Binary logic

[Art Vandelay]

Then don't do it that way. That's like saying a pencil's not helpful for working sums 'cuz I can't keep the columns straight when I write on my backside. So I quit using it that way. For arithmetic, anyway.

The point is that there's a perfectly natural way of using the hands in base 10, and a horribly awkward one in base 2. So of course the former was used.

 

[rockoon]

The point is that there's a perfectly natural way of using the hands in base 10, and a horribly awkward one in base 2. So of course the former was used.

Is it awkward because of something fundamental, or because you simply arent used to it?

I find myself doing quite a bit of binary math on my hands.

 

[Basilio]

Sorry, Im just a sucker for a Rolls, but I would miss good disc brakes

basilio

 

[Art Vandelay]

Is it awkward because of something fundamental, or because you simply arent used to it?

Our fingers aren't designed to be moved independently.

I find myself doing quite a bit of binary math on my hands.

How? Do you use both hands?

 

[Big Al]

Well, with the "new ruthlessly drives out the old" paradigm*, why don't we all do binary arithmetic instead of decimal?

Well, just imagine giving someone your phone number in binary. Or punching it out on your (now much smaller) keypad. And I'm just musing on the joys of using exponential notation in binary for very large numbers.

Fractions are useful for much algebra, and many can be converted into decimal with a glance: 9/20 easily mentally converts to .45. However, 1001/10101 doesn't quite have that same gestalt property, and it's not just a question of familliarity. A relatively large radix has the advantage of providing readily recognised distinct symbols, which is the real advantage of computer programmers working in hex: 4F is clearer than 1001111.

*Do I win the Buzzword Bingo?

 

[DavidS]

The point is that there's a perfectly natural way of using the hands in base 10, and a horribly awkward one in base 2. So of course the former was used.

My points are

(a) the existence of a horribly awkward technique is not evidence of the absence of a perfectly natural one. Consider, for example, various techniques for using a barometer to measure the height of a building.

(b) numerical representation, counting,and arithmetic are but one application of binary logic.

 

[rockoon]

Our fingers aren't designed to be moved independently.

Well I do it all the time and you do it too in your 'natural' base-10 counting-on-the-fingers method.

How? Do you use both hands?

Each hand holds a value between 0 and 31.

For adding or subtracting, I place one hand on top of the other and iterate across.

Both hands combined hold a value between 0 and 1023.

Its only awkward if you insist on converting to and from base-10 but thats a little dog chasing the tail reasoning.

 

[Almo]

I'm not convinced that the number of fingers we have has anything to do with our current use of base 10, it is simply coincidence.

Last I heard, we use 10 cause Pythagoras was a Numerologist, and 1 + 2 + 3 + 4 = 10 had some mystic signifigance.

 

[Big Al]

Last I heard, we use 10 cause Pythagoras was a Numerologist, and 1 + 2 + 3 + 4 = 10 had some mystic signifigance.

Yet it was Arab mathematicians who introduced decimal arithmetic to Europe. We use their notation. The words zero, algebra and algorithm are all derived from Arabic.

It's certainly caught on - decimal arithmetic is used around the world, even in historically isolated countries like Japan. There does seem to be something specially attractive about it.

 

[MortFurd]

Positional noation is the key to the whole thing, and positional notaion happened to have been invented by folks who used base ten. The notational form caught on for its praticality, and the folks who started using it went along with base ten because they came together.

Here's the catch to positional notation:
10 can mean different things depending on the base.
10 base two is two.
10 base ten is ten.
10 base sixteen (hexadecimal) is sixteen.
10 base twelve is twelve.

It doesn't really matter what base you use. You just need the same number of symbols as your base.

 

[Cuddles]

In addition to MortFurd's point, I think another important point is that most of us don't count on our fingers. While apparently a few people in this thread still do, most people stop doing this some time in primary school, since not only are large numbers hard to count on fingers, but writing things down can make it much faster and simpler. Since virtually all advance in maths have been accompanied by advances in notation (such as positional notation and a symbol for zero), it seems rather odd to suggest that people were working these things out on their hands first , and then writing it down. It also seems odd to suggest that mathematical advances have been driven by a technique used primarily by small children, especially since at the time children would probably not have been taught maths at all.

