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Let (x, y) stands for a complex number z E C where x = Re(z) and y = Im(z). Now, let us define a new binary operation * on C by (X₁, Y₁) *k (X2, Y2) def ( x1x2 + Y1Y2, X2Y1 — X1Y2 ). (a) Determine if this binary operation * is commutative on C. Justify your answer. (b) Determine if this binary operation is associative on C. Justify your answer. (c) For (x₁, y₁) # (0,0), there is an unique (x2, y2) such that (x1, y₁) *k (X2, Y2) = (1, 0). Write (x2, y2) in terms of (x₁, y₁).

Let (x, y) stands for a complex number z E C where x = Re(z) and y = Im(z). Now, let us define a new binary operation * on C by (X₁, Y₁) *k (X2, Y2) def ( x1x2 + Y1Y2, X2Y1 — X1Y2 ). (a) Determine if this binary operation * is commutative on C. Justify your answer. (b) Determine if this binary operation is associative on C. Justify your answer. (c) For (x₁, y₁) # (0,0), there is an unique (x2, y2) such that (x1, y₁) *k (X2, Y2) = (1, 0). Write (x2, y2) in terms of (x₁, y₁).

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Let (x, y) stands for a complex number z EC where x = Re(z) and y = Im(z). Now, let us define a new binary operation * on C by (X1, Y₁) *k (X2, Y2) def (X1X2 + Y1Y2, X2Y1 — X1Y2 ). - (a) Determine if this binary operation * is commutative on C. Justify your answer. (b) Determine if this binary operation is associative on C. Justify your answer. (c) For (x₁, y₁) # (0, 0),there is an unique (x2, y2) such that (x₁, y₁) *k (X2, Y2) = (1, 0). Write (x2, 92₂) in terms of (x₁, y₁).
Transcribed Image Text:Let (x, y) stands for a complex number z EC where x = Re(z) and y = Im(z). Now, let us define a new binary operation * on C by (X1, Y₁) *k (X2, Y2) def (X1X2 + Y1Y2, X2Y1 — X1Y2 ). - (a) Determine if this binary operation * is commutative on C. Justify your answer. (b) Determine if this binary operation is associative on C. Justify your answer. (c) For (x₁, y₁) # (0, 0),there is an unique (x2, y2) such that (x₁, y₁) *k (X2, Y2) = (1, 0). Write (x2, 92₂) in terms of (x₁, y₁).
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