Does zero have a 'value'

[chance]

Interesting - I think Zero only has meaning when compared with some quantity of something. I.e can zero stand alone?

e.g “Hey john, I have zero quantity”

reply - “zero what?”

“Zero of any of those ponies over there”

 

[Metullus]

Interesting - I think Zero only has meaning when compared with some quantity of something. I.e can zero stand alone?

e.g “Hey john, I have zero quantity”

reply - “zero what?”

“Zero of any of those ponies over there”

Or how about "I am a total zero."

 

[69dodge]

from the preface of the book "Concrete Mathematics: A Foundation for Computer Science" by Graham, Knuth, and Patashnik:

Students always know better than their teachers, so we have asked the first students of this material to contribute their frank opinions, as "graffitti" in the margins. Some of these marginal markings are merely corny, some are profound; some of them warn about ambiguities or obscurities, others are typical comments make by wise guys in the back row; some are positive, some are negative, some are zero. But they all are real indications of feelings that should make the text material easier to assimilate.

 

[Dubium]

I'm not sure I agree with that, but I do agree that context is important. Can you tell us more about how your discussion at work started? Was it just an abstract discussion about numbers, or was there some practical situation that gave rise to it?

We work for the Australian Tax Office (pity us)!!

A 'Pay as you go' statement is required to be lodged if an amount of tax is to be remitted to the commissioner. A nil amount is still an amount for these purposes.

So this begged the question...the commish can say an event has happened that hasn't actually happened, that a payment has happened that hasn't actually happened, and now enters the world of hypermathematics...go Commish!!

 

[Soapy Sam]

Lost thread marker.

For the benefit of posterity. The above post marks the loss of 1 week's posts related to a forum upgrade.
During this hiatus, Soapy Sam disproved the validity of number theory using his newly invented "Patagonian variable model". Sadly, the argument is now lost for all time.

 

[Beerina]

I think you miss my meaning.

Crimresearch said
'Hey! I wanted 4 ponies for Christmas, an all I got was 3!'

In a normal context of conversational English, that would usually mean "3 ponies", as you say, but in the context of a question about mathematics, might mean just "three" -ie the numerical concept.

"Three ponies" is a quantity,- a physical group of ponies.
"Three" is not a quantity. It's a value, an abstraction.

I see a fundamental difference here.

Zero is an oddity. I agree it has a value, but is that value also a quantity?

If so, how would we distinguish between "Zero garages" and "Zero invisible unicorns"?

The tall woman with waist-length jet black hair tossed back the last of her Manhattan, uncrossed her legs and slowly stood up from the bar.

"Meet me in my hotel room. Number 3."


Now something tells me you lads'll have little trouble figuring out she meant the 3rd hotel room, and not the 3rd McDonald's down the road...

 

[Schneibster]

This is an interesting question, and marks the division of arithmetic from algebra.

The ancient Greeks and the Romans didn't have zero in their numbering system. The Arabs invented the idea of having a numeral for zero, and also invented the idea of giving numerals a value based on their place within a number; zero signified places for which there was no value needed to define the number, just as they do today. 10 is the most obvious example, signifying one ten and zero ones. Technically, it is ....000010, signifying zero hundreds, zero thousands, zero ten-thousands, and so forth, but since we know that all the rest to the left are zero, but we have to specify all the places to the right to make sure we know how to interpret the non-zero places, we can see how this considerably simplifies the representation of numbers.

The Arabs eventually invented a new kind of mathematics from this; if one could abstract the concept of "zero ones," could not one abstract the entire idea of "number" itself and define the equations using place-holders for the numbers, and then relate the equations to one another? This is precisely what they did, and it led to a revolution in the sciences, first in the Arab culture, and then in the Renaissance in Europe in Western culture when European people finally figured out what to do with it.

So, because zero is used as a placeholder to make other numerals within the representation of a number meaningful, I contend that it is indeed a number in and of itself, and has a value.

Let's follow another train of thought. The result of any equation that is not an undefined result must be a number. Certainly it is not a letter, or an orange, or an orangutan. So what is the result of the equation 5 - 5 = x? What, in other words, is the value of x? It is, of course, zero. Thus, zero is the representation of the null value; and it is both a numeral and a number.

 

[epepke]

So let's say that I have no horses. This system permits me to represent "zero horses" by using a stick with no marks on it. But it just as equally permits me not to represent the number zero, simply by not using a stick at all in that case. This is a clear example where the concept "none" (no stick) is different from the representation for "zero" (an unmarred stick). I actually consider it to be more likely that if I were to hand said primitive an unmarred stick as a representation of how many horses I have, he wouldn't know what the hell I was on about.

We LISP hackers to this day make a distinction between no stick and a stick with no marks.

 

[jj]

For the benefit of posterity. The above post marks the loss of 1 week's posts related to a forum upgrade.
During this hiatus, Soapy Sam disproved the validity of number theory using his newly invented "Patagonian variable model". Sadly, the argument is now lost for all time.

Would that take us from Peano to Peanissimo?

 

[Taffer]

Does zero have a mathematical value? Just something we were discussing at work today. I said I'd ask. It sounds really simple but when you think about it... maybe not.

I'd have to say "yes". Its 'value' is less then 1, and more the -1.

 

[Soapy Sam]

The difference between having no measles and having no money is considerable.
But is it a quantitative difference?

 

[DrMatt]

value of zero

Syntactically, 0 is just a symbol which may be combined with other symbols to create numerals. Remember numerals?

The question "Does 0 have a value" has a well-understood meaning in mathematics. It asks "What is the semantics for 0".

There are several more or less equivalent answers.

