PDA

View Full Version : Just ones and zeros

Spiny Norman
02-02-2005, 09:30 PM
WARNING: Some may interpret this as a "troll". Well, it sort of is, but then again, its not really, I'm just having a bit of fun because I like logic and mathematics.

As I drove to work this morning in some of the worst summer weather that I have ever seen in Melbourne (lowest February maximum temperature since they started keeping records, pelting rain, more rain, and yet more rain), I pondered some of the recent complicated discussions about genetics and evolution, the nature of the universe, whether God exists, etc. Fascinating stuff. Yet my mind turns inexorably to the subject of mathematics ... and this is something that I know is of particular interest to a number of other board denizens.

Now maths has always been the love of my life (present spouse not withstanding!). Back when I did high school it was still split out into Pure and Applied. Lord only knows what they've done to it nowadays! Applied maths was always my strength. I loved the problem solving bits such as:

Bill (not our Bill) weighs 75 kilos. He climbs a ladder set at an angle of NNN degrees to the perpendicular which is leaning against a wall. The coefficient of friction is blah blah blah, etc ... how high can Bill climb the ladder before it slips?

But all of that is all too difficult for a bear of very little brain such as myself, so I have resolved in my own mind to post the following questions and invite your responses. I have decided to restrict myself to only ask questions about ONES and ZEROS. Anything else would be far too complicated and, therefore, not much fun!

Q1. Where did the concept of ZERO come from?

You see, I can see ONE apple. But I can't see ZERO apples. Quite irrational really. So it strikes me as particularly abstract. My recollection is that, once upon a time, there was no concept of zero but someone "invented it". Perhaps it was one of those fundamentalist Greeks? No matter whether I start with ONE apple and then take that apple away, or whether I start without any apples and add ONE apple, it still doesn't make sense when someone says I have ZERO apples .... I keep telling them, plain as the quite substantial nose on my face, that I can't have ZERO apples because I clearly haven't got any apples at all, so I just challenge them to show me the ZERO apples. At that point they start throwing things at me and calling me names.

Q2. Can it be proven that ZERO exists?

I can't see it, so how can I be sure that its not a conspiracy of scientists or mathematicians leading me astray.

Q3. What about dividing ONE by ZERO?

I know that when you divide ONE by ever smaller numbers, the resulting answer gets larger and larger ... but when I divide ONE by ZERO they tell me that I don't get a number, I get an answer that isn't really a number after all ... I really get a "concept" called infinity.

Q4. Where did this concept of ONE divided by ZERO come from?

I'm pretty confident in suggesting that nobody has ever seen anything that is infinite, because everything that I've ever seen has existed within the constraints of three dimensions (four if you count time).

Don't get me started on the square root of MINUS ONE or my head will explode, because all of a sudden you start getting answers that are a bit wishy-washy (e.g. the answer is PLUS or MINUS ONE ... I mean, hello, would someone please make up their mind, that's just not rational)!

All very confusing. I'm sure someone will be able to help me out.....

Kaitlin
02-02-2005, 09:52 PM
Q1. Where did the concept of ZERO come from?

You see, I can see ONE apple. But I can't see ZERO apples. Quite irrational really. So it strikes me as particularly abstract. My recollection is that, once upon a time, there was no concept of zero but someone "invented it". Perhaps it was one of those fundamentalist Greeks? No matter whether I start with ONE apple and then take that apple away, or whether I start without any apples and add ONE apple, it still doesn't make sense when someone says I have ZERO apples .... I keep telling them, plain as the quite substantial nose on my face, that I can't have ZERO apples because I clearly haven't got any apples at all, so I just challenge them to show me the ZERO apples. At that point they start throwing things at me and calling me names.

This one I will have a go at, the others I will have to think lots more on. I think Zero probably come from measuring stuff rather than counting stuff if that makes sense. Like you say Frosty no-one would think mmm I would like no apples :whatthe:. But if they were measuring stuff, like takeing steps to measure mmm whatever they measured ( I cant think of anything mm maybe a garden or something) and you lost count you would have to tell the person taking the steps to go back to the start and the start wouldn' have been 1.