 

[Big Al]

Positional noation is the key to the whole thing, and positional notaion happened to have been invented by folks who used base ten. The notational form caught on for its praticality, and the folks who started using it went along with base ten because they came together.

Here's the catch to positional notation:
10 can mean different things depending on the base.
10 base two is two.
10 base ten is ten.
10 base sixteen (hexadecimal) is sixteen.
10 base twelve is twelve.

It doesn't really matter what base you use. You just need the same number of symbols as your base.

However, the lower the base, the greater the number of digits in a given quantity. Imagine your new friend's phone number is 164-355-41709. In binary, that would be 10100100-101100011-1010001011101101.

A little hard to scribble on the back of your hand, and very easy to make a mistake! Also, imagine reading it out to someone!

The higher the base, the greater the number of symbols needed. This certainly trims down the length of numbers (in hex, the above example is A4-163-A2ED), and it's easier to memorise correctly.

However, 10 is a handily sized base for the sorts of quantities with which we're familiar in our everyday lives. (Well, OK, I'm an electronics engineer, so I use exponential notation a lot, and I use hex frequently, but that's not everyday life for most people).

I think that's why decimal notation has caught on so well. It's not uniquely useful (duodecimal and hexadecimal would be easy for kids to learn), but I do feel it's more useful than binary in everyday life.

If we dealt routinely in quintillions rather than tens, hundreds and thousands, decimal would be cumbersome and another radix, say base 1000, would be more useful.

Binary's useful if you deal in 2's, like computers. But we don't as a matter of course, and the number lengths for humans are cumbersome and prone to copying error. That's just why octal and hex were invented by computer programmers, rather than them using straight binary.

A computer doesn't see FE - its sees 11111110. However, programmers needed to add a level of complexity to make the numbers manageable.

 

[cyborg]

If we dealt routinely in quintillions rather than tens, hundreds and thousands, decimal would be cumbersome and another radix, say base 1000, would be more useful.

Except then the sheer number of symbols makes the base cumbersome. (Just look at the problems the modern hieroglyphic based languages have with things like the keyboard - hell, just the problems of learning all the symbols).

 

[Big Al]

Except then the sheer number of symbols makes the base cumbersome. (Just look at the problems the modern hieroglyphic based languages have with things like the keyboard - hell, just the problems of learning all the symbols).

Yes, the sheer number of symbols would get to be an inconvenience. Try counting in traditional Japanese (check out http://greggman.com/edit/editheadlines/2003-01-26.htm)!

The name of the number depends on the type of thing you're counting - and there are hundreds of categories of "counting things", each with its own number-words.

 

[MortFurd]

Yes, the sheer number of symbols would get to be an inconvenience. Try counting in traditional Japanese (check out http://greggman.com/edit/editheadlines/2003-01-26.htm)!

The name of the number depends on the type of thing you're counting - and there are hundreds of categories of "counting things", each with its own number-words.

Worse than French. I gave up on French when I was learning the numbers. I got as far 70, and realized the whole business was insane.

It goes sixty eight, sixty nine, sixty ten and one, sixty twelve, sixty thirteen, etc.

I saw the stuff about numbers in Japanese here a while back on TV. I really hope they have a "numbers for math" way to express numbers.

 

[Big Al]

It goes sixty eight, sixty nine, sixty ten and one, sixty twelve, sixty thirteen, etc.

Then you get four-twenties, for the eighties and four-twenties ten to nineteen for the nineties! So ninety-nine is quatre-vingts dix-neuf.

I saw the stuff about numbers in Japanese here a while back on TV. I really hope they have a "numbers for math" way to express numbers.

They do: Hitotsu, futotsu, mittsu... will take you up to a million or so, but it's still not as straight forward as Arabic decimal. http://www.thejapanesepage.com/no1.htm refers.

However, it's no use if you want to order ten pencils, six plates, a dozen packets of screws for 9:10 on February 9, for the fourth floor of building 20. All those have different counting systems.

 

[MortFurd]

Do you know how (or why) the Japanese developed such a cumbersome system?

 

[drkitten]

Do you know how (or why) the Japanese developed such a cumbersome system?

