The Golden Number and Countable Infinities

2017-07-05

I came across something disturbing today, although it turned out to be a false alarm. But by the point I realized that, I had already written most of this post, so you get it anyway. Also, it's about mathematics, so probably most people would not have been affected anyway.

It its well known that there are lots of place value number systems. Famous ones include unary (tally marks), binary, and decimal (the normal one). These are based on integers, in this case 1, 2, and 10. In decimal, for example, a number like 234.5 is interpreted to mean:

234.5 = 2 * 10^2 + 3 * 10^1 + 4 * 10^0 + 5 * 10^-1
      = 2 * 100 + 3 * 10 + 4 * 1 + 5 * 0.1
      = 200 + 30 + 4 + 0.5

Similarly, in binary, the number 1101.01 means

1101.01 = 1 * 2^3 + 1 * 2^2 + 0 * 2^1 + 1 * 2^0 + 0 * 2^-1 + 1 * 2^-2
        = 1 * 8 + 1 * 4 + 0 * 2 + 1 * 1 + 0 * 0.5 + 1 * 0.25
        = 8 + 4 + 1 + 0.25

You can even use use non-integer numbers as bases. For example, in base 2.5, the number 121.2 would mean

121.2 = 1 * 2.5^2 + 2 * 2.5^1 + 1 * 2.5^0 + 2 * 2.5^-1
      = 1 * 6.25 + 2 * 2.5 + 1 * 1 + 2 * 0.4
      = 6.25 + 5 + 1 + 0.8

However, something really odd seems to happen if you use the golden number as the base. For those of you who haven't heard the term, the golden number (φ) is (1 + sqrt(5)) / 2. It has a good number of interesting mathematical properties (e.g. 1/φ = 1 + φ), and is useful for solving some strange problems (like efficiently packing an unknown number of objects). Probably because of this, it shows up in nature a lot, and has spawned a lot of weird Internet sites. So many, in fact, that finding explanations of the mathematical significance in now difficult.

Now for the disturbing part. It seems that, if you use a golden-ratio-based numbering system, it is possible to encode any real number in this format. I mistakenly took this to mean "in a finite number of digits", which would implied that they are countable. In turn, this would have implied that there are the same number (aleph-null) of real numbers as integers. To me, this resulted in a serious "my world is falling apart" moment.

Luckily for me, it all turned out to be a misunderstanding, and, once again, "God is in His Heaven, and all is well with the world". Math is fun.

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