Posts about mathematics

2013-10-04 13:57

Hardcore Finger Counting

Earlier today, @elnomoteta mentioned in twitter that if you count with your fingers you are in trouble, because at least you have overflows.

That got me thinking. Not about whether there is an overflow, since there is always an overflow, even if you count by electron quantum states using the whole universe, because infinity, dude, but about how high you can finger-count.

Sure, the naïve answer is ten, but that's trivial to improve. For example, here is a very simple system to count to 99 but even that is very simplistic.

If you are a computer nerd, you may think you are clever by now saying \(2^{10}-1\) but really, how unimaginative is that? It's unimaginative enough that it has its own wikipedia page.

One thing I do (and I recognize it as one of my most annoying traits) is to consider unorthodox answers to questions. Because often they will show that the one asking the question has only a very vague idea of what he is asking, and exposes a ton of unexpressed assumptions.

So, computer geek, \(2^{10}-1\), that is 1023. Congratulations, you have done much worse than the Venerable Bede, who in 710AD described in De Computo vel Loguela per Gestum Digitorum a system to express numbers up to 9999 using both hands.

So, let's think about the unexpressed assumptions here.

Is a finger a bit?

Hell no. A finger is a finger. Sure, it can express a bit, but it can also (in some cases) express more. For example, I have 6 fingers I can bend independently in more than one place (thumb, index, pinky).

So, I could use those to have a ternary digit (if you pardon the pun), and count to \(3^6 2^4-1\) (or 11663)

Is finger-counting just about fingers?
If we consider it hand counting instead it's much better. For example, I could hold each hand palm-up or palm-down for 2 extra bits. That's \(3^6 2^6-1\) (or 46655)
Is finger-order relevant?
So, suppose I put my left hand to the right of my right hand. Since I can tell which hand is which, because fingers are not all the same, I can count that as an extra bit, counting to 93311.
How long can I take before I show you the number?
I could say: "if a fingernail is long, that's a 3 (or a 4) depending on which finger". Sure, it will then take me days to express a number, but I just raised the number I can count to, using my fingers, to a really large number I won't bother calculating (2985983)
Do I have to keep my fingers still?
Because with one finger I could tap morse code for any number given patience, a hard surface, a resilient finger and knowledge of morse.
Can't I just keep on adding bits?
Of course. I could bite on the back of my left hand and leave a mark. I can use different hand positions other than palm up/down and straight/crossed. I could tattoo a number on the palm. I could express a URL to a site that contains a number. This is because the amount of information on a person's hand is huge.

So, sure, you can count to ten, or 99, or 1023, or 2985983. The tradeoff is, the higher your system goes, the harder it is to read, and the more previously agreed knowledge you need between the one expressing the number and the one reading it.

That's why you still count with your fingers just to 10. Because it's obvious.

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