My Math Forum  

Go Back   My Math Forum > High School Math Forum > Algebra

Algebra Pre-Algebra and Basic Algebra Math Forum


Reply
 
LinkBack Thread Tools Display Modes
May 16th, 2016, 10:47 AM   #1
Senior Member
 
Joined: Oct 2015
From: Antarctica

Posts: 128
Thanks: 0

Exclamation Do Transcendental Numbers Guarantee Certain Sequences?

One of my friends made that argument that, just because a number repeats with no pattern (with certain transcendental numbers being a good example of this) then any given series of digits is guaranteed to appear somewhere within the number. For example:

"Since Pi goes on forever and has no pattern, then somewhere in that string of digits lies the binary code for the Microsoft Word program."

I find it difficult to either accept or reject this statement. Obviously it would take an unreasonable amount of digits for us to actually find something useful like that, but is that assumption even true in the first place? Just because a number is transcendental and with no pattern, does that guarantee that any given string of numbers will appear somewhere in the decimal approximation, regardless of how many digits it takes to find it? Or is this just a naive assumption that isn't always true? Furthermore, how can we know this with certainty?

Last edited by John Travolski; May 16th, 2016 at 10:52 AM.
John Travolski is offline  
 
May 16th, 2016, 11:20 AM   #2
Math Team
 
Joined: Dec 2013
From: Colombia

Posts: 7,640
Thanks: 2624

Math Focus: Mainly analysis and algebra
The number
0.090090009000090000090000009...
is transcendental (I think) but clearly omits most finite strings of digits.

In fact we can prove that there are transcendental numbers that contain only the numerals 0 and 9 in their decimal expansion. There are uncountably many binary sequences - by Cantor's diagonal argument. Replace every 1 with a 9 and precede the sequence by "0." to create a decimal between zero and 1. Since the algebraic numbers are countable, not all of the numbers we created can be algebraic.

Last edited by v8archie; May 16th, 2016 at 11:38 AM.
v8archie is offline  
July 25th, 2016, 12:24 PM   #3
Senior Member
 
Joined: Oct 2015
From: Antarctica

Posts: 128
Thanks: 0

I'm sorry for such a late reply. Thanks for the explanation!
John Travolski is offline  
Reply

  My Math Forum > High School Math Forum > Algebra

Tags
guarantee, numbers, sequences, transcendental



Thread Tools
Display Modes


Similar Threads
Thread Thread Starter Forum Replies Last Post
What percentage of real numbers are transcendental? mick7 Elementary Math 6 August 21st, 2015 09:18 PM
Cardinality of transcendental numbers topsquark Abstract Algebra 54 August 17th, 2015 07:46 AM
About some transcendental numbers Jimmy57 Number Theory 1 April 5th, 2013 06:13 AM
to guarantee a win westworld Elementary Math 3 February 20th, 2012 02:04 PM
transcendental numbers john_145 Number Theory 7 February 8th, 2012 02:20 AM





Copyright © 2019 My Math Forum. All rights reserved.