Infinity x Infinity
Infinity x Infinity
infinity x infinity
What is infinity x infinity? The problem of defining the infinite does not stop people asking questions about it. One such question which crops up regularly is “What is infinity x infinity?”
Some answer infinity squared which is to beg the question, some say it is 1 and others 0 which begs the question – what are they doing?
As maths deals with clearly defined objects and the infinite is not clearly defined then we have to be prepared from the outset for a range of potential answers and try to sift them down to one which seems to work best in the most circumstances. The first task we have is to find a suitable definition for the type of infinity we are going to use. It would have to be the infinity N as understood by the natural numbers or infinity R as understood by the Real numbers. As these are two different infinities we may find that we have two different results for the problem, infinity x infinity.
We know infinity R is a higher order infinity than infinity N. That is, there are more members in infinity R than in infinity N. Just to recap from earlier posts, the number of even numbers is the same as the number of even + odd numbers (both infinite sets), whilst the number of Real points on a line between 1 and 0 is the same as the number of points on a line between 0 and 100, or between 0 and a million for that matter. These apparently different infinite sets are actually equal in size, yet the set N and the set R are different in size and can be shown to be so.
Cantors proof that Set R is bigger than set N
First of all we need to put every Real decimal between 0 and 1 into a form that it has infinite length. So 0.5 becomes 0.500000… for example. We can also assume decimal parts such as pi – 3 which would give us 0.1415…, the decimal part of root 2 which is 0.4142… and recurring rationals such as 1/3 which is 0.3333…..
Now given an infinite list of such decimals between 0 and 1 we can begin to pair them off with the Natural numbers.
So,
0 – 0.3333…
1 – 0.1415…
2 – 0.5000…
3 – 0.4142…
. .
. .
. .
Take a diagonal line through the decimal parts, then extract this line to create another decimal. Add 1 to each digit of this decimal and we have generated a new decimal which we have not yet listed. Though we have paired an infinite number of decimals with the infinite Natural numbers, there are still decimal numbers out there for which we haven’t accounted. This tells us that the real numbers are greater than the naturals.
So, in considering the question, infinity x infinity we must first ask, which infinity are we looking at. Infinity N or infinity R?
If infinity x infinity is referring to N then we can see that the answer is infinity N.
Proof.
Consider the powers of primes. We know there are an infinite number of prime numbers and we also know that each power of primes numbers will not replicate a number in any other prime power list.
2,4,8,16….
3,9,27,81…
5,25,125,625….
7,49,343,2401..
We can now generate a table of p and powers of p which shows and infinity x infinity.
P = prime number , n = natural number.
|
P |
PxP |
PxPxP |
PxPxPxP |
Pn |
|
2 |
4 |
8 |
16 |
… |
|
3 |
9 |
27 |
81 |
… |
|
5 |
25 |
125 |
625 |
… |
|
7 |
49 |
343 |
2401 |
… |
|
… |
… |
… |
… |
… |
So infinity x infinity = infinity.
The acceptance of such a conclusion is no more problematic than accepting there are the same number of Natural numbers as there are Even numbers.
If we now consider the same problem, infinity x infinity, but with reference to the infinity of the Real numbers we find the result to be the same.
It is far less intuitive however, and a lot more interesting.
Assume a line containing an infinite number of points which is perpendicular to a similar line such as an x,y axes. Now we know that there are the same number of points in a line no matter its length. We also know that there are the same number of points in an area no matter its size, the question we are now faced with is, are there the same number of points on a line as there are in an area? Answering this question will give us an answer to the problem of infinity x infinity where the infinity in question is that of R.
The answer it appears is yes from a proof provided by Cantor after work by Peano. It is a highly technical proof and those further interested can find it at wikipedia.
