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### Appendix A.7 : Types of Infinity

Most students have run across infinity at some point in time prior to a calculus class. However, when they have dealt with it, it was just a symbol used to represent a really, really large positive or really, really large negative number and that was the extent of it. Once they get into a calculus class students are asked to do some basic algebra with infinity and this is where they get into trouble. Infinity is NOT a number and for the most part doesn’t behave like a number. However, despite that we’ll think of infinity in this section as a really, really, really large number that is so large there isn’t another number larger than it. This is not correct of course but may help with the discussion in this section. Note as well that everything that we’ll be discussing in this section applies only to real numbers. If you move into complex numbers for instance things can and do change.

So, let’s start thinking about addition with infinity. When you add two non-zero numbers you get a new number. For example, \(4 + 7 = 11\). With infinity this is not true. With infinity you have the following.

\[\begin{align*}\infty + a & = \infty \hspace{0.25in}{\mbox{where }}a \ne - \infty \\ \infty + \infty & = \infty \end{align*}\]

In other words, a really, really large positive number (\(\infty \)) plus any positive number, regardless of the size, is still a really, really large positive number. Likewise, you can add a negative number (*i.e.* \(a < 0\)) to a really, really large positive number and stay really, really large and positive. So, addition involving infinity can be dealt with in an intuitive way if you’re careful. Note as well that the \(a\) must NOT be negative infinity. If it is, there are some serious issues that we need to deal with as we’ll see in a bit.

Subtraction with negative infinity can also be dealt with in an intuitive way in most cases as well. A really, really large negative number minus any positive number, regardless of its size, is still a really, really large negative number. Subtracting a negative number (*i.e.* \(a < 0\)) from a really, really large negative number will still be a really, really large negative number. Or,

\[\begin{align*} - \infty - a & = - \infty \hspace{0.25in}{\mbox{where }}a \ne - \infty \\ - \infty - \infty & = - \infty \end{align*}\]

Again, \(a\) must not be negative infinity to avoid some potentially serious difficulties.

Multiplication can be dealt with fairly intuitively as well. A really, really large number (positive, or negative) times any number, regardless of size, is still a really, really large number we’ll just need to be careful with signs. In the case of multiplication we have

\[\begin{array}{c}\left( a \right)\left( \infty \right) = \infty \hspace{0.25in}{\mbox{if }}a > 0\hspace{0.75in}\left( a \right)\left( \infty \right) = - \infty \hspace{0.25in}{\mbox{if }}a < 0\\ \\ \left( \infty \right)\left( \infty \right) = \infty \hspace{0.5in}\left( { - \infty } \right)\left( { - \infty } \right) = \infty \hspace{0.75in}\left( { - \infty } \right)\left( \infty \right) = - \infty \end{array}\]

What you know about products of positive and negative numbers is still true here.

Some forms of division can be dealt with intuitively as well. A really, really large number divided by a number that isn’t too large is still a really, really large number.

\[\begin{align*}\frac{\infty }{a} & = \infty & \hspace{0.25in} & {\mbox{if }}a > 0,a \ne \infty & \hspace{0.75in}\frac{\infty }{a} & = - \infty & \hspace{0.25in}{\mbox{if }}a < 0,a \ne - \infty \\ \frac{{ - \infty }}{a} & = - \infty & \hspace{0.25in} & {\mbox{ if }}a > 0,a \ne \infty & \hspace{0.75in}\frac{{ - \infty }}{a} & = \infty & \hspace{0.25in}{\mbox{ if }}a < 0,a \ne - \infty \end{align*}\]

Division of a number by infinity is somewhat intuitive, but there are a couple of subtleties that you need to be aware of. When we talk about division by infinity we are really talking about a limiting process in which the denominator is going towards infinity. So, a number that isn’t too large divided an increasingly large number is an increasingly small number. In other words, in the limit we have,

\[\frac{a}{\infty } = 0\hspace{0.5in}\hspace{0.5in}\hspace{0.25in}\frac{a}{{ - \infty }} = 0\]

So, we’ve dealt with almost every basic algebraic operation involving infinity. There are two cases that that we haven’t dealt with yet. These are

\[\infty - \infty = {\mbox{?}}\hspace{0.5in}\hspace{0.5in}\frac{{ \pm \,\infty }}{{ \pm \,\infty }} = ?\]

The problem with these two cases is that intuition doesn’t really help here. A really, really large number minus a really, really large number can be anything (\( - \infty \), a constant, or \(\infty \)). Likewise, a really, really large number divided by a really, really large number can also be anything (\( \pm \infty \) – this depends on sign issues, 0, or a non-zero constant).

