# (1 + 1) = 1? **URL:** \ **Category:** Factual Questions\ **Created:** [September 14, 2015, 1:27am UTC](https://boards.straightdope.com/t/1-1-1/731166 "2015-09-14T01:27:43Z")\ **Posts on this page:** 20\ **Page:** 2
**Author:** ![SpyOne](https://avatars.discourse-cdn.com/v4/letter/s/82dd89/32.png) [@SpyOne](https://boards.straightdope.com/u/SpyOne)\ **Post date:** [September 14, 2015, 2:07pm UTC](https://boards.straightdope.com/t/1-1-1/731166/21 "2015-09-14T14:07:13Z")
Um, I just noticed that this one doesn’t hide the division by zero as cleverly as I expected. If x=y, then you can never divide by (x-y). In that case, one can quickly see that it is invalid for all values of x. ☹ ---
**Author:** ![Chronos](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/chronos/32/134_2.png) [@Chronos](https://boards.straightdope.com/u/Chronos)\ **Post date:** [September 14, 2015, 2:18pm UTC](https://boards.straightdope.com/t/1-1-1/731166/22 "2015-09-14T14:18:08Z")
> [@](#): > > Quoth **robert\_columbia** : > > The second, Pope-related argument also displays the problem of equivocation, in which “one” is taken to mean both the number one as well as a term meaning “the same person”. No, those are the same meaning. If you want to make it clearer, say “the cardinality of the set containing the Pope and I is 2” and “the cardinality of the set containing the Pope and I is 1”. Though, of course, the cardinality of the Pope is 1, and the cardinality of myself is 0. And I’ve also seen puzzles like this that are based on a missing constant of integration. Usually, integrals are expressed in such a way that the constant of integration is the value of the integral at 0, and so students have a tendency to assume that that’s always true, but sometimes it can’t be, like with the integral of 1/x. ---
**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:** [September 14, 2015, 2:20pm UTC](https://boards.straightdope.com/t/1-1-1/731166/23 "2015-09-14T14:20:28Z")
> [@j\_sum1](#): > > Not always the case. I forget the details but I came across a similar one where the overlooked detail was that log(z) has more than one value in the domain of complex numbers. > > Another good one is a nice geometric proof that all triangles are isosceles: which relies on a slightly distorted diagram that implies a certain point is contained within the triangle; whereas the actual construction means that it is always exterior. The point to get from that one is that good geometric reasoning can stand without needing a diagram (and is bloody difficult to do.) I can spot the error with the diagram, but if it was presented to me as text only I would have a very hard time of it. > > I think these kind of conundrums have value in that they emphasis the need for precision and rigour in communication and they force a person to reflect properly on what they think they know. There is an interesting point there and I know one such that is not obvious from a diagram. The plain fact is that Euclid’s axioms are badly incomplete. The big thing you have to add is the relation of “betweenness”. Given any three points on a line, one of them is between the other two. Then several axioms relate betweenness to the other concepts. Another problem is the undefined "principle of superposition that is used in a hand-waving fashion to “prove” SAS. The mess of Euclid was finally cleaned up once and for all by David Hilbert around the beginning of the 20th century. It took something like 22 axioms, one of which I could abbreviate as SASAS: if two triangles have all three sides equal and two angles, they are congruent. Then all the rest, including SAS, ASA, and SSS can be rigorously proved. And the proper use of betweenness will tell you whether a point you have constructed will be inside or outside of a triangle, which will fix the problem alluded to in the quote. ---
**Author:** ![septimus](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/septimus/32/410_2.png) [@septimus](https://boards.straightdope.com/u/septimus)\ **Post date:** [September 14, 2015, 3:21pm UTC](https://boards.straightdope.com/t/1-1-1/731166/24 "2015-09-14T15:21:04Z")
Anyone want to see my simplish proof of the Four Color Map Theorem? 🆒 ---
