The Largest Estimated Number of Things

Mathematical model

Using "scientific notation," we can describe an arbitrarily large (or arbitrarily small) number.
n = b×10^d, where b is an integer between 0 and 10, and d is a positive (or negative) integer.

Implementation

Beyond a certain size, there are no names for large numbers, so you can create a new number and call it whatever you like. For example: mybignum = 10^10101 .

Extension

Use the following exercise as a completely open-ended question. In the suggested answers, explore some possible outcomes for this exercise.

Exercise 1.7.1: “The biggest number”

Write a short essay on the subject “The Biggest Number in the World.”

Suggested answer to Exercise 1.7.1: “The biggest number”

On 27 January 1998, the team of Roland Clarkson, George Woltman, and Scott Kurowski discovered a new record prime: 2^3021377 - 1.

The Mathematica function PrimeQ tests an integer for primality. For example, you can show that not all numbers of the form 2^2^n + 1 are prime.

Table[{n, 2^2^n + 1, PrimeQ[2^2^n + 1]}, {n, 0, 6}]//TableForm

0 3 True
1 5 True
2 17 True
3 257 True
4 65537 True
5 4294967297 False
6 18446744073709551617 False

The result shows that 2^2^5 + 1 and 2^2^6 + 1 are not primes. Students can see how far Mathematica will go in testing a large number, and how much time it takes.

A googol is defined as 10^100 . It is possible to make conjectures about things you cannot actually construct. For example, the web site: www.googol.com describes a googolhedron. What can you say about such a figure? Although you would have a hard time constructing a googolhedron, you can say it could not satisfy the criteria for being a regular (Platonic) solid.

For calculations involving large numbers of things that may actually exist, a web site maintained by the National Aeronautics and Space Administration an excellent resource. Here is an example.
“How many galaxies are there in the universe? We do not know exactly. Within the part of the universe we can observe there seem to be at least 100 billion, but this could be an under estimate if you include dwarf galaxies that are too far away. In the entire universe, if the universe is closed, there may not be more than perhaps a few trillion galaxies. If the universe is infinite, then taking our portion as typical, there are an infinite number of galaxies!”

An advanced student can readily explain the difference between “the largest number” and “the largest number that has a name.” When children ask, “What is the largest number?” they should be congratulated and encouraged to think of what adding 1 would do to the largest number.

Here is a list of number names:

million 1,000,000
billion 1,000,000,000
trillion 1,000,000,000,000
quadrillion 1,000,000,000,000,000
quintillion 1,000,000,000,000,000,000
sextillion 1,000,000,000,000,000,000,000
septillion 1,000,000,000,000,000,000,000,000
octillion 1,000,000,000,000,000,000,000,000,000
nonillion 1,000,000,000,000,000,000,000,000,000,000
decillion 1,000,000,000,000,000,000,000,000,000,000,000
undecillion (1 followed by 36 zeros)
duodecillion (1 followed by 39 zeros)
tredecillion (1 followed by 42 zeros)
quattuordecillion (1 followed by 45 zeros)
quindecillion (1 followed by 48 zeros)
sexdecillion (1 followed by 51 zeros)
septendecillion (1 followed by 54 zeros)
octodecillion (1 followed by 57 zeros)
novemdecillion (1 followed by 60 zeros)
vigintillion (1 followed by 63 zeros)

From , by Philip Davis.

For a discussion of the difference between the U.S. and the British billion, see the web site Ask Dr. Math.

The system used above, which was evidently borrowed from the French, is to start with a root value of 1,000 and then add “-llions” as sets of three zeros. Thus a “quadrillion” takes 1,000 and then adds twelve zeros.

Created by Mathematica  (April 8, 2003)