1

We are to solve $(x-1)^3+8=0$ in the complex set to do so we bring 8 to the RHS and cube root on both sides $$(x-1)=2(-1)^{1/3}$$

The RHS is of three different forms $(-1,\omega,\omega^2)$

But the LHS is just written in a single form why do we do that ?

I'm nothing close to a mathematician so forgive me for my mathmatical illiteracy .

Glowingbluejuicebox
  • 423
  • It is difficult to respond positively, even though you are not a mathematician, because your query has serious defects, re mathSE standards. For discussion of these defects, please see this. – user2661923 Mar 02 '21 at 09:55
  • Well... if the RHS is one of $-2,2\omega, 2\omega^2$, then the LHS is (being equal to the RHS) also one of $-2,2\omega, 2\omega^2$, thus $x$ is one of $-1,2\omega+1, 2\omega^2+1$. I don't see anything disputable here. Writing $(-1)^{1/3}$ as a shorthand for a set ${-1,\omega,\omega^2}$ is acceptable if you know what you are doing (similar to using $\pm$ w.r.t. square roots). –  Mar 02 '21 at 10:04
  • Is there any relation to real analysis (as mentioned in the title) here? – xxxxxxxxx Mar 02 '21 at 10:07

1 Answers1

1

Let's pick another, less complex example. When solving $x^2 = 1$, we conclude $x = \pm 1$. So, why does the left hand side have a single value, whereas the right has two values?

When you solve an equation like $x^2 = 1$, you are finding all possible values of $x$ that satisfy this equation. You start with the assumption that $x$ is a single value that satisfies the equation $x^2 = 1$. We can then conclude, logically, from this assumption, that $x = 1$ or $x = -1$. It's not that $x$ has both values, it has one value or the other (but not both).

Now, in your example, there is an issue where complex exponentiation can produce multiple answers. In real numbers, we understand that $x^{1/3}$ refers to the one and only cube root of $x$, i.e. the unique number $y \in \Bbb{R}$ such that $y^3 = x$. In the complex numbers, there is no unique $y \in \Bbb{C}$ such that $y^3 = x$ (for $x \neq 0$). As such, you should interpret $2(-1)^{1/3}$ as one of three possible complex numbers, much like you would treat $\pm 1$ as one of two real numbers.

