Zero

[Feelingpink]

I found out this morning that 0 (zero) is an integer (please don't ask why this was being discussed on a Friday morning - it's too sad). Does anyone know why? I always had something in my head about integers being whole numbers ... and 0 somehow doesn't seem to fit the bill (apart from the wits who might say it's a whole lot of nothing). An internet search just seems to bring up "integer sequences" and companies with that name.


TIA

[LMC]

An integer is any whole number (which isn't a fraction)

So the real argument is whether zero is a number...

[David Franklin]

I found out this morning that 0 (zero) is an integer (please don't ask why this was being discussed on a Friday morning - it's too sad). Does anyone know why? I always had something in my head about integers being whole numbers ... and 0 somehow doesn't seem to fit the bill (apart from the wits who might say it's a whole lot of nothing). An internet search just seems to bring up "integer sequences" and companies with that name.It's difficult to answer this on a level other than "it just is", or "by definition" without quickly getting into very abstract (and surprisingly complicated) mathematics.

One way of looking at it that might be helpful is to consider the integers as the numbers you can get to starting from 1, and using either addition or subtraction.

So, 2 = 1 + 1, 3 = 1 + 1 + 1 = 2 + 1, and 0 = 1 - 1.

For that matter -1, -2, etc are integers as well.

[El Salsero Gringo]

See, for instance: http://en.wikipedia.org/wiki/Integer

The set of integers without Zero would not form a Group (with the addition operator: '+' ) since it would not have an identity element, a necessary part of a Group.

In plain-speak, the set of integers including zero has properties that are desirable and interesting to study which is not the case if zero is excluded.

[qjohn]

Eh looks like I've b******d up the edit function here :o

[qjohn]

An integer is any whole number (which isn't a fraction)

So the real argument is whether zero is a number...
Zero is a number, just not a Natural number. It is a Whole number (a set which allows you to subtract any two numbers) and therefore is presumably an integer.

My head hurts now :tears:

[David Franklin]

See, for instance: http://en.wikipedia.org/wiki/Integer

The set of integers without Zero would not form a Group (with the addition operator: '+' ) since it would not have an identity element, a necessary part of a Group.But this is kind of begging the question. You're pulling the set of integers out of nowhere, noticing it contains zero, and then saying "if we didn't have zero, bad things would happen". But where did the zero come from in the first place?

[But the "simplest" way I know of defining the integers treats them as equivalence classes of pairs of natural numbers under the operation (a, b) ~ (c, d) iff a + d = b + c. You then find 0 is the equivalence class (a, a). Of course, you've still got to define the natural numbers - I never did that bit...]

[bigdjiver]

In simple speak an integer is a number that does not have a fractional part, and zero does not have a fractional part.

[Clive Long]

In simple speak an integer is a number that does not have a fractional part, and zero does not have a fractional part.
Nice idea - but wrong.

Clive

[El Salsero Gringo]

But this is kind of begging the question. You've already got this set of integers, which including zero, and you say "if we didn't have zero, bad things would happen". But where did the zero come from in the first place?True. But FP doesn't ask where Zero comes from. She asks why is it an integer? And as you rather succinctly pointed out, the reason it's an integer is because if it wasn't, bad things would happen!

For a definition of integers, I prefer one based on Z being the infinite cyclic group and zero being it's identity element.

[DavidY]

I'm surprised by the number of people debating pure mathematics on this forum which is ostensibly about dancing. :eek:

[killingtime]

I'm surprised by the number of people debating pure mathematics on this forum which is ostensibly about dancing. :eek:

We are hoping to find moves that we can put in-between zero beats; so it's totally related :D.

I've never had an issue with zero being an integer. Mainly because I work a lot with computers and the concept of not having zero as a useable integer is alien to me.

[Clive Long]

I've never had an issue with zero being an integer. Mainly because I work a lot with computers and the concept of not having zero as a useable integer is alien to me.
Here's one I never could understand.

(Simply) values in computer memory are defined by "strings of" ones and zero - assuming the computer uses binary representation.