What we’ve got to remember here is that there are really, really large numbers and then there are really, really, really large numbers. In other words, some infinities are larger than other infinities. With addition, multiplication and the first sets of division we worked this wasn’t an issue. The general size of the infinity just doesn’t affect the answer in those cases. However, with the subtraction and division cases listed above, it does matter as we will see.

Here is one way to think of this idea that some infinities are larger than others. This is a fairly dry and technical way to think of this and your calculus problems will probably never use this stuff, but it is a nice way of looking at this. Also, please note that I’m not trying to give a precise proof of anything here. I’m just trying to give you a little insight into the problems with infinity and how some infinities can be thought of as larger than others. For a much better (and definitely more precise) discussion see,

http://www.math.vanderbilt.edu/~schectex/courses/infinity.pdf

Let’s start by looking at how many integers there are. Clearly, I hope, there are an infinite number of them, but let’s try to get a better grasp on the “size” of this infinity. So, pick any two integers completely at random. Start at the smaller of the two and list, in increasing order, all the integers that come after that. Eventually we will reach the larger of the two integers that you picked.

Depending on the relative size of the two integers it might take a very, very long time to list all the integers between them and there isn’t really a purpose to doing it. But, it could be done if we wanted to and that’s the important part.

Because we could list all these integers between two randomly chosen integers we say that the integers are *countably infinite*. Again, there is no real reason to actually do this, it is simply something that can be done if we should choose to do so.

In general, a set of numbers is called countably infinite if we can find a way to list them all out. In a more precise mathematical setting this is generally done with a special kind of function called a bijection that associates each number in the set with exactly one of the positive integers. To see some more details of this see the pdf given above.

It can also be shown that the set of all fractions are also countably infinite, although this is a little harder to show and is not really the purpose of this discussion. To see a proof of this see the pdf given above. It has a very nice proof of this fact.

Let’s contrast this by trying to figure out how many numbers there are in the interval \( \left(0,1\right) \). By numbers, I mean all possible fractions that lie between zero and one as well as all possible decimals (that aren’t fractions) that lie between zero and one. The following is similar to the proof given in the pdf above but was nice enough and easy enough (I hope) that I wanted to include it here.

To start let’s assume that all the numbers in the interval \( \left(0,1\right) \) are countably infinite. This means that there should be a way to list all of them out. We could have something like the following,

\[\begin{align*}{x_1} & = 0.692096 \cdots \\ {x_2} & = 0.171034 \cdots \\ {x_3} & = 0.993671 \cdots \\ {x_4} & = 0.045908 \cdots \\ \vdots \,\, & \hspace{0.6in} \vdots \end{align*}\]

Now, select the \(i\)^{th} decimal out of \({x_i}\) as shown below

\[\begin{align*}{x_1} & = 0.\underline 6 92096 \cdots \\ {x_2} & = 0.1\underline 7 1034 \cdots \\ {x_3} & = 0.99\underline 3 671 \cdots \\ {x_4} & = 0.045\underline 9 08 \cdots \\ \vdots \,\, & \hspace{0.6in} \vdots \end{align*}\]

and form a new number with these digits. So, for our example we would have the number

\[x = 0.6739 \cdots \]

In this new decimal replace all the 3’s with a 1 and replace every other numbers with a 3. In the case of our example this would yield the new number

\[\overline x = 0.3313 \cdots \]

Notice that this number is in the interval \( \left(0,1\right) \) and also notice that given how we choose the digits of the number this number will not be equal to the first number in our list, \({x_1}\), because the first digit of each is guaranteed to not be the same. Likewise, this new number will not get the same number as the second in our list, \({x_2}\), because the second digit of each is guaranteed to not be the same. Continuing in this manner we can see that this new number we constructed, \(\overline x \), is guaranteed to not be in our listing. But this contradicts the initial assumption that we could list out all the numbers in the interval \( \left(0,1\right) \). Hence, it must not be possible to list out all the numbers in the interval \( \left(0,1\right) \).

Sets of numbers, such as all the numbers in \( \left(0,1\right) \), that we can’t write down in a list are called *uncountably* infinite.

The reason for going over this is the following. An infinity that is uncountably infinite is significantly larger than an infinity that is only countably infinite. So, if we take the difference of two infinities we have a couple of possibilities.

\[\begin{align*}\infty \left( {{\mbox{uncountable}}} \right) - \infty \left( {{\mbox{countable}}} \right) & = \infty \\ & \\ \infty \left( {{\mbox{countable}}} \right) - \infty \left( {{\mbox{uncountable}}} \right) & = - \infty \end{align*}\]

Notice that we didn’t put down a difference of two infinities of the same type. Depending upon the context there might still have some ambiguity about just what the answer would be in this case, but that is a whole different topic.