Theo Bendit
  • 50,900
  • "Let's pick another, less complex example": Pun intended, I assume? ;) –  Mar 02 '21 at 10:05
  • I am trying to raise the overall standard of quality for mathSE queries. In order to do this, some way must be found to encourage query posters to put much more effort into their queries. When a mathSE reviewer answers a low quality query, it works against mathSE. – user2661923 Mar 02 '21 at 10:06
  • @StinkingBishop Of course! :-) – Theo Bendit Mar 02 '21 at 10:06
  • 2
    @user2661923 I disagree that it's a low-quality query. It's a question of interest to people without much mathematical sophistication, posed in a way that such people might ask it, but yet is clear enough in its intent that it can be answered definitively. Moreover, the asker has shown evidence of their own consideration of the problem, which has culminated in what they see as a paradox. I have no qualms about answering such queries. – Theo Bendit Mar 02 '21 at 10:08
  • 1
    Fair rebuttal. I definitely think that the math is of general interest, and was not expressing an opinion on that aspect of the query. I also agree that each reviewer should be free to decide the issue for themself. My (highly subjective) view is that there is a significant distinction between showing thought and showing work. Further, if you examine the link in the comment that I left the OP, you will see that there is a great deal of work/effort that the OP could have made, and questions that he could have answered that he didn't. ...see next comment. – user2661923 Mar 02 '21 at 10:12
  • 1
    Personally, when I first came to mathSE, my attitude was, who cares what work they show; lets just be of service. However, I had to modify my view because mathSE doesn't belong to me. Each of us has to decide for themself if the way that they are responding is consistent with the overall will of the majority. – user2661923 Mar 02 '21 at 10:14
  • @user2661923 I don't want to tell you or other reviewers what to think about this; I was just giving my opinion about the question. Inevitably, there will be disagreements. I have a piece of advice for you (feel free to ignore it): don't lose your own perspective in the furore in meta. The voices in favour of "question quality" tend to be more strident and more vocal than their opponents. I remain unconvinced that their end goal is even useful to people in general (a repository searchable only by people who know of the third party tool Approach0 and who know MathJax)... continued... – Theo Bendit Mar 02 '21 at 10:29
  • @user2661923 ... let alone worth the discouragement we deliver every day to the more elementary mathematics students who make up the majority of our questions. But again, that's my perspective. Make sure you develop your own, and try not to let people make you feel bad for helping others. – Theo Bendit Mar 02 '21 at 10:30
  • 1
    To both commentators here, first class, magnificent . You have my admiration for retaining civility and mutual respect in conversation. Full marks. Disagreements do not mean end of relationship, civility and humanity. I have seen these spiral out of control, often frustrations spill out onto other posts, and a bad mood always spreads fast. I request, however your opinions may change, that such civility is retained on site. I'd like both of you to be examples to new users in this regard. Role models. – Sarvesh Ravichandran Iyer Mar 02 '21 at 10:33
  • 1
    Cheers to that @TeresaLisbon! – Theo Bendit Mar 02 '21 at 10:34
  • 1
    outstanding 2nd rebuttal. The thing is, why do so many talented math professionals frequent mathSE? I suspect that there is a causal relation to the (often regarded as torturous) query standards. As I stated elsewhere, it has gotten to the point that it is common that a math pro will leave a comment that has more elegance/insight than the answer that I would have left. – user2661923 Mar 02 '21 at 10:35
  • @user2661923 I can see you make a very conscious effort to hold to site standards! Let me tell you how much I appreciate you. But unfortunately, things are two-ended, and sometimes we have to realize that there are gray areas, and loosen those tortuous query standards to accommodate some contributions. I also request you ; if you are finding the list tortuous then make a simpler list that is easier to follow yet retains some stringency, and argue by it. If you are being suffocated by site rules, then I can feel your pain and wish you release it, on me if required. – Sarvesh Ravichandran Iyer Mar 02 '21 at 10:39
  • @TeresaLisbon I don't really regard them as torturous, but most new users do. What it typically means for a new user is that instead of posting their query in 10 minutes, it will take them 90 minutes. Personally, I don't regard the standards as onerous, because I feel that a new user should be willing to get down in the mud, and forgo the path of least resistance. That is what it comes down to. – user2661923 Mar 02 '21 at 10:41
  • @user2661923 Agreed! Yes, so very true. New users really need to work hard. We must change our attitude in this regard. How would you do this? Let me tell you what I would do :be a little more lenient and hands-on. Maybe go for a promising new user each week and help him/her bed onto the site and be a role model? I leave comments on new users' posts, telling them to edit their post and call me back, or giving tips on bettering their post etc. but moist importantly, I do not insist that all eight bullet points be followed, rather just a few key ones in the start. – Sarvesh Ravichandran Iyer Mar 02 '21 at 10:45
  • @user2661923 It seems plausible that there is a connection. I have spoken at length about the connection between low quality questions and elementary students, and how it is essentially inevitable. If the professional mathematicians have lost their patience with such students, my advice to them would be to ignore tags like [tag:algebra-precalculus], [tag:calculus], or possibly even [tag:analysis] and [tag:linear-algebra]. It's important that MSE be for askers and answerers. – Theo Bendit Mar 02 '21 at 10:46
  • @user2661923 We must realize that a balance is required : not all users are the same, some will agree to us and some will not. Merely rejecting those that do not agree is not going to work, it makes us as self-righteous and stubborn as those people. We are the ones who have to be flexible, with our rules and judgement of quality contributions. Which is why a rule set cannot be strictly followed, but we must also use intuition to gauge a user's ability/capacity/interest before we can converse. Developing that will be of so much more help than telling them the rules! We can do it together! – Sarvesh Ravichandran Iyer Mar 02 '21 at 10:48
  • Thank you this really helped .Sorry for all the commotion I may have caused .I do think I could make cleaner queries now onwards. Thanks for understanding this one and for the succinct answer. – Glowingbluejuicebox Mar 02 '21 at 14:36
  • @Glowingbluejuicebox Don't worry about it; this debate has been going on for longer than I've been here. As I said before, I think your question was just fine for the site, but if you think you can improve your questions in the future, don't let me stop you! – Theo Bendit Mar 02 '21 at 15:26