Stupid Numbers, Fun Irrationals

I promised in a previous post (Does pi follow a pattern?) that I would have more to say in a future post about what I call “fun” irrational numbers; Numbers that one of my professors once called “stupid.”  Here it is:

A fun irrational is an irrational number whose decimal expansion follows a nice pattern.  Here is the example that I gave previously:

0.2202002000200002000002000000200…

Irrationals are easy to make.  You just have to be sure that the decimal expansion doesn’t terminate (that it goes on forever) and that it doesn’t ever turn into a repeating block of digits.

Let me first give you some examples of rational numbers:

0.43789238972359872234324333333333333333333333333333333…

The one above starts out messy, but then settles down into nice repeating 3’s.  If the 3’s go on like that forever, then this is a bonefide rational number.  Here is another:

0.25825894759827594956349563495634956349563495634956349563…

My point here is that the repeating part doesn’t have to be a single digit.  It can be a whole bunch of them (in this case 49563).

So you can see that

0.2202002000200002000002000000200…

is irrational, because it never turns into a block of repeating digits.

Here is my newest recipe for making irrational numbers:  Start with your favorite irrational (mine is pi).  Now write it out as a decimal expansion:

pi = 3.1415926535…

Make a new irrational (call it mu) by starting just to the right of the decimal point.  We have a 1 there in pi.  In our new irrational we will put that many zeros followed by a 1:

mu = 0.01….

Next in pi we have a 4, so let’s put 4 zeros next in mu followed by a 1:

mu = 0.0100001…

Next will be 1 zero followed by a 1 and so on:

mu = 0.01000010100000100000000100100000010000010001000001…

I should say that I prefer to think of the mu I get as being written here in base 2.  It seems to make more sense.  Of course you can think of your mu as being written in base 10 here.  Then our two mu’s would be different, but both irrational.

That’s all for tonight.

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