A Subtle Error

Algebra Level 2

I will attempt to prove that π = − π \pi = -\pi . In which of these steps did I make a flaw in my logic?

Step 1: Using a famous theorem , e i π = − 1. e^{i\pi } = -1 . Step 2: Reciprocate both sides of the equation: e − i π = 1 − 1 = − 1. e^{-i\pi} = \dfrac{1}{-1} = -1 . Step 3: Equating both equations in the above two steps gives e i π = e − i π . \large e^{i \pi} = e^{-i \pi} . Step 4: Since the bases are the same, i π = − i π . i \pi = -i \pi . Step 5: Canceling the imaginary number yields π = − π . \pi = -\pi .

1 2 3 4 5 None, this is correct

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Viki Zeta by Viki Zeta , 19, India

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1 solution

Relevant wiki: Euler's Formula

Note that the following equation holds :

e 2 i π + z = e 2 i π e z = ( − 1 ) 2 e z = e z e^{2i\pi + z} = e^{2i\pi}e^z = (-1)^2 e^z = e^z

We conclude that the exponential function is not one-one (for complex z and in particular for imaginary z).

Hence, although the base is same, the exponents need not be. This means that the 4th step is incorrect.

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Great solution +1.

Prince Loomba - 4 years, 7 months ago

@Vicky Vignesh . WHat is your RMO score?

Md Zuhair - 4 years, 7 months ago

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Viki Zeta - 4 years, 7 months ago

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