181
edits
Changes
Jump to navigation
Jump to search
Line 59:
Line 59: + +
→Fundamental theorem of algebra: Can say a little more about that.
One can verify that
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>
<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>
For a given field, the field containing it (strictly speaking, the smallest such field) that is algebraically closed is called its algebraic closure. The field of real numbers is not algebraically closed; its closure is the field of complex numbers.
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.
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.