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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)−1
Solution
(1 + i)(1 − i)−1
= `(1 + "i")/(1 - "i")`
= `((1 + "i")(1 + "i"))/((1 - "i")(1 + "i")`
= `(1 + 2"i" + "i"^2)/(1 - "i"^2)`
= `(1 + 2"i" - 1)/(1 - (-1))` ...[∵ i2 = −1]
= `(2"i")/2`
= i
∴ (1 + i)(1 − i)−1 = 0 + i
∴ a = 0 and b = 1
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