Quotes About Complex numbers

It is important to bear in mind that this is a property of single photons. Each individual photon must be considered to feel out both routes that are open to it, but it remains one photon; it does not split into two photons in the intermediate stage, but its location undergoes the strange kind of complex-number weighted co-existence of alternatives that is characteristic of quantum theory.

But, as powerful mathematicians like Minkowski and Hilbert found striking harmony between their pure mathematical results and the workings of the physical world, many found the claims for such a harmony hard to resist. Thus, in the early years of the twentieth century, we begin to see why Minkowski's application of complex numbers to the description of space and time was hailed by one physicist as 'one of the greatest revolutions in our accepted views'.

Indeed, if a mathematician is asked to justify his interest in complex numbers, he will point, with some indignation, to the many beautiful theorems in the theory of equations, of power series, and of analytic functions in general, which owe their origin to the introduction of complex numbers. The mathematician is not willing to give up his interest in these most beautiful accomplishments of his genius.

Complex numbers can be written in standard form In this form, they can be combined using the operations defined for real numbers, together with the definition of the imaginary unit i: . The conjugate of a complex number z is denoted . If , then

Leonhard Euler

After exponential quantities the circular functions, sine and cosine, should be considered because they arise when imaginary quantities are involved in the exponential.

In particular, the three roots to a cubic polynomial either must all be real, or there must be one real root and one conjugate pair.

The breakthrough for came not from quadratic equations, but rather from cubics which clearly had real solutions but for which the Cardan formula produced formal answers with imaginary components.