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I am having trouble identify the smallest positive integer $n$ such that $(\frac{1+i}{1-i})^n = 1$

Can someone please throw on approach?

(Also, please correct the equation in the form of Tex/Latex format).


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closed as off-topic by Jonas Meyer, Sujaan Kunalan, MPO, 5xum, Najib Idrissi Mar 24 at 8:10

This question appears to be off-topic. The users who voted to close gave this specific reason:

  • "This question is missing context or other details: Please improve the question by providing additional context, which ideally includes your thoughts on the problem and any attempts you have made to solve it. This information helps others identify where you have difficulties and helps them write answers appropriate to your experience level." – Jonas Meyer, Sujaan Kunalan, MPO, 5xum, Najib Idrissi
If this question can be reworded to fit the rules in the help center, please edit the question.

LaTeX is not introduced, did I interpreted your question appropriately? –  Quixotic Dec 13 '11 at 13:27
@MaX: $\left( \frac{1+n}{1-n} \right)^n = 1 $ has no positive integer solutions –  Daniel Freedman Dec 13 '11 at 13:28
I would also like to know that many users find questions posted in the imperative ("Show that", "Prove", "Do", "Find") unpleasant and somewhat rude. –  Quixotic Dec 13 '11 at 13:29
I am sorry Max, I would just update the post. –  Nikhil Mulley Dec 13 '11 at 13:30
This is real-analysis?! –  Quixotic Dec 13 '11 at 13:34

1 Answer 1

EDIT: Scrap what's below the line, the original question has been changed several times.

If your questions means "Find the smallest positive integer $n$ such that $\left(\frac{1+i}{1-i}\right)^n = 1$, where $i$ is the imaginary unit", then I would first make the denominator of your fraction real (by multiplying top and bottom by $1+i$). After simplification, we find that we are looking for the smallest $n$ such that $ i^n = 1 $.

Alternatively, as GEdgar suggests above, we could convert to polar coordinates. What moduli do $1+i$ and $1-i$ have? What are their arguments? So what is the modulus and argument of $\frac{1+i}{1-i}$?

If your question meant "Find the smallest positive integer $n$ such that $\left(\frac{1+n}{1-n}\right)^n = 1$", then the answer is "there are no solutions". Clearly we'd need $\frac{1+n}{1-n} = 1$ if this were to work, but then we're forced to have $n = 0$.

If your question meant "Find the smallest positive integer $n$ such that $\left(1+ \frac{n}{1-n}\right)^n = 1$", you can use the same logic as above to arrive at the answer.

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