**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:** [September 14, 2015, 3:42pm UTC](https://boards.straightdope.com/t/1-1-1/731166/25 "2015-09-14T15:42:51Z")
> [@Hari\_Seldon](#): > > There is an interesting point there and I know one such that is not obvious from a diagram. The plain fact is that Euclid’s axioms are badly incomplete. The big thing you have to add is the relation of “betweenness”. Given any three points on a line, one of them is between the other two. Then several axioms relate betweenness to the other concepts. Another problem is the undefined "principle of superposition that is used in a hand-waving fashion to “prove” SAS. In all fairness to Euclid, he wasn’t writing THE book to axiomize geometry for all time to a hyper-critical level of proof. He was writing a textbook for his students. How many modern day geometry textbooks suffer the exact same ommissions we criticize _Elements_ for? ---
**Author:** ![Chronos](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/chronos/32/134_2.png) [@Chronos](https://boards.straightdope.com/u/Chronos)\ **Post date:** [September 14, 2015, 4:30pm UTC](https://boards.straightdope.com/t/1-1-1/731166/26 "2015-09-14T16:30:12Z")
And in even more fairness to Euclid, it took millennia for anyone to do better. ---
**Author:** ![Stoneburg](https://avatars.discourse-cdn.com/v4/letter/s/df705f/32.png) [@Stoneburg](https://boards.straightdope.com/u/Stoneburg)\ **Post date:** [September 14, 2015, 5:52pm UTC](https://boards.straightdope.com/t/1-1-1/731166/27 "2015-09-14T17:52:30Z")
It also makes sense spiritually, since everything is One, obviously x + x = 1 no matter what x is. ---
**Author:** ![Leo\_Bloom](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/leo_bloom/32/10377_2.png) [@Leo\_Bloom](https://boards.straightdope.com/u/Leo_Bloom)\ **Post date:** [September 14, 2015, 5:59pm UTC](https://boards.straightdope.com/t/1-1-1/731166/28 "2015-09-14T17:59:14Z")
> [@Thudlow\_Boink](#): > > It can indeed be used in teaching math. What lesson this example teaches depends on how specific you want to get. > > At the lowest level, it’s “Be careful when dividing: watch out for division by zero.” I’ve seen students “solve” equations like > x[sup]2[/sup] = 6x > by dividing both sides by x and getting the solution x = 6 (thereby missing the other solution x = 0). > > More generally, it teaches, as **j\_sum1** said, “the need for precision and rigour,” and that a false assumption or invalid step in an otherwise sound argument (mathematical or otherwise) can lead to a ludicrous conclusion. I take your point, but thank goodness that when I learned how to think early on in school, when I got “x=6” I had solved the problem, not “solved” it. ---
**Author:** ![Indistinguishable](https://avatars.discourse-cdn.com/v4/letter/i/90ced4/32.png) [@Indistinguishable](https://boards.straightdope.com/u/Indistinguishable)\ **Post date:** [September 14, 2015, 7:12pm UTC](https://boards.straightdope.com/t/1-1-1/731166/29 "2015-09-14T19:12:02Z")
> [@Chronos](#): > > Though, of course, the cardinality of the Pope is 1, and the cardinality of myself is 0. 😃 ---
**Author:** ![RTFirefly](https://avatars.discourse-cdn.com/v4/letter/r/c77e96/32.png) [@RTFirefly](https://boards.straightdope.com/u/RTFirefly)\ **Post date:** [September 14, 2015, 7:21pm UTC](https://boards.straightdope.com/t/1-1-1/731166/30 "2015-09-14T19:21:41Z")
> [@Chronos](#): > > the cardinality of the Pope is 1, and the cardinality of myself is 0. The first time a professor of mine said he was going to lecture about infinite cardinals, my first thought was, “does the Vatican know about this?” 😃 > [@septimus](#): > > Anyone want to see my simplish proof of the Four Color Map Theorem? 🆒 Will it fit in the margin of this thread? 😉 ---
