An x and its reciprocal walk into a bar...

Algebra Level pending

...and vanish!

If x + x − 1 = 0 x + x^{-1} = 0 , what is the sum of the absolute values of all x x ?

Details & Assumptions :

  • x − 1 x^{-1} is the multiplicative inverse of x x .
  • Solve over the complex numbers.


The answer is 2.

Alex Delhumeau by Alex Delhumeau , 20, USA

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1 solution

Alex Delhumeau
Apr 17, 2015

Multiplying x x through x + 1 x = 0 x + \frac {1}{x} = 0 , we get that x 2 + 1 = 0 x^2 + 1 = 0 , which factors to ( x + i ) ⋅ ( x − i ) (x+i) \cdot (x-i) .

Therefore, the solutions are i i and − i ⇒ ∣ i ∣ + ∣ − i ∣ = 1 + 1 = 2 -i \Rightarrow |i| + |-i| = 1 + 1 = \boxed{2} .

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