Understanding the Complexities of Square Roots in Equations

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SUMMARY

The discussion centers on the complexities of square roots in equations, specifically addressing the equation (-1)/1=1/(-1). The user incorrectly applies the property sqrt(a/b)=sqrt(a)/sqrt(b), leading to the erroneous conclusion that i squared equals 1. The key issue identified is that the square root function is not uniquely defined for negative numbers, as every non-zero number has two square roots of opposite signs, complicating the validity of the property in certain contexts.

PREREQUISITES
  • Understanding of complex numbers, particularly the imaginary unit i.
  • Familiarity with properties of square roots and their definitions.
  • Basic knowledge of algebraic manipulation and equations.
  • Awareness of mathematical conventions regarding principal square roots.
NEXT STEPS
  • Study the properties of square roots in complex numbers.
  • Learn about the principal square root and its implications in equations.
  • Research the implications of using square roots in algebraic fractions.
  • Explore the concept of multi-valued functions in mathematics.
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Mathematics students, educators, and anyone interested in deepening their understanding of complex numbers and the properties of square roots in equations.

madness
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I have come up with a simple "paradox" (obviously I am doing something wrong). Sorry I can't use latex:

(-1)/1=1/(-1), take square root of both sides, so sqrt((-1)/1)=sqrt(1/(-1))
use sqrt(a/b)=srt(a)/sqrt(b) *, so i/1=1/i, implies i squared =1.

I think the problem is the part marked *, but I'm not sure in what circumstances this is valid. Can anybody help?
 
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The problem is that every number except 0 actually has two square roots with opposite sign. You can arbitrarily choose one for each number, but then the formula sqrt(a/b)=srt(a)/sqrt(b) is not always true (as you demonstrated).

See also http://en.wikipedia.org/wiki/Square_root#Notes"
 
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Ok that helps a lot. The example on wikipedia was pretty much exactly the one I posted. Thanks.
 

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