Another negative one equals one proof

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SUMMARY

The forum discussion centers around the invalid proof that attempts to equate the imaginary unit i with the real number 1 through flawed substitutions and exponent manipulations. The critical error lies in the misuse of exponentiation, particularly in the step where i^4 = 1^4 is incorrectly simplified to i = 1. Participants emphasize that dropping exponents without proper justification leads to incorrect conclusions, as demonstrated by the example of (-1)^2 = 1^2, which does not imply -1 = 1.

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  • Understanding of complex numbers, specifically the imaginary unit i.
  • Familiarity with exponentiation rules and properties.
  • Knowledge of algebraic manipulation and equivalence relations.
  • Basic comprehension of mathematical proofs and logical reasoning.
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This discussion is beneficial for mathematics students, educators, and anyone interested in understanding the complexities of complex numbers and the importance of rigorous proof techniques.

Ajgrinds
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Hey guys! I need help proving why this proof is wrong. I know it's wrong, but I can't figure out why. Anyway:
i = sqrt -1
i^4 = 1
1^4 = 1
Substution: i^4 =1^4
i = 1
1 = sqrt -1
1^2 = -1
1 = 1^2
1= -1

If you have any questions, feel free to ask.
 
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Ajgrinds said:
Substution: i^4 =1^4
i = 1
That step is not valid. Just dropping exponents doesn't even work for real numbers.
 
-1^2 = 1^2 therefore -1 = 1. Same idea. Do you see where this fails? No need to drag i into it.
 
Vanadium 50 said:
-1^2 = 1^2 therefore -1 = 1. Same idea. Do you see where this fails? No need to drag i into it.
Yeah, thanks
 
mfb said:
That step is not valid. Just dropping exponents doesn't even work for real numbers.
4rt them both...
 
Ajgrinds said:
4rt them both...
##(-1)^4 = 1^4##, but obviously ##-1 \neq 1##.
This is the one-step version of post 1.
 
Vanadium 50 said:
-1^2 = 1^2 therefore -1 = 1.
But -1^2 ≠ 1^2, as I'm sure you know...

However, (-1)^2 does equal 1^2.
 

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