For instance how do we know that something weird doesn’t happen at some insanely large number?
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> For instance how do we know that something weird doesn’t happen at some insanely large number?
This is both the reason why mathematics wants formal proofs instead of heuristics, intuition or induction, as well as why it sometimes is incredibly hard to really proof things.
Take f(x) = x, as it is the simplest example. “y goes to infinity” means that y gets at some point and onward larger than any finite bound. Then “f(x) goes to infinity when x goes there” is almost tautological, as it says: “whenever x gets and stays larger then any finite bound, then x gets and stays larger then any finite bound”. Or without the fluff “if A then A”, which is absolutely always true.
Now for other functions like x² , e^x , log(x) , or x+100·sin(x) the argument might be more convoluted. Lets try the last one:
“Whenever x gets and stays larger then any finite bound, then x+100·sin(x) gets and stays larger then any finite bound”.
We can note that sin(x) is between -1 and 1, therefor we know that x+100·sin(x) is at least as big as x-100. Looking as x-100 as a kind of “worst case” simplifies things:
“Whenever x gets and stays larger then any finite bound, then x-100 gets and stays larger then any finite bound”.
No more sine to deal with! If we now want x-100 to be (and stay) larger than a finite bound (lets say 1,000,000), then we guarantee that by choosing x itself to be bigger than that bound plus 100 (here: 1,000,100). This proves what we wanted!
Never did we have to check an infinite number of cases by hand, nor is there any wiggly room, despite x+100·sin(x) wiggling up and down all the time (draw a graph if you want to see this). We argued purely by reason and logic, which goes further than simple calculation.
For some way questions, showing that something grows without bound can become seriously complicated; too complicated for even modern mathematicians. Only time will show if we are able to absolutely prove such things, but the chance for failure is always there. Heck, it is even an unsolved problem if there is a way to always and definitely decide (an “algorithm”) if two elementary expressions describe the same function.
> For instance how do we know that something weird doesn’t happen at some insanely large number?
This is both the reason why mathematics wants formal proofs instead of heuristics, intuition or induction, as well as why it sometimes is incredibly hard to really proof things.
Take f(x) = x, as it is the simplest example. “y goes to infinity” means that y gets at some point and onward larger than any finite bound. Then “f(x) goes to infinity when x goes there” is almost tautological, as it says: “whenever x gets and stays larger then any finite bound, then x gets and stays larger then any finite bound”. Or without the fluff “if A then A”, which is absolutely always true.
Now for other functions like x² , e^x , log(x) , or x+100·sin(x) the argument might be more convoluted. Lets try the last one:
“Whenever x gets and stays larger then any finite bound, then x+100·sin(x) gets and stays larger then any finite bound”.
We can note that sin(x) is between -1 and 1, therefor we know that x+100·sin(x) is at least as big as x-100. Looking as x-100 as a kind of “worst case” simplifies things:
“Whenever x gets and stays larger then any finite bound, then x-100 gets and stays larger then any finite bound”.
No more sine to deal with! If we now want x-100 to be (and stay) larger than a finite bound (lets say 1,000,000), then we guarantee that by choosing x itself to be bigger than that bound plus 100 (here: 1,000,100). This proves what we wanted!
Never did we have to check an infinite number of cases by hand, nor is there any wiggly room, despite x+100·sin(x) wiggling up and down all the time (draw a graph if you want to see this). We argued purely by reason and logic, which goes further than simple calculation.
For some way questions, showing that something grows without bound can become seriously complicated; too complicated for even modern mathematicians. Only time will show if we are able to absolutely prove such things, but the chance for failure is always there. Heck, it is even an unsolved problem if there is a way to always and definitely decide (an “algorithm”) if two elementary expressions describe the same function.
Cos maths is a human created language based on logical consistency, so even when the weird shit happens its because of other logical consistencies
Cos maths is a human created language based on logical consistency, so even when the weird shit happens its because of other logical consistencies
A way to prove numbers are unbounded is through proof by contradiction. Let’s assume that an upper bound to numbers exists and that upper bound is B. We can then add 1 to B, so that upper bound is now B+1. We can then wash, rinse, repeat. So, numbers are unbounded.
A way to prove numbers are unbounded is through proof by contradiction. Let’s assume that an upper bound to numbers exists and that upper bound is B. We can then add 1 to B, so that upper bound is now B+1. We can then wash, rinse, repeat. So, numbers are unbounded.
In maths, there’s no observation to do. It’s just a language at the end of the day. It does nothing weird like that because it is just a description and iteration of logic
In physics, this absolutely could be and probably is true. We can only be certain of the laws of physics in the place where we observe them. There’s some interesting theories about how we might be living in a void and the rest of space might have a different speed of light or strength of gravitational force. We also think that *a lot* of weird things happen on event horizons of black holes, and we have no idea what kind of things could be beyond the observable universe and no reason to suspect that there’s nothing else out there
In maths, there’s no observation to do. It’s just a language at the end of the day. It does nothing weird like that because it is just a description and iteration of logic
In physics, this absolutely could be and probably is true. We can only be certain of the laws of physics in the place where we observe them. There’s some interesting theories about how we might be living in a void and the rest of space might have a different speed of light or strength of gravitational force. We also think that *a lot* of weird things happen on event horizons of black holes, and we have no idea what kind of things could be beyond the observable universe and no reason to suspect that there’s nothing else out there
I had a calculus teacher explain an infinity graph like this and it has always helped.
Imagine you are standing in an endless hallway. You have a flashlight and shine it directly on the wall immediately next to you. This is the “lowest” point on the graph of “how far does the beam go before touching the wall”. Now start rotating to the left or right and the flashlight beam will still be hitting the wall, but at accelerating distances compared to your constant rotation. The point on the graph of “how far until the beam touches the wall” increases exponentially. There will come a point where your beam suddenly no longer touches the wall, but instead travels to infinity. Nothing special, just a mathematical representation showing that the beam will not touch. You continue to rotate and the beam comes back to you in the same pattern on the other wall.
It has been a lot of years since that class, but I THINK that describes an asymptote.
Now, if you’re asking what happens on the infinity side of where the light travels to, like what is way over there that we can’t know about, I don’t know. Science fiction explores that topic pretty thoroughly.
I had a calculus teacher explain an infinity graph like this and it has always helped.
Imagine you are standing in an endless hallway. You have a flashlight and shine it directly on the wall immediately next to you. This is the “lowest” point on the graph of “how far does the beam go before touching the wall”. Now start rotating to the left or right and the flashlight beam will still be hitting the wall, but at accelerating distances compared to your constant rotation. The point on the graph of “how far until the beam touches the wall” increases exponentially. There will come a point where your beam suddenly no longer touches the wall, but instead travels to infinity. Nothing special, just a mathematical representation showing that the beam will not touch. You continue to rotate and the beam comes back to you in the same pattern on the other wall.
It has been a lot of years since that class, but I THINK that describes an asymptote.
Now, if you’re asking what happens on the infinity side of where the light travels to, like what is way over there that we can’t know about, I don’t know. Science fiction explores that topic pretty thoroughly.
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