Are there sizes to infinity?

[Sir Shfvingle]

This is a question my Algebra 2 teacher posed to our class just to tick us off. He exampled it by saying:

How many numbers are there between 0 and 5? Infinity right? (1.5, 1.51 etc)
How many numbers are between 0 and 100? Infinity right?
So, is that 0-100 infinity larger in mathematical value than the 0-5 infinity?
Inversely, can you add more infinity to infinity?
Succeed your answer with your reasoning.

So, there. I thought about this and looked it up on google. I can't find anything decisive. Could I get some brainiacs' help on this? Are there values to Infinity?

edit: I don't know if I can clarify as my teacher didn't. So, interpret in anyway. Thanks

[Sphere]

Are there values to Infinity?

No, infinity is value less. I think your teacher is simply trying to get you away from thinking of infinity as just a really big number as it is not, and does not behave like one.

For example ∞+∞ = ∞ not 2∞. And ∞/∞ doesn't equal one and ∞-∞ doesn't equal zero.

What gets interesting though is when you can use the concept of infinity but still get a discrete result; the simplest I can come up with is 1/∞=0 and not just really close to 0 (as if infinity was just a large number) but rather it is exactly 0.

This will lead you into limits, which will lead you into calculus, which leads to infinite summations and integrals, and that is were the real fun is. You get stuff like this

4 - 4/3 + 4/5 - 4/7 + 4/9 -4/11 ... = π (and not just really close to pi, exactly pi)

[Jack04]

Infinity is just putting a word to a number that cannot possibly be well defined or well understood. I also like to imagine it's the sound made by a mathematician knocking their head against a wall, but that's just hope.

The hotel analogy is probably the best, certainly worth googling if you want to understand the mind boggling nature of infinity.

In the real world, infinity normally means something has gone horribly wrong. It's a good pointer towards the fact that general relativity and quantum physics aren't the entire story, for example.

[thegsg]

No, infinity is value less. I think your teacher is simply trying to get you away from thinking of infinity as just a really big number as it is not, and does not behave like one.

For example ∞+∞ = ∞ not 2∞. And ∞/∞ doesn't equal one and ∞-∞ doesn't equal zero.

What gets interesting though is when you can use the concept of infinity but still get a discrete result; the simplest I can come up with is 1/∞=0 and not just really close to 0 (as if infinity was just a large number) but rather it is exactly 0.

This will lead you into limits, which will lead you into calculus, which leads to infinite summations and integrals, and that is were the real fun is. You get stuff like this

4 - 4/3 + 4/5 - 4/7 + 4/9 -4/11 ... = π (and not just really close to pi, exactly pi)

You're just awesome

[Plant]

There are different kinds of infinty, which are concepts to be used in real applications.
There is not 20 times more infinty between 0-100 than for 0-5.
There are an infinite set of numbers for both.
It is the same kind of infinity.

What gets interesting though is when you can use the concept of infinity but still get a discrete result; the simplest I can come up with is 1/∞=0 and not just really close to 0 (as if infinity was just a large number) but rather it is exactly 0.

Which method did you use? When I do it using limits, I get an approach to 0. How did you recieve exactly 0?

[Time Commander Bob]

Some infinite sets are indeed "larger" than others, the set of real numbers is "larger" than the set of rational numbers (technically it is better to say that the set of real numbers is uncountably infinite and the set of rational numbers is countably infinite). Look up the diagonal theorem for the easiest explanation of this as it is not as obvious as simply saying that the set of real numbers includes the set of rational numbers, since the set of even numbers has the same size as the set of positive integers.

[Musthavename]

What Time Commander says.

If you consider the set of Fractions (The Rationals) and whole Numbers (The Integers, though the same holds for the Natural Numbers, strictly positive integers), on face value it looks like there's infinitely more Rational numbers than Integers, as between any two integers is an infinite amount of rationals. Yet, you can cook up a map that maps every integer to one rational, and you'll cover the entire set of rationals in the process (so essentially, no matter what obscure fraction you gave me, I could tell you what integer maps to it, even if that integer is insanely large). Hence, these sets are deemed to be "countably" infinite. However, if you start adding the irrational numbers (numbers like pi, sqrt(2), e) to get the set of Real Numbers, you can't do this, and you have an "uncountably" infinite set.

Now, this is usually the point where Simm comes into the thread and expands upon every little detail that's been mentioned but yes, you can say that certain infinite sets are bigger than one another.

[Gaidin]

How many numbers are there between 0 and 5? Infinity right? (1.5, 1.51 etc)
How many numbers are between 0 and 100? Infinity right?
So, is that 0-100 infinity larger in mathematical value than the 0-5 infinity?
Inversely, can you add more infinity to infinity?
Succeed your answer with your reasoning.

So, there. I thought about this and looked it up on google. I can't find anything decisive. Could I get some brainiacs' help on this? Are there values to Infinity?

edit: I don't know if I can clarify as my teacher didn't. So, interpret in anyway. Thanks

Well...You also have to pay attention to your domain.

