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.

An alien is thinking of a number…

…and you have to guess what it is.

You were captured by a malevolent alien, who would really like to eat you for dinner. He told you his name, but since it is unpronounceable to humans, you like to call him Evil Eddie. Evil Eddie has a morbid sense of fairness and so, before roasting you with an apple stuck in your mouth and serving you with a side of glowing, purple, French-cut string beans, has locked you in a chamber carved out of the bedrock of a strange world. Before the polycarbide door slides shut with a hiss, Eddie tells you that you he is thinking of a number. If you can guess it, he will reluctantly release you. If not, well…let’s just hope that you can guess the number.

As you look around the cavern you see, in the dim and flickering light (does it come from torches? You don’t see any.), an enormous stone bowl, filled to the brim with glass marbles. How many are there? Thousands at least. Tens of thousands, perhaps. On the other side of the chamber is an identical bowl, but this one is empty. In the middle of the room, hanging from the ceiling is a rope. You hear Eddie’s voice thundering into the room “when you think that you know the number, put that many marbles in the empty bowl. Then pull the rope.”

Fortunately you have a friend. In an adjacent chamber is a benevolent alien. You don’t know his name, so you decide to call him Manvel. By some contrived plot device that the author has not taken the time to think of, you know that Manvel:

1) can read Evil Eddie’s mind (hence, knows the number),

2) is going to try to transmit this number to you, and

3) is going to do so by tapping out the digits (through the wall) in some base (could be base 10 (that’s what we normally use), could be base 2 (called binary. We think of computers as using base 2), could be any whole number base.

You listen and hear …tap… long pause …tap tap tap tap tap… long pause …tap tap tap… then silence.

You have the digits! They are 153. As stated above, you now know the number is:

You just don’t know what is. (I’ve glossed over the issue of big endian versus little endian order, but we are assuming, by aforementioned plot device, that big endian order has been used by Manvel).

So…how many marbles do you put in the bowl?

to be continued…

Synchronicity

I first heard the word synchronicity when I was sitting in an airport, waiting for a plane, working on this math problem (for fun) and over-hearing a man’s cell phone conversation. I thought that it was a good word for this phenomenon. Here it is:

Have you noticed that 3 times 7 is 21 and if you expand 7 base 3 it is 21. That is:

This seems like an interesting coincidence. Could it be that this happens for any other numbers? Maybe it happens infinitely often, maybe never. Asking such questions is where the adventure begins. Of course we are implicitly working base 10 here. That is, in the first line the string of numerals “21″ is interpreted as a base 10 number. In the second one it is (explicitly) interpreted base 3. You could allow 10 to be replace by other bases also.

If we restrict our attention to base 10 (in the first line), I claim that, by exhaustive computer search, there are only five other instances of this phenomenon. They are:

and

and

and

and

and