Complex Number Plane

Key Questions

  • The complex plane is a Cartesian-like plane where each complex number is a point, the x coordinate being the real part of the complex number and the y coordinate the imaginary part. In other words, #z=a+bi# in the complex plane is the point #(a,b)# in the Cartesian plane.

  • In simple terms the modulus of a complex number is its size.

    If you picture a complex number as a point on the complex plane, it is the distance of that point from the origin.

    If a complex number is expressed in polar coordinates (i.e. as #r(cos theta + i sin theta)#), then it's just the radius (#r#).

    If a complex number is expressed in rectangular coordinates - i.e. in the form #a+ib# - then it's the length of the hypotenuse of a right angled triangle whose other sides are #a# and #b#.

    From Pythagoras theorem we get: #|a+ib|=sqrt(a^2+b^2)#.

  • Answer:

    #1 = (1, 0)# and #i = (0, 1)#

    Explanation:

    The complex number plane is usually considered as a two dimensional vector space over the reals. The two coordinates represent the real and imaginary parts of the complex numbers.

    As such, the standard orthonormal basis consists of the number #1# and #i#, #1# being the real unit and #i# the imaginary unit.

    We can consider these as vectors #(1, 0)# and #(0, 1)# in #RR^2#.

    In fact, if you start from a knowledge of the real numbers #RR# and want to describe the complex numbers #CC#, then you can define them in terms of pairs of real numbers with arithmetic operations:

    #(a, b) + (c, d) = (a+c, b+d)" "# (this is just addition of vectors)

    #(a, b) * (c, d) = (ac-bd, ad+bc)#

    The mapping #a -> (a, 0)# embeds the real numbers in the complex numbers, allowing us to consider real numbers as just complex numbers with a zero imaginary part.

    Note that:

    #(a, 0) * (c, d) = (ac, ad)#

    which is effectively scalar multiplication.

Questions