We could also do something similar for quotients of infinities.

\[\begin{align*}\frac{{\infty \left( {{\mbox{countable}}} \right)}}{{\infty \left( {{\mbox{uncountable}}} \right)}} & = 0\\ & \\ \frac{{\infty \left( {{\mbox{uncountable}}} \right)}}{{\infty \left( {{\mbox{countable}}} \right)}} & = \infty \end{align*}\]

Again, we avoided a quotient of two infinities of the same type since, again depending upon the context, there might still be ambiguities about its value.

So, that’s it and hopefully you’ve learned something from this discussion. Infinity simply isn’t a number and because there are different kinds of infinity it generally doesn’t behave as a number does. Be careful when dealing with infinity.

## FAQs

### What are the 3 types of infinity? ›

Three main types of infinity may be distinguished: **the mathematical, the physical, and the metaphysical**. Mathematical infinities occur, for instance, as the number of points on a continuous line or as the size of the endless sequence of counting numbers: 1, 2, 3,….

**Are there different levels of infinity? ›**

**There are actually many different sizes or levels of infinity**; some infinite sets are vastly larger than other infinite sets. The theory of infinite sets was developed in the late nineteenth century by the brilliant mathematician Georg Cantor.

**What are the two types of infinity in math? ›**

Two common types of infinities that people are aware of are **countably infinite (like natural numbers) and uncountably infinite (like real numbers)**. A lot of people are not aware of the fact that there are actually infinite types of infinities.

**What is infinity in calculus? ›**

INFINITY, along with its symbol ∞, is not a number and it is not a place. When we say in calculus that a function becomes "infinite," we simply mean that **there is no limit to its values**.

**What are the 7 indeterminate forms of infinity? ›**

**Indeterminate Form**

- Infinity over Infinity.
- Infinity Minus Infinity.
- Zero over Zero.
- Zero Times Infinity.
- One to the Power of Infinity.

**How many infinities are there? ›**

The set of real numbers (numbers that live on the number line) is the first example of a set that is larger than the set of natural numbers—it is 'uncountably infinite'. There is more than one 'infinity'—in fact, there are **infinitely-many** infinities, each one larger than before!

**Is Aleph bigger than infinity? ›**

Aleph is the first letter of the Hebrew alphabet, and **aleph-null is the first smallest infinity**.

**What is the highest level of infinity? ›**

The Absolute Infinite (symbol: Ω) is an extension of the idea of infinity proposed by mathematician Georg Cantor. It can be thought of as a number that is bigger than any other conceivable or inconceivable quantity, either finite or transfinite.

**Is there a difference between infinity and positive infinity? ›**

Numbers approach negative infinity as they move left on a number line and positive infinity as they move right on a number line. Negative infinity is always less than any number, and **positive infinity is greater than any number**.

**What is bigger than infinity? ›**

Infinity is endless and therefore cannot be reached. The expressions “beyond infinity” or “to infinity and beyond" simply represent **limitless possibilities**.

### What is infinity over infinity called? ›

BUT in Mathematics infinity divided by infinity is actually **undefined**.

**What is 2 infinity infinity? ›**

**Multiplying infinity by infinity will result in infinity**.

**Why ∞ is not a number? ›**

Infinity is not a number, but if it were, it would be the largest number. Of course, such a largest number does not exist in a strict sense: **if some number n n n were the largest number, then n + 1 n+1 n+1 would be even larger, leading to a contradiction**. Hence infinity is a concept rather than a number.

**Is infinity plus 1 bigger than infinity? ›**

Yet even this relatively modest version of infinity has many bizarre properties, including being so vast that it remains the same, no matter how big a number is added to it (including another infinity). So **infinity plus one is still infinity**.

**What is 5 divided infinity? ›**

Answer: Evaluate the value of 5 divided by infinity. Hence, 5 divided by infinity is **0**. Alternatively, we know that any number divided by 5 is equal to 0.

**Is ∞ ∞ an indeterminate form? ›**

Product: ∞ ⋅ ∞ \infty \cdot \infty ∞⋅∞ is **not indeterminate**; the limit is ∞ \infty ∞.

**Are there only 7 indeterminate forms? ›**

**There are seven indeterminate forms** which are typically considered in the literature: , and it is easy to construct similar examples for which the limit is any particular value.

**What is bigger than Omega? ›**

**ABSOLUTE INFINITY** !!! This is the smallest ordinal number after "omega". Informally we can think of this as infinity plus one.

