# Equations with No Solution **URL:** \ **Category:** Factual Questions\ **Tags:** science-math\ **Created:** [June 8, 2025, 3:14pm UTC](https://boards.straightdope.com/t/equations-with-no-solution/1019330 "2025-06-08T15:14:26Z")\ **Posts on this page:** 20\ **Page:** 3
**Author:** ![Saint\_Cad](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/saint_cad/32/18907_2.png) [@Saint\_Cad](https://boards.straightdope.com/u/Saint_Cad)\ **Post date:** [June 9, 2025, 10:05pm UTC](https://boards.straightdope.com/t/equations-with-no-solution/1019330/41 "2025-06-09T22:05:14Z")
> [@Ynnad](#): > > I think **LSLGuy** paraphrased my question quite well. I am just not satisfied with the answer, “Aristotle said so and that’s how we’ve always done it.” I want to know **why**. Do you want to know why we use the Law of the Excluded Middle? Or do you want to know why the Law of the Excluded Middle is correct? The first is the history of us using Aristotleian Logic in Western Europe. and the idea that there is no such thing as a mathematical statement that is neither/both true and false. The second is that the LotEM is neither true or not true but rather a choice we make due to our choice of logic systems and there are logic systems in which it is not true. ---
**Author:** ![OldGuy](https://avatars.discourse-cdn.com/v4/letter/o/3bc359/32.png) [@OldGuy](https://boards.straightdope.com/u/OldGuy)\ **Post date:** [June 9, 2025, 10:16pm UTC](https://boards.straightdope.com/t/equations-with-no-solution/1019330/42 "2025-06-09T22:16:50Z")
> [@Saint\_Cad](#): > > Can you give us an example? It certainly cannot be a continuous one. You can find continuous even functions with no roots f(x) = 1 is a simple example. ---
**Author:** ![Saint\_Cad](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/saint_cad/32/18907_2.png) [@Saint\_Cad](https://boards.straightdope.com/u/Saint_Cad)\ **Post date:** [June 9, 2025, 10:36pm UTC](https://boards.straightdope.com/t/equations-with-no-solution/1019330/43 "2025-06-09T22:36:05Z")
> [@OldGuy](#): > > You can find continuous even functions with no roots f(x) = 1 is a simple example. I don’t think that’s a difficult one. For an odd function to not have a root, it is necessary that f(0) does not exist. ---
**Author:** ![Andy\_L](https://avatars.discourse-cdn.com/v4/letter/a/c67d28/32.png) [@Andy\_L](https://boards.straightdope.com/u/Andy_L)\ **Post date:** [June 9, 2025, 10:38pm UTC](https://boards.straightdope.com/t/equations-with-no-solution/1019330/44 "2025-06-09T22:38:47Z")
> [@Saint\_Cad](#): > > The second is that the LotEM is neither true or not true but rather a choice we make due to our choice of logic systems and there are logic systems in which it is not true. Yes. One of the surprises of modern (post-1800, say) mathematics is that some axioms are unnecessary - things that Aristotle or Euclid thought were obviously true, or at least so useful that if they weren’t true you couldn’t do anything, turned out to be unnecessary. You can prove things that are usually proved with the LotEM without it, and you can actually assume that Euclid’s 5th postulate is _false_ and produce a consistent geometry (something that Kant (who was pretty smart about math) thought was impossible). ---
