# Math Question **URL:** \ **Category:** Factual Questions\ **Created:** [January 31, 2012, 3:42am UTC](https://boards.straightdope.com/t/math-question/611267 "2012-01-31T03:42:53Z")\ **Posts on this page:** 17\ **Page:** 2
**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:** [January 31, 2012, 4:33pm UTC](https://boards.straightdope.com/t/math-question/611267/21 "2012-01-31T16:33:21Z")
> [@Jas09](#): > > Hell, sometimes they may produce roots that aren’t even _real_. That is likely for a later course… To avoid confusion, it bears reminding to people unfamiliar with the words that, in math, ‘real’ doesn’t mean ‘actual’, just as ‘imaginary’ doesn’t mean ‘not actual’. “Imaginary numbers” are every bit as real[sup]1[/sup] as “real numbers[sup]2[/sup]”. [sup]1[/sup]vernacular sense of the word [sup]2[/sup]math jargon sense of the term [QUOTE=Johnny L.A.] Is ±0 valid? I don’t know, since zero isn’t signed. [/QUOTE] In our standard number system, the number zero is unsigned. You can come up with your own system that includes signed 0s, but it won’t mean much in the “standard” (whatever “standard” means these days) math you might see in school. ---
**Author:** ![standingwave](https://avatars.discourse-cdn.com/v4/letter/s/9de0a6/32.png) [@standingwave](https://boards.straightdope.com/u/standingwave)\ **Post date:** [January 31, 2012, 5:26pm UTC](https://boards.straightdope.com/t/math-question/611267/22 "2012-01-31T17:26:51Z")
Google will now plots functions as well. Put _ **x^2 + 3x - 18** _ into Google and it will produce [this plot](http://i44.tinypic.com/2ijoh8y.png) complete with a slider to show you the evaluation at each point. The question is asking for where it’s equal to zero. The only downside to using Google is that they will begin to profile you as a nerd. 😛 ---
**Author:** ![Johnny\_L.A](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/johnny_l.a/32/1084_2.png) [@Johnny\_L.A](https://boards.straightdope.com/u/Johnny_L.A)\ **Post date:** [January 31, 2012, 5:31pm UTC](https://boards.straightdope.com/t/math-question/611267/23 "2012-01-31T17:31:47Z")
> [@standingwave](#): > > Google will now plots functions as well. Put _ **x^2 + 3x - 18** _ into Google and it will produce a plot complete with a slider to show you the evaluation at each point. I’m not seeing it. Just links to various pages. Is it supposed to pop up at the top of the page like their maps do? ---
**Author:** ![MikeS](https://avatars.discourse-cdn.com/v4/letter/m/919ad9/32.png) [@MikeS](https://boards.straightdope.com/u/MikeS)\ **Post date:** [January 31, 2012, 5:41pm UTC](https://boards.straightdope.com/t/math-question/611267/24 "2012-01-31T17:41:05Z")
> [@Chessic\_Sense](#): > > The only solution is 0, unless you want to get silly and say “The solutions are 0 and 0,” in which case you might as well say “The solutions are 0 and 0 and 0 and 0 and 0 and 0…” This is derailing quite a bit from the original question, but just to clear this up: any polynomial of degree n (i.e. x[sup]n[/sup] is the highest power of x present) can be uniquely rewritten in the form (x - a[sub]1[/sub])_(x - a[sub]2[/sub])_…\*(x - a[sub]n[/sub]) = 0 The values a[sub]1[/sub], a[sub]2[/sub], … , a[sub]n[/sub] are the “roots” of the polynomial (and are not necessarily real numbers.) Now for some polynomials, two or more of these roots (the values of a[sub]1[/sub], a[sub]2[/sub], … , a[sub]n[/sub]) will be equal. For example, consider the polynomial x[sup]2[/sup] - 2x + 1. This can be rewritten as (x - 1)(x - 1); so a[sub]1[/sub] = a[sub]2[/sub] = 1, and we say that 1 is a double root of this polynomial (or, if you prefer, that “the roots are 1 and 1”. It’s not as accurate to say that “the roots are 1 and 1 and 1”, because x[sup]2[/sup] - 2x + 1 is not the same polynomial as (x - 1)(x - 1)(x - 1). Oh, and to use your example: x[sup]2[/sup] = (x - 0)(x - 0), so zero is a double root of the polynomial (rather than a triple or quadruple or quintuple root.) ---
