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Agreed. The jump from real to complex numbers is about as difficult to explain as the jump from natural numbers to integers. Integers are the numbers you need f
by ThrustVectoring 10y ago
Agreed. The jump from real to complex numbers is about as difficult to explain as the jump from natural numbers to integers. Integers are the numbers you need for "subtracting numbers gives you a number". Complex numbers are the numbers you need for "factoring polynomials with numeric coefficients gives you numeric roots". There's a similar argument for real numbers that you hinted at, but I don't understand it well enough to give a simple explanation for it, so I pretty much always hand-wave over it.
- j2kun 10y agoFor reals, it's "Real numbers are the numbers you need to ensure every convergent sequence of rational numbers has a terminating point" Convergence is determined in the "Cauchy" sense by having a vanishing distance between subsequent sequence entries, so as not to rely on the (potentially nonexistent) limit.
- chadcmulligan 10y agoI thought you could have calculus using 'computable numbers' that is reals that have a rule. It's been on my list for a while to look into it - there's some notes here for the curious http://math.stackexchange.com/questions/963061/can-the-set-of-computable-numbers-be-used-as-a-theoretical-basis-for-calculus http://math.stackexchange.com/questions/963061/can-the-set-o...