**Author:** ![Hari\_Seldon](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/hari_seldon/32/5173_2.png) [@Hari\_Seldon](https://boards.straightdope.com/u/Hari_Seldon)\ **Post date:** [June 9, 2025, 10:39pm UTC](https://boards.straightdope.com/t/equations-with-no-solution/1019330/45 "2025-06-09T22:39:30Z")
There are various groups of mathematicians who do not accept Aristotelian logic. Let me call them notA’s. The most interesting of them (IMHO) was Everett Bishop who is deceased. I would really like to know what his answer would have been. His school was the constructivists. Here is an argument that he explicitly rejected. Consider the statement. There are irrational numbers a,b such that a^b is rational. The argument that most of us would accept, but not Bishop, goes as follows. Consider \sqrt{2}^{\sqrt{2}}. If that number happened to be rational that would be the answer with a=b=\sqrt{2}. If not, let a=\sqrt{2}^{\sqrt{2}} and b=\sqrt{2}. It is immediate that a^b=2. Bishop rejected this argument because I cannot tell you what a and b are and that is the only solution he would have accepted. Another school, the finitists, accepts arguments only using finite sets. Then there are the ultrafinitists who accept only numbers that we can comprehend, say up to a million. Can you really say you understand the concept of a billion? Since they don’t understand what a billion means, perhaps they might accept the possibility that 5x = 5x +5 might be possible when x is a billion. Certainly that it possible when x is infinite. I am a classical mathematician and accept all the usual rules of logic. But if you reject (some of) them, then what you can prove breaks down. (Just for the record, it is known that \sqrt{2}^{\sqrt{2}} is irrational, but Bishop likely would not have accepted the proof of that fact.) ---
**Author:** ![Saint\_Cad](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/saint_cad/32/18907_2.png) [@Saint\_Cad](https://boards.straightdope.com/u/Saint_Cad)\ **Post date:** [June 9, 2025, 11:16pm UTC](https://boards.straightdope.com/t/equations-with-no-solution/1019330/46 "2025-06-09T23:16:18Z")
> [@Hari\_Seldon](#): > > Let me call them notA’s. But if you are not notA, does that mean you are an A? ---
**Author:** ![Great\_Antibob](https://avatars.discourse-cdn.com/v4/letter/g/e47c2d/32.png) [@Great\_Antibob](https://boards.straightdope.com/u/Great_Antibob)\ **Post date:** [June 9, 2025, 11:20pm UTC](https://boards.straightdope.com/t/equations-with-no-solution/1019330/47 "2025-06-09T23:20:51Z")
I don’t know if anybody else has caught on, but it looks like the OP is fully invested in the question, i.e. every response is getting a “why”? That can go on ad infinitum. If we’re positing a person who is not yet capable of accepting a consistent logical framework, there is no satisfactory answer. The mind has to develop to a certain point first, and some 5 year olds aren’t there yet. I don’t know if the thread is supposed to be quite so meta, but there may be no solution to it. ---
**Author:** ![Saint\_Cad](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/saint_cad/32/18907_2.png) [@Saint\_Cad](https://boards.straightdope.com/u/Saint_Cad)\ **Post date:** [June 10, 2025, 4:00am UTC](https://boards.straightdope.com/t/equations-with-no-solution/1019330/48 "2025-06-10T04:00:05Z")
> [@Great\_Antibob](#): > > I don’t know if anybody else has caught on, but it looks like the OP is fully invested in the question, i.e. every response is getting a “why”? That can go on ad infinitum. Especially as it has been answered. ---
**Author:** ![LSLGuy](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/lslguy/32/5813_2.png) [@LSLGuy](https://boards.straightdope.com/u/LSLGuy)\ **Post date:** [June 10, 2025, 11:44am UTC](https://boards.straightdope.com/t/equations-with-no-solution/1019330/49 "2025-06-10T11:44:17Z")
I can certainly see someone saying “When I was a kid in school, nobody taught us about LEM. Instead they just assumed it was so, the definition of the words “true” & “false” being all the basis this idea needed. Now we’re talking about algebra and my little 5x-5x = 5 formula and nobody has yet explained the need of LEM to get to the obvious result.” IOW, “My personal informal set of math axioms that I use to reason with don’t knowingly include LEM. What now, and why should I include it? Seems like everybody else got the memo except me”. It also seems that unlike many of us SDMB addicts, the OP stops by every couple of days, not every couple of hours. That can be a frustrating slog for both parties as we’re waiting around for them to finally return, and meanwhile we’ve had 40 posts of tangents and professional-level details (read “red herrings”) for the OP to process upon their eventual return. ---
