Associated Topics || Dr. Math Home || Search Dr. Math

### Transcendental Numbers

```
Date: 5/31/96 at 14:24:27
From: David Lewis
Subject: Transcendental Numbers

I am doing a school project and one of the questions is:

Joesph Louville's candidate for a transcendental number was
a=.110001000000000000000001..... Where is the next 1?

My partner and I have looked all over and have not been able to come
up with the answer.  Can you help us?
```

```
Date: 5/31/96 at 15:21:19
From: Doctor Darrin
Subject: Re: Transcendental Numbers

The next 1 would be in the 120th place after the decimal. Liouville
proved that numbers of the form:

1/n+1/n^2+1/n^6+1/n^24+1/n^120+1/n^720+...

are trancendental (where the exponents are factorials (usually written
with an exclamation point):
1!=1
2!=2*1=2
3!=3*2*1=6
4!=4*3*2*1=24
5!=5*4*3*2*1=120
...
n!=n*(n-1)*(n-2)*...*2*1

The number that you gave is the case of the formula above where n=10.
So you would have 1 in the 1st, 2nd, 6th, 24th, 120th, 720th, 5040th,
etc, places and zeroes everywhere else.

-Doctor Darrin,  The Math Forum
Check out our web site!  http://mathforum.org/dr.math/
```
Associated Topics:
High School Transcendental Numbers

Search the Dr. Math Library:

 Find items containing (put spaces between keywords):   Click only once for faster results: [ Choose "whole words" when searching for a word like age.] all keywords, in any order at least one, that exact phrase parts of words whole words

Submit your own question to Dr. Math
Math Forum Home || Math Library || Quick Reference || Math Forum Search