This division will fail if and only if <math>c^2 + d^2 = 0\,</math>, that is, <math>c\,</math> and <math>d\,</math> are both zero, that is, the complex denominator <math>z\,</math> is exactly zero (both components zero). This is exactly analogous to the rule that real division fails if the denominator is exactly zero.
==Fundamental theorem of algebra==
The complex numbers form an [[algebraic closure|algebraically closed]] field. This means that any <math>n^{th}\,</math> degree polynomial can be factored into n first degree (linear) polynomials. Equivalently, such a polynomial has n roots (though one has to count all multiple occurrences of repeated roots. This statement is the Fundamental Theorem of Algebra, first proved by [[Gauss|Carl Friedrich Gauss]] around 1800. The theorem is ''not true'' if the roots are required to be real. But when the roots are allowed to be complex, the theorem applies even to polynomials with complex coefficents.
The simplest polynomial with no real roots is <math>x^2 + 1\,</math>, since -1 has no real square root. But if we look for roots of the form <math>a + bi\,</math>, we have:
:<math>(a + bi)^2 + 1 = a^2 - b^2 + 1 + 2abi = 0\,</math>
For the imaginary part to be zero, one of a or b must be zero. For the real part to be zero, a must be zero and <math>b^2\,</math> must be 1. This means that <math>b = \pm 1\,</math>, so the roots are <math>(0, 1)\,</math> and <math>(0, -1)\,</math>, or <math>\pm{}i\,</math>.
These two numbers, <math>i\,</math> and <math>-i\,</math>, are the square roots of -1.
Similar analysis shows that, for example, 1 has three cube roots:
*<math>1\,</math>
*<math>-\frac{1}{2} + \frac{\sqrt{3}}{2}i\,</math>
*<math>-\frac{1}{2} - \frac{\sqrt{3}}{2}i\,</math>
One can verify that
<math>(x-1) (x + \frac{1}{2} - \frac{\sqrt{3}}{2}i) (x + \frac{1}{2} + \frac{\sqrt{3}}{2}i) = x^3 - 1\,</math>
It is a common belief that complex numbers have a weaker connection to physical reality than real numbers. Observables in [[Physics]] for example weight, energy, pressure etc. are usually represented as [[real]] [[numbers]], and the SI system of units relies on real numbers. However, the transformation between a SI base unit, e.g. an inductance/capacitance/resistance value and a complex impedance is arbritrary and set by convention, and the "natural" representation depends on the measurement method. As a matter of fact, a number of measurement devices (network analysers, lock in amplifiers) directly output real and imaginary component (where the imaginary component is obviously a real voltage/current value). Also, the [[index of refraction]] is often expressed as a complex number whose imaginary component indicates [[absorption]] loss as light propagates through the medium.