I did honours math back in the day bare with me I'm watching an x files n been smoking weed but wouldn't that mean a circle isn't normal unless it's constantly being re-evaluated for it's shape?
So firstly pi isn't infinite, it's just that its decimal representation is of infinite length. It is a perfectly defined real number. The rational number 1/9 is also a perfectly defined real number whose decimal representation is of infinite length.
I view the idea of a circle as a simplification of real life, no real perfect circle exists, but the simple 'concept' of a collection of points in the plane that are all the same distance from a center point is natural and easy for us to consider. We naturally identify real approximately circular objects with this simplified concept. When we explore this concept mathematically we encounter the single well-defined transcendental number pi and can represent it as a decimal to very high precision. By the way, nobody knows what the 101010 th digit of pi is, but we have a set recipe to get there if we ever obtained enough computing power to carry it out. The number 1/9 is also a mathematical concept that's a simplification of real life - I don't think anything is perfectly 1/9 th of a whole object.
@SilverBanana suggested redefining the basis of the number system: this would make the decimal representation of pi finite, but unfortunately would make the decimal representation of 1 (the radius of the circle with circumference pi) infinitely long.
PS: Seriously, I think you should go easy on the weed. It might be good for inspiring your sense of wonder, but I'm pretty sure it's bad for developing mathematical comprehension.
So firstly pi isn't infinite, it's just that its decimal representation is of infinite length. It is a perfectly defined real number. The rational number 1/9 is also a perfectly defined real number whose decimal representation is of infinite length.
I view the idea of a circle as a simplification of real life, no real perfect circle exists, but the simple 'concept' of a collection of points in the plane that are all the same distance from a center point is natural and easy for us to consider. We naturally identify real approximately circular objects with this simplified concept. When we explore this concept mathematically we encounter the single well-defined transcendental number pi and can represent it as a decimal to very high precision. By the way, nobody knows what the 10^10^10 th digit of pi is, but we have a set recipe to get there if we ever obtained enough computing power to carry it out. The number 1/9 is also a mathematical concept that's a simplification of real life - I don't think anything is perfectly 1/9 th of a whole object.
@SilverBanana suggested redefining the basis of the number system: this would make the decimal representation of pi finite, but unfortunately would make the decimal representation of 1 (the radius of the circle with circumference pi) infinitely long.
PS: Seriously, I think you should go easy on the weed. It might be good for inspiring your sense of wonder, but I'm pretty sure it's bad for developing mathematical comprehension.
well okay if it constantly redefines a circle there's a problem with our idea of a circle... I'm still smokin the ganja but I don't think drawing a circle represents a true circle.... it wouldn't make sense because it can never be perfect right?