**Author:** ![Senegoid](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/senegoid/32/6606_2.png) [@Senegoid](https://boards.straightdope.com/u/Senegoid)\ **Post date:** [September 14, 2015, 8:30pm UTC](https://boards.straightdope.com/t/1-1-1/731166/31 "2015-09-14T20:30:56Z")
> [@robert\_columbia](#): > > - The system _will_ support. Meaning: . . . \ > > - The system _shall_ support. Meaning: . . . \ Are the two words _will_ and _shall_ used the same ways in British English vs. American English? ISTM I’ve read that they are not. In American English, “will \” is a simple future tense, indicating factually that something is going to happen. “shall \” is giving an order or mandate, indicating some action that the subject of the sentence is required to perform. It is used this way extensively in legalese, especially in the wording of laws and regulations. Actually, it’s even more nuanced than that. [See here for discussion.](https://law.utexas.edu/faculty/wschiess/legalwriting/2005/05/shall-vs-will.html) I think in British English (some Brit help me out here if necessary!), _shall_ can be simply a statement of something that will happen: “I shall dream about a thousand pounds to-night, I know I shall!” from _Through The Looking-Glass_. ---
**Author:** ![Senegoid](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/senegoid/32/6606_2.png) [@Senegoid](https://boards.straightdope.com/u/Senegoid)\ **Post date:** [September 14, 2015, 8:37pm UTC](https://boards.straightdope.com/t/1-1-1/731166/32 "2015-09-14T20:37:27Z")
> [@Saint\_Cad](#): > > Those proof seem to always center around a number having two swuare roots, a positive and a negative one. Ultimately they boil down to: > 1 = 1 > sqrt(1) = sqrt(1) > therefore -1 = 1 Another fertile source of fallacies, using squirts of negative numbers, involves falsely relying on the well-known rule: ```auto (√a)(√b) = √(ab) ``` which doesn’t work when both a and b are negative numbers. ---
**Author:** ![Senegoid](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/senegoid/32/6606_2.png) [@Senegoid](https://boards.straightdope.com/u/Senegoid)\ **Post date:** [September 14, 2015, 8:45pm UTC](https://boards.straightdope.com/t/1-1-1/731166/33 "2015-09-14T20:45:15Z")
> [@Thudlow\_Boink](#): > > It can indeed be used in teaching math. What lesson this example teaches depends on how specific you want to get. > > At the lowest level, it’s “Be careful when dividing: watch out for division by zero.” I’ve seen students “solve” equations like > x[sup]2[/sup] = 6x > by dividing both sides by x and getting the solution x = 6 (thereby missing the other solution x = 0). I have an old college-level algebra book, titled simply _College Algebra_ by Britton and Snively, published in 1948 (older than I am!) that was laying around the house when I was a young child, which I’ve kept to this day. It is the best algebra textbook I have ever seen, and goes into detail on these kinds of things. Their take: If, in the process of solving an equation, you ever need to multiply or divide both sides by any expression containing the unknown, you need to be careful. If you _multiply_ both sides by any expression containing the unknown, you _might_ introduce a new solution that wasn’t there before. This is easy to catch: Just be sure, after you find all the solutions, to check every solution in the _original_ equation, to discard any superfluous ones. If you _divide_ both sides by any expression containing the unknown, you _might_ lose a solution that the original equation had. To find these, create an auxiliary equation in which you equate the expression (that you divided both sides by) to 0, and solve that. If any roots were lost, they will be found among the roots of this auxiliary equation. (If anyone wants actual examples, I think I have that book someplace more-or-less accessible, and I can dig up some.) ---
**Author:** ![Anny\_Middon](https://avatars.discourse-cdn.com/v4/letter/a/bc79bd/32.png) [@Anny\_Middon](https://boards.straightdope.com/u/Anny_Middon)\ **Post date:** [September 14, 2015, 9:11pm UTC](https://boards.straightdope.com/t/1-1-1/731166/34 "2015-09-14T21:11:51Z")
Let x=1.999… (That is, 1.999 with 9s extending to infinity.) Then 10x = 19.999… 10x - x = 9x 19.999… - 1.999… = 18 If 9x = 18, then 1.999… = 2 ---