:classic: kewl that does even makes sense.

Rincewind
02-02-2005, 09:59 PM
Q1.

I believe the Arabic mathematical school was responsible for much of the Greek mathematical knowledge. Including the concept of zero. Not sure if it can be attributed to an individual but it is an ancient idea.

Q2.

Zero is an idea it no more exists than one exists. Yes you can have one apple, but that is just an instance of the idea of one being applied to a physical object. The existence of ideas can't be measured by the same yardstick as objects.

If you accept that 1 exists, that addition exists (1+1 = 2) then you can quickly say that something called subtraction also exists and therefore 1-1 should have an answer. We call that answer zero.

Q3.

The division of anything by zero just doesn't make sense. This is because division is the inverse of multiplication. When you say what is 1/0 you are really asking, what number do I multiply 0 by to get to 1. Or going back to addition, how many times do I add zero to itself to make up one. There is obviously no answer. Not all questions have answers.

Strictly speak one shouldn't say "1/0 = infinity" for a couple of reasons. Firstly infinity is not a number, per se and so should not appear in equations. Also because 1/0 is not defined for the reason I said in the previous paragraph.

The correct way of expressing this relationship is as follows

Let y = 1/x, then as x tend to zero from the positive side then y tends to positive infinity. Of course if x approached zero from the negative side then y would approach negative infinity.

You can see from this illustration that the sign of infinity is determined by the direction by which you approach zero. This is another reason why 1/0 = infinity is not considered correct.

Q4.

The idea of infinity is an interesting one. I've not heard who was the first person to deal with it. It was certainly known ot the ancient Greeks who formulated Zeno's paradox (or borrowed it from the Arabs) which is in short a sum of an infinite positive series which never equals 1.

1/2 + 1/4 + 1/8 + ...

As each term is added the number gets larger. However it is bounded above and always less than one.

Cat
02-02-2005, 10:48 PM
Q1.

I believe the Arabic mathematical school was responsible for much of the Greek mathematical knowledge. Including the concept of zero. Not sure if it can be attributed to an individual but it is an ancient idea.

Q2.

Zero is an idea it no more exists than one exists. Yes you can have one apple, but that is just an instance of the idea of one being applied to a physical object. The existence of ideas can't be measured by the same yardstick as objects.

If you accept that 1 exists, that addition exists (1+1 = 2) then you can quickly say that something called subtraction also exists and therefore 1-1 should have an answer. We call that answer zero.

Q3.

The division of anything by zero just doesn't make sense. This is because division is the inverse of multiplication. When you say what is 1/0 you are really asking, what number do I multiply 0 by to get to 1. Or going back to addition, how many times do I add zero to itself to make up one. There is obviously no answer. Not all questions have answers.

Strictly speak one shouldn't say "1/0 = infinity" for a couple of reasons. Firstly infinity is not a number, per se and so should not appear in equations. Also because 1/0 is not defined for the reason I said in the previous paragraph.

The correct way of expressing this relationship is as follows

Let y = 1/x, then as x tend to zero from the positive side then y tends to positive infinity. Of course if x approached zero from the negative side then y would approach negative infinity.

You can see from this illustration that the sign of infinity is determined by the direction by which you approach zero. This is another reason why 1/0 = infinity is not considered correct.

Q4.

The idea of infinity is an interesting one. I've not heard who was the first person to deal with it. It was certainly known ot the ancient Greeks who formulated Zeno's paradox (or borrowed it from the Arabs) which is in short a sum of an infinite positive series which never equals 1.

1/2 + 1/4 + 1/8 + ...

As each term is added the number gets larger. However it is bounded above and always less than one.

Very good Barry. In fact zero arrived in Europe from the Arabic world in the 13th century. It revolutionised accounting practices and this single event hugely influenced European trade. Prior to this Roman numerals were used as the predominant form of accounting.

Isn't it amazing, despite all we know, mathematics still can't explain how an arrow flies through the air?

