What are the three solutions to x^3 = -0.5?

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Discussion Overview

The discussion revolves around the solutions to the equation x3 = -0.5, exploring both real and complex solutions. Participants examine the nature of fractional powers of negative numbers and the implications for identifying roots in different number systems.

Discussion Character

  • Exploratory, Technical explanation, Debate/contested, Mathematical reasoning

Main Points Raised

  • Some participants note that there are two square roots of -1, represented as +i and -i.
  • One participant questions whether positive fractional powers of negative numbers, such as (-0.5)1/3, are possible.
  • Another participant claims that there is one real solution to x3 = -0.5, approximately -0.7937, and two complex solutions, approximately 0.3968 + 0.6874i and 0.3969 - 0.6974i, while expressing uncertainty about the arithmetic.
  • A participant discusses the distinction between square roots in the real number system and the lack of such ordering in complex numbers.

Areas of Agreement / Disagreement

Participants generally agree that there is one real solution and two complex solutions to the equation, but there is some uncertainty regarding the arithmetic and the nature of fractional powers of negative numbers.

Contextual Notes

There are unresolved questions about the arithmetic involved in finding the complex solutions and the implications of defining fractional powers in different number systems.

phymatter
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what is (-1)1/2
 
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There are two square roots of -1, denoted +i and -i.
 


actually i missed the point , i wanted to ask that is the positive fractional power of a negative number possible , like (-0.5)^1/3
 


phymatter said:
actually i missed the point , i wanted to ask that is the positive fractional power of a negative number possible , like (-0.5)^1/3

The point is that there are two answers to your first question and three to your second.
 


ONE of the three numbers, x, such that [itex]x^3= -0.5[/itex], is a real number (it is approximately -0.7937), the other are two complex numbers, 0.3968+ 0.6874i and 0.3969- 0.6974i, approximately (and assuming I have done the arithmetic correctly).

In the real number system we can distinguish between two square roots in that one is positive and the other negative. And we define [itex]a^{1/2}[/itex] to be the positive number, x, such that [itex]x^2= a[/itex]. The complex numbers, however, cannot be made into an ordered field so there is no way of distinguishing one of the two square roots.
 
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