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Question: 1 the complex plane instead of writing a point pr2...

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1) The Complex Plane: Instead of writing a point LaTeX: P\in\mathbb R^2 P∈R2 as an ordered pair of real numbers LaTeX: (x,y)(x,y) we replace the unit vector along the x-axis LaTeX: (1,0)(1,0) by the number LaTeX: 11 and the unit vector LaTeX: (0,1)(0,1) along the y-axis by the complex number LaTeX: ii. Then

LaTeX: P=(x,y) = x.(1,0) + y.(0,1)P=(x,y)=x.(1,0)+y.(0,1)

is denoted by

LaTeX: P=z=x.1 + y.i = x+iyP=z=x.1+y.i=x+iy

The set of all such points LaTeX: \{z=x+iy|x,y\in {\mathbb R}\} {z=x+iy|x,y∈R} is called the complex plane LaTeX: {\mathbb C} C. It has more structure than the plane LaTeX: {\mathbb R}^2 R2 because not only can we add points using the standard parallelogram rule for vector addition

LaTeX: z+z' = (x+iy) + (x'+iy') = (x+x') + i (y+y'),z+z′=(x+iy)+(x′+iy′)=(x+x′)+i(y+y′),

we can also multiply points using the rule

LaTeX: i^2=-1i2=−1

so that

LaTeX: z z' = (x+iy)(x'+iy') = xx'-yy' + i(xy'+yx')zz′=(x+iy)(x′+iy′)=xx′−yy′+i(xy′+yx′)

Lets study some elements of Euclidean geometry using complex numbers:

  • Use the Taylor series formula for the exponential function to show that LaTeX: \cos \theta + i \sin\theta = e^{i\theta} cos⁡θ+isin⁡θ=eiθ.
  • Define LaTeX: \bar z =\overline{x+iy}=x-iy z¯=x+iy¯=x−iy and show that the map LaTeX: z\mapsto \bar z z↦z¯ is a reflection about the x-axis.
  • Show that the Euclidean length of a vector is given by LaTeX: |z|=\sqrt{z\bar z} |z|=zz¯
  • Show that the map LaTeX: z\mapsto e^{i\theta} z z↦eiθz does not change the length of vectors.
  • Explain why this map is a rotation by angle LaTeX: \thetaθ. Is it clockwise or anticlockwise?
  • Explain why LaTeX: z\mapsto z+ w z↦z+w (where LaTeX: ww is a fixed complex number) is a translation.
  • Discuss how to write a glide transformation in the complex plane.
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