UTC](https://boards.straightdope.com/t/knowledge-without-proof/326040/187 "2005-10-22T01:35:26Z")
> [@Kozmik](#): > > First I find the cube root of 12 and 18 > Cube root of 12 = 2.28942849 > Cube root of 18 = 2.62074139 Those are of course approximations to the true values. But fair enough; let’s run with that. > [@](#): > > Then I **add** the two cube roots > Cube root of 12 \* Cube root of 18 = 6 You’re _multiplying_ them actually — which is good, as that’s what was in my example. > [@](#): > > Therefore, > Cube root of 12 \* Cube root of 18 = Cube root of 216 Certainly true, though I would have passed through this equality on my way to the one above, rather than vice-versa. > [@](#): > > Can you please explain this: > > 2.28942849 \* 2.28942849 \* 2.62074139 \* 2.62074139 \* 2.62074139 = 216 It doesn’t equal that, actually. Probably you left out a factor of 2.28942849 on the left. ---
**Author:** ![bitwise](https://avatars.discourse-cdn.com/v4/letter/b/ea5d25/32.png) [@bitwise](https://boards.straightdope.com/u/bitwise)\ **Post date:** [October 22, 2005, 3:18am UTC](https://boards.straightdope.com/t/knowledge-without-proof/326040/188 "2005-10-22T03:18:25Z")
I think someone should get Kozmik a computer algebra system like Mathematica, so that he can see that computers are perfectly capable of symbolic manipulation and are not limited to computations in decimal notation. ---
**Author:** ![matt](https://avatars.discourse-cdn.com/v4/letter/m/a6a055/32.png) [@matt](https://boards.straightdope.com/u/matt)\ **Post date:** [October 22, 2005, 8:22am UTC](https://boards.straightdope.com/t/knowledge-without-proof/326040/189 "2005-10-22T08:22:01Z")
From **Koskik’s** link: _“In computability theory and computer programming, a crucial part of the definition of an algorithm is that it must terminate—that is, produce its answer after running for a finite number of steps. Whether this number is or isn’t too large for practical execution of the algorithm on a real computer is the subject of computational complexity theory.”_ > [@Kozmik](#): > > Why is this a problem? It isn’t! What I was referring to was the Turing Halting Problem, which is linked to from the link you gave. > **[Halting problem](https://en.wikipedia.org/wiki/Halting_problem)** > > In computability theory, the halting problem is the problem of determining, from a description of an arbitrary computer program and an input, whether the program will finish running, or continue to run forever. Alan Turing proved in 1936 that a general algorithm to solve the halting problem for all possible program-input pairs cannot exist. > For any program f that might determine if programs halt, a "pathological" program g, called with some input, can pass its own source and its input to f and t... The Turing Halting Problem applies to certain algorithms, whereby it is impossible to predict whether or not they will terminate for a particular input. The only way to get a partial answer is to put an input in and let it run. If it terminates, then you know it will terminate for that input. If it doesn’t terminate, you DON’T know whether or not it will eventually if you only wait a little bit longer, and there is no way to find out. However long you wait, just because it hasn’t terminated doesn’t mean that it definitely won’t if you have a couple of millenia more patience! Note that this is not the same as being stuck in a loop - the algorithm isn’t repeating itself. This is a true example of an **uncomputable** problem. > [@Kozmik](#): > > Because π - π = π / π - 1 is intractable. No it isn’t! You’re getting hung up on the idea that “π” is irrational and so can only be approximately represented using integers, and so “can’t” be represented in a computer! Numbers in a computer are arrays of charged capacitors, or patches of magnetised material. They only stand for integers because we chose that they do. They could just as easily stand for irrationals, or negative numbers, or imaginary numbers, or the abstract “x”. Just because you can’t write π as the ratio of two integers doesn’t make the subtraction or irrationals intractable, no more than the lack of negative numbers in Roman numerals would make negative algebra intractable. Truly intractable problems are interesting, but π - π = π / π - 1 IS NOT one of them! (Fun fact, 22 π[sup]4[/sup] = 2143. Exactly. Who’d a thunk it? Substitute the fourth root of 2143/22 for π into π - π = π / π - 1 and it all becomes so easy…) ---
**Author:** ![Cabbage](https://avatars.discourse-cdn.com/v4/letter/c/f07891/32.png) [@Cabbage](https://boards.straightdope.com/u/Cabbage)\ **Post date:** [October 22, 2005, 9:49am UTC](https://boards.straightdope.com/t/knowledge-without-proof/326040/190 "2005-10-22T09:49:24Z")
> [@matt](#): > > Fun fact, 22 π[sup]4[/sup] = 2143. Exactly. Actually, it’s not exact. Pi is a transcendental number, which means it is not the root of any polynomial with integer coefficients. The fourth root of 2143/22, on the other hand, is algebraic (the opposite of transcendental), being a root of the polynomial 22x[sup]4[/sup]-2143. ---
**Author:** ![matt](https://avatars.discourse-cdn.com/v4/letter/m/a6a055/32.png) [@matt](https://boards.straightdope.com/u/matt)\ **Post date:** [October 22, 2005, 6:48pm UTC](https://boards.straightdope.com/t/knowledge-without-proof/326040/191 "2005-10-22T18:48:44Z")
> [@Cabbage](#): > > Actually, it’s not exact. Well that’s spoilt my day! Pesky Ramanujan and his approximations… you did provoke me into reading about transcendental numbers though. More edumacation for me! ---
**Author:** ![Cabbage](https://avatars.discourse-cdn.com/v4/letter/c/f07891/32.png) [@Cabbage](https://boards.straightdope.com/u/Cabbage)\ **Post date:** [October 23, 2005, 6:45am UTC](https://boards.straightdope.com/t/knowledge-without-proof/326040/192 "2005-10-23T06:45:43Z")
Hey, sorry about that, **matt**! To make it up to you, here’s another approximation involving pi (and e): Interestingly, e[sup]pi\*sqrt(163)[/sup] is very nearly an integer; it is approximately 262,537,412,640,768,743.99999999999925. ---
**Author:** ![42fish](https://avatars.discourse-cdn.com/v4/letter/4/b487fb/32.png) [@42fish](https://boards.straightdope.com/u/42fish)\ **Post date:** [October 25, 2005, 3:56am UTC](https://boards.straightdope.com/t/knowledge-without-proof/326040/193 "2005-10-25T03:56:13Z")
> [@Liberal](#): > > Maybe it’s time we fessed up. It’s obvious he isn’t going to fall for any more of our coy tricks. We may as well admit what he alreadys knows we know — numbers are magical. Isn’t revealing this a violation of the mathematicians’ code? (For my next trick, I’ll raise e to the i pi power – without a net!) [Previous page](https://boards.straightdope.com/t/knowledge-without-proof/326040.md?page=9)