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

1) The Complex Plane: Instead of writing a point P∈R2 as an ordered pair of real numbers (x,y) we replace the unit vector along the x-axis (1,0) by the number 1 and the unit vector (0,1) along the y-axis by the complex number i. Then

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

is denoted by

P=z=x.1+y.i=x+iy

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

z+z′=(x+iy)+(x′+iy′)=(x+x′)+i(y+y′),

we can also multiply points using the rule

i2=−1

so that

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 cos⁡θ+isin⁡θ=eiθ.
• Define z¯=x+iy¯=x−iy and show that the map z↦z¯ is a reflection about the x-axis.
• Show that the Euclidean length of a vector is given by |z|=zz¯
• Show that the map z↦eiθz does not change the length of vectors.
• Explain why this map is a rotation by angle θ. Is it clockwise or anticlockwise?
• Explain why z↦z+w (where w is a fixed complex number) is a translation.
• Discuss how to write a glide transformation in the complex plane.