Why must the circumference and diameter of a circle be related in such a way by two integers precisely?
IE: Why are you so confident in “proving” that these two values are related to integers? Especially if you’re a modern mathematician who knows about irrational numbers (aka: can never be represented by a ratio of two integers) or imaginary numbers (which truly appear in electricity: phasors and the like. Just because the name is “imaginary” doesn’t mean that they’re not real!!!)
I don’t know that the common proof by contradiction is even remotely straightforward for this community. Niven’s proof relies on way more shit than you’d expect someone asking the question this way to know. I’m honestly not sure there is a simple proof because even Lambert’s relies on continued fractions.
Lets reverse it.
Why must the circumference and diameter of a circle be related in such a way by two integers precisely?
IE: Why are you so confident in “proving” that these two values are related to integers? Especially if you’re a modern mathematician who knows about irrational numbers (aka: can never be represented by a ratio of two integers) or imaginary numbers (which truly appear in electricity: phasors and the like. Just because the name is “imaginary” doesn’t mean that they’re not real!!!)
I don’t know that the common proof by contradiction is even remotely straightforward for this community. Niven’s proof relies on way more shit than you’d expect someone asking the question this way to know. I’m honestly not sure there is a simple proof because even Lambert’s relies on continued fractions.