Kevin Bonham
03-02-2005, 02:56 AM
To confuse poor Frosty still further, we should also mention that there isn't just one concept called "infinity" but rather there are lots of different infinities and some of them are bigger than others!

antichrist
03-02-2005, 03:31 AM
Very good Barry. In fact zero arrived in Europe from the Arabic world in the 13th century. It revolutionised accounting practices and this single event hugely influenced European trade. Prior to this Roman numerals were used as the predominant form of accounting.

Isn't it amazing, despite all we know, mathematics still can't explain how an arrow flies through the air?

I was under the impression that the Arabs borrowed the zero from the Indians, that they sort of combined Indian and Greeks maths.

In the wrong thread has anyone checked out the correct spelling of Bishop Usher or Ussher, I don't have any references nearby.

Rincewind
03-02-2005, 07:43 AM
To confuse poor Frosty still further, we should also mention that there isn't just one concept called "infinity" but rather there are lots of different infinities and some of them are bigger than others!

Fundamentals of mathematics is not really my area but I know of at least 2 sorts of infinity. One sort is the number of integer, not sure what this is called but lets call it an enumerated infinity. The interesting thing about this set is that members can be adjacent. We can say 1 is next to 2 and we are certain there are no integers in between. Another type of infinity is the idea of a continuum. For example, there is an infinite number of real numbers between 1 and 2. In a continuum sort of infinity, individual members have no adjancency.

Georg Cantor produced a famous proof that the continuum sort of infinity is "bigger" in some sense to the enumerated sort of infinity. It really is a bolt from the blue sort of original idea. Well worth looking into if you are interested in the idea of infinity.

Ian Rout
03-02-2005, 09:44 AM
You might be interested in the thoughts of Theo Rout. He is no relation that I know of (though people occasionally ask, and I have have met him once without at the time knowing who he was) but he apparently has some radical views about zero.

http://www.ratbags.com/rsoles/strange/routhighcourt.htm

Rincewind
03-02-2005, 10:21 AM
You might be interested in the thoughts of Theo Rout. He is no relation that I know of (though people occasionally ask, and I have have met him once without at the time knowing who he was) but he apparently has some radical views about zero.

http://www.ratbags.com/rsoles/strange/routhighcourt.htm

Thanks for the link, very interesting. However, if I were you, I'd be genetically distancing myself from Mr T Rout as far as possible. :eek:

Don_Harrison
03-02-2005, 05:54 PM
According to what I have read, without the concept of zero, a negative number is a logical impossibility. Astounding stuff when you think about that. My old text books indicate that zero came from the hindu (sunya) and then arabic (cifr) numeric systems. Cifr became "cipher" which apparently means empty and refered to the empty column in an abacus. I think zero became more widely used in Europe in the 13th Century.

Bill Gletsos
03-02-2005, 06:04 PM
The following comes from Wikipedia.

The numeral or digit zero is used in numeral systems where the position of a digit signifies its value. Successive positions of digits have higher values, so the digit zero is used to skip a position and give appropriate value to the preceding and following digits.

By the mid second millennium BC, Babylonians had a sophisticated sexagesimal positional numeral system. The lack of a positional value (or zero) was indicated by a space between sexagesimal numerals. By 300 BC a punctuation symbol (two slanted wedges) was co-opted as a placeholder in the same Babylonian system.

The Ancient Greeks were unsure about the status of zero as a number: they asked themselves "how can 'nothing' be something?", leading to interesting philosophical and, by the Medieval period, religous arguments about the nature and existence of zero and the vacuum. The paradoxes of Zeno of Elea depend in large part on the uncertain interpretation of zero.

By 130 Ptolemy, influenced by Hipparchus and the Babylonians, had begun to use a symbol for zero (a small circle with a long overbar) within a sexagesimal system otherwise using alphabetic Greek numerals. Because it was used alone, not as just a placeholder, this Hellenistic zero was the first true zero in the Old World. In later Byzantine manuscripts of his Syntaxis Mathematica (Almagest), the Hellenistic zero had morphed into the Greek letter omicron (usually meaning 70).

