Exploring the Impact of Different Numbering Systems on Scientific Calculations

[g33kski11z]

.. just a question that came up in a convo between some friends of mine..

What (if anything) would change or be different if we used a different numbering system. Would using a system based on 12 (or whatever) maybe yield different results when calculating large physics problems? (like black holes and such)

 

[Andre]

A numerical system always has 10 as base, Think about that.

 

[g33kski11z]

But why? wasn't that just arbitrarily assigned?

I think this is what I'm asking.. ..if we used this system of numbering (http://en.wikipedia.org/wiki/Duodecimal) would any equations come out different?

 

[Jimmy Snyder]

My last DIY project at home didn't go as smoothly as I had anticipated. As a result, I would consider it a big favor if everyone would switch over to a base 9 system. Thanks in advance.

 

[jobyts]

Ulam's Rose may look different.

 

[Antiphon]

Results would be the same. Bases are like coordinate systems. The values don't change, just the representation.

 

[Chi Meson]

Results would be the same. Bases are like coordinate systems. The values don't change, just the representation.

correct answer. It's almost the same as if you could get a different answer if doing a calculation in a different language.

The only real difference would be due to things like rounding errors, and the like. Under some bases, a certain decimal would round up, where in others, the same amount would round down. In the pure mathematics the values would be the same

 

[Chi Meson]

A numerical system always has 10 as base, Think about that.

I'm thinking about that...

...

eh

 

[g33kski11z]

correct answer. It's almost the same as if you could get a different answer if doing a calculation in a different language.
The only real difference would be due to things like rounding errors, and the like. Under some bases, a certain decimal would round up, where in others, the same amount would round down. In the pure mathematics the values would be the same

Thanks.. I get it now..

So, (off topic) if, for example, aliens came down, they too would use a base 10 system?

 

[Jimmy Snyder]

Thanks.. I get it now..

So, (off topic) if, for example, aliens came down, they too would use a base 10 system?

There are 10 kinds of people, those that understand binary notation, and those that don't.

 

[Ivan Seeking]

Thanks.. I get it now..

So, (off topic) if, for example, aliens came down, they too would use a base 10 system?

Maybe if they had ten fingers

I do a lot of work in base 16 [Hexadecimal]. This is common for some types of computer systems. My theory is that the fathers of industrial computers had sixteen fingers.

 

[g33kski11z]

so where did the base 10 come from? why do we use it? who 'deemed' it correct?

 

[Ivan Seeking]

Count your fingers

 

[Ivan Seeking]

I would add that some older computer systems used base 8. There are still systems that address all of the inputs and outputs in octal.

 

[g33kski11z]

Count your fingers

wait, so seriously, if we had 12 fingers, it would be different??

 

[Ivan Seeking]

wait, so seriously, if we had 12 fingers, it would be different??

That's the way I understand it.

 

[Klockan3]

wait, so seriously, if we had 12 fingers, it would be different??

Of course, you can count to 10 on your fingers before you need a new hand. So some old rudimentary number systems like roman had III... etc then V for a full hand and X for a full hand pair. How else would you design a number system as a math illiterate living 3000 years ago when the only way people knew how to count was with their fingers?

 

[Andre]

I'm thinking about that...

...

eh

Every system is based on ten. Remember there are ten kinds of people, those who understand binary systems and those who don't. Right? Ten kinds. Ten is base, even if you call it binary.

 

[Ivan Seeking]

Every system is based on ten. Remember there are ten kinds of people, those who understand binary systems and those who don't. Right? Ten kinds. Ten is base, even if you call it binary.

In general, if b is the base, we write a number in the numeral system of base b by expressing it in the form a_nbⁿ + a_{n − 1}b^{n − 1} +a_{n − 2}b^{n − 2} + ... + a₀b⁰ and writing the enumerated digits a_na_{n − 1}a_{n − 2} ... a₀ in descending order.
- wiki

We are playing word games here.

 

[Redbelly98]

My last DIY project at home didn't go as smoothly as I had anticipated. As a result, I would consider it a big favor if everyone would switch over to a base 9 system. Thanks in advance.

That reminds me of my brother, who has worked around machinery most of his entire working life. I'll have to ask him how that base 9-2/3 system is working out for him.

Every system is based on ten. Remember there are ten kinds of people, those who understand binary systems and those who don't. Right? Ten kinds. Ten is base, even if you call it binary.