**Author:** ![Johnny\_L.A](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/johnny_l.a/32/1084_2.png) [@Johnny\_L.A](https://boards.straightdope.com/u/Johnny_L.A)\ **Post date:** [January 31, 2012, 5:44pm UTC](https://boards.straightdope.com/t/math-question/611267/25 "2012-01-31T17:44:37Z")
Thank you, **MikeS**! ---
**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:** [January 31, 2012, 6:38pm UTC](https://boards.straightdope.com/t/math-question/611267/26 "2012-01-31T18:38:00Z")
> [@Chessic\_Sense](#): > > Solve the following: x[sup]2[/sup] = 0 > The only solution is 0, unless you want to get silly and say “The solutions are 0 and 0,” in which case you might as well say “The solutions are 0 and 0 and 0 and 0 and 0 and 0…” No, it really does make sense to say “the solutions are 0 and 0.” Another way to put it is to say that 0 is a root of multiplicity 2. See **MikeS** ’s explanation. For reasons that are not immediately obvious, but which become more and more obvious the more you study polynomials, algebra, etc., it makes sense to say that a polynomial equation of degree 2 (or, more generally, _n_) has _exactly_ 2 (or _n_) solutions, _counting multiplicity_ (which means that some of those solutions may be repeated—they may occur more than once). ---
**Author:** ![drewtwo99](https://avatars.discourse-cdn.com/v4/letter/d/d07c76/32.png) [@drewtwo99](https://boards.straightdope.com/u/drewtwo99)\ **Post date:** [January 31, 2012, 7:01pm UTC](https://boards.straightdope.com/t/math-question/611267/27 "2012-01-31T19:01:00Z")
> [@Thudlow\_Boink](#): > > No, it really does make sense to say “the solutions are 0 and 0.” Another way to put it is to say that 0 is a root of multiplicity 2. See **MikeS** ’s explanation. For reasons that are not immediately obvious, but which become more and more obvious the more you study polynomials, algebra, etc., it makes sense to say that a polynomial equation of degree 2 (or, more generally, _n_) has _exactly_ 2 (or _n_) solutions, _counting multiplicity_ (which means that some of those solutions may be repeated—they may occur more than once). For one, the shape of the graph that gets plotted will behave differently at a root that has a multiplicity of 1 vs 2 vs 3, etc. With a keen eye and a well plotted graph, it’s possible to tell what multiplicity a root is by inspection! Cool stuff 🙂 ---
**Author:** ![standingwave](https://avatars.discourse-cdn.com/v4/letter/s/9de0a6/32.png) [@standingwave](https://boards.straightdope.com/u/standingwave)\ **Post date:** [January 31, 2012, 7:54pm UTC](https://boards.straightdope.com/t/math-question/611267/28 "2012-01-31T19:54:12Z")
> [@Johnny\_L.A](#): > > I’m not seeing it. Just links to various pages. Is it supposed to pop up at the top of the page like their maps do? Yeah, like any other their other gadgets, like calculator results. Be sure you’re phrasing it as a function and not an equation. There’s a tutorial [here](http://googlesystem.blogspot.com/2011/12/googles-graphing-calculator.html). ---
**Author:** ![Indistinguishable](https://avatars.discourse-cdn.com/v4/letter/i/90ced4/32.png) [@Indistinguishable](https://boards.straightdope.com/u/Indistinguishable)\ **Post date:** [January 31, 2012, 7:58pm UTC](https://boards.straightdope.com/t/math-question/611267/29 "2012-01-31T19:58:44Z")
> [@Johnny\_L.A](#): > > I can’t think of an equation right now, but a 3rd-power equation might have roots of -1, -1, and 5. So there are three roots, but two of them are the same. The equation is simple: (x + 1)(x + 1)(x - 5) = 0. ---