But the late Olmec had already begun to use a true zero (a shell glyph) several centuries before Ptolemy in the New World (possibly by the fourth century BC but certainly by 40 BC), which became an integral part of Maya numerals. Another true zero was used in tables alongside Roman numerals by 525 (first known use by Dionysius Exiguus), but as a word, nulla meaning nothing, not as a symbol. When division produced zero as a remainder, nihil, also meaning nothing, was used. These medieval zeros were used by all future computists (calculators of Easter). An isolated use of their initial, N, was used in a table of Roman numerals by Bede or a colleague about 725, a true zero symbol.

The first decimal zero was introduced by Indian mathematicians about 300. An early study of the zero by Brahmagupta dates to 628. By this time it was already known in Cambodia, and it later spread to China and the Islamic world, from where it reached Europe in the 12th century.

The word zero (as well as cipher) comes from Arabic sifr, meaning "empty".

Spiny Norman
04-02-2005, 07:30 PM
Zero is an idea it no more exists than one exists. Yes you can have one apple, but that is just an instance of the idea of one being applied to a physical object. The existence of ideas can't be measured by the same yardstick as objects.

This is in danger of heading down a philosphical path ... ;) ... its all part of my fiendish plan.

If you accept that 1 exists, that addition exists (1+1 = 2) then you can quickly say that something called subtraction also exists and therefore 1-1 should have an answer. We call that answer zero.

I do agree that 1 + 1 = 2 ... except for very LARGE values of 1.

The division of anything by zero just doesn't make sense. This is because division is the inverse of multiplication. When you say what is 1/0 you are really asking, what number do I multiply 0 by to get to 1. Or going back to addition, how many times do I add zero to itself to make up one. There is obviously no answer. Not all questions have answers.

Indeed, I do agree.

Strictly speak one shouldn't say "1/0 = infinity" for a couple of reasons. Firstly infinity is not a number, per se and so should not appear in equations. Also because 1/0 is not defined for the reason I said in the previous paragraph. The correct way of expressing this relationship is as follows:

Let y = 1/x, then as x tend to zero from the positive side then y tends to positive infinity. Of course if x approached zero from the negative side then y would approach negative infinity.

You can see from this illustration that the sign of infinity is determined by the direction by which you approach zero. This is another reason why 1/0 = infinity is not considered correct.

There are a number of things that "look different" depending on which side of the equation one approaches them. But I'll leave that for the "Does God Exist" thread I think.

The idea of infinity is an interesting one. I've not heard who was the first person to deal with it.

Eccl. 3:11 ;)

Spiny Norman
04-02-2005, 07:32 PM
According to what I have read, without the concept of zero, a negative number is a logical impossibility. Astounding stuff when you think about that. My old text books indicate that zero came from the hindu (sunya) and then arabic (cifr) numeric systems. Cifr became "cipher" which apparently means empty and refered to the empty column in an abacus. I think zero became more widely used in Europe in the 13th Century.

Yet we managed to somehow end up with B.C. and A.D. without a corresponding year zero .... someone's stuffed up the calendar! :eek:

Spiny Norman
04-02-2005, 07:39 PM
You might be interested in the thoughts of Theo Rout.

Indeed I am interested. I would love to know more about it. Unfortunately he appears to be from another planet and does not speak English. :eek:

antichrist
04-02-2005, 08:19 PM
Yet we managed to somehow end up with B.C. and A.D. without a corresponding year zero .... someone's stuffed up the calendar! :eek:

I had this debate when working out ten years ago when the world was "6,000" years old, I got around it by saying that as it was calculated retrospectively the Gregorians could work it out any which way.

Rincewind
09-02-2005, 09:04 PM
A book with the title of Philosophy of Mathematics (Selected Readings) has come my way. I haven't had a chance to more than begin the 30 page introduction yet but the TOC looks very interesting. There are pieces by luminaries such as von Neumann, Frege, Russell, Hilbert, Godel, Poincare and Wittgenstein (among others). Many dealing with subjects recently discussed in this thread and others. One I look forward to in particular is called "What is Cantor's continuum problem?" by Godel. Fascinating stuff.

Of course, some may say I need to get out more.

