Peano Postulates

[cheesehead]

Originally Posted by olmari

Well, gotta mark that nothing in some way.

But before the invention of zero, people got along without it. Look at Roman numerals -- there's no zero! Generally, folks just said "there's none" or something like that, which was common sense, and their number systems didn't require placeholders because they didn't have places in numbers as we do now.

It's even possible to do multiplication and division with Roman numerals, and perform complex civil engineering without any zeros.

There were, of course, certain disadvantages to this situation. So eventually people adopted number systems with places and zero (but not necessarily with the average person comprehending exactly why places and zeros made those systems superior).

[ET_]

Trying to semantically define "nothing" may lead to a "strange ring", or worse to a (endlessly recursive) tautology; you need to offer a meta-semantic to describe the meaning of the meaning of your concept.

It was the starting point of the idea about passive and active infinite (and maybe of Cantor's thoughts on infinite).

May I suggest the reading of Douglas Hofstadter's "Goedel, Escher, Bach: an eternal golden braid"?

Luigi

[ewmayer]

Originally Posted by cheesehead

But before the invention of zero, people got along without it. Look at Roman numerals -- there's no zero!

Look at the Roman empire - oh wait, there is no Roman empire. (Is that the same as saying that "there is zero Roman empire?")

Perhaps it was the awkward nature of their numeral system that caused the Romans to underestimate the strength of the barbarian hordes at their gates ... "Oh divine emperor, according to the military's latest calculations there are roughly (((I))) barbarians massing on the northern border...oh wait, maybe I added one too many sets of () there...let's try it without those damn parentheses ... so should it be IIM or IIIM? Better safe than sorry, my Caesar... let's call it the latter...whoopsie, running out of parchment here...well, at least I got the 'III' in there. Send this to the supreme commander north immediately!"

[xilman]

Originally Posted by R.D. Silverman

As for zero being unsigned, the DEFINITION of positive means greater than 0.
The DEFINITION of negative means less than 0. 0 is the unique real number
x such that x = -x. 0 is neither positive nor negative.

Going very slightly off-topic, in that it's closer to engineering than mathematics, the IEEE committee that specified a very widely used floating point format defined a signed-zero in certain circumstances. In some formats, +0 is not the same as -0. Very occasionally this turns out to be useful when one wants to preserve the sign of a quantity that has underflowed, whether by arithmetic on appropriately (or should that be inappropriately?) sized numbers or by manipulation of things that are not numbers, such as division of a number by a signed infinity.


Paul

[mfgoode]

I beg to differ with you Cheesehead as you are widely off the mark.
The Romans like all other early civilisations made extensive use of some form of the abacus which I would term crude analog computers. I quote Georges Ifrah from his 3 vol. work ‘The Universal History of numbers’
“Like the preceding systems The Roman numerals allowed arithmetical calculations only with the greatest of difficulty.
To be convinced of this let us try to do an addition in these figures without translating into our own system. It is very difficult, if not impossible to succeed.
The example which is most often cited is the following.
232 + 413 + 123 + 1,852.” Unquote.
He gives the values in both theirs and our systems. Can any one give the correct answer in Roman numerals? (Leave alone multiplication and division!)
To continue “ It is remarkable that a people who, in the course of a few centuries attained a very high technical level, should have preserved throughout that time a system which was needlessly complicated, unusable and downright obsolete in concept”
“That is why Roman accountants and calculators of the middle ages after, used the abacus with counters for arithmetical work”
Personally I have seen even today money changers in Hong Kong and Tokyo using the famous Chinese ‘suan pan’ and Japanese ‘soroban’ respectively who calculated as fast as I on my calculator
Mally.

[garo]

Mally,
It is not that hard to do addition with Roman numerals. The basic idea is the same as that of the abacus which ironically you cite in your last post. I went ahead and did the conversion to Roman numerals and added up the four numbers above. I got
MMDCXX
with only a little more time than with ordinary long addition.
The basic idea is to add like symbols together, starting with the smallest.
The numbers are:
232 = CC XXX II
413 = CCCC X III
123 = C XX III
1852= M D CCC L II

Take all units you get X.
Take all X's, you get 7 of them => L XX.
Take the two L's and you get a C.
Take all C's, you get 11 of them => M C.
Take the lone D.
Take the M and add the carried over M => MM.

Final answer => MM D C XX.
Not so bad, I'm not saying that zero does not make things easy or faster, but that it is possible to do addition and subtraction with Roman numerals and it is as easy as the abacus!!

