i = 1 π i=\frac { 1 }{ \pi } ?

Algebra Level 2

In which line was the first mistake made? The line below is considered line 1.

e i × π = 0 e^{ i\times \pi }=0

i × π = ln ⁡ ( 0 ) i\times \pi =\ln { (0)}

i = 1 π i=\frac { 1 }{ \pi }

Clarification: i = − 1 i = \sqrt{-1} represents the imaginary unit.

2 1 No mistake was made. 3

Ron Lauterbach by Ron Lauterbach , 18, Germany

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4 solutions

Munem Shahriar
Nov 18, 2017

e i π = cos ⁡ π + i sin ⁡ π = − 1 e^{i \pi} = \cos \pi + i \sin \pi = -1

Hence the first mistake was made in the first line.

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John Jo
Dec 7, 2017

Even if we don't know the exact value of the expression, we cannot make a logarithm with argument 0 since e^n will never be 0 (as far as my limited, highschool knowledge goes.

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Sumukh Bansal
Dec 5, 2017

e i × π + 1 = 0 e^{i\times\pi}+1=0 So, e i × π = − 1 e^{i\times\pi}=-1 not 0 0 . Hence mistake is in first line.

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Considering Euler's Identity to be true, let's plug e π i = 0 e^{\pi i} = 0 in the equation.

e π i + 1 = 0 e^{\pi i} + 1 = 0

0 + 1 = 0 0 + 1 = 0

Clearly, 1 ≠ 0 1 \neq 0 .

So, it is not possible that e π i = 0 e^{\pi i} = 0 .

e π i = − 1 e^{\pi i} = -1 makes the identity true. Therefore the mistake is in statement 1 \boxed {1} .

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