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Ralsina.Me — Roberto Alsina's website

You know more math than you think: non-decimal numbers

Yes, you do. If you are a frequent reader of this blog, then you probably already know about binary numbers, hexadecimal numbers, and sundry non-decimal numbers. You know, the kind we nerds know about. The ones that make us confuse thanksgiving and christmas because oct(31) == dec(25).

But how about nor­mal peo­ple (or as I like to call them: peo­ple)? Well, they may look at you con­fus­ed­ly if you tell them that they use way more ex­ot­ic things ev­ery day.

Let's start with the time. When you say "it's 10:30? well, that's a base-60 num­ber.

If we add days, it gets harder, because days are base-24. So "2 days, 10 hours and 30 minutes" is just a difficult way to say 2*24*60 + 10*60 +30 minutes. It's a numerical system with two different bases.

Sure, it does­n't do the cutesy thing hex does of hav­ing ex­tra sym­bol­s, like A mean­ing 10, but it's ex­act­ly that, ex­cept 20 is writ­ten "20" or "8P­M".

And how about Jan­u­ary 11th, at 5:20 PM? Well, that is al­so an­oth­er way to ex­press a num­ber of min­utes, in an even more com­pli­cat­ed mixed-base sys­tem!

January = 0 months = 0 days = 0 hours = 0 minutes
11th = 11 days = 251 hours = 15060 minutes
5PM  = 17 hours = 1020 minutes
20 = 20 minutes

Total: 16100 minutes

That way to ex­press a date us­es a mix of base 60, base 24, and base 365 (if we can, please, ig­nore leap years) or maybe base 60, base 24, base ~30 and base 12

I don't know if nu­mer­i­cal sys­tems with non-­fixed bases have a name in math­e­mat­ic­s, yet you use them, ran­dom non-­math­-per­son!

And you can even do arith­metic on them! Yes, you! You know what ex­act time it will be at "Jan 9th 2:10 + 12:15". You can even do mul­ti­-base arith­metic in your head.

And I have not men­tioned sec­onds (base 60 again), years (mul­ti­ple base 10 dig­it­s) and sec­ond frac­tion­s.

Yet, when hex and bi­na­ry are ex­plained to peo­ple in school, it's in­cred­i­bly hard to make them "get it". And once they get that if you try to ex­plain, say base-3 num­ber­s, it's con­fus­ing again.

fisadev / 2012-01-11 21:13:

y la gente se sorprende cuando uno tiene un reloj binario, como si una mancha más al tigre lo haría tanto más manchado... jeje

Roberto Alsina / 2012-01-11 23:19:


Guest / 2012-01-11 21:45:

y la "hora de internet" tenia segundos o era la "hora atomica oficial" alguien se acuerda?

Roberto Alsina / 2012-01-11 23:19:
Pablo A. Costesich / 2012-01-11 21:48:

Once you get through the concept of bases and powers (you can explain them using coins and stacks, rocks, or even fingers) explaining how a specific base works follows naturally. I think other concepts are even more complicated, like modulo arithmetic, but they become more obvious to people if you use the right analogies and strip them from all the noise.

Roberto Alsina / 2012-01-11 23:18:

I have succesfully explained octal numbers to 5-year olds using floor tiles.

patricio / 2012-01-11 23:11:

Bueno, tampoco se usa de manera correcta la base 24. Cuántas veces escuchaste que nos juntamos "a las 12:30 de la noche" (24:30 es menos común escuchar).

Roberto Alsina / 2012-01-11 23:19:

Eso es porque son unos quesos para aplicar módulo :-)

Lucio / 2012-01-20 18:55:

"Well-optimized modern definitions have unexpected advantages. They give access to material that is not (as far as we know) reflected in the physical world. A really “good” definition often has logical consequences that are unanticipated or counterintuitive. A great deal of modern mathematics is built on these unexpected bonuses, but they would have been rejected in the old, more scientific approach. Finally, modern definitions are more accessible to new users. Intuitions can be developed by working directly with definitions, and this is faster and more reliable than trying to contrive a link to physical experience...rank and-file mathematicians can use the new methods confidently and effectively, while success with older methods was mostly limited to the elite"
-- A Revolution in Mathematics? What Really Happened a Century Ago and Why It Matters Today

Roberto Alsina / 2012-01-20 19:10:

Sí. La idea de entender definiciones y sistemas axiomáticos como separados de la realidad y sólo sujetos a su lógica interna esunamuy buena idea.

No te ayuda a calcular cuando tenés que poner para el asado, pero es una buena idea.