A number's perspective

A number's perspective, inspired by Edwin A. Abbott's book called Flatland.

You've probably already have heard about binary numbers or hexadecimal numbers. The most common numbers used in this world by humans are base ten, presumably for obvious reasons related to human anatomy.  

Binary numbers are numbers built from powers of two like how base ten (decimal) numbers are built from powers of ten.  They are traditionally written with two symbols such as zero and one.  In base ten, a number represented as 1142 means 2 units, 4 tens, 1 hundred, and one thousand, all added together.  Especially if you've ever dealt with scientific notation, we see that thousands, hundreds, tens, and units (ones) are all powers of ten: a thousand is ten to the third power, a hundred is ten to the second power, ten is ten to the first power, and one is ten to the zero power.

The overall pattern remains the same in other bases.     When you see something like 1011 and if we want to convert binary to base ten, we must first recall two to various powers:

2^0, two to the zero power, is one, 2^1=2, 2^2=4, 2^3=8.

1011 = 1 + 1 x 2^1 + 0 x 2^2 + 1 x 2^3 = 1+2+0+8 = 11 in base ten.

For example, if we were talking about any base z, the meaning is still implied:

(abcd) in base z would be d + c x z^1 + b x z^2 + a x z^3.

What about having other types of numbers as bases?  Some numbers are called rational and others irrational.  The Greeks might have thought a number that repeats a sequence ad infinitum (or, in simpler cases a number whose digits terminate entirely) is more rational than a number whose decimal expansion involves a tail in which no set sequence of digits repeat, i.e., an irrational number.

There are some really famous constants that are irrational: pi, the square root of two, the golden ratio which is somewhat close to the ratio 16:9, and maybe even Euler's constant e.

There is one fact I want to focus on.  If we are not told what the base is, it is pointed to by the representation 10.  This is so because of the expression mentioned earlier.  In the instance of 10, we have 10 = 0 + 1 x z^1 which simplifies to the desired formula for the base z, namely 10 = z which may look nicer if we say z = 10.  In base ten, lo and behold that 10 is the representation, base ten, of ten.  But it is also true in binary.   10 in binary means 0 + 1 x 2^1 which simplifies to 10 = 2 and 2 is the base in binary.

Let's say a few words about base pi.  In accordance with the above paragraph, in base pi, 10 = pi.  In base ten, 3.14 approximates pi but since pi is irrational, in base ten, it's digits will never repeat a set sequence, i.e., it's expansion in base ten is not repeating, in contrast to the rational number 1/3, which involves repeating 3 "forever."

If irrationality means "has no repeating sequence of digits," what about pi's representation in base pi?  In base pi, pi is represented by 10 as we saw earlier.

What's so special about this? Well, 10(base pi) is a terminating representation.  Numbers with terminating representation are called rational.

Pi is irrational in base ten and it is rational in base pi.

So there must be something other that what I've stated that truly characterizes what the difference between a rational and an irrational number is.