**Author:** ![Dr.Drake](https://avatars.discourse-cdn.com/v4/letter/d/ad7895/32.png) [@Dr.Drake](https://boards.straightdope.com/u/Dr.Drake)\ **Post date:** [September 14, 2015, 9:31pm UTC](https://boards.straightdope.com/t/1-1-1/731166/35 "2015-09-14T21:31:11Z")
> [@Johnny\_L.A](#): > > The first four steps work. The fifth step contains division by zero, and so will not work. Why does step four work? (x + y)(x - y) = y(x - y) That suggests that y = x + y, which could only work if X = 0, in which case Y also = 0. And which also means that neither X nor Y can equal 1. ---
**Author:** ![Senegoid](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/senegoid/32/6606_2.png) [@Senegoid](https://boards.straightdope.com/u/Senegoid)\ **Post date:** [September 14, 2015, 9:43pm UTC](https://boards.straightdope.com/t/1-1-1/731166/36 "2015-09-14T21:43:55Z")
> [@Anny\_Middon](#): > > Let x=1.999… (That is, 1.999 with 9s extending to infinity.) > > Then 10x = 19.999… > > 10x - x = 9x > > 19.999… - 1.999… = 18 > > If 9x = 18, then 1.999… = 2 This, or something similar (0.999… = 1) was touched upon in [this earlier](http://boards.straightdope.com/sdmb/showthread.php?t=32760) thread, among others. ---
**Author:** ![Indistinguishable](https://avatars.discourse-cdn.com/v4/letter/i/90ced4/32.png) [@Indistinguishable](https://boards.straightdope.com/u/Indistinguishable)\ **Post date:** [September 14, 2015, 9:51pm UTC](https://boards.straightdope.com/t/1-1-1/731166/37 "2015-09-14T21:51:17Z")
> [@Dr.Drake](#): > > Why does step four work? > > (x + y)(x - y) = y(x - y) > > That suggests that y = x + y, which could only work if X = 0, in which case Y also = 0. And which also means that neither X nor Y can equal 1. It only suggests that if you divide out the “(x - y)” factor, which is zero. This is the very division which has been noted as problematic in the next step. Go ahead and plug in equal values for x and y; x = y = 1, say. You will indeed find that (x + y)(x - y) = y(x - y), as both sides will be zero. ---
**Author:** ![Senegoid](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/senegoid/32/6606_2.png) [@Senegoid](https://boards.straightdope.com/u/Senegoid)\ **Post date:** [September 14, 2015, 9:52pm UTC](https://boards.straightdope.com/t/1-1-1/731166/38 "2015-09-14T21:52:25Z")
> [@Dr.Drake](#): > > Why does step four work? > > (x + y)(x - y) = y(x - y) > > That suggests that y = x + y, which could only work if X = 0, in which case Y also = 0. And which also means that neither X nor Y can equal 1. In some sense, it works _too well._ If x = y ( and thus, (x-y)=0 ), then the above amounts to (x + y)(0) = y(0) which is all too true no matter what x and y are. Suppose, for example that x = y = 123. Then (x + y)(0) = y(0) becomes (123 + 123)(0) = 123(0) which is still indubitably true and correct and valid. Now, “cancel” the factor of 0 from both sides and you get (123 + 123) = 123, or 246 = 123 which is where the error pops up. This is what happens when you multiply both sides of an equation by zero. For a simpler example, 2(0) = 3(0). It’s mathematically correct, but multiplying both sides by zero, reducing the equation to 0 = 0 has eliminated whatever information the equation formerly had. The next step, “dividing out” the 0 from both sides, obscures the fact that your equation had no useful information in it, installing some new (and wrong) information that wasn’t there before. ---
**Author:** ![cmyk](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/cmyk/32/3353_2.png) [@cmyk](https://boards.straightdope.com/u/cmyk)\ **Post date:** [September 14, 2015, 10:03pm UTC](https://boards.straightdope.com/t/1-1-1/731166/39 "2015-09-14T22:03:43Z")
> [@septimus](#): > > Anyone want to see my simplish proof of the Four Color Map Theorem? 🆒 _looks around_ That might be me. 😉 ---
**Author:** ![Smeghead](https://avatars.discourse-cdn.com/v4/letter/s/f1d935/32.png) [@Smeghead](https://boards.straightdope.com/u/Smeghead)\ **Post date:** [September 14, 2015, 10:23pm UTC](https://boards.straightdope.com/t/1-1-1/731166/40 "2015-09-14T22:23:53Z")
> [@Dr.Drake](#): > > Why does step four work? > > (x + y)(x - y) = y(x - y) > > That suggests that y = x + y, which could only work if X = 0, in which case Y also = 0. And which also means that neither X nor Y can equal 1. (2)(0) = 1(0) True. [Previous page](https://boards.straightdope.com/t/1-1-1/731166.md?page=1) [Next page](https://boards.straightdope.com/t/1-1-1/731166.md?page=3)