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If we have the relationship x^2 = -25 then x = SQRT(-25) which is equivalent to x = 5i giving (5i)^2 = -25. Could another form of the complex relationship be described as -1*(x^2) = -25 so that like i being separated out we can use -1 as a post multiplier? Then -1*x = -5 becomes another representation so that -1 becomes implicit in the definition of imaginary numbers in a slightly different way. Would this make any difference to calculations and the solutions of equations?

-1*(x^2) = -25

Quote from: jeffreyH-1*(x^2) = -25This is the same as solving x^{2} = -25, for which the solution is the simple x=5, or the slightly less obvious x=-5.

Actually if you have x^2 = 25 it follows that the complete solution is

x^{2} + 25 = 0 has the solutions:x = 5e^{iθ}, θ = π/2 + k*π; k is an integer number.for k = 0: x =5i, because e^{iπ/2} = i (Euler's formula).for k = 1: x = -5i, because e^{3iπ/2} = -i.the other values of k gives, periodically, the same two up here.--lightarrow

Why do you say that?

I thought it was the case that quarternions were out of favour

Quote from: lightarrow on 07/01/2015 15:48:52x^{2} + 25 = 0 has the solutions:x = 5e^{iθ}, θ = π/2 + k*π; k is an integer number.for k = 0: x =5i, because e^{iπ/2} = i (Euler's formula).for k = 1: x = -5i, because e^{3iπ/2} = -i.the other values of k gives, periodically, the same two up here.--lightarrowI think Richard Feynman summed it up by calling Euler's formula the jewel of physics.