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#1 2006-02-07 11:18:17

siva.eas
Member
Registered: 2005-09-17
Posts: 166

Number Systems

I have two questions.

What is the total number of number systems that have been established today not counting decimal, binary, octal and hexadecimal number system?

And what are they?

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#2 2006-02-07 13:25:03

ryos
Member
Registered: 2005-08-04
Posts: 394

Re: Number Systems

If you're just talking about bases, you can base a number system on any arbitrary base you want. It just depends on what's convenient for you. Computers use base-2 not because it's awesome, but because they don't have much of a choice considering they only have two states. (Whoever thought up binary and applied it to computers was a genius...)

The Babylonians used a base-60 number system. Yikes...how did they get anywhere with base 60?


El que pega primero pega dos veces.

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#3 2006-02-07 13:35:01

Ricky
Moderator
Registered: 2005-12-04
Posts: 3,791

Re: Number Systems

(Whoever thought up binary and applied it to computers was a genius...)

It came naturally because transistors only have two states.

But by far, the coolest base is -2:

0110 0110: 34 or 1*(-2)^1 + 1*(-2)^2 + 1*(-2)^5 + 1*(-2)^6


"In the real world, this would be a problem.  But in mathematics, we can just define a place where this problem doesn't exist.  So we'll go ahead and do that now..."

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#4 2006-02-08 10:35:33

mathsyperson
Moderator
Registered: 2005-06-22
Posts: 4,900

Re: Number Systems

ryos wrote:

The Babylonians used a base-60 number system. Yikes...how did they get anywhere with base 60?

Clocks seem to cope well enough.


Why did the vector cross the road?
It wanted to be normal.

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#5 2006-02-08 10:49:52

Ricky
Moderator
Registered: 2005-12-04
Posts: 3,791

Re: Number Systems

Clocks use base 10...


"In the real world, this would be a problem.  But in mathematics, we can just define a place where this problem doesn't exist.  So we'll go ahead and do that now..."

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#6 2006-02-12 20:09:43

Zmurf
Member
Registered: 2005-07-31
Posts: 49

Re: Number Systems

Ricky wrote:

(Whoever thought up binary and applied it to computers was a genius...)

It came naturally because transistors only have two states.

But by far, the coolest base is -2:

0110 0110: 34 or 1*(-2)^1 + 1*(-2)^2 + 1*(-2)^5 + 1*(-2)^6

its not about transistors, its about charge in magnetic tape, there millions of seperate magnetic bits on a tape and the point either north or south, north being 0 south being 1. And its read by the computer.

Last edited by Zmurf (2006-02-12 20:12:01)


"When subtracted from 180, the sum of the square-root of the two equal angles of an isocoles triangle squared will give the square-root of the remaining angle squared."

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#7 2006-02-12 20:25:33

MathsIsFun
Administrator
Registered: 2005-01-21
Posts: 7,552

Re: Number Systems

siva.eas wrote:

What is the total number of number systems that have been established today not counting decimal, binary, octal and hexadecimal number system?

Imperial: 12 inches in a foot, 3 feet in a yard, 6 feet in a fathom, 660 feet in a furlong, 5280 feet in a mile (6080 in a nautical mile), not to mention pints, gallons, quarts, pecks, bushels, roods, poles, perches et al.

And old British money required great arithmetic skill: 12 pennies in a shilling, 20 shillings in a pound. And a guinea was 1 pound and one shilling, wasn't it? Clerks, shopkeepers and accountants had to deal with this without thinking.

So ... lots!


"The physicists defer only to mathematicians, and the mathematicians defer only to God ..."  - Leon M. Lederman

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#8 2006-02-12 20:28:06

MathsIsFun
Administrator
Registered: 2005-01-21
Posts: 7,552

Re: Number Systems

Ricky wrote:

But by far, the coolest base is -2:

0110 0110: 34 or 1*(-2)^1 + 1*(-2)^2 + 1*(-2)^5 + 1*(-2)^6

Genius! I never thought of a negative base.

Does it have any wonderful uses (apart from confounding humans)?


"The physicists defer only to mathematicians, and the mathematicians defer only to God ..."  - Leon M. Lederman

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#9 2006-02-12 21:12:40

Zmurf
Member
Registered: 2005-07-31
Posts: 49

Re: Number Systems

wots the difference between base 2 and base -2, bar the 1*(02)^1 its identical because 1*(-2)^2 = 4 just like 1*(2)^2, is there sumthin' im missing?


"When subtracted from 180, the sum of the square-root of the two equal angles of an isocoles triangle squared will give the square-root of the remaining angle squared."

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#10 2006-02-12 23:40:49

MathsIsFun
Administrator
Registered: 2005-01-21
Posts: 7,552

Re: Number Systems

Every odd power ends up negative ... !  So 1*(-2)^1 + 1*(-2)^2 + 1*(-2)^5 + 1*(-2)^6 = -2 + 4 -32 + 64 = 34

I have this idea in the back of my head that there might be a more "natural" number system than base10. Maybe we should have base "e"?


"The physicists defer only to mathematicians, and the mathematicians defer only to God ..."  - Leon M. Lederman

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#11 2006-02-13 06:50:56

ryos
Member
Registered: 2005-08-04
Posts: 394

Re: Number Systems

Oh yeah, and then we'd need a calculator to count to 10. Natural logs would be a breeze, though!


El que pega primero pega dos veces.

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#12 2006-02-13 08:24:02

Ricky
Moderator
Registered: 2005-12-04
Posts: 3,791

Re: Number Systems

its not about transistors, its about charge in magnetic tape, there millions of seperate magnetic bits on a tape and the point either north or south, north being 0 south being 1. And its read by the computer.

You are talking about rom.  The first computers didn't even have rom, all they had was ram, which was not in the form of magnetic tape.

Last edited by Ricky (2006-02-13 08:24:35)


"In the real world, this would be a problem.  But in mathematics, we can just define a place where this problem doesn't exist.  So we'll go ahead and do that now..."

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