Infinity: Are 1/0 and 2/0 same or we have different infinities ?

[juvekaradheesh]

We know that anything divided by 0 is infinity.
eg. 1/0 = ∞ , since, if you were to divide 1 into zero pieces, you would need to divide an infinite number of times.

similarly2/0 = ∞ .

So,
1. Are 1/0 and 2/0 same or we have different infinities ?
2. 2/0-1/0 = 1/0 , so we still have infinity so is 2/0 bigger infinity than 1/0 ?

 

[Deleted member 4993]

We know that anything divided by 0 is infinity.
eg. 1/0 = ∞ , since, if you were to divide 1 into zero pieces, you would need to divide an infinite number of times.

similarly2/0 = ∞ .

So,
1. Are 1/0 and 2/0 same or we have different infinities ?
2. 2/0-1/0 = 1/0 , so we still have infinity so is 2/0 bigger infinity than 1/0 ?

Are these questions for your homework at school - or are these your "privete" query?

 

[juvekaradheesh]

Are these questions for your homework at school - or are these your "privete" query?

It's my private query but does it matter ?

 

[pka]

It's my private query but does it matter ?

Yes, it matters very much. We do not do homework or help with test questions

The philosopher Kermit Scott said to me "infinity is where mathematicians hide their ignorance."
As a mathematician, I say that infinity is where here the general public shows its ignorance of mathematics.

If you learn nothing else about this topic, learn this: INFINITY IS NOT A NUMBER.
No one can divide by a non-number. Division is the inverse of multiplication on a well-defined number system.
The adjective infinite is used to describe the numerosity of a set. Sets are either finite or infinite. A finite set is one that can be listed using a bounded set of counting numbers, (technically known as an initial sequence in the natural numbers). A set is infinite if it is not finite.

The expression \(\displaystyle \Large\dfrac{2}{\infty}\) is totally meaningless.

 

[Ishuda]

It's my private query but does it matter ?

Of course pka is correct but as someone who likes to play around with numbers, I take shortcuts sometimes so that, yes is some esoteric sense but maybe only sometimes
\(\displaystyle \frac{2}{0}\, -\, \frac{1}{0}\, =\, \frac{1}{0}\, =\, \infty\)
BUT MAKE SURE YOU KNOW YOU ARE PLAYING AROUND when you are talking about manipulating the idea of infinity like this. Again, infinity is an idea, it is not a number. But here is an interesting, at least to me, concept: there is an infinity of infinities each of which is larger than the last.

 

[ksdhart2]

I'd also like to point out that the notion of n/0 is undefined. We say that the 'answer' is infinity, but what we really mean involves taking limits, such that:

\(\displaystyle \displaystyle \lim _{x\to 0}\left(\frac{n}{x}\right)=\infty \) where n is some constant (e.g. 2, 3, 4, etc.)

In other words, at the precise value of x=0, n/x is undefined, but we can observe a pattern as x gets closer and closer to 0. And that pattern is that n/x grows without bounds, thus we say the limit is infinity.

 

[stapel]

We know that anything divided by 0 is infinity.

You're saying that "infinity" is itself a number. It is not.

eg. 1/0 = ∞ ,

No. Division by zero is not defined, by nature of division. For instance, 12/3 = 4 because 4*3 = 12. But zero "multiplied" by infinity is... maybe zero? maybe infinity? But certainly not 1. So the division doesn't make sense, because the multiplication doesn't make sense.

Instead, we say that, as n gets arbitrarily close to zero, the value of 1/n gets arbitrarily large. It only "equals" infinity in the limit.

1. Are 1/0 and 2/0 same or we have different infinities?

Yes. Each of:

. . . . .\(\displaystyle \displaystyle \lim_{n\, \rightarrow \, 0}\, \dfrac{1}{n}\, =\, \infty\)

...and:

. . . . .\(\displaystyle \displaystyle \lim_{n\, \rightarrow \, 0}\, \dfrac{2}{n}\, =\, \infty\)

...has the same limit value. But we also have different "size" infinities.