Rincewind
28-02-2005, 11:57 AM
Nothing from the book I mention above but I did find the following on the web today. It's a good article for those interested in the question of infinity and the significance of Cantor's contribution. It also contains something on the bigness of countable vs continuum infinities as alluded to earlier in this thread.

pax
28-02-2005, 03:41 PM
To confuse poor Frosty still further, we should also mention that there isn't just one concept called "infinity" but rather there are lots of different infinities and some of them are bigger than others!

Infinitely many, even.

pax
28-02-2005, 03:47 PM
Fundamentals of mathematics is not really my area but I know of at least 2 sorts of infinity. One sort is the number of integer, not sure what this is called but lets call it an enumerated infinity. The interesting thing about this set is that members can be adjacent. We can say 1 is next to 2 and we are certain there are no integers in between. Another type of infinity is the idea of a continuum. For example, there is an infinite number of real numbers between 1 and 2. In a continuum sort of infinity, individual members have no adjancency.

Georg Cantor produced a famous proof that the continuum sort of infinity is "bigger" in some sense to the enumerated sort of infinity. It really is a bolt from the blue sort of original idea. Well worth looking into if you are interested in the idea of infinity.

You can also prove that the number of rational numbers (the numbers that can be expressed as fractions) is equal to the number of whole numbers. That is, you can construct a one to one and onto mapping between the set of rational numbers and the set of positive integers. This is even though the set of rational numbers is dense (that is, there are infinitely many rational numbers contained in any finite open interval).

pax
28-02-2005, 03:53 PM
Q2. Can it be proven that ZERO exists?

I can't see it, so how can I be sure that its not a conspiracy of scientists or mathematicians leading me astray.

In an additive number system, zero is defined to be the number which when added to any other number gives that number. It is also known as the additive identity. In most number systems, the existence of the identity element (zero in this case) is assumed, not proved.

Rincewind
28-02-2005, 04:01 PM
You can also prove that the number of rational numbers (the numbers that can be expressed as fractions) is equal to the number of whole numbers. That is, you can construct a one to one and onto mapping between the set of rational numbers and the set of positive integers. This is even though the set of rational numbers is dense (that is, there are infinitely many rational numbers contained in any finite open interval).

Thanks for that clarification, pax. I was not aware of this at the time of of writing the post you quoted (although I might have read it earlier but forgot it). However, by the time of your reply it was no longer news (as its proof is contained in the article I link to above).

BTW I have certainly revised my opinion on the effect of adjacentness (a characteristic of a non-dense set) on cardinality.

Don_Harrison
28-02-2005, 09:47 PM
With some trepidation, and now I know it was with good reason, I went to Mr T Rout's web-site. My head hurts.

pax
01-03-2005, 08:13 AM
Thanks for that clarification, pax. I was not aware of this at the time of of writing the post you quoted (although I might have read it earlier but forgot it). However, by the time of your reply it was no longer news (as its proof is contained in the article I link to above).

BTW I have certainly revised my opinion on the effect of adjacentness (a characteristic of a non-dense set) on cardinality.

Well actually, there is still the concept of adjacentness for any countable set (the rationals, integers, positive integers etc are all countable sets). It's just that you have to order the numbers in a special way.

e.g for the rationals you can have:

0 -> -1 -> 1 -> 1/2 -> -1/2 -> 2/1 -> -2/1 -> 1/3 -> -1/3 -> 3/1 -> -3/1 -> 2/3 -> -2/3 -> 3/2 -> -3/2 etc...

Rincewind
01-03-2005, 08:35 AM
0 -> -1 -> 1 -> 1/2 -> -1/2 -> 2/1 -> -2/1 -> 1/3 -> -1/3 -> 3/1 -> -3/1 -> 2/3 -> -2/3 -> 3/2 -> -3/2 etc...

Yes, of course. So perhaps the density is just an artifact of reordering the set as above and putting into a distance from zero order that we are used to.

pax
01-03-2005, 02:39 PM
Yes, of course. So perhaps the density is just an artifact of reordering the set as above and putting into a distance from zero order that we are used to.

Yes. Density or not of a set is essentially a property of the metric used to measure distance. I can think of lots of metrics that make the set of rational numbers non-dense, but with the metric we are used to (Euclidean distance), they are dense. Density and adjacency (or countability) are independent concepts.