**What is the smallest infinity? ›**

The smallest version of infinity is **aleph 0** (or aleph zero) which is equal to the sum of all the integers. Aleph 1 is 2 to the power of aleph 0. There is no mathematical concept of the largest infinite number.

**What is Aleph in math? ›**

In mathematics, particularly in set theory, the aleph numbers are **a sequence of numbers used to represent the cardinality (or size) of infinite sets that can be well-ordered**. They were introduced by the mathematician Georg Cantor and are named after the symbol he used to denote them, the Semitic letter aleph ( ).

### What comes before absolute infinity? ›

Answer and Explanation: **There is no number before infinity**. It is possible to represent infinity minus one as a mathematical expression, but it does not actually equal anything or have any real mathematical value.

**What is Aleph 1? ›**

Aleph-1 is **the set theory symbol for the smallest infinite set larger than**. **(Aleph-0)**, which in turn is equal to the cardinal number of the set of countable ordinal numbers.

**Can infinity have a beginning? ›**

As the number line is endless, it is infinite, but as it also **does not have a beginning**. As such, in the number line, we represent two types of infinity: a positive infinity, indicating that the line has no end, and a negative infinity, which indicates that the line representing the numbers has no beginning either.

**Is Omega the smallest infinity? ›**

Yes, there is a proof that **ω is the smallest infinite cardinality**. It all goes back to some very precise definitions.

**Is infinity bigger than Googolplexian? ›**

Is Googolplex bigger than infinity? Nope. A googolplex is a number, a very big number, but one that is fixed in size. **Infinity is more of a concept than a number**.

**Is Omega the largest infinity? ›**

Is the omega bigger than infinite? Since you tagged this mathematics, you're presumably asking about ω, the number symbolized by the lowercase letter omega. The primary definition of ω is as the first transfinite* ordinal number. This number is infinite, but **it's the smallest infinity**.

**What is infinity beyond? ›**

Infinity is endless and therefore cannot be reached. Thus the expression "To infinity and beyond!" would simply represent **limitless possibilities**.

**Is God absolute infinity? ›**

**The fundamental principle of the philosophy of Benedict de Spinoza is the necessary and absolute infinity of God**. He defined God as an absolutely infinite being.

**Can infinity have an end? ›**

**Infinity has no end**

So we imagine traveling on and on, trying hard to get there, but that is not actually infinity. So don't think like that (it just hurts your brain!). Just think "endless", or "boundless". If there is no reason something should stop, then it is infinite.

**What is 3 divided by infinity? ›**

Any number divided by infinity is equal to **0**.

### Is there a difference between infinity and absolute infinity? ›

Infinity is something which is considered never ending but may end. Lets go to the sky! To infinity and beyond! **Absolute infinity is omnidimensional**.

**What is beyond infinity? ›**

Infinity is the idea or concept of something that has no end. Infinity is endless and therefore cannot be reached. The expressions “beyond infinity” or “to infinity and beyond" simply represent **limitless possibilities**.

**What is 0 multiplied by infinity? ›**

Multiplying 0 by infinity is the equivalent of **0/0** which is undefined.

**What is 5 over infinity? ›**

Hence, 5 divided by infinity is **0**. Alternatively, we know that any number divided by 5 is equal to 0. Therefore, 5 divided by 0 is 0.

**Can infinity be divided by 0? ›**

Answer and Explanation: **We cannot really divide infinity by zero** because infinity is not a number and we do not divide by zero.

**Is anything bigger than absolute infinity? ›**

Answer: The concept of infinity varies accordingly. Mathematically, if we see infinity is the unimaginable end of the number line. As no number is imagined beyond it(no real number is larger than infinity).

**Why is infinity a paradox? ›**

The paradox arises from one of the most mind-bending concepts in math: infinity. **Infinity feels like a number, yet it doesn't behave like one**. You can add or subtract any finite number to infinity and the result is still the same infinity you started with.

**Is Pi bigger than infinity? ›**

Q: Is pi bigger than infinity? A: **No**. Infinity is a value greater than any countable number.

**Can you double infinity? ›**

The Double Infinity Meaning

**The double infinity symbol is often used in jewelry and is very popular in the tattoo world**. It is created by intertwining two infinity symbols together, one on top of the other. These two infinity symbols together signify perfection.

**Does infinity ever end? ›**

**Infinity has no end**

So we imagine traveling on and on, trying hard to get there, but that is not actually infinity. So don't think like that (it just hurts your brain!). Just think "endless", or "boundless". If there is no reason something should stop, then it is infinite.