Another negative one equals one proof

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    Negative Proof
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Discussion Overview

The discussion revolves around the validity of a proof that claims to show that -1 equals 1 through manipulations involving complex numbers and exponents. Participants are examining the logical steps taken in the proof and identifying where errors may occur, particularly in the context of complex numbers and real number properties.

Discussion Character

  • Debate/contested
  • Mathematical reasoning

Main Points Raised

  • One participant presents a proof involving the imaginary unit i, claiming that it leads to the conclusion that 1 equals -1.
  • Another participant argues that the substitution step i^4 = 1^4 is invalid, stating that dropping exponents is not a valid operation even for real numbers.
  • Several participants point out that the reasoning used in the proof is flawed, specifically highlighting the incorrect assumption that -1^2 equals 1^2 without proper consideration of parentheses.
  • One participant reiterates the idea that the proof's logic fails without needing to involve complex numbers, suggesting that the error is more fundamental.
  • Another participant emphasizes that while (-1)^2 equals 1^2, the expression -1^2 does not equal 1^2, indicating a misunderstanding in the original proof's steps.

Areas of Agreement / Disagreement

Participants generally agree that the proof is flawed, but there is no consensus on the precise nature of the errors or the best way to articulate them. Multiple competing views on the validity of certain mathematical manipulations remain present.

Contextual Notes

Participants have not fully resolved the implications of exponent manipulation in both real and complex number contexts, and there are unresolved discussions about the proper treatment of negative numbers in squared terms.

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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