Large Numbers and Infinity

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#### What's the largest number?

There is no largest number! Why? Well, 1,000,000,000 (1 billion) can't be the largest number because 1 billion + 1 is bigger - but that is true for any number you pick. You can choose any big number and I can make a bigger one just by adding 1 to it.

#### What's a googol?

A googol is a 1 with a hundred zeroes behind it. We can write a googol using exponents by saying a googol is 10^100.

The biggest named number that we know is googolplex, ten to the googol power, or (10)^(10^100). That's written as a one followed by googol zeroes.

#### How do we name large numbers?

There's some disagreement in the English language about how to name large numbers. There are two systems, the American and the English:
```                                              Scientific
American:                    Number            Notation

Thousand                             1,000        10^3
Million                          1,000,000        10^6
Billion                      1,000,000,000        10^9
Trillion                 1,000,000,000,000       10^12
Quintillion      1,000,000,000,000,000,000       10^18

English:

Thousand                             1,000        10^3
Million                          1,000,000        10^6
Thousand Million             1,000,000,000        10^9
Billion                  1,000,000,000,000       10^12
Thousand Billion     1,000,000,000,000,000       10^15
Trillion         1,000,000,000,000,000,000       10^18
```
And see: Decillion, Vigintillion, Trigintillion...

And, in particular:
Names for Large Numbers - Russ Rowlett
Large Numbers - Robert Munafo

#### What is infinity?

Infinity is not a number; it is the name for a concept. Most people have sort of an intuitive idea of what infinity is - it's a quantity that's bigger than any number. This is sort of correct, but it depends on the context in which you're using the concept of infinity (see below).

There are no numbers bigger than infinity, but that does not mean that infinity is the biggest number, because it's not a number at all. For the same reason, infinity is neither even nor odd.

The symbol for infinity looks like a number 8 lying on its side:.1

Now for the fun part! Even though infinity is not a number, it is possible for one infinite set to contain more things than another infinite set. Mathematicians divide infinite sets into two categories, countable and uncountable sets. In a countably infinite set you can 'number' the things you are counting. You can think of the set of natural numbers (numbers like 1,2,3,4,5,...) as countably infinite. The other type of infinity is uncountable, which means there are so many you can't 'number' them. An example of something that is uncountably infinite would be all the real numbers (including numbers like 2.34.. and the square root of 2, as well as all the integers and rational numbers). In fact, there are more real numbers between 0 and 1 than there are natural numbers (1,2,3,4,...) in the whole number line! See One infinity larger than another? for more information.

Lastly, please realize that the concept of infinity varies depending on what mathematical subject area you're talking about. If you're simply counting things up, then the picture looks something like this:

If you're talking about the number line, then the picture looks something like this:

If you're talking about the number of elements in a set, here's the picture:

If you're doing fancy geometry, you might have something like this (this is called Projective Geometry):

Finally, just for fun, here's another kind of geometry (Finite Projective Geometry) in which EVERY point can be called a point at infinity!2 Here the "lines" are the straight lines and the circle.

#### On the Web:

1Note: In his book A History of Mathematical Notations, Florian Cajori credits the English mathematician, John Wallis, with inventing the modern notation for Infinity, citing Wallis' works Arithmetic infinitorum (1655) and De Sectionibus Conicis: "Cum enim primus terminus in serie Primanorum sit 0, primus terminus in serie reciproca erit vel infinitus." It has been conjectured that Wallis, who was a classical scholar, adopted this sign from the late Roman symbol for 1000.

2Note: When viewed as the projective completion of an affine plane, any line can be considered "the line at infinity," and the points on it are "the points at infinity."