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Posts about mathematics

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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