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What is complex math?
Complex math covers how to do operations on complex numbers. Complex numbers include real numbers, imaginary numbers, and the combination of real+imaginary numbers.
What is a polynomial that does not factor over the real numbers referred to as?
There is no specific term for such polynomials. They may be referred to as are polynomials with only purely complex roots.
What happens when you square root a negative number?
The square root of a negative number is not real. However, there is a field of numbers known as the complex number field which contains the reals and in which negative numbers have square roots. Complex numbers can all be expressed in the form a+bi where a and b are real and i is the pure imaginary such that i2=1. Please see the related links for more information about complex and imaginary numbers.
What is the significance of complex roots?
One significant feature of complex numbers is that all polynomial equations of order n, in the complex field, have n solutions. When multiple roots are Given any set of complex numbers {a(0),  … , a(n)}, such that at least one of a(1) to a(n) is non-zero, the equation a(n)*z^n + a(n-1)*z^(n-1) + ... + a(0) has at least one solution in the complex field. This is the Fundamental Theorem of Algebra and establishes the set of Complex numbers as a closed field. [a(0), ... , a(n) should be written with suffices but this browser has decided not to be cooperative!] The above solution is the complex root of the equation. In fact, if the equation is of order n, that is, if the coefficient a(n) is non-zero then, taking account of the multiplicity, the equation has exactly n roots (some of which may be real).
When were imaginary numbers developed?
Imaginary numbers were first recognised in the first century CE by Heron of Alexandria but development was slow because "the establishment" did not consider these to be proper numbers. Gerolamo Cardano, in his work on finding roots of cubic equations in early 16th century CE, set out some of the rules for manipulating complex numbers. Rafael Bombelli set down the rules for multiplication of complex numbers later in that century. However there was no serious work done on these numbers for a long time: their name did not help. It was not until two of the giants of mathematics, Leonhard Euler and Carl Friedrich Gauss in the 18th century worked on them that they were accepted as worthy of attention by serious mathematicians! And the rest, as they say, is history!
