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May 16th, 2016, 10:47 AM  #1 
Senior Member Joined: Oct 2015 From: Antarctica Posts: 128 Thanks: 0  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. 
May 16th, 2016, 11:20 AM  #2 
Math Team Joined: Dec 2013 From: Colombia Posts: 7,674 Thanks: 2654 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. 
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!


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guarantee, numbers, sequences, transcendental 
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