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Publicaciones sobre 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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