2. 2/0-1/0 = 1/0 , so we still have infinity so is 2/0 bigger infinity than 1/0 ?

No, these are the same infinity. This is part of the weirdness of infinity.

 

[juvekaradheesh]

What different sizes of infinity we have? can you give me some examples to understand ?

 

[Deleted member 4993]

What different sizes of infinity we have? can you give me some examples to understand ?

A good book for beginner to read would be "One, Two, Three, ... , Infinity" by George Gammow. There is very good description of different types of infinities in that book.

 

[eighsse]

A good book for beginner to read would be "One, Two, Three, ... , Infinity" by George Gammow. There is very good description of different types of infinities in that book.

Also Roads to Infinity by John Stillwell is great -- I never finished it but what I read I enjoyed.

 

[Ishuda]

What different sizes of infinity we have? can you give me some examples to understand ?

A very simple explanation [in this case meaning not rigorous proof] for the difference between two infinities: Let's start with a set which contains a finite set of members, e.g. S₀ = {}. The number of element in S₀ is 0. The number of subsets of S₀ is 2⁰=1, i.e. the set {}. For S₁={0} the number of elements is 1 and the number of subsets is 2¹=2, i.e. the sets {} and {0}. In general, we note that the number of subsets of a set with n elements is 2ⁿ and certainly, for n>1, 2ⁿ>n . Thus the of number of elements in the set {1,2,3,...} which is equal to \(\displaystyle \infty\) is less than the number of subsets of that set which is equal to \(\displaystyle 2^{\infty}\,=\, \infty\). So one type of \(\displaystyle \infty\) is less than another type of \(\displaystyle \infty\).

 

[Mates]

Is aleph 0 a subset of infinity?

 

[stapel]

Is aleph 0 a subset of infinity?

Aleph-naught is a cardinality; it's a size of infinity. It isn't actually an infinity itself.

 

[Mates]

Aleph-naught is a cardinality; it's a size of infinity. It isn't actually an infinity itself.

Thanks!

 

[pka]

Is aleph 0 a subset of infinity?

Here is a very good summary.

 

[Mates]

Here is a very good summary.

On that Wikipedia site, it uses aleph 0 as an exponent with a base of 2. If aleph 0 is not a number, then how does 2^(aleph 0) have any meaning?

 

[stapel]

On that Wikipedia site, it uses aleph 0 as an exponent with a base of 2. If aleph 0 is not a number, then how does 2^(aleph 0) have any meaning?

Aleph-naught is a number; it is the number that counts the size of a particular type of infinity, just as "2" is the number that is the cardinality of the set {@, #}. Aleph-naught is not itself "infinity"; it gives the size of a kind of infinity.

 

[pka]

On that Wikipedia site, it uses aleph 0 as an exponent with a base of 2. If aleph 0 is not a number, then how does 2^(aleph 0) have any meaning?

@Mates, You are correct the notation \(\displaystyle 2^{\aleph_0}\) is standard notation in set theory.
In fact, given a set \(\displaystyle \mathcal{A}\), then \(\displaystyle 2^{\mathcal{A}}\) is the cardinality of the power set \(\displaystyle \mathcal{P(A)}\)
The origin is quite natural, \(\displaystyle 2^{n}\) is the number of subsets of a finite set with exactly \(\displaystyle n\) elements.

 

[enhandle]

Are these questions for your homework at school - or are these your "privete" query?

We know that anything divided by 0 is infinity.
eg. 1/0 = ∞ , since, if you were to divide 1 into zero pieces, you would need to divide an infinite number of times.

similarly2/0 = ∞ .

So,
1. Are 1/0 and 2/0 same or we have different infinities ?
2. 2/0-1/0 = 1/0 , so we still have infinity so is 2/0 bigger infinity than 1/0 ?

you can not divide by zero. the answer is not infinity. the answer is undefined because the operation is impossible.