Complex Numbers

Comp.145  December 1996  Dr. A.J. Mestel

(1) No real number can equal \sqrt{−1}, and we denote this imaginary number by i. A complex number z, can be written as z = x + iy, where x and y are real numbers. The real part and imaginary part of z are x and y respectively. We write x = ℜe(z) and y = ℑm(z). If ℑm(z) = 0 then z is real, while if ℜe(z) = 0, we say that z is purely imaginary, or just imaginary.

(2) Addition, subtraction and multiplication by a real number are defined using the usual rules of algebra, so that for example α((a + ib) + (c + id)) = α(a + c) + iα(b + d).

(3) Multiplication of two complex numbers is similar, except that the product i^{2} is replaced by −1, so that (a + ib)(c + id) = (ac − bd) + i(ad + bc).

(4) The complex conjugate of a complex number z is written \overline{z} (or sometimes z^{∗}), and has the same real part as z, but an oppositely signed imaginary part. So if z = x + iy then \overline{z} = x − iy. Note that z = \overline{z} iff z is real, and z = −\overline{z} iff z is purely imaginary.

(5) A complex number z_{1} = a + ib can be divided by another non-zero complex number z_{2} = c + id. The quotient can be expressed in standard form by multiplying top and bottom by the complex conjugate of z_{2}:

\frac{z_{1}}{z_{2}} = \frac{a + ib}{c + id} = \frac{(a + ib)(c − id)}{(c + id)(c − id)} = \frac{ac + bd}{c^{2} + d^{2}} + i (\frac{bc − ad}{c^{2} + d^{2}})

(6) The modulus of a complex number z = x + iy is written as |z|. It is defined by |z| = (x^{2} + y^{2})^{1/2}. Note that z\overline{z} = |z|^{2}. The argument of z is written Arg(z) and defined by Arg(z) = tan^{−1}(y/x), with −π < Arg(z) ⩽ π. If we write x = r cos θ, y = r sin θ, then

z = x + iy = r(cos θ + i sin θ) ,  r = |z| ,  θ = Arg(z) .

(7) Complex numbers are frequently regarded as lying in the complex plane or Argand diagram. The number z = x + iy can be thought of as the position 2-vector of the point (x, y). The modulus and argument of z correspond to a representation of z in polar coordinates. Note that as with most angles, there is an ambiguity in the definition, in that we can add an integer multiple of 2π to Arg(z) without affecting the value of z.

(8) A very important identity is that

e^{iθ} ≡ cos θ + i sin θ .

This can be shown to be true by considering power series expansions, or by differentiating with respect to θ. A consequence of this is De Moivre’s theorem:

(cos θ + i sin θ)^{n} = (e^{iθ})^{n} = e^{inθ} = cos(nθ) + i sin(nθ) .

Note that e^{iπ} = −1, and e^{2kπi} = 1 for any integer k. As e^{−iθ} = cos θ − i sin θ, we also have the identities

cos x ≡ \frac{e^{ix} + e^{−ix}}{2} sin x ≡ \frac{e^{ix} − e^{−ix}}{2i} .

(9) De Moivre’s theorem can be used in conjunction with the binomial theorem to obtain expressions for cos(nθ) is terms of cos θ. For example, writing c = cos θ, s = sin θ,

cos 4θ = ℜe[(c + is)^{4}] = ℜe[c^{4} + 4ic^{3}s − 6c^{2}s^{2}− 4ics^{3}+ s^{4}] = c^{4}− 6c^{2}(1 − c^{2}) + (1 − c^{2})^{2}

= 8 cos^{4} θ − 8 cos^{2} θ + 1

(10) The modulus/argument representation is useful when performing multiplications and divisions, or taking powers of complex numbers. If z_{1} = r_{1}e^{iθ_{1}} and z_{2} = r_{2}e^{iθ_{2}}, then z_{1}z_{2} = (r_{1}r_{2})e^{i(θ_{1}+θ_{2})}, while z_{1}/z_{2} = (r_{1}/r_{2})e^{i(θ_{1}−θ_{2})}, and z_{1}^{n} = r_{1}^{n}e^{inθ_{1}}. We therefore see that |z_{1}z_{2}| = |z_{1}||z_{2}| and Arg(z_{1}z_{2}) = Arg(z_{1}) + Arg(z_{2}).

(11) When complex numbers are taken into account, there are n roots of a polynomial equation of degree n. (Sometimes these roots are repeated, just as for quadratic equations.) If the coefficients of the polynomial are all real, then any complex roots must occur as complex conjugate pairs. As a result, any polynomial of odd order must have at least one real root.

(12) For example, consider the nth roots of unity, defined by the equation z^{n} = 1. We write 1 = e^{2kπi}, where k is any integer. Then taking the nth root, we have z = e^{2kπi/n}. This gives n different complex numbers as k takes the n values 0, 1, 2, … (n − 1). (Other values of k do not give different values of z.) Note that all these values have modulus 1, and are evenly spaced around a circle of radius 1 (the unit circle) in the complex plane.

(13) To solve the equation z^{n} = z_{0}, for given z_{0}, we first write z_{0} = r_{0}e^{i(θ_{0}+2kπ)}. For example, suppose z^{3} = 1 + i = \sqrt{2}(1/\sqrt{2} + i/\sqrt{2}) = \sqrt{2}e^{i(π/4+2kπ)}. Then z = 2^{1/6}e^{iπ/12}, or z = 2^{1/6}e^{i(9π/12)} or z = 2^{1/6}e^{i(17π/12)}.

(14) Various curves can be represented in terms of complex numbers. For example,

(a) |z − z_{0}| = 4, where z_{0} is a complex constant but z = x + iy is a variable, represents a circle of radius 4 centred on the point z = z_{0}.

(b) |z − 1| = |z + i| is the perpendicular bisector of the points z = 1 and z = −i, i.e. the line x + y = 0

(c) |z − 1| = 2|z + i| is less obvious. Writing z = x + iy, and squaring, we have

\begin{matrix}|(x − 1) + iy|^{2} = 4|x + i(y + 1)|^{2} & or (x − 1)^{2} + y^{2} = 4(x^{2} + (y + 1)^{2}) \\ or 3x^{2} + 3y^{2} + 8y + 2x + 5 = 0 & or (x + \frac{1}{3})^{2} + (y + \frac{4}{3})^{2} = \frac{8}{9} \end{matrix}

This is a circle centre (−1/3 − 4i/3), and radius \sqrt{8}/3.

(d) Arg(z − 1) = \frac{1}{4}π is the straight half-line from the point z = 1 inclined at π/4 to the real axis, i.e. y = x − 1 for y > 0.