# (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:** 14\ **Page:** 4
**Author:** ![Indistinguishable](https://avatars.discourse-cdn.com/v4/letter/i/90ced4/32.png) [@Indistinguishable](https://boards.straightdope.com/u/Indistinguishable)\ **Post date:** [September 16, 2015, 12:47am UTC](https://boards.straightdope.com/t/1-1-1/731166/61 "2015-09-16T00:47:03Z")
The principal square root function under discussion is certainly not from R to R, since it sends negative inputs to “imaginary” outputs. Which is vital in the example **Senegoid** gives which **Saint Cad** responds to. And this is “ill behavior” insofar as it is counter-intuitive that the normally reasonable (and desirable) pattern (√a)(√b) = √(ab) should go out the window. The reason we are discussing this example at all is precisely because it behaves “pathologically”. You may consider it irrelevant dick measuring, but we are both responding to a post where **Saint Cad** desired to remark upon a natural sense in which we needn’t consider this pattern broken. Dick measuring or not, it seemed to me therefore quite relevant to back **Saint Cad** up with my agreement in perspective. ---
**Author:** ![VunderBob](https://avatars.discourse-cdn.com/v4/letter/v/839c29/32.png) [@VunderBob](https://boards.straightdope.com/u/VunderBob)\ **Post date:** [September 16, 2015, 12:57am UTC](https://boards.straightdope.com/t/1-1-1/731166/62 "2015-09-16T00:57:18Z")
> [@robert\_columbia](#): > > That type of thinking is also encountered in _formal_ software requirements specifications (e.g. SRS’s). There is a critical difference between the following clauses: > > - The system _will_ support. Meaning: The customer, owner, or someone other than the developer/engineer guarantees that this statement will be true and the developer/engineer may justly rely on this statement being true if necessary to guarantee that something else will work. E.g. if the system _will_ support 120v mains power input, then the customer is guaranteeing that they will buy, make, or install whatever 120v power input interface is required. > - The system _shall_ support. Meaning: The developer/engineer is required to implement or otherwise ensure, as part of their job, that this statement becomes true for the system. E.g. if the system _shall_ support 120v mains power input, then the developer/engineer must find some way to make that a reality, for example via design, implementation, and installation of a 120v mains power input interface. If the supplied power is defined in the software requirements, you need to fire your system requirements staff. ---
**Author:** ![friedo](https://avatars.discourse-cdn.com/v4/letter/f/8edcca/32.png) [@friedo](https://boards.straightdope.com/u/friedo)\ **Post date:** [September 16, 2015, 12:57am UTC](https://boards.straightdope.com/t/1-1-1/731166/63 "2015-09-16T00:57:37Z")
> [@Indistinguishable](#): > > The principal square root function under discussion is certainly not from R to R, since it sends negative inputs to “imaginary” outputs. Well, the function as defined by [Mathworld](http://mathworld.wolfram.com/PrincipalSquareRoot.html) has a domain of x \>=0, so I suppose it’s fair to say it’s not really R -\> R. But still the point is that the radical symbol has a well-known conventional meaning, and if you’re going to generalize it to negative or complex inputs then you have to say so. As written in **Saint** ’s post, it is correct to reject √(1) = -1 as fallacious. ---
**Author:** ![Indistinguishable](https://avatars.discourse-cdn.com/v4/letter/i/90ced4/32.png) [@Indistinguishable](https://boards.straightdope.com/u/Indistinguishable)\ **Post date:** [September 16, 2015, 1:01am UTC](https://boards.straightdope.com/t/1-1-1/731166/64 "2015-09-16T01:01:49Z")
[On edit: I wrote this before seeing the last post from **friedo** ; this was not in response to that, but rather a continuation of my last post.] > [@friedo](#): > > Throwing the conventional definition out because you think the graph is ugly… Mathematics generally is not so interested in arbitrary, ugly definitions. Generally, it is more interested in clean definitions leading to nice patterns. For example, having first been told that x[sup]y[/sup] means the product of y many xes when y is