TYPES OF INFINITY : Mostly the students have run across infinity at several points in previous time to a calculus class. Though, when they have dealt along with this, this was just a symbol used to show a really, really large positive or negative number and which was the extent of this. Once they find in a calculus class students are asked to do some fundamental algebra along with infinity and it is where they find in trouble. Infinity is NOT a number and for the most part does not behave as a number. Though, despite this we'll think of infinity into this section like a really, really large number which is more large there isn't the other number larger than this. It is not correct obviously, but may assist with the discussion in this section. Consider as well that everything which we'll be discussing in this section applies just to real numbers. If you go in complex numbers for illustration things can and do change.
Therefore, let's start thinking about addition along with infinity. While you add two non-zero numbers you find a new number. For illustration, 4 + 7 = 11. With infinity it is not true. With infinity you get the following.
∞ + a = ∞ here a ≠ - ∞
∞ + ∞ = ∞
However, very large positive number (∞) plus any positive number, regardless of the size, is even a really large positive number. Similarly, you can add a negative number (that is a = 0) to a really, really large positive number and stay really very large and positive. Hence, addition including infinity can be dealt along with in an intuitive way if you are careful. Consider as well that the a should NOT be negative infinity. If this is, there are several serious matters that we require to deal with as we will consider in a bit.
Subtraction along with negative infinity can also be dealt with in an intuitive way in most cases finally. A really large negative number minus any positive number, regardless of its size, is even a really, very large negative number. Subtracting a negative number that is a = 0) from a really, very large negative number will even be a really, really large negative number. Otherwise,
- ∞ - a = - ∞ here a ≠ - ∞
- ∞ - ∞ = - ∞
Again, a should not be negative infinity to ignore some potentially serious difficulties.
Multiplication can be dealt along with fairly intuitively additionally. A really, very large number that may be positive or negative times any number, regardless of size, is even a really, very large number we will just require to be careful with signs. In the case of multiplication we contain,
(a)(∞) = ∞ if a > 0 (a) (∞) = - ∞ if a < 0
(∞)(∞) = ∞ (-∞)(-∞) = ∞ (-∞)(-∞) = -∞
What you identify about products of positive and negative numbers is even true now.
Several forms of division can be dealt along with intuitively as well. A really, very large number divided with a number that is not larger is even a really, very large number.
∞/a = ∞ if a > 0, a ≠ ∞ ∞/a = -∞ if a < 0, a-≠ -∞
-∞/a = -∞ if a > 0, a ≠ ∞ -∞/a = ∞ if a < 0, a-≠ -∞
Division of a number through infinity is somewhat intuitive, although there are a couple of subtleties which you require to be aware of. While we talk about division through infinity we are really talking regarding to a limiting process wherein the denominator is going towards infinity. Therefore, a number which isn't more large divided an increasingly large number is a more and more small number. Conversely, in the limit we get,
a/∞ = 0 a/-∞ = 0
Therefore, we've dealt with almost every fundamental algebraic operation involving infinity. There are two cases that which we have not dealt with until now. These are as,
∞ - ∞ = ? ±∞/±∞ = ?
The problem along with these two cases is that intuition does not really assist here. A really, very large number minus a really, very large number can be anything (-∞, a constant, or ∞). Similarly, a really, very large number divided through a really, very large number can also be something (± ∞ - this based on sign matters, 0, or a non-zero constant).
What we have got to learn now is that here are really, very large numbers and then there are really, very large numbers. Conversely, some infinity is larger than other infinities. Along with addition, multiplication and the firstly sets of division we worked that wasn't a matter. The common size of the infinity just does not influence the answer in those cases. Though, with the subtraction and division cases listed above, this does issue as we will consider.
Now there is one way to think of this concept that some infinity is larger than others. It is a fairly dry and technical method to think of this and your calculus problems will possibly never use such stuff, but this is a good way of looking at this. Also, please consider that I'm not trying to provide a precise proof of anything now. I'm just trying to provide you a little insight in the problems along with infinity and how several infinities can be thought of as larger than others.
Let's start by searching how many integers here are. Obviously I hope, there are an infinite number of them, although let's try to find a better grasp on the "size" of this infinity. Therefore, pick any two integers totally at random. Start at the smaller of the two and list, in rising order, all the integers which come after that. Finally we will reach the larger of the two integers which you picked.
Depending upon the relative size of the two integers this might take a very long time to list all the integers among them and there is not really a reason to doing this. But, it could be done if we needed to and that's the significant part.
Since we could list all these integers among two randomly chosen integers we say that the integers are countably infinite. Again, here is no real purpose to actually do it; this is simply something which can be done if we must choose to do so.
Generally a set of numbers is termed as countably infinite if we can get a way to list them all out. In a more exact mathematical setting it is usually done with a particular kind of function termed as a bijection which associates each number in the set with precisely one of the positive integers.
This can also be shown that the set of all fractions are also countably infinite, though it is a little harder to demonstrate and is not actually the purpose of this discussion.
Let's contrast that by trying to understand how loads of numbers there are in the interval (0,1). By numbers, I mean all probable fractions which lie in between zero and one including all possible decimals which are not fractions, which lie between zero and one. The subsequent is same to the proof as given above, but was nice sufficient and easy adequate (I hope) that I needed to comprise this there.
To start let's suppose that all the numbers in the interval (0,1) are countably infinite. It means that there must be a way to list all of them out. We could contain something as the following,
x1 = 0.692096.....
x2 = 0.171034.....
x3 = 0.993671.....
x4 = 0.045908......
Now, choose the ith decimal out of xi as demonstrated below
x1 = 0.692096.....
x2 = 0.171034.....
x3 = 0.993671......
x4 = 0.045908.....
and by a new number with these digits. Therefore, for our instance we would have the number
x = 0.6739.....
In the new decimal replace all the 3's along with a 1 and replace each other numbers with a 3. In the case of our illustration it would yield the new number x = 0.3313....
Consider that that number is in the interval (0,1) and also consider that specified how we decide the digits of the number it number will not be equivalent to the first number in our list, x1, since the first digit of each is guaranteed to not be similar. Similarly, this new number will not find similar number as the second in our list, x2, since the second digit of each is guaranteed to not be similar. Continuing in this way we can consider that such new number we constructed x, is guaranteed to not be in our listing. Although this contradicts the initial assumption as we could list out all the numbers in the interval (0,1). Therefore, this should not be possible to list out all the numbers into the interval (0,1).
Sets of numbers, that as all the numbers in (0,1), which we cannot write down into a list are termed as uncountably infinite.
The purpose for going over it is the following. An infinity it is uncountably infinite is considerably larger than an infinity which is only countably infinite. Therefore, if we take the difference of two infinities we get a couple of possibilities.
∞(uncountable) - ∞(countable) = ∞
∞(countable) - ∞( uncountable) = - ∞
Consider that we didn't give a difference of two infinities of similar type. Depending on the context that might still have some ambiguity about as, what the answer would be here, but it is a completely different topic.
We could do something same for quotients of infinities.
(∞(countable))/ (∞(uncountable)) = 0
(∞(uncountable))/ (∞(countable)) = ∞
Again, we ignored a quotient of two infinities of the same type since, again depending upon the context; there might even be ambiguities about its value.
Therefore, that's it and hopefully you have learned something from this discussion. Infinity simply is not a number and as there are different kinds of infinity it generally doesn't behave as a number does. Be careful while dealing with infinity.