Seventy-twelve. Impossible, Imaginary, Useful Complex Numbers. By:Daniel Fulton. Eleventeen. Why imagine the imaginary. Where did the idea of imaginary numbers come from Descartes, who contributed the term "imaginary" Euler called sqrt(-1) = i Who uses them
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Impossible, Imaginary, UsefulComplex Numbers
Why imagine the imaginary
As Cardano had stated “ is either +3 or –3, for a plus [times a plus] or a minus times a minus yields a plus. Therefore is neither +3 or –3 but in some recondite third sort of thing.
Leibniz said that complex numbers were a sort of amphibian, halfway between existence and nonexistence.
So Is There A Real Solution to this equation
I say let us try x = 4
He used plus of minus for adding a square root of a negative number, which finally gave us a way to work with these imaginary numbers.
Solves quadratic equations and allows for the possibility of negative solutions.
General solution to cubic equations
Uses these square roots of negative numbers
Shows a way to represent complex numbers geometrically.
Infinite series formulations of ex, sin(x) and cos(x), and deducing
the formula, eix = cos(x) + i sin(x)
Gives the first clear theory of functions of a complex variable.
Relates the rules of real numbers and complex numbers
Introduces a formal algebra of real number couples using rules
which mirror the algebra of complex numbers
Algebra of complex numbers as number pairs (x + iy)