a cardinal number, why then do we choose in generalizing this to say that x[sup]-1[/sup] should mean the reciprocal of x? Why not say “x[sup]y[/sup] is the product of y many xes when y is a cardinal number, and is 7 otherwise”? Well, because the choice of x[sup]-1[/sup] as the reciprocal of x allows us a clean pattern; it lets us continue to say that x[sup]a + b[/sup] = x[sup]a[/sup] \* x[sup]b[/sup] and so on. This focus on maintaining clean patterns comes up over and over. To re-examine a conventional definition, point out its warts, and consider what would be in many contexts a less ugly perspective instead is entirely in keeping with the manner in which mathematics is developed. Mathematics isn’t about definitions handed down on stone tablets from God to be received unquestioningly; it is about seeking the greatest clarity of understanding. In this example, I believe attaching undue significance to the arbitrarily defined principal square root function generally obscures more than it clarifies. ---
**Author:** ![Indistinguishable](https://avatars.discourse-cdn.com/v4/letter/i/90ced4/32.png) [@Indistinguishable](https://boards.straightdope.com/u/Indistinguishable)\ **Post date:** [September 16, 2015, 1:11am UTC](https://boards.straightdope.com/t/1-1-1/731166/65 "2015-09-16T01:11:35Z")
> [@friedo](#): > > Well, the function as defined by [Mathworld](http://mathworld.wolfram.com/PrincipalSquareRoot.html) has a domain of x \>=0, so I suppose it’s fair to say it’s not really R -\> R. And I agree that the square root function from non-negative reals to non-negative reals is perfectly well-behaved, as I noted before. Note that this function continues to satisfy the pattern (√a)(√b) = √(ab). It does not furnish a counter-example to this rule. Mathworld claims “The concept of principal square root cannot be extended to real negative numbers”. Others instead say “The principal square root is defined at all complex numbers; for negative inputs, it is chosen as a positive multiple of i, while for other inputs, it is chosen to have non-negative ‘real’ component”. I find this latter messy function not frequently useful or natural, and thus am more sympathetic to the perspective Mathworld takes. ---
**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 16, 2015, 2:40am UTC](https://boards.straightdope.com/t/1-1-1/731166/66 "2015-09-16T02:40:13Z")
My point was that I always considered the square root operator to be open to equivocation. sqrt4) = 2 [principal root] sqrt(4) = sqrt(-2 x -2) [so -2 is a square root of 4] = sqrt(-2) x sqrt(-2) = sqrt(2)i x sqrt(2)i [but now we only consider the prinicipal root?!] = sqrt(2 x 2) x (i x i) = 2 x -1 = -2 Therefore 2 = -2 To me Senegoid’s statement showed that ignoring that a square root really has two solutions leads to contradictions that don’t really exist. ---
**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 16, 2015, 6:40am UTC](https://boards.straightdope.com/t/1-1-1/731166/67 "2015-09-16T06:40:55Z")
> [@SmartAlecCat](#): > > Ha… The next thing you’ll be telling me is that the sum of all natural numbers 1+2+3+… is -1/12… Oh no, not me! But you can read all about it in [this thread from May 2007](http://boards.straightdope.com/sdmb/showthread.php?t=420045) and [this thread from January 2014](http://boards.straightdope.com/sdmb/showthread.php?t=712713). ---
**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 16, 2015, 6:57am UTC](https://boards.straightdope.com/t/1-1-1/731166/68 "2015-09-16T06:57:54Z")
I think the usefulness of defining the square-root-function as the _principal_ square root is this: It provides us with the vocabulary (that is, the notation) to specify whichever kind of square root(s) we want to deal with for the problem at hand. If we want only the positive square root, then √x provides us the notation we need to say that. If we wanted the negative square root, then we can write -√x to specify that. If we want to talk about both of the square roots, then we can write ±√x to indicate that. This is used, for example, in the quadratic formula as **friedo** mentioned. ---
