How to solve complex numbers using geometry?
Answers
Several features of complex numbers make them extremely useful in plane geometry. For example, the simplest way to express a spiral similarity in algebraic terms is by means of multiplication by a complex number. A spiral similarity with center at c, coefficient of dilation r and angle of rotation t is given by a simple formula
f(z) = r(z - c)(cos(t) + i·sin(t)) + c.
CircleA particularly simple equation is that of a circle:
{z: |z - a| = r},
is the circle with radius r and center a. By squaring that equation we obtain
(z - a)(z' - a') = r²
or
zz' - (za' + z'a) + (aa' - r²) = 0.
and finally
zz' - (za' + z'a) + s = 0,
where s is a real number. The circle is centered at a and has the radius r = √aa' - s, provided the root is real.
This representation of the circle is more convenient in some respects. For example, we may immediately check that the transformation w = f(z) = 1/z maps circles onto circles. Indeed, substituting z = 1/w we get
1/w × 1/w' - (a'/w + a/w') + s = 0
which, if multiplied by ww', leads to
ww' - (wb' + w'b) + t = 0,
where b = a'/s and t = 1/s, an equation in the same form.
Letting a = α + iβ yields yet another form of essentially same equation:
zz' - α(z + z') - iβ(z - z') + s = 0,
where α and β are both real. Yet the most general form of the equation is this
Azz' + Bz + Cz' + D = 0,
which represents a circle if A and D are both real, whilst B and C are complex and conjugate. For A = 0, the equation represents a straight line.