[xilman]

Originally Posted by mfgoode

I beg to differ with you Cheesehead as you are widely off the mark.
The Romans like all other early civilisations made extensive use of some form of the abacus which I would term crude analog computers.

I beg to differ with you mfgoode as you are widely off the mark. An abacus is undoubtedly a crude digital calculator. If you want an example of a crude analog calculator, I would suggest that you consider the slide rule.

Originally Posted by mfgoode

I quote Georges Ifrah from his 3 vol. work ‘The Universal History of numbers’
“Like the preceding systems The Roman numerals allowed arithmetical calculations only with the greatest of difficulty.
To be convinced of this let us try to do an addition in these figures without translating into our own system. It is very difficult, if not impossible to succeed.
The example which is most often cited is the following.
232 + 413 + 123 + 1,852.” Unquote.
He gives the values in both theirs and our systems. Can any one give the correct answer in Roman numerals? (Leave alone multiplication and division!)

Addition is very easy in Roman numerals. I will take the time to convert those numbers into Roman numerals, and then give a detailed exposition of how a Roman computer would have evaluated the sum.

You want: CCXXXII + CCCCXIII + CXXIII + MDCCCLII.

First, concatenate all these symbols and sort to numerical order. The result is: MDCCCCCCCCCLXXXXXXIIIIIIIIII.

Now, convert each string of CCCCC into D, each XXXXX into L and each IIIII into V. We get MDDDLLXVV.

Now convert DD to M, each LL to C and each VV to X, and get MMDCXX

What was difficult about that? It's about as easy as addition with "Arabic" (really Hindu) numerals.

Subtraction is equally easy, though instead of borrowing from the next column, we split the next higher symbol.

Multiplication is straightforward. Instead of memorizing a 10*10 table, all we have to remember is that V*V is XXV, V*X is L, V*C is D, V*D is MMD, X*X is C, X*L is D, X*C is M and L*L is MMD. As larger numbers, M*M for instance, hardly ever ocurred in real life, there was no need to deal with them. Having memorized this small table, multiply each symbol in one operand by each symbol in the other and write down the result as a string of symbols. Then take the result and use exactly the same regrouping algorithm as for addition.

Here is an example: VII * XI * X111.

First, VII*XI = LV XI XI = LXXVII --- here, only reordering is needed.

LXXVIII* XIII = DLLL CXXX CXXX LVVV XIII XIII

This is D CC LLLL XXXXXXXX VVV IIIIII

which in turn is

D CC CC LXXX XV VI

which is D CCCC L XXXX X I

which is D CCCC LL I

which is D CCCCC I

which is DD I

which is MI, the correct answer,

Note that the above calculation can be, and would have been, streamlined greatly. For example, an accomplished computer would have enough experience to recognize CCCCLL equals D without having to go via CCCCC.

All in all, multiplication of Roman numerals is no more difficult than multiplying Arabic numerals --- easier in that less needs to be memorised, but the "additions with carries" may be a bit more long-winded.


I fully concede that division of Roman numerals is rather difficult!


Paul

[Wacky]

Originally Posted by garo

The numbers are:
232 = CC XXX II
413 = CCCC X III
123 = C XX III
1852= M D CCC L II

Is this correct?
I was taught that "4" is written as "1" "5" rather than "1" "1" "1" "1".
And "9" is written "1" "10"
Thus, We begin I, II, III, IV, V, VI, VII, VIII, IX, X
413 would be "100" "500" "10" "3" or CDXIII

[garo]

@Wacky: It doesn't really matter. You can use both notations. IV is marginally more compact but ancient romans used IIII as well.

@xilman: Thanks for your exposition on multiplication. BTW, it's Arabic (really Indian) and not (really Hindu). There were probably some Budhhists and Jains in the mix as well :)

[Wacky]

Originally Posted by garo

@Wacky: It doesn't really matter. You can use both notations. IV is marginally more compact but ancient romans used IIII as well.

I agree that it is only marginally more compact. However, I think that it will significantly increase the complexity of arithmetic. If you use the IIII notation, addition is almost trivial in that you simply treat the various symbols in a polynomial fashion, add like terms and then "simplify" the result. However in the more compact notation, you must give a coefficient of (-1) to any symbol that occurs "out of order". The corresponding simplification rules are not so simple.

[cheesehead]

Originally Posted by mfgoode

I beg to differ with you Cheesehead as you are widely off the mark.

Off which mark am I widely?

Abaci? I neither mentioned nor denied that Romans used abaci. Difficulty of calculation? I mentioned possibility of calculation, but said nothing about difficulty or ease.

I don't see where we differ.