If you are talking about the natural numbers there are four numbers between 0 and 5. The reals, and there are an uncountably infinite amount of numbers.

And no, one is not strictly larger than the other when it comes to set size, especially when you are considering in the same set. You also can't add infinity to infinity, strictly speaking.

The fact that infinity isn't a number, but a concept, might help you with this reasoning.

[♔Goodguy1066♔]

Well...You also have to pay attention to your domain.

If you are talking about the natural numbers there are four numbers between 0 and 5. The reals, and there are an uncountably infinite amount of numbers.

And no, one is not strictly larger than the other when it comes to set size, especially when you are considering in the same set. You also can't add infinity to infinity, strictly speaking.

The fact that infinity isn't a number, but a concept, might help you with this reasoning.

Exactly, since it is a concept, both the 0-5 infinity and the 0-100 infinity should equal each other! Infinity is infinity is infinity; i.e. without an end. You can never say that 0-100 holds more infinity than 0-5, as infininty is a concept. If you wrote all the numbers from 0-5 and stacked them up they would never ever stop, and the same goes for 0-100. However, we know instinctively that there are more numbers between 0-100 than there are 0-5, which is where the paradox lies.

[Lord of Lost Socks]

Basically the reason why I've sometimes wondered how I can go anywhere at all. No matter how many steps I take toward my destination, there's still an infinity left.

[Plant]

Now, this is usually the point where Simm comes into the thread and expands upon every little detail that's been mentioned but yes, you can say that certain infinite sets are bigger than one another.

Someone should invite him. If someone knows well what they are talking about and can explain the majority of the concepts of infinity then I'll be happy for him to do so.

Exactly, since it is a concept, both the 0-5 infinity and the 0-100 infinity should equal each other! Infinity is infinity is infinity; i.e. without an end. You can never say that 0-100 holds more infinity than 0-5, as infininty is a concept. If you wrote all the numbers from 0-5 and stacked them up they would never ever stop, and the same goes for 0-100. However, we know instinctively that there are more numbers between 0-100 than there are 0-5, which is where the paradox lies.

I don't see a paradox though.

Basically the reason why I've sometimes wondered how I can go anywhere at all. No matter how many steps I take toward my destination, there's still an infinity left.

That makes no sense. An infinity left of what?

[Amorphos]

Are there values to Infinity?

no

To mathematical infinities apparently there are, though they can change according to the sets. I don’t believe infinity can have cardinality, so hit your teacher with defining how it can! it is by definition ‘unlimited’ where any absolute value is a ‘limit’, indeed perhaps any value whatsoever attempts to give infinity some manner of cardinality and is therefore a metaphor at best.

I don’t think math can describe infinity or supposed infinities, generally it seams as though the math is a way to put large collections and variables in a bag. But this is my philosophical perception, I am no mathematician.

its a bit like asking 'how long is a piece of string' or 'if you cut an orange in half, is it half an orange or a whole object'.

If you wrote all the numbers from 0-5 and stacked them up they would never ever stop, and the same goes for 0-100.

The thing is you are taking real amounts 0-5 and adding an imaginary infinity between them, how can we denote a segment of infinity ~ by what do we define the edges of 1,2,3,4,…. What is the space between each integer, we could say that is infinite too.

No matter how many steps I take toward my destination, there's still an infinity left.

Except you take steps relative to your quantum makeup which is finite, if you were an imaginary person then you may be right, but I would still question the logic.

[mp0295]

My teacher said 1/∞ is not possible as it would be putting ∞ in an operation (division) which is not allowed. Infinity is a concept, not a number so it wouldn't make sense to put it in a numbers operation. he said something to the effect of it makes as much sense as 1/banana

And I've heard of the decimal paradox thing before, friend did math project on it. Interesting stuff. Does .9999 repeating = 1 tie into this?

[Time Commander Bob]

My teacher said 1/∞ is not possible as it would be putting ∞ in an operation (division) which is not allowed. Infinity is a concept, not a number so it wouldn't make sense to put it in a numbers operation. he said something to the effect of it makes as much sense as 1/banana

And I've heard of the decimal paradox thing before, friend did math project on it. Interesting stuff. Does .9999 repeating = 1 tie into this?

Use of infinity in operations is actually quite common for proofs but it is often written as a certain quantity tends to infinity rather than putting in infinity directly. For example:
As x tends to infinity 1/x tends to zero though it can be written directly.
0.999... = 1 is an equivalence through infinite decimal expansion and is the summation of 9/(10^n) for an infinite number of terms. This is different approach to infinity to that in set theory, because we aren't concerned about the cardinality (size) of the infinity quantity.

