Dr.Peterson
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As always, you must link to what you "saw", so we can see the context. Please stop doing this.
Yes, that's almost the only thing that matters in this discussion.
Complex no
- Thread starter Saumyojit
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Multivalued function - Wikipedia
Which statements are you referring to ?
I think you had told every thing carefully i.e "when we will use negative base we deal with complex no " in post 28
What other statement that you made are u referring to?
Does (a^m)^n Always Equal a^(mn)?
Is the exponent law (a^m)^n = a^(mn) always true? Maybe not! A look at some cases where the law needs some careful interpretation.
www.nctm.org
What is the fuss about the ^1/2 notation.
I saw that when the no or base is negative the law broke and also by talking the principal root of the radicand .I think we discussed about this law and its invalidity in this post if you go back.
But I don't get the point what is the connection if I don't take 1/2 as double valued with the post in link
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Dr.Peterson
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So, what is your current question about multivalued functions? I am annoyed by having to go back to the previous page of the thread to try to figure out what you are asking.
In any case, that is a topic that is far beyond you, and I have no intention of pursuing it any further.
In any case, that is a topic that is far beyond you, and I have no intention of pursuing it any further.
See post 42 and post 44 .
I think Jeffm said the right thing in post 32 about this √ and ^ 1/2 means same
And -√ and - ^1/2 means the same .
Please clear this up by giving example and counter example
Thanks btw.
Suppose I consider square root as a multivalued function .
Can I say √4 = -2 directly (as now sqrt is not single valued but mutlivalued so I don't have to give a minus sign before!)
or I have to do it like this -√4 = - 2 ..
Some where in the previous posts u said that we consider Exponention as double valued . Ok if we consider 4^1/2 as double valued ={2,-2} that means naturally square root has also become double valued.
4^1/2 or √4 ( both has become double valued)
To the power 1/2 and root 2 ( using surds notation) are the same thing but different ways of representation
Right??
That arises the doubt that I can also directly write
4^1/2 = -2 without using the minus sign before 4^1/2 .
I am getting confused in those notation thing and the context of seeing expo and root as single valued and multivalued function .
I think Jeffm said the right thing in post 32 about this √ and ^ 1/2 means same
And -√ and - ^1/2 means the same .
Please clear this up by giving example and counter example
Thanks btw.
Suppose I consider square root as a multivalued function .
Can I say √4 = -2 directly (as now sqrt is not single valued but mutlivalued so I don't have to give a minus sign before!)
or I have to do it like this -√4 = - 2 ..
Some where in the previous posts u said that we consider Exponention as double valued . Ok if we consider 4^1/2 as double valued ={2,-2} that means naturally square root has also become double valued.
4^1/2 or √4 ( both has become double valued)
To the power 1/2 and root 2 ( using surds notation) are the same thing but different ways of representation
Right??
That arises the doubt that I can also directly write
4^1/2 = -2 without using the minus sign before 4^1/2 .
I am getting confused in those notation thing and the context of seeing expo and root as single valued and multivalued function .
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We did not resolve the issue because he no longer wishes to discuss it.
As far as I am concerned, 80% of the utility of the whole concept of a function is that it never means one to many. It is always single-valued. I would not worry about multi-valued functions.
The simple fact is that once we get into roots, we have more than one for each number other than zero. As a result, we have essentially two choices. First, we can say that exponentiation by a non-integer is not a function. Second, we can say AS AN ALTERNATIVE that exponentiation is a function by specifying which root is the one that the function points too. A way to have your cake and eat it too is to treat one of the notations as being a function and the other notation not being a function.
My belief, subject to correction, is that the general view among mathematicians is:
[MATH]a \in \mathbb R, \ c \in \mathbb Z, \ a \ge 0, \text { and } c \ne 0 \implies a^{1/c} \equiv \sqrt[c]{a}[/MATH] and
[MATH]a \in \mathbb R, \ c \in \mathbb Z, a \ge 0, \ c \ne 0, \text { and } d = 2c \implies a^{1/d} \equiv \sqrt[d]{a} \ge 0.[/MATH]
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You told about treating the Exponention 4^1/2 as double valued in post 31 ...from there I needed to go to wiki page of multivalued function to get a idea (only in context to Exponention and root function).
I am not sure why you say " that is a topic far beyond you" when I only wants to know in respect with Exponention and root. I didn't want to know other aspects of multivalued.
(I know where to limit myself)
I don't want to get into complex analysis but just want to know in terms with Exponention and root
I got the idea that if I consider Exponention and root function as double valued then some laws will hold true .
Right??.
If the confusion regarding post 32 of JeffM could have been cleared I could have moved on. But what I feel post 32 is right .
I think it's the context .When it is required to consider Exponention and root as a mutlivalued function we consider it then so that the laws holds true
Right?
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