The algebraic answer is based on the definition of a field, say, F. To be a field, F must have a binary operation, +, and an element 0, such that for any element e of F, e + 0 = 0 + e = e. Furthermore, F must have an element 1 and a binary operation * such that for any e in F, e * 1 = 1 * e = e. A theorem follows that e * 0 = 0 * e = 0. NB that this theorem follows without stating the relationship between + and *, and indeed without requiring them to be addition or multiplication. My ex pawned my algebra book and I'm a bit rusty after 20 years, but I think somebody else here can fill in the details.

The set theory model builds equivalents of the counting numbers from sets containing the empty set, and declares those sets to actually be the numbers. One such approach starts by defining
0 = {}
1 = {{}}
2={ {}, {{}} }
3={ {}, {{}}, {{},{{}}} }
4={ {}, {{}}, {{},{{}}}, {{},{{}},{{},{{}}}}} }
...
successor(N)= N U { N}

That's just the first two mathematical values-of-zero that I can think of. Others that I thought about, such as the role of zero in distance formulas, the triangle inequality, and metrics, all depend indirectly on the algebraic definitions of fields, as I recall. Again, it's been a while since I did this stuff--I'm just a musician, fergoshsakes--but I think a discussion of mathematics ought to contain some genuine mathematics and not just philosophical history.

 

[10001]

0, nothing, zero, none, non existant, empty? etc

aren't they all the same?

of course, as society evolved as did its ablity to see things, the way we explain how the uiverse around us becam more complex. so you might think quanty is different from value and zero is different from nothing. but the fat is that what it is trying to differentiate is something that exist and something that does not.

i would imagine that as we evolved, we didn't try to explain what doesn't exsist in a quanty or value. but rather we tried to communicate how many or how much of the things we were talking about.

then again you can see atemps of explanation for so called nothing or zero ness, in eastern philosophies and early mathmatitions. ofcourse this attempt has grown very scientific as we approached the present day.

personally, i would say that zeroor nothingness is like a gateway. a gateway that is every where and in every time. it could even be a whole universe of its own.

one things for sure if there wasn't "nothing or zero" the "exsistance' as we know would also not exist. i know this sentance doe't make sense to mny people. but there isn't much a better way to put it. if you can please reply.

now that i read back to my writing... i sound a bit arrogant..
oh well...

if nothing or zero seems weird then what about a negative? what does that qantify or value? and how?
mathmatically? ofcourse... but in real terms

 

[drkitten]

aren't they all the same?

No.

Is a stick with no marks on it the same as no stick?

 

[Nick Bogaerts]

The Arabs invented the idea of having a numeral for zero, and also invented the idea of giving numerals a value based on their place within a number; zero signified places for which there was no value needed to define the number, just as they do today.

Schneibster is propagating a myth; the Arabs did not invent either the zero, or the place-value notation. They first started using the Greek numbering system, which started at alpha (1), beta (2), and so on including the archaic digamma, until theta (9), then iota (10), kappa (20) until the archaic koppa (90) excluding san, rho (100) to omega (800) and the archaic san (900). Up until the twelfth century, the Arabs used this system, either by transposing it on their own letters or directly using Greek characters. By the ninth century one variant started to gain popularity, with the numbers up to 400 (from the 22 Hebrew or Syriac letters) mapped onto the alphabet with 500, 600, ..., 1000 from the Byzantines.

This system was purely additive. for instance, 472 would have been expressed ta (400) ayn (70) ba (2). This had a major drawback; it was difficult to express numbers larger than 1000. For 10,000 the letter ghayn would have to be repeated ten times. To get over that, a hybrid system (again possibly borrowed from the greeks) was devised, with a letter in front of the ghayn denoting a multiplication, hence dal (4) ghayn (1000) meaning 4 x 1000, and thus dal ghayn ta ayn ba would be 4 x 1000 + 400 + 70 + 2. This can also still be found in English, the Gettysburg Address being the classic example: 'Four score and seven years ago...' (4 x 20 + 7).

This was the system most Arabs used, though in learned circles another system was introduced by Mohammed ibn Musa al-Khowarizmi. His book survived only in the Latin translation, where he was called Algorismi. (Another book of his, Al-jabr w'al-muqabalah, gave rise to the word algebra). This was where our place-value notation comes from. The Arabs did not invent it, though they did introduce it to the Europans. This system used nine symbols, eka (1), dvi (2), tri (3), catur (4), panca (5), sat (6), sapta (7) asta (8), nava (9), with which they could express arbitrarily large numbers, going with powers of ten. This, however, required columns, which could be eliminated if a symbol was added to denote an empty column, which was either a circle or a dot.

Originally, this symbol represented merely an empty column, and only later did the Indians used it to denote a null value.

The Babylonians, Chinese and the Mayans also independently came up with a place-value notation, but the Chinese lacking a symbol for 0, which was introduced to them by the Indians.

(Source: Georges Ifrah, From One to Zero. Highly recommended)

 

[DrMatt]

Um, I think i got it wrong, the 0* e=0 must be an axiomatic requirement for F to be a field, otherwise I'd be able to string-substitute and "1 * e = 1" would be equally legitimate.

 

[RapmasterT]

This is an interesting question

I disagree.

 

[DrMatt]

Not one mathematician to help out? I'm just a musician, but my answer so far appears to be the only one that addresses the question.

 

[Corpse Cruncher]

I have always had the understanding that zero is neutral between numbers. It is to the value of plus or minus 1 when it stands on it's own.

 

[gnome]

Not one mathematician to help out? I'm just a musician, but my answer so far appears to be the only one that addresses the question.

Mathematician here (if having a degree qualifies me)... I think you've touched on definition rather than value--though two important definitions, one in abstract algebra and the other in set theory.