For instance how do we know that something weird doesn’t happen at some insanely large number?
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You say a sequence goes to infinity if for any number you pick, the sequence has a larger number.
So let’s use natural numbers, 1,2,3,4….
You pick a any number you want, say 999. I can then say at position 1000, in the sequence is 1000 which is greater than 999.
You would generalise it to be for any number e, there is a number e+1 in the sequence that is greater than it. So no matter what number you can think of, there is always a number in the sequence greater than it
You say a sequence goes to infinity if for any number you pick, the sequence has a larger number.
So let’s use natural numbers, 1,2,3,4….
You pick a any number you want, say 999. I can then say at position 1000, in the sequence is 1000 which is greater than 999.
You would generalise it to be for any number e, there is a number e+1 in the sequence that is greater than it. So no matter what number you can think of, there is always a number in the sequence greater than it
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.
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
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.
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
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