البوابة
مُعاينة البيانات الشخصية
Proof that 1 = −1
Version 1
Start with the identity
Convert both sides of the equation into the vulgar fractions
Apply square roots on both sides to yield
Multiply both sides by
to obtain
Any number's square root squared gives the original number, so
Q.E.D.
The proof is invalid because it applies the following principle for square roots incorrectly:
This is only true when x and y are positive real numbers, which is not the case in the proof above. Thus, the proof is invalid.
------------------------------------------------------------------------
Version 2
By incorrectly manipulating radicals, the following invalid proof is derived:
Q.E.D.
The rule
is generally valid only if at least one of the two numbers x or y is positive, which is not the case here. Alternatively, one can view the square root as a 2-valued function over the complex numbers; in this case both sides of the above equation evaluate to {1, -1}.
------------------------------------------------------------------------
Version 3
By crossing into and out of the realm of complex numbers, the following invalid proof is derived:
Q.E.D.
The equation abc = (ab)c, when b and/or c are fractions, is generally valid only when a is positive, which is not the case here, leading to an invalid proof.
-------------------------------------------------------------------------
Version 4
Start with the Pythagorean identity
Raise both sides of the equation to the 3/2 power to obtain
Let x = π
Q.E.D.
In this proof, the fallacy is in the third step, where the rule (ab)c = abc is applied without ensuring that a is positive.
Version 1
Start with the identity
Convert both sides of the equation into the vulgar fractions
Apply square roots on both sides to yield
Multiply both sides by
to obtain Any number's square root squared gives the original number, so
Q.E.D.
The proof is invalid because it applies the following principle for square roots incorrectly:
This is only true when x and y are positive real numbers, which is not the case in the proof above. Thus, the proof is invalid.
------------------------------------------------------------------------
Version 2
By incorrectly manipulating radicals, the following invalid proof is derived:
Q.E.D.
The rule
is generally valid only if at least one of the two numbers x or y is positive, which is not the case here. Alternatively, one can view the square root as a 2-valued function over the complex numbers; in this case both sides of the above equation evaluate to {1, -1}.------------------------------------------------------------------------
Version 3
By crossing into and out of the realm of complex numbers, the following invalid proof is derived:
Q.E.D.
The equation abc = (ab)c, when b and/or c are fractions, is generally valid only when a is positive, which is not the case here, leading to an invalid proof.
-------------------------------------------------------------------------
Version 4
Start with the Pythagorean identity
Raise both sides of the equation to the 3/2 power to obtain
Let x = π
Q.E.D.
In this proof, the fallacy is in the third step, where the rule (ab)c = abc is applied without ensuring that a is positive.
عدل سابقا من قبل mezooo125 في الجمعة 25 أبريل 2008, 5:36 pm عدل 1 مرات
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