Had you not explained it, I think I never would have gotten it.

 

[BobG]

wait, so seriously, if we had 12 fingers, it would be different??

Perhaps. Humans have always had 10 fingers, but not all human numbering systems use 10 as a base. In fact, many human numbering systems use more than one base.

Sumerian and Babylonian used a dual base system - 10 and 60. And their numbering system is still reflected in the way we measure time and angles. If it wasn't such an awkwardly large number, 360 would have been a good base, since there's approximately 360 days in a year (yes, they probably knew there were actually 365 days in a year, but building a numbering system using 365 would have sucked). As is, the stars shift approximately 1 degree per night (.99 degress per night on average).

Five is a rather natural base for a numbering system based on a repeating symbol for more than 1. It's hard to instantly recognize how many marks there are when there are more than 4. If you put a diagonal slash across the marks to finish off the group with the 5th item (or some other special symbol), you can then step up to counting the number of groups instead of individual marks.

The Romans used a dual base system - 5 and 10. Using that combination for a dual base system was kind of rare.

Base 20 is a fairly common numbering base. Most base 20 systems are dual base systems - 5 and 20. Why wasn't it more common to go straight base 5, where the next order of magnitude would be 25 instead of 20? Perhaps because humans have 20 fingers and toes? I don't know, but most groups of 5 were grouped by 4's instead of 5's. In fact, in Western culture, 4 groups of 5 would be a 'score' of items, whether it be sheep, stones, or years (as in 4 score and 7 years ago).

If your low numerals are merely a series of marks (Roman numerals I, II, III, etc), base 5 and 20 wind up being very common as the numbering system is built up to accommodate higher numbers. (Roman numerals would be an exception to that generalization, however.)

If you use symbolic notation for your lower numerals (1, 2, 3, 4, etc), base 10 winds up being very common - kind of. Egyptians and Greeks kind of used a base 10 system, but they didn't use a system as simple as ours, where you only had 10 numeric symbols that would be reused with their location in the number specifying whether that '1' was a 1, 10, 100 etc. They had unique symbols for 10, 20, 30, 40, etc; unique symbols for 100, 200, etc., which made for a lot of memorization. Base 10 had some significant successes, but it wasn't very common due to having so many symbols to memorize.

The idea of using the type of base 10 system we use today required something that was very rare in numberings systems: a zero. The Hindus were probably the first to use 0 and invented the first true base 10 system using only 10 symbols, relying on the symbols location within the number to indicate whether the '1' was a 1, 10, 100, etc. The Arabs borrowed the idea from the Hindus and eventually lent the idea to Europe, as well.

 

[Chi Meson]

Every system is based on ten. Remember there are ten kinds of people, those who understand binary systems and those who don't. Right? Ten kinds. Ten is base, even if you call it binary.

I got it, I got it while driving my kids to swimming. Too late to edit...

 

[Jack21222]

This thread reminds me of this Tom Lehrer song:

https://www.youtube.com/watch?v=UIKGV2cTgqA

Base 8 halfway through.

 

[mugaliens]

Yes, base 10 was arbitrary, based on our fingers. Count 'em, unless you're missing a digit or two. Base 2, 3, 7, 60, whatever - it really doesn't matter. It's all notation, and variable math algebra and beyond) would remain the same, as it's not based on the number system anyway.

In practical and quite modern applications, it remains the basis of our navigation systems, worldwide.

 

[dlgoff]

I do a lot of work in base 16 [Hexadecimal]. This is common for some types of computer systems. My theory is that the fathers of industrial computers had sixteen fingers.

Still working with PDP8s?

I would add that some older computer systems used base 8. There are still systems that address all of the inputs and outputs in octal.

Kidding aside, were these Digital Equipment Corp. computers that predated the PDPs?

 

[Proton Soup]

This thread reminds me of this Tom Lehrer song:

https://www.youtube.com/watch?v=UIKGV2cTgqA

Base 8 halfway through.

that old math seems confusing, and I'm >35.

so, 3 from 2 is 9? they're still doing a carry, just weirdly.

 

[Jack21222]

that old math seems confusing, and I'm >35.

so, 3 from 2 is 9? they're still doing a carry, just weirdly.

The song came out in 1965, so you'd have to be >50 to understand the old math, I guess. The "new math" made perfect sense to me, including base 8, but back then I guess it was whacky.