√(−1) – The Equation That Changed The World
By Vasos L Pavlika, on 20 October 2021
This is arguably one of the greatest equations in all of Mathematics, in fact Richard Feynman (1918-1988) as a 13-year-old stated that “it was the most amazing equation in Math” this was just after he stated that when he tried to self-teach himself circular trigonometry functions he was left confused. Of course, the equation introduces the use of the square root of negative one, however the existence of this number was known much earlier in fact it was known to the great Italian algebraists including: Scipione del Ferro (1465-1526), Gerolamo Cardano (1501-1575) and Niccolò Fontana Tartaglia (the stammerer) (1499-1557) who whilst solving the general cubic equation realised that the path of the solution led to numbers of the form a+b* root(-1) . In fact, they had “Mathematical” duals between themselves on who could solve cubic equations posed by the other, however this discussion is for another day. The square root of -1 is unfortunately today called imaginary and this is very inconvenient for Mathematics and the numbers created by using it along with another real number are called complex and we have Rene Descartes (1596-1650) to thank for that. Rene showed (and it is very easy to do so by reduction ad absurdum) that i is not greater than zero, it is not less than zero and it is not equal to zero thus consequently he decided to call it imaginary, and we are stuck with it.
Interest in root(-1) was ignited again by the Prnce of Mathematics J.C.G.Gauss (1777-1855) who at the age of 17 showed that an algebraic equation of order n has n solutions (taking into account repeated roots) so if one looks at x^2+4=0, then either Gauss was wrong (and I find such a statement tantamount to blasphemy) or we need a “larger” set of numbers. Of course, these numbers were not accepted immediately into Mathematics but now their axiomatic foundations are well understood.
Euler’s result which leads nicely to what we call de Moivres (1667-1754) theorem is such a beautiful theorem, I often wonder why it is not called Euler’s theorem as I doubt he would have missed it (even thought he was blind for the last 17 years of his life). Even in his blindness Euler became the most prolific Mathematician in history with Jabobi (1801-1854) of the Jacobian determinant fame being second. Euler also has another amazing formula (his polyhedral formula) which states that for a polyhedron with no holes: V – E + F = 2; where V is the number of vertices, E is the number of edges and F is the number of faces, this formula gave birth to graph theory and network analysis but alas “this page is too narrow to contain it” (for my Mathematical friends that was a reference to Fermat (1607-1665)).
There are descendants of Euler still alive today who decided not to study Mathematics citing that there would be too much pressure on them with the surname Euler if they studied Mathematics (I don’t blame them).
One of Switzerland’s finest sons by a country mile!!