Comment on Why don't we define “imaginary” numbers for every “impossibility”? (2012)parentComments−afiori3ylet ϴ = 0/0 then 1*ϴ = ϴ = 0/0 = (0*0)/0 = 0*(0/0) = 0*ϴ it follows 1 = 0 and thus x = x * 1 = x * 0 = 0 = y * 0 = y * 1 = y for all x and y−xigoi3yThis is assuming that Θ interacts with arithmetic operations the usual way (that is, ℝ ∪ {Θ} is a field), which the person you're replying to did not say.−afiori3yTrue, but the point of giving a "value" to 0/0 is to use it somehow.For example in the context of limits you define a whole lot of number like values like 0+ or 0- that are useful wrt operations on limits.I was trying to give an example of how ℝ ∪ {Θ} has almost no advantages compared to just ℝ−tshaddox3ySure, but the whole "problem" we were trying to solve was that zero doesn't interact with arithmetic operations the usual way.
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let ϴ = 0/0 then 1*ϴ = ϴ = 0/0 = (0*0)/0 = 0*(0/0) = 0*ϴ it follows 1 = 0 and thus x = x * 1 = x * 0 = 0 = y * 0 = y * 1 = y for all x and y
This is assuming that Θ interacts with arithmetic operations the usual way (that is, ℝ ∪ {Θ} is a field), which the person you're replying to did not say.
True, but the point of giving a "value" to 0/0 is to use it somehow.
For example in the context of limits you define a whole lot of number like values like 0+ or 0- that are useful wrt operations on limits.
I was trying to give an example of how ℝ ∪ {Θ} has almost no advantages compared to just ℝ
Sure, but the whole "problem" we were trying to solve was that zero doesn't interact with arithmetic operations the usual way.