**Author:** ![Chessic\_Sense](https://avatars.discourse-cdn.com/v4/letter/c/7c8e57/32.png) [@Chessic\_Sense](https://boards.straightdope.com/u/Chessic_Sense)\ **Post date:** [January 31, 2012, 8:08pm UTC](https://boards.straightdope.com/t/math-question/611267/30 "2012-01-31T20:08:24Z")
> [@drewtwo99](#): > > For one, the shape of the graph that gets plotted will behave differently at a root that has a multiplicity of 1 vs 2 vs 3, etc. > > With a keen eye and a well plotted graph, it’s possible to tell what multiplicity a root is by inspection! Cool stuff 🙂 …which is what I’m imagining in my head. Johnny’s example of, say, 1, 1, and 3 would have an inflection point at x=1. I’m calling that the same root. Now if you’re telling me that a root is defined as a[sub]1[/sub] in (x-a[sub]1[/sub]), then I guess I’ll take your word for it, but it doesn’t make much sense to me to count the same point twice, and [Wiki](http://en.wikipedia.org/wiki/Root_of_a_function) doesn’t seem to agree with that definition. Even if a root has multiplicities, it’s still one root. ---
**Author:** ![Indistinguishable](https://avatars.discourse-cdn.com/v4/letter/i/90ced4/32.png) [@Indistinguishable](https://boards.straightdope.com/u/Indistinguishable)\ **Post date:** [January 31, 2012, 8:16pm UTC](https://boards.straightdope.com/t/math-question/611267/31 "2012-01-31T20:16:21Z")
> [@Chessic\_Sense](#): > > …which is what I’m imagining in my head. Johnny’s example of, say, 1, 1, and 3 would have an inflection point at x=1. You are misusing the term “inflection point” here. In standard mathematical terminology, a root is an inflection point just in case it has multiplicity 3 or greater. For example, x = 0 is not an inflection point of x^2, but it is an inflection point of x^3. > [@](#): > > I’m calling that the same root. Now if you’re telling me that a root is defined as a[sub]1[/sub] in (x-a[sub]1[/sub]), then I guess I’ll take your word for it, but it doesn’t make much sense to me to count the same point twice, and [Wiki](http://en.wikipedia.org/wiki/Root_of_a_function) doesn’t seem to agree with that definition. For some purposes, “root” might as well just mean “point where the function hits 0”. But for other purposes, what is most interesting is “point around which the function behaves the way x^n behaves around x = 0, counted n many times”, and the term “root” (counted with multiplicity) is standardly used for this latter, more informative sense as well. > [@](#): > > Even if a root has multiplicities, it’s still one root. Well, what does “multiplicity” mean but “counted more than once”? ---
**Author:** ![tmarcl](https://avatars.discourse-cdn.com/v4/letter/t/a6a055/32.png) [@tmarcl](https://boards.straightdope.com/u/tmarcl)\ **Post date:** [January 31, 2012, 9:01pm UTC](https://boards.straightdope.com/t/math-question/611267/32 "2012-01-31T21:01:13Z")
> [@standingwave](#): > > Google will now plots functions as well. Put _ **x^2 + 3x - 18** _ into Google and it will produce [this plot](http://i44.tinypic.com/2ijoh8y.png) complete with a slider to show you the evaluation at each point. The question is asking for where it’s equal to zero. The only downside to using Google is that they will begin to profile you as a nerd. 😛 The amusing thing is that there were over 34 million results for that particular equation. I think I’m starting to understand this a lot better, especially after watching the video friedo posted. Thanks again, guys. Marc ---
