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Whether a negative number is an extraneous root or not can be context dependent. Y'all have probably heard those word problems like the three guys and a monkey on a desert island. They gather a pile of coconuts and agree to divide them up equally among themselves. The effort of gathering the coconuts has worn them out so they agree to divide them up in the morning, but guy 1 gets up in the middle of the night and counts out the coconuts into three piles with one left over. He takes his third and stashes them and gives the one left over to the monkey. The second guy does the same thing a little later, dividing the pile into three (with one left over), taking his third and giving the one left over to the monkey. The third guy does the same thing and in the morning they find the pile still has one left over after dividing into three. The question is how many coconuts there were to start with and somebody found a solution with a negative number of coconuts.
Note: This problem is dredged out of my JDM and may not, in fact, have any solution so doen't try to solve it. It is only here as an example.
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Say Faldage, if explaining why there can be no solution to the coconut and monkey puzzle is the solution, then here is the solution... The requirement to have a coconut left to give the monkey after the third division precludes the first division because the monkey's accumulative coconuts totaling three would have been divided during that first division. Negative numbers or naught...that's the truth.
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Dang it, Milo! I tole you an I tole you: don' take the parblem too serious like, thas what I tole you. It done come out my JDM®!
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Ok, ok, I'm sorry a thousand. You doesn't have to put up an unhappy face.
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"Minus times minus gives plus - the root was there all the time and is not extraneous."
Regarding the extraneous root, I was referring not to the derivation of Golden Ratio, but to the other problem given on my page:
Sqrt(1 + Sqrt(1 + Sqrt(1 + ...)))
In this case, there is only plusses. The square root is therefore positive. Here's an vastly simplified example of what I'm talking about:
Say we way have y=sqrt(5). In this case, the notation means y is equal to the positive sqrt of 5 only. We know that we can square both sides, so we do this:
y^2 = 5, BUT the moment we did this we introduced an extraneous root, -sqrt(5)!
Now, in general, this is not extraneous, but in this particular case, we know that the result is ONLY the positive square root (by definition).
I very clearly understand your point on the triangles. Sometimes it is appropriate to consider negative lengths, and other times it is not. It depends on the particulars of the situation. Generally speaking, one doesn't consider negative lengths for the sides of a triangle. Often, I think of distances being negative, but lengths being only positive, but I'm not sure that's correct.
Unfortunatley, while I use algebra, geometry, trig, analytic geometry almost every day at work, I don't often put a lot of thought into why I reject some solutions. The bottom line is I look at it and I just know whether it makes sense or not. I don't have a firm and formal grasp on why I reject. The only criteria is - does it make sense in the context of the particular problem?
k
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referring not to the derivation of Golden Ratio, but to the other problem given on my page:
Sqrt(1 + Sqrt(1 + Sqrt(1 + ...)))
In this case, there is only plusses. The square root is therefore positive.
Hi FF. I'm afraid I still don't agree that there are only plusses in this! You have an overarching square root and that could be either positive or negative.
...that said, I don't use algebra, geometry, trig, analytic geometry almost every day at work. My maths these days is more about how much the groceries cost and how much is in my purse! I suspect the key difference here may be that you are working from an approach of daily applied maths (where lots of theoretically possible 'solutions' are clearly wrong or impossible in the 'real world') and I am working from my (hopefully correctly but it's been a long time!) remembered pure maths.
Friends?
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FF, I will not look at your links since I know my brain will explode if I do. But to make them clickable for those braver than I (hi Bridget! ), use url in your brackets instead of html.
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FF,
Yup, I think we have narrowed down what I meant - and as said before, none of it was criticising your solution, just querying it being the 'only' one.
I must say my initial reaction to your comment on 'convention' of positive only was 'I've never heard that'. And I was gobsmacked to find it written in the revision notes for A level maths - clearly UK, and one of the maths exams I sat way back when, so how did I miss this convention.
Then I thought about it some more. Whenever I used a square root in physics, it was automatically the positive. When I used it in solving problems around eg angular momentum or geometrical questions, it was automatically the positive. But in algebra and calculus, I don't think it was, and I know I did roots of negative numbers and for some reason I associate them with quadratic equations. So I guess I had some awareness of this rule as applying to 'real life', but was also subliminally driven by the quadratic to thinking on a more theoretical mathematical plane.
I've learnt something about my assumptions. Thank you!
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Well, thank you, as well.
Like I said, I use this stuff nearly every day and I think I have a pretty good handle on some of it, but there's a lot of bits and pieces - like this extraneous root business - that I haven't thought through very well. I just use it without thinking about it in any detail. I enjoy going through it and trying to figure out why I think the thinks I think.
k
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