Questions around the concept of infinity

[Skyun]

Hi everyone,

I am posting here because I felt like my question didn't fit any other category (I might be wrong).

Let me start off by saying that I understand that infinity is not a real number, and that any real number has a finite amount of digits before the decimal (to the left). So these questions may actually sound very stupid as the scenario might not even be correct to begin with, but here they are:

• Suppose we are not talking about real numbers, is it possible to have a number with infinite digits before the decimal ? (e.g.: ...11111111 or ...82828282 or ...54307209; as long as it's not just a bunch of zeroes)

• If yes, isn't that number the same as infinity ? If not, how do we distinguish such a number from infinity ?

• If it's not possible to have a number with infinite digits (that are not just zeroes) to the left, isn't there an inconsistency with limits ?

e.g.:
Limit x→∞ (x) = ∞
Does that imply that the number will increase forever to the point where it tends towards infinity without ever reaching an infinite number of digits before the decimal ? Isn't there a contradiction there ? That would mean the number of digits are finite but it tends towards infinity... I feel like there must be an answer but I'm unaware of it, which is why I would like to clarify the concept.

This question came to my mind as I was trying to grasp the concept of eternity as people were commenting that they would rather be immortal and live forever than die now in a "Would you rather?" post. Pretty much any number you can think of is technically closer to 0 than infinity (or am I wrong ?), so technically, if you lived until the theoretical heat death of the universe in 10^100 years and were to draw a timeline of your "lifetime" it would actually be pretty much non-visible on the drawing as eternity (AKA infinite amount of time) would just dwarf this number, actually is it correct to say that considering infinity is not a real number ? Are all my statements and ideas wrong because it's not possible to make use of non-real number in this way ?

Regards.

 

[Steven G]

There can not be a finite number that has an infinite number of digits to the left of the decimal. Suppose that was not true. Then you can surely give me a number that is larger than this number. You can argue that that can't happen.

 

[Skyun]

There can not be a finite number that has an infinite number of digits to the left of the decimal. Suppose that was not true. Then you can surely give me a number that is larger than this number. You can argue that that can't happen.

So considering that it is not possible to have a number with an infinite number of digits to the left of the decimal, isn't it wrong to say that anything tends towards infinity ? Aren't all the numbers closer to 0 than infinity ? For example when someone says that something gets so big (or increases/expands so much) that it eventually tends towards infinity.

 

[Cubist]

is it possible to have a number with infinite digits before the decimal ?

You seem to have come up with a thought experiment along the lines of y=10^x, where y is a number that has a 1 digit followed by "x" zero digits (if x is a +ve integer). If x is finite, then y is also finite (and vice-versa). Thinking of their magnitudes, for a large value of x, say 1,000,000 then y will be MUCH larger - but they will both still be finite. Then, think about...

[math] \lim_{x\to\infty}{10^x} = \,[/math]?

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The words "tends towards infinity", usually referring to a variable, can help people to visualize what happens to an expression containing that variable as the variable becomes ever larger and larger in value. In the limit as a variable tends towards infinity then the variable ceases to be a finite number and switches to become the concept that we call infinity. At this point you should not do any further numerical operations/ methods using the symbol ∞. For example never try to work out ∞ - ∞ or ∞/∞ etc. Always do math on the variable before taking the limit. And while taking the limit we can then apply knowledge to standard sub-expressions like 1/x which becomes 0 in the limit as x tends towards infinity.

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FYI: In certain branches of mathematics there exists the concept of "Aleph numbers" which represent different types of infinity (I think!). I have never studied this myself so I won't comment any further of this ? !

 

[JeffM]

It is perfectly possible to define infinity as a number. One way, Cantor’s way, is to define it as the number of positive integers. That number is a transfinite number called Aleph null. Another way that appeals to me personally is the one-point compactification of the real numbers, where infinity is the quotient of any non-zero real number and zero.

But if you are talking about integers or real numbers, then infinity is not a number at all. When we talk about the limit of a function at a being infinity, we mean that no real number bounds the function as we approach a. Whatever real number we pick, if you get close enough to a, the absolute value of f(a) will exceed that number. In other words, the limit being infinity does not constitute a claim that infinity is a number at all.