**Author:** ![Indistinguishable](https://avatars.discourse-cdn.com/v4/letter/i/90ced4/32.png) [@Indistinguishable](https://boards.straightdope.com/u/Indistinguishable)\ **Post date:** [January 31, 2012, 9:06pm UTC](https://boards.straightdope.com/t/math-question/611267/33 "2012-01-31T21:06:12Z")
> [@tmarcl](#): > > The amusing thing is that there were over 34 million results for that particular equation. That’s because you didn’t put quotes around it, and furthermore, Google doesn’t care about the ^, +, or - for web search. Anything with x, 2, 3x, and 18 somewhere on the page would’ve come up. (Also, “x^2 + 3x - 18” isn’t an equation; “x^2 + 3x - 18 = 0” is) ---
**Author:** ![Cabbage](https://avatars.discourse-cdn.com/v4/letter/c/f07891/32.png) [@Cabbage](https://boards.straightdope.com/u/Cabbage)\ **Post date:** [January 31, 2012, 9:17pm UTC](https://boards.straightdope.com/t/math-question/611267/34 "2012-01-31T21:17:53Z")
> [@Indistinguishable](#): > > You are misusing the term “inflection point” here. In standard mathematical terminology, a root is an inflection point just in case it has multiplicity 3 or greater. For example, x = 0 is not an inflection point of x^2, but it is an inflection point of x^3. Don’t you mean “ODD multiplicity of 3 or greater”? x = 0 is not an inflection point of x^4 or x^6 or x^8… ---
**Author:** ![Indistinguishable](https://avatars.discourse-cdn.com/v4/letter/i/90ced4/32.png) [@Indistinguishable](https://boards.straightdope.com/u/Indistinguishable)\ **Post date:** [January 31, 2012, 9:40pm UTC](https://boards.straightdope.com/t/math-question/611267/35 "2012-01-31T21:40:59Z")
Yes, good catch. ---
**Author:** ![standingwave](https://avatars.discourse-cdn.com/v4/letter/s/9de0a6/32.png) [@standingwave](https://boards.straightdope.com/u/standingwave)\ **Post date:** [January 31, 2012, 9:46pm UTC](https://boards.straightdope.com/t/math-question/611267/36 "2012-01-31T21:46:27Z")
> [@Indistinguishable](#): > > That’s because you didn’t put quotes around it, and furthermore, Google doesn’t care about the ^, +, or - for web search. Anything with x, 2, 3x, and 18 somewhere on the page would’ve come up. > > (Also, “x^2 + 3x - 18” isn’t an equation; “x^2 + 3x - 18 = 0” is) Right. To invoke google’s graphing, it needs to be 1) a function, not an equation (no equals sign) and 2) no quotes around said function. Otherwise, it works great. ---
**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:** [January 31, 2012, 9:54pm UTC](https://boards.straightdope.com/t/math-question/611267/37 "2012-01-31T21:54:06Z")
> [@Cabbage](#): > > Don’t you mean “ODD multiplicity of 3 or greater”? x = 0 is not an inflection point of x^4 or x^6 or x^8… That’s right, if there is a root of “multiplicity n”, then: – for odd n, the graph crosses the x-axis; for odd n \>= 3, that would be an inflection point. – for even n, the graph is tangent to the x-axis but doesn’t cross it; in this case you have a relative min or max there. That is for real roots only, of course. For the unreal roots the graph doesn’t come near the x-axis. (I keep wondering what it would look like to graph an equation y=f(x) in three dimensions, using one dimension for real x inputs and the other two for complex y outputs. Or even graph in 4 dimensions, to show real and unreal values for both x and y.) ETA: I meant the other way around. Imaginary x inputs with real y outputs. ETA: Note to OP: How ya doing with this discussion? Are you following all the chatter about multiplicities and inflection points and imaginary roots and all that? [Previous page](https://boards.straightdope.com/t/math-question/611267.md?page=1)