**Author:** ![Indistinguishable](https://avatars.discourse-cdn.com/v4/letter/i/90ced4/32.png) [@Indistinguishable](https://boards.straightdope.com/u/Indistinguishable)\ **Post date:** [September 16, 2015, 7:29am UTC](https://boards.straightdope.com/t/1-1-1/731166/69 "2015-09-16T07:29:06Z")
Sure. The notions of “positive” and “negative” square roots make natural sense when the input is positive. And, as I said, the square root function from (semi)positives to (semi)positives is perfectly well-behaved. But when the input is negative, the notions of “positive” and “negative” square roots make rather less natural sense. There are few contexts in which one will want to distinguish one of the complex square roots of a negative number as more relevant than the other. Thus, I don’t consider it generally valuable to distinguish principal square roots for negative inputs. [In some particular niche situations, there may be reason for this, though the appropriate way to do so will then depend on the niche] This does mean understanding the appropriate interpretation of square roots is context-sensitive. But so it is, with everything, forever and always. That bothers some, but them’s the breaks; that’s the reality of things. [Do negatives even have square roots?: That depends on the context: should we consider only “real” values, or complex values as well, or even values of some other kind, such as square matrices? How many square roots does -1 have?: Over the reals, 0; over the complex numbers, 2; over quaternions or 4 by 4 integer matrices more generally, infinitely many. Does 2 have a square root?: Over the reals, yes; over the rationals, no. Everything is context-dependent, inevitably, whether we like it or not. But in very few contexts is there natural call to think of square roots of reals as principally defined as single complex numbers in general in such a way as that √1 is uniquely 1 and √(-1) is uniquely i, breaking the pattern (√a)(√b) = √(ab); though it is convenient for teachers to sidestep this sort of discussion, nonetheless, that particular notion of “principal square root” is rarely the clearest way to think of things.] ---
**Author:** ![Indistinguishable](https://avatars.discourse-cdn.com/v4/letter/i/90ced4/32.png) [@Indistinguishable](https://boards.straightdope.com/u/Indistinguishable)\ **Post date:** [September 16, 2015, 7:33am UTC](https://boards.straightdope.com/t/1-1-1/731166/70 "2015-09-16T07:33:07Z")
(I’m sure everyone’s tired of me talking about this, so I’ll stop doing so now.) ---
**Author:** ![Digital\_is\_the\_new\_Analog](https://avatars.discourse-cdn.com/v4/letter/d/a88e4f/32.png) [@Digital\_is\_the\_new\_Analog](https://boards.straightdope.com/u/Digital_is_the_new_Analog)\ **Post date:** [September 16, 2015, 8:55pm UTC](https://boards.straightdope.com/t/1-1-1/731166/71 "2015-09-16T20:55:44Z")
> [@cmyk](#): > > _looks around_ > > That might be me. 😉 Sorry, I think it’s a different set of four colors. 😉 ---
**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 17, 2015, 1:47am UTC](https://boards.straightdope.com/t/1-1-1/731166/72 "2015-09-17T01:47:25Z")
Come to think of it, the same method that was used to untie the Gordian knot can also be used to color any map with fewer than four colors, and Alexander made a pretty good attempt at that one, too. ---
**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 17, 2015, 2:06am UTC](https://boards.straightdope.com/t/1-1-1/731166/73 "2015-09-17T02:06:41Z")
> [@Senegoid](#): > > Oh no, not me! But you can read all about it in [this thread from May 2007](http://boards.straightdope.com/sdmb/showthread.php?t=420045) and [this thread from January 2014](http://boards.straightdope.com/sdmb/showthread.php?t=712713). Read those and they don’t explain why because of infintesimals 1 =/= 0.999999999… ---
**Author:** ![Indistinguishable](https://avatars.discourse-cdn.com/v4/letter/i/90ced4/32.png) [@Indistinguishable](https://boards.straightdope.com/u/Indistinguishable)\ **Post date:** [September 17, 2015, 2:20am UTC](https://boards.straightdope.com/t/1-1-1/731166/74 "2015-09-17T02:20:01Z")
> [@Indistinguishable](#): > > How many square roots does -1 have?.. over quaternions or 4 by 4 **integer** matrices more generally, infinitely many. Not that it matters, but the word “integer” here should have been replaced with “real”. [Previous page](https://boards.straightdope.com/t/1-1-1/731166.md?page=3)