Finite Infinity?

Question

I'm just a middle school student who needs your help.

If it is impossible to reach infinity and -infinity from zero, then it is also impossible to "get back" to zero from infinity and -infinity, assuming there are something located at both "points of infinity".

What I'm trying to say here is that if someone wants to travel from one point to another on a real number line, then both points must be rational and "reachable", so that he/she is able to stand on that point. Thus, the "points of inifinity" do not exist.

Am I right? Am I making sense?

Answer 1

you are right, there is NO such a thing as
points of infinity.
infinity is a limit of numbers, but it is NOT a number itself.
numbers do not need to be rational to be "reachable".

Answer 2

First of all, infinity (or negative infinity) is NOT a number. It is not something which you can count, or add or subtract or something. It is just a concept, more precisely, it is a limiting concept. Best to wait until Calculus one because there is just a certain degree of mathematical maturity required.

Second, be careful with your words. Remember that, strictly mathematically speaking, you cannot "reach" a point either. This has nothing do with the point's rationality. You can never "reach" an integer either.

Think of like this, this notion of exactness is only in our minds. It can never be in the real physical world. For example, if you start from zero, you can NEVER EVER EVER reach the number one. You can never measure something exactly either. You can never ever ever cut a piece of wood that is EXACTLY one foot long. It will be always too long or too short. Even if you get a piece that is exactly one foot long, you will never know because the ruler which you use won't be good enough.

Just try it! Get a piece of paper and measure it using a ruler. And then measure again and again. The same piece of paper and the ruler won't give you the same reading everytime.

That is why statistics is a HUGE field in mathematics.

So, if the point is "nice" like positive one, it is still unreachable. You will never stand on that point. Because a point is "infinitely" small. One is EXACTLY 1.0000000.....with "infinite" number of zeros after the decimal point.

There are different kinds of infinities too. Some infinities are bigger than other. For example, we say that there are more irrational numbers than there are rational numbers...but they are both infinite...and also it is quite easy to prove that between any two real numbers, there is always a rational and an irrational number. But somehow, there is no contradiction.

And also remeber, that real numbers have this property that between ANY two real numbers, there is an infinite amount of numbers between them and the real numbers are also "well-ordered" meaning any two real numbers you give me, I can put < or > or = between them.

With complex numbers you cannot do that.
What would you put between 1+i & 1-i?

Okay, getting too abstract! You just have to wait until you take some advanced calculus, numerical analysis, abstract algebra, and complex analysis.

Answer 3

Both points need not be 'rational' on the real number line, the set of real numbers contains both the set of rational numbers and the set of irrational numbers.
The usual point at infinity is defined for the neighbourhood(of the point) such that modulus z > 1/E for sufficiently small E>0, though E approaches 0 it never gets there, it's'journey'is what is really infinite.
Another way of definition is by a parameter t such that z= 1/t
and let t=0 correspond to the point at infinity.
Limits usually are employed in defining from where the reach for infinity arises from, but this is no problem for stereographic projections.
As I said the 'journey' of approach is infinite but it is axiomatic to set the reached value and evaluate a limit taking definition into account.
(Try L'Hopital's rule if you get into trouble with limits)

Answer 4

Points do not need to be rational to be stepped on.

You CAN get to infinity but NOT in a finite time....

infinity is the limit of the natural numbers {1,2,3,... }

But you could not go back. Why because at infinity there is no number before so there is no place to start your trip back.

Points at infinity DO exist. Exactly in the same way as 'i' exists, trancendental numbers exists, and so on. Lack of a physical model is not proof it does not exist. (EDIT - it is not a 'real' number... yet it is a 'number' in other senses like when one compactifies C)

EDIT

The guy below me has been smoking up..

There ARE orderings to C... just not natural orderings.... ill give you one.

We say x < y if either |x| < |y| or if |x|=|y| then the angle of x (in polar form) is smaller than the angle of y. If those are equal they are the same.

This is an ordering. but there are many other orderings as well.

As well... if your saying that 'infinity' is not a number well then neither is 'i' but thats all bullshit... its just not a 'natural number' which is wher eyour counting comes from... and it is not a 'real number'.

And comparing sizes of sets its a little off topic

Answer 5

You are incorrect - the line keeps going, indefinitely, in each direction. So you can choose what point you want to go from and want to go to, but you must choose a number, rather than a concept ie you cannot choose the point of infinity as it is not a point per se.

Answer 6

Go back infinitly in time. Then come back here. It would take an infinite amount of time to get here. Therefore, we are not here.

The problem here being the assumption that time is continuous infinitely.

Maybe space is not continuous.

Who said "points of infinity"? Did your math teacher say that? I don't think there are "points of infinity", although there are an infinite amount of points.

Quit the math and hit the philosophy books. Who cares about that Cartesian coordinate plane, DesCartes real stuff is in Cogito, Ergo Sum.

Answer 7

Infinity is a number, but a number in a different class than real numbers. It is one of a set of "trans-finite" numbers as described by the mathemetician Georg Cantor. There are an infinite number of "infinities" (trans-finite numbers).

Answer 8

think about this and you'll figure it out: even between 1 and 2 there are an infinite number of points.

Answer 9

YES AND nO.
BOTH ARE CONCEPTS NOT WHOLLY GRASPED, AND OUR NUMBERING MAKES SENSE....FOR US.
IT IS ARBITRARY, BASED ON OUR DIGITS.
WHEN YOU FEEL REALLY MELLOW, LOOK UP BOOLEAN LOGIC!!