Fractional exponents of negative numbers?

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Discussion Overview

The discussion revolves around the evaluation of fractional exponents of negative numbers, particularly focusing on the expression y=x^2.5 for x=-2. Participants explore the implications of complex numbers in this context and consider the graphical representation of functions transitioning between different exponent values.

Discussion Character

  • Exploratory
  • Technical explanation
  • Conceptual clarification
  • Debate/contested
  • Mathematical reasoning

Main Points Raised

  • One participant questions the validity of the expression y=x^2.5 for negative x, noting that their calculator indicates invalid input.
  • Another participant explains that x^2.5 can be expressed as x^2 * x^0.5, leading to the square root of a negative number.
  • There is uncertainty about the evaluation of complex numbers, with one participant attempting to calculate the expression and arriving at a result involving imaginary numbers.
  • A later reply clarifies that the expression is valid in the complex number system, suggesting that calculators may not handle complex inputs.
  • One participant introduces the concept of multivalued results for negative bases raised to fractional exponents, proposing that the principal value is the positive one.
  • Another participant discusses the continuity of graphs transitioning from x^2 to x^3, suggesting that the transition involves passing through imaginary space rather than a discontinuous jump.
  • A mathematical formulation using exponential notation is presented to describe the behavior of negative bases raised to fractional exponents, noting the significance of the rational exponent.
  • One participant expresses a desire to graph the transition from x^2 to x^3 using real numbers as exponents, indicating an interest in visualizing the relationship between these functions.

Areas of Agreement / Disagreement

Participants do not reach a consensus on the evaluation of fractional exponents of negative numbers, with multiple competing views on how to interpret and graph these expressions, particularly regarding the transition between real and imaginary values.

Contextual Notes

There are limitations in the discussion regarding the assumptions made about the properties of complex numbers and the handling of multivalued functions. The mathematical steps involved in the evaluation of the expressions are not fully resolved.

Who May Find This Useful

This discussion may be of interest to those exploring complex numbers, fractional exponents, and the graphical representation of mathematical functions, particularly in the context of transitioning between different exponent values.

DaveC426913
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I was just playing around in my head. I wanted to plot this graph:

y=x^2.5; x=-2

This is valid right? My calc says it's invalid input.
 
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Oh I see.

x^2.5 is the same as x^2 * x^.5

So you're taking the square root of a negative number.

Uh, I don't know quite enough about imaginary arithmetic to figure out the answer but I'll take a stab.

-2^2 * -2^.5
= 4 * 2i
= 8i ?
 
It's valid, but it's complex. So if you can't evaluate complex numbers on your calculator that would explain why the calculator says it's invalid.
 
DaveC426913 said:
Oh I see.

x^2.5 is the same as x^2 * x^.5

So you're taking the square root of a negative number.

Uh, I don't know quite enough about imaginary arithmetic to figure out the answer but I'll take a stab.

-2^2 * -2^.5
= 4 * 2i
= 8i ?

(-2)^.5 is actually either plus or minus 2.5i
 
Well, technically, it's multivalued:

[tex] (-2)^{2.5} = \pm 4 i \sqrt{2}[/tex]

I think the principal value is the one with the + sign.
 
OK, I think that actually sort of answers the original question I was going to ask.

The graph of x^2 is a parabola, never crossing below the x-axis, yet the graph of x^3 does. Since the range from 2 to 3 is a continuum, you should be able to draw a sequence of graphs that shows where and how the "negative x" portion of one graph flips about the X-axis to the other graph.

So, it seems that answer is that it doesn't discontinuously jump from one the other, it actually passes through imaginary space to get there.

If this is true, then I have managed to, just through my own logic, discover the 3D space wherein real numbers and imaginary numbers exist together...

I wish I'd gone on to post-secondary math...
 
In case you're curious:

(-2)t := exp(t ln (-2))
= exp(t (ln |-2| + i arg -2)) = exp(t ln 2) exp(t i (pi + 2 pi n))
= 2t ( cos(t pi) + i sin(t pi) ) ( cos(2 pi n t) + i sin(2 pi n t))

The principal value occurs when n = 0. (or maybe n=-1... but I think it's n=0) Note that usually there are infinitely many values to the exponential; that 2.5 is rational makes it special.
 
Ultimately what I want to do is graph the change from x^2 to x^3 with the real numbers as the exponent.
 

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