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Question
Express the following in the form of a + ib, a, b∈R i = `sqrt(−1)`. State the values of a and b:
`((1 + "i")/(1 - "i"))^2`
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Solution
`((1 + "i")/(1 - "i"))^2 = (1 + "i")^2/(1 -"i")^2`
= `(1 + 2"i" + "i"^2)/(1 - 2"i" + "i"^2)`
= `(1 + 2"i" - 1)/(1 - 2"i" - 1)` ...[∵ i2 = – 1]
= `(2"i")/(-2"i")`
= – 1
= – 1 + 0·i
This is of the form a + bi, where a = – 1 and b = 0.
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