**Author:** ![The\_Other\_Waldo\_Pepper](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/the_other_waldo_pepper/32/12370_2.png) [@The\_Other\_Waldo\_Pepper](https://boards.straightdope.com/u/The_Other_Waldo_Pepper)\ **Post date:** [June 10, 2025, 11:53am UTC](https://boards.straightdope.com/t/equations-with-no-solution/1019330/50 "2025-06-10T11:53:31Z")
> [@Ynnad](#): > > If we are asked to solve the equation for x we end up with the equation 0 = 5 which is false. Therefore, we conclude that the equation 5x – 5x = 5 has no solution. Explain this to me like I’m five years old. “You stated that solving this would mean that you’d end up saying 0=5. _I_ didn’t state that; _you_ stated it. Feel free to state it again; I’ll then ask you: are you willing to say that 0=5? Do you believe that 0=5?” ---
**Author:** ![DPRK](https://avatars.discourse-cdn.com/v4/letter/d/4491bb/32.png) [@DPRK](https://boards.straightdope.com/u/DPRK)\ **Post date:** [June 10, 2025, 12:37pm UTC](https://boards.straightdope.com/t/equations-with-no-solution/1019330/51 "2025-06-10T12:37:40Z")
As others and I have pointed out, zero not equalling five requires, depending on what the OP is looking for, somewhat more explanation than “proof by intimidation”. It is not true over the integers modulo 5 which is a perfectly good field, as has been pointed out. ---
**Author:** ![Ynnad](https://avatars.discourse-cdn.com/v4/letter/y/4491bb/32.png) [@Ynnad](https://boards.straightdope.com/u/Ynnad)\ **Post date:** [June 10, 2025, 1:19pm UTC](https://boards.straightdope.com/t/equations-with-no-solution/1019330/52 "2025-06-10T13:19:34Z")
The original question was “What is the logical reasoning behind this process [proof by contradiction] and how do we know the reasoning is sound?” Based solely on the responses in this thread, it appears the answer is we don’t “know” the process is valid, instead we just assume it is valid. Does anyone disagree? ---
**Author:** ![Pleonast](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/pleonast/32/1183_2.png) [@Pleonast](https://boards.straightdope.com/u/Pleonast)\ **Post date:** [June 10, 2025, 1:22pm UTC](https://boards.straightdope.com/t/equations-with-no-solution/1019330/53 "2025-06-10T13:22:32Z")
> [@Ynnad](#): > > Based solely on the responses in this thread, it appears the answer is we don’t “know” the process is valid, instead we just assume it is valid. > > Does anyone disagree? We “know” the process is valid the same way we “know” any scientific facts: the evidence indicates it works. ---
**Author:** ![Ynnad](https://avatars.discourse-cdn.com/v4/letter/y/4491bb/32.png) [@Ynnad](https://boards.straightdope.com/u/Ynnad)\ **Post date:** [June 10, 2025, 1:23pm UTC](https://boards.straightdope.com/t/equations-with-no-solution/1019330/54 "2025-06-10T13:23:25Z")
> [@The\_Other\_Waldo\_Pepper](#): > > I’ll then ask you: are you willing to say that 0=5? Do you believe that 0=5?” I stated that 0 = 5 is false in the second sentence of the OP. ---
