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The least positive integer n for which ( {1+i}{1-i} )^{n} re

Question

The least positive integer $n$ for which $(\frac{1 + i}{1 - i})^{n}$ represents a real number is

A

4

B

8

C

12

D

2

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Eleanor Discussed

The least positive integer

n

for which

(\frac{1 + i}{1 - i})^{n}

represents a real number is

7 mins ago

Discuss this question LIVE

7 mins ago

Text solutionVerified

(\frac{( 1 + i ) ( 1 + i )}{( 1 - i ) ( 1 + i )})^{n}

(Dividing and multiplying by

(1 + i)

)

=

\frac{1}{2 ^{n}}

.

(1 + i)^{2 n}

-(1)

Now,

1 + i

=

2

(

\frac{1}{2}

+

\frac{i}{2}

) -(2)

From (1) and (2),

\frac{1}{2 ^{n}}

.

(1 + i)^{2 n}

=

(\frac{1}{2} + \frac{i}{2})^{n}

=

cos

nπ /2

+ i s in

nπ /2

To be a real number, an imaginary part should be

0

∴

n = 2

so that

s in

component becomes

0.

Hence, option D is correct.

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Question Text	The least positive integer $n$ for which $(\frac{1 + i}{1 - i})^{n}$ represents a real number is
Answer Type	Text solution:1
Upvotes	150