Wikipedia:Reference desk/Archives/Mathematics/2011 July 20
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July 20
editPolynomials
editJust read the article of Polynomials and saw something that surprised me, namely, the properties that an expression is not a polynomial if it contains division or non-integer exponents. Anyone care to explain the reasons of those properties?190.24.186.224 (talk) 04:00, 20 July 2011 (UTC)
- These things have to be excluded for the concept of polynomials to be applicable to general rings. Also, one of the defining characteristics of polynomials is that they have a 0 higher-order derivative.
- For functions that have division, but no fractional powers, you have rational functions. And for functions that do have fractional powers, you have, well, "functions" - you need to limit it somehow if you want to give it a special name. (Also power functions are a specific case). -- Meni Rosenfeld (talk) 04:11, 20 July 2011 (UTC)
- Exactly. The reason for the naming is to do with the history of mathematics. It's like anything. They only had the natural numbers, then they included zero, then the included minus numbers, then came the rationals and the irrations; finally the complex numbers. (The timeline might be off, but you get the idea.) You can imagine a similar evolution in the types of equations, and functions, that mathematicians would consider. I'd imagine that they only came across non-integer powers as solutions to polynomials like x3 + 1 = 0. So, polynomials are in some sense the simplest functions you can think of. — Fly by Night (talk) 14:30, 20 July 2011 (UTC)
- As an account of history, that is false. Negative numbers came before rational and irrational numbers???? That is obviously false. Next you'll tell us Abraham Lincoln assassinated Julius Caesar. Michael Hardy (talk) 01:29, 21 July 2011 (UTC)
- I did include the disclaimer: "The timeline might be off, but you get the idea." — Fly by Night (talk) 18:00, 21 July 2011 (UTC)
- Come on, Fly by Night. You know better than to respond to assholes. --72.179.51.84 (talk) 21:18, 21 July 2011 (UTC)
- Yes, he did say the timeline might be off, but to suggest that the purpose for which number systems were extended was to include solutions of algebraic equations is to make a crackpot of oneself. And calling someone an asshole because he correctly points out crackpothood when he sees it is erroneous. And to do it anonymously is cowardice. Michael Hardy (talk) 04:19, 25 July 2011 (UTC)
- Michael, once again your lust for deprecation highlights your lack of regard for, and interest in, others. Where did I say, or for that matter imply, that "the purpose for which number systems were extended was to include solutions of algebraic equations"? If you take the time to read what people write, and more importantly understand their sentiment, then you might understand a little better, and might not make such obviously false statement. Learn to relax Michael. Learn to be good to yourself and those around you. What do you gain by attacking someone's good intentions? Does it make you feel big? Does it make you feel special? — Fly by Night (talk) 20:07, 25 July 2011 (UTC)
- I don't have any lust for deprecation. But sometimes my time is limited and I don't expand at length on things. It's irresponsible to write like that about something you don't know about. OK, look at how number systems got extended: People wanted to measure things that were not an integer number of gallons, etc., hence fraction. Then they wanted to consider the ratio of lengths of the diagonal of a square to the side, and found out that that is not rational. Centuries later, accountants (1400? 1500? something like that) needed to deal with debts and liabilities. Hence negative numbers. Complex numbers came from the fact that they couldn't be avoided when solving a third-degree equation whose solutions are real. Michael Hardy (talk) 00:07, 26 July 2011 (UTC)
- .....and I do think it helps to address people with expressions of disaproval when they use anonymity to run from people they personally attack. Michael Hardy (talk) 00:10, 26 July 2011 (UTC)
- I can't help it. I'm just too desensitized to moronic behavior by the existence of morons in this world to give a fuck whether my own behavior is moronic or not. --72.179.51.84 (talk) 04:04, 26 July 2011 (UTC)
- (edit conflict) You might say that you don't have a lust for deprecation; your edit history says otherwise. To accuse me of being irresponsible for using a marginally incorrect analogy, while stating that it might be anachronistic, is simple melodrama. You complain about an IP attacking you, while you attack me and call me a crackpot. Where in WP:CIVIL does it say that it's fine to insult people provided you're signed in? Michael, please, just be nice to people. It isn't hard. Even if you think someone is a total crackpot, and a jackass, you don't need to say so. There are three reasons for that: 1) You might be wrong. 2) You yourself might come across as a crackpot/jackass. 3) You might hurt people's feelings. — Fly by Night (talk) 04:14, 26 July 2011 (UTC)