**Author:** ![Saint\_Cad](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/saint_cad/32/18907_2.png) [@Saint\_Cad](https://boards.straightdope.com/u/Saint_Cad)\ **Post date:** [June 10, 2025, 1:25pm UTC](https://boards.straightdope.com/t/equations-with-no-solution/1019330/55 "2025-06-10T13:25:37Z")
First of all, refresh my memory. Were you the one that started the math thread about frisbees and zero? > [@Ynnad](#): > > Based solely on the responses in this thread, it appears the answer is we don’t “know” the process is valid, instead we just assume it is valid. That’s not what we said. What we said was choosing a logical system that includes the Law of the Excluded middle (and not working with the field **Z** mod 5) the process is completely valid. Choosing one that does not have the Law of the Excluded Middle, the process may not be. We often choose Aristotleian Logic which includes LEM. Now please do not ask about the Axiom of Choice. ---
**Author:** ![Jasmine](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/jasmine/32/2964_2.png) [@Jasmine](https://boards.straightdope.com/u/Jasmine)\ **Post date:** [June 10, 2025, 1:31pm UTC](https://boards.straightdope.com/t/equations-with-no-solution/1019330/56 "2025-06-10T13:31:21Z")
> [@Ynnad](#): > > Let us consider the equation 5x – 5x = 5 where 5 is a real number and x is a variable for any real number. Do we consider an incorrect equation an “equation with no solution”? Your example has no solution because it is incorrect. No matter what the value of “X”, five times “X” minus 5 times itself has to equal zero. It cannot equal 5. So the equation has a solution: 5X - 5X = 0 ---
**Author:** ![Ynnad](https://avatars.discourse-cdn.com/v4/letter/y/4491bb/32.png) [@Ynnad](https://boards.straightdope.com/u/Ynnad)\ **Post date:** [June 10, 2025, 1:36pm UTC](https://boards.straightdope.com/t/equations-with-no-solution/1019330/57 "2025-06-10T13:36:21Z")
> [@Saint\_Cad](#): > > Were you the one that started the math thread about frisbees and zero? I never started a thread about frisbees. Would the Axiom of Choice allow me to choose to do so? ---
**Author:** ![DPRK](https://avatars.discourse-cdn.com/v4/letter/d/4491bb/32.png) [@DPRK](https://boards.straightdope.com/u/DPRK)\ **Post date:** [June 10, 2025, 1:37pm UTC](https://boards.straightdope.com/t/equations-with-no-solution/1019330/58 "2025-06-10T13:37:46Z")
The equation was 5x-5x=5, not 5x-5x=0. So assuming there is a solution leads to a contradiction. (And proof by contradiction is justified by classical logic, as has been discussed.) ---
**Author:** ![Thudlow\_Boink](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/thudlow_boink/32/320_2.png) [@Thudlow\_Boink](https://boards.straightdope.com/u/Thudlow_Boink)\ **Post date:** [June 10, 2025, 1:49pm UTC](https://boards.straightdope.com/t/equations-with-no-solution/1019330/59 "2025-06-10T13:49:37Z")
> [@DPRK](#): > > So assuming there is a solution leads to a contradiction. That’s the way some people are looking at it: 5x-5x=5 cannot have a solution (that is, cannot be true), because if we assume that it _is_ true, this leads to a contradiction (0 = 5). But that’s not the way I’ve been looking at it (in my earlier posts #11 and #22). I’m just saying that 5x-5x=5 is equivalent to 0=5, and since the latter has no solution, the former also has no solution. I don’t _think_ there’s any need to invoke the LEM—is there? ---
**Author:** ![Jasmine](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/jasmine/32/2964_2.png) [@Jasmine](https://boards.straightdope.com/u/Jasmine)\ **Post date:** [June 10, 2025, 1:49pm UTC](https://boards.straightdope.com/t/equations-with-no-solution/1019330/60 "2025-06-10T13:49:38Z")
Well, all I can say is that my Algebra I teacher wouldn’t have told me that. My Algebra I teacher would have said, "Your solution for 5x - 5x = ? is incorrect. Find the correct solution. He wouldn’t have apologized to me for giving me an equation that has no solution. [Previous page](https://boards.straightdope.com/t/equations-with-no-solution/1019330.md?page=2) [Next page](https://boards.straightdope.com/t/equations-with-no-solution/1019330.md?page=4)