The hitotsu, futatsu, mittsu,... system was more or less home-grown; the ichi, ni, san system is almost a direct transplant from the Chinese language and culture.

This kind of syncrenism isn't that uncommon; for example, English uses Anglo-Saxon (germanic) words for counting in small numbers -- one, two, three, ... but then switches to Latin roots for higher numbers (million, billion, trillion). Basically, the Anglo-Saxons never had to count up much higher than 1000, so they never developed numbers for that. (Frankly, I'm impressed that the Anglo-Saxons had a word for "thousand.") The various native groups in the Americas tend to have adopted Spanish number systems for their higher numbers; in Quechua, for example, I believe they have a native system for numbers up to 99 but then use Spanish for 100+. (Although you might want to check on that before citing it.)

One key reason for this kind of synchrenism is that the new numbers usually arrive with new technology.

Having said this, it is a little unusual -- but not unheard of -- for the new system and the old system to persist side by side. But again English has the same pattern... we can count events as happening once, twice, thrice (although few people know thrice any more), or we can count them as happening one time, two times, three times, four times. I believe that there used to be an extension of "once, twice, thrice,...." to four, five, six, and so forth, but only a few specialist linguists know the terms any more.

 

[rockoon]

Obviously we all (hopefully) memorized single digit number operations, even multiplication.

But at least when I use my fingers, each finger does not represent 1. Each finger is a digit (aha.. get it?) in a different position so has a different value. A hand holds a 5 digit number, not a number up to 5.

Only people schooled in very basic mathematics would assume that the fingers of the hand could only represent 6 (0 to 5) unique values. Yes thats a vague dig at somebody in return for his.

32 unique values is trivial, and even more (243) is possible with base 3 (unextended, half-extended, and fully-extended .. or perhaps on a surface you would have unextended, extended, and extended and touching)

With base-3, two hands can hold a value between 0 and 59048. I personally don't use a base-3 system but it is certainly possible.

Why not go to paper and pencil and use base-10? Some operations are non-trivial in base-10, such as bitwise boolean operations such as And, Or, Xor, and so forth.

As far as arguments that the fingers arent meant to move individualy, tell that to people who play musical instruments such as guitar, piano, saxaphone, flute, etc.. its just a matter of practice just like everything else.

 

[Art Vandelay]

Well, with the "temporally proximate highly mission focused supplants outmoded" paradigm

I've fixed your post to be more buzzword-inclusive.

(a) the existence of a horribly awkward technique is not evidence of the absence of a perfectly natural one.

Well, yeah, but you should be able to tell me what they are.

(b) numerical representation, counting,and arithmetic are but one application of binary logic.

Yes, but that was what was being discussed.

Well I do it all the time and you do it too in your 'natural' base-10 counting-on-the-fingers method.

In base ten, the fingers aren't moved independently. And Inever really used my fingers for counting.

Last I heard, we use 10 cause Pythagoras was a Numerologist, and 1 + 2 + 3 + 4 = 10 had some mystic signifigance.

Close, but actually, it comes from bowling.

Positional noation is the key to the whole thing, and positional notaion happened to have been invented by folks who used base ten.

As I said before, binary notation doesn't work very well without positional notation, and number systems (including base 10) developed long before positional notation.

The name of the number depends on the type of thing you're counting - and there are hundreds of categories of "counting things", each with its own number-words.

That actually would help in math. We use capital, bold, greek, script, etc., to denote different types of variables. Having a few hundred different categories would be useful.

Then you get four-twenties, for the eighties and four-twenties ten to nineteen for the nineties! So ninety-nine is quatre-vingts dix-neuf.

I think that a more appropriate translation would be "four score and 19".

Having said this, it is a little unusual -- but not unheard of -- for the new system and the old system to persist side by side. But again English has the same pattern... we can count events as happening once, twice, thrice (although few people know thrice any more), or we can count them as happening one time, two times, three times, four times. I believe that there used to be an extension of "once, twice, thrice,...." to four, five, six, and so forth, but only a few specialist linguists know the terms any more.

http://www.askoxford.com/asktheexperts/faq/aboutwords/once
There are also several other systems:
one, two, three (cardinal)
first, second, third... (ordinal)
primary, secondary, tertiary... (Latin)
mono, bi, tri (Greek)