- .....and I do think it helps to address people with expressions of disaproval when they use anonymity to run from people they personally attack. Michael Hardy (talk) 00:10, 26 July 2011 (UTC)
- I don't have any lust for deprecation. But sometimes my time is limited and I don't expand at length on things. It's irresponsible to write like that about something you don't know about. OK, look at how number systems got extended: People wanted to measure things that were not an integer number of gallons, etc., hence fraction. Then they wanted to consider the ratio of lengths of the diagonal of a square to the side, and found out that that is not rational. Centuries later, accountants (1400? 1500? something like that) needed to deal with debts and liabilities. Hence negative numbers. Complex numbers came from the fact that they couldn't be avoided when solving a third-degree equation whose solutions are real. Michael Hardy (talk) 00:07, 26 July 2011 (UTC)
- Michael, once again your lust for deprecation highlights your lack of regard for, and interest in, others. Where did I say, or for that matter imply, that "the purpose for which number systems were extended was to include solutions of algebraic equations"? If you take the time to read what people write, and more importantly understand their sentiment, then you might understand a little better, and might not make such obviously false statement. Learn to relax Michael. Learn to be good to yourself and those around you. What do you gain by attacking someone's good intentions? Does it make you feel big? Does it make you feel special? — Fly by Night (talk) 20:07, 25 July 2011 (UTC)
- Yes, he did say the timeline might be off, but to suggest that the purpose for which number systems were extended was to include solutions of algebraic equations is to make a crackpot of oneself. And calling someone an asshole because he correctly points out crackpothood when he sees it is erroneous. And to do it anonymously is cowardice. Michael Hardy (talk) 04:19, 25 July 2011 (UTC)
- Come on, Fly by Night. You know better than to respond to assholes. --72.179.51.84 (talk) 21:18, 21 July 2011 (UTC)
- I did include the disclaimer: "The timeline might be off, but you get the idea." — Fly by Night (talk) 18:00, 21 July 2011 (UTC)
- As an account of history, that is false. Negative numbers came before rational and irrational numbers???? That is obviously false. Next you'll tell us Abraham Lincoln assassinated Julius Caesar. Michael Hardy (talk) 01:29, 21 July 2011 (UTC)
- Exactly. The reason for the naming is to do with the history of mathematics. It's like anything. They only had the natural numbers, then they included zero, then the included minus numbers, then came the rationals and the irrations; finally the complex numbers. (The timeline might be off, but you get the idea.) You can imagine a similar evolution in the types of equations, and functions, that mathematicians would consider. I'd imagine that they only came across non-integer powers as solutions to polynomials like x3 + 1 = 0. So, polynomials are in some sense the simplest functions you can think of. — Fly by Night (talk) 14:30, 20 July 2011 (UTC)
- Other nice properties of polynomials I can think of that don't extend to rational functions or functions with fractional exponents: Polynomials give rise to the Fundamental theorem of algebra which says that polynomials over the complex numbers always have a root, and therefore they can be uniquely factored into linear terms. Polynomial functions are well-defined everywhere and are continuous and infinitely differentiable everywhere, i.e. they are analytic functions (and in particular the derivatives eventually become 0 as Meni Rosenfeld mentioned). Rckrone (talk) 17:45, 20 July 2011 (UTC)
- What about the constant polynomial g(z) = 1? That has no roots over C. I know that you know this already Rckrone, and just forgot to mention it. But for the OP: The FTA says that every non-constant, single-variable polynomial with complex coefficients has at least one complex root. All irreducible rational functions have the same property of having at least one complex root. Just apply the FTA to the numerator and use the fact that the numerator and denominator share no common factors (a polynomial is a rational function). — Fly by Night (talk) 21:02, 20 July 2011 (UTC)
The definition of polynomials can be looked at this way: start with numbers and variables. Then close under addition and multiplication. What you get is polynomials. Michael Hardy (talk) 01:30, 21 July 2011 (UTC)