Exactly, since it is a concept, both the 0-5 infinity and the 0-100 infinity should equal each other! Infinity is infinity is infinity; i.e. without an end. You can never say that 0-100 holds more infinity than 0-5, as infininty is a concept. If you wrote all the numbers from 0-5 and stacked them up they would never ever stop, and the same goes for 0-100. However, we know instinctively that there are more numbers between 0-100 than there are 0-5, which is where the paradox lies.

The set of real numbers from 0-5 and 0-100 can be mapped one-to-one and thus both infinite sets have the same size. The fact that this can be done means that it cannot be said that there are more real numbers from 0-5 as there is 0-100, even if it seems "intuitive" to say otherwise. There is no paradox here when one considers scalar quantities such as multiplying by 20 does not effect infinite sets.

[Vagn]

My teacher said 1/∞ is not possible as it would be putting ∞ in an operation (division) which is not allowed. Infinity is a concept, not a number so it wouldn't make sense to put it in a numbers operation. he said something to the effect of it makes as much sense as 1/banana

And I've heard of the decimal paradox thing before, friend did math project on it. Interesting stuff. Does .9999 repeating = 1 tie into this?

I'm fairly sure 1/∞ is allowed, as it is key to one form of improper integrals.

[Plant]

I'm fairly sure 1/∞ is allowed, as it is key to one form of improper integrals.

When I inputted
f(x) = 1/x
I got ∞ as the answer. I'm probably doing it wrong though.

How did you input 1/∞ and what answer did you receive?

[mp0295]

Eh whatever, I guess he was trying to simplify it for us

[handsome pete]

I've studied a fair bit of physics and math, I'll give a brief explanation of some of the confusions that are going on in this thread.

Firstly let's explain the debate about 1/infinity. In physics we would often just write 1/infinity and I knew a lot of physicists who would often do it. But it's about convention and definition. When Newton and Liebniz invented calculus they used the concept of an infinitesimal, an infinitesimal is where you say that a number is not equal to zero, but no matter how many times you divide your number you can never approach it. An infinitesimal is a concept, just like infinity is a concept. But after Newton and Liebniz the idea of the Real Number system was formalised, wherebye a number was only something you could specifically measure, like five. Five has a value whereas infinity and infinitesimals do not and are not numbers. 1/infinity is like saying if i kept adding to the denominator, the value of the equation keeps changing but there is a value which it approaches, though it never reaches and yet this value is unique. But the way different formalised definitions of infinity get there is not the same (1,2,3,4 vs, 2,4,6,8,10). 1/infinity is a short hand way of dealing with this but actually is not the right nomenclature used by mathematicians. Mathematicians call the number which the concept approaches, a limit, and instead of writing 1/infinity you are supposed to write limn->infinity 1/n. But a rose is still a rose by any other name. Where you get in to trouble, and why mathematicians write it this way is when you have more than one infintesimal/infinity. Often in physics you divide one infinity by another and get a definite number. The reason this happens is that the two infinities are defined as ongoing concepts relative to a common time dimension. So if every time you calculate an equation part of the equation is the natural numbers (1,2,3,4,5) and the other is the even numbers (2,4,6,8) no matter how long you do this for, as both parts of the equation approach infinity 1/2 = 2/4= 3/6=4/8 approaches a half and in this case also equals it for each number. Hence why people sometimes write 1/infinity and sometimes write out each part of the equation in limit form and then combine the limits in to one final limit. And although each infinity is valueless they can be said to have a relative value to each other (because the two concepts are counting in the same equation or aka the same universe).

What's interesting is that there are two types of infinities, commonly referred to as the inwards infinity and the outwards infinity. Though in mathematics they are called countable infinities and non-countable infinities. An example of a countable infinity is 1,2,3,4,5..., Whereas consider the decimal numbers, pick any two numbers you like as close as you like, say 1 and 2, and there are an infinite number of decimals in between these two numbers. There is no way of counting through these numbers and being able to say that you've covered all the numbers lower than the number your up to. If you say .1 .2 .3, i would say what about .05, and if you say .05 .1 .15 I would say what about .06.

Continuing on from MustHaveName, it's also possible for one uncountable infinity to bigger than another uncountable infinity. An example is the real numbers and the rational numbers. The rational numbers x/y are not countable (like the decimals) but neither are the irrational (the square root of 2 for example cannot be expressed as x/y and is an irrational number, for every x/y there is a unique square root), the real numbers contains both of these uncountable infinities. Therefore the real numbers (an uncountable infinity) is bigger than either the rational or the irrational numbers (both of which are also uncountable infinities).

[Bucken]

1/∞=0

Can someone explain the sence in that.

Also If that is true shoulnt

0*∞= 1

[Jack04]

1/∞=0

Can someone explain the sence in that.

Also If that is true shoulnt

0*∞= 1

It isn't technically true, as (IIRC) 1/∞ cannot really be defined. Nor can 1/0.

Generally, there isn't too much point looking to make sense out of infinity, since it is (almost by definition) beyond comprehension.