bability and the application of negative numbers used today in IQ quadra- ture modulation. 1550-1617 - John Napier and Henry Briggs (1561-1631) work on logarithms. 1596-1650 - Rene Descartes - works on merging algebra and Euclidean geometry, giving us the Cartesian coordinates still used as the basis for modern constellation diagrams. 1601-1665 - Fermat and his theorem. Pierre De Fermat of Toulouse works on an- alytic geometry and number theory and differential calculus, determining the rate of change of a quantity. He produces irritating problems for other mathematicians such as why a cube cannot be divided into two cubes. 1642-1726 - Isaac Newton develops a geometric approach for calculating Pi as well as that business with the apple and gravity and his work on optical theory. 1707-1783 - Leonard Euler proves that complex numbers, the extension of a real number by the inclusion of an imaginary number, can be related directly to the real trigonometric functions of sines and cosines - the basis of the Fourier base and Fourier transform used in all present OFDM radio systems. 1768-1830 - Jean Baptiste Joseph Fourier narrowly avoids being guillotined in the French revolution and produces a paper in 1807 on the use of sinusoids to represent temperature distribution. He uses the technique in a study on heat propa- gation that finally gets published in 1822, the Theorie Analytique De La. Chaleur. In the meantime he claims that any continuous periodic signal can be repre- sented as the sum of properly chosen sinusoidal waves. Challenged by Lagrange (1736-1813) and Laplace (1749-1827), Lagrange asserts that sinusoids cannot be used to describe signals with corners, for example square waves. Lagrange is (sort of) right but in practice over time it is shown that sinusoids can be used to the point where the difference between the two has zero energy (the Gibbs effect). This becomes the basis for translating signals from the time domain to the fre- quency domain and back again, taking simple elements in the frequency domain