Assuming (I define) the coding for zero is the value "0" assigned to a bit or "00000000" in an 8-bit byte - how do I represent the "absence of value" - the "null value"?

I suspect it something to do with Turing machines but makes no sense to a chap like me. :confused:

CRL

[Rhythm King]

Saw something on BBC Breakfast recently, about Pi and the fact that the decimal places can be calculated to infinity. Apparently a computer is running the calculations in Switzerland. Sooner or later, every telephone number in the World will appear in the sequence of numbers.

[Clive Long]

Saw something on BBC Breakfast recently, about Pi and the fact that the decimal places can be calculated to infinity. Apparently a computer is running the calculations in Switzerland. Sooner or later, every telephone number in the World will appear in the sequence of numbers.
And more than that, the sequences within that sequence (say you pick every other digit (or is it integer :) ) or every 500th or every squillionth) the resulting sequences are non-recurring.

Maffs. The best thing since Marmite on buttered toast.

Clive

[David Franklin]

And more than that, the sequences within that sequence (say you pick every other digit (or is it integer :) ) or every 500th or every squillionth) the resulting sequences are non-recurring.References? I thought this was still unknown (same for fact that every digit sequence appears). Although the statistical evidence strongly implies Pi is normal, in which case both conclusions would follow.

[killingtime]

Here's one I never could understand.

(Simply) values in computer memory are defined by "strings of" ones and zero - assuming the computer uses binary representation.

Assuming (I define) the coding for zero is the value "0" assigned to a bit or "00000000" in an 8-bit byte - how do I represent the "absence of value" - the "null value"?

I suspect it something to do with Turing machines but makes no sense to a chap like me. :confused:

CRL

You don't represent the absence of value in an integer. Floating point numbers have the concept of "not a number" and "infinity" built in (if you are really interested in this check out http://stevehollasch.com/cgindex/coding/ieeefloat.html).

Computers also tend to have divide by zero error codes. Some times we use an unassigned value. For example if you have an signed integer (so you can have negatives) but aren't actually using the signed components than you can use -1 or any negative value to represent "null" but you are wasting a lot of possible numbers on one flag (it beats checking another variable though unless you really need those high value numbers).

[El Salsero Gringo]

References? I thought this was still unknown (same for fact that every digit sequence appears). Although the statistical evidence strongly implies Pi is normal, in which case both conclusions would follow.Pi is irrational and so has no repeating sequences.

A normal number is one in whose decimal expansion all digits are equally likely to appear.

Is the claim about telephone numbers equivalent to being a normal number, or is irrationality sufficient?

[David Franklin]

Pi is irrational and so has no repeating sequences.

A normal number is one in whose decimal expansion all digits are equally likely to appear.

Is the claim about telephone numbers equivalent to being a normal number, or is irrationality sufficient?It's not equivalent - you don't need the sequences to be equally likely; I think you can even have sequences having zero 'probability' as long they appear at least once (c.f. 'set of measure zero'). Irrationality (nor transcendentality) is definitely not sufficient - Liouville's number sum_i=0^inf 10^(-i!) is transcendental, but obviously doesn't have any digitis other than 1 and 0 in it, so no finding 123 anywhere etc...

[Gus]

I OFFICIALY declare this thread to be the Greatest discussion about absolutely NOTHING .... EVER. :wink:

[clevedonboy]

As part of my degree course, I was taught at great lengths of the conceptual importance of 0, how the arabs had intoduced it to the west. that the fundamental calculation (+1) + (-1) = 0 rocked thinking, how 0!=1 is a contradion , 1/0 is infinity, Y to the power of 0 is 1 regardless of the value of Y. Without doubt for pure mathematicans 0 is cool. So how soul destroying was it to find this article about a Parrot who can uinderstand the concept of 0?


An African Grey Parrot at Brandeis University appears to understand a numerical concept akin to zero. In a series of counting experiments, Alex, a 28-year-old parrot who lives in a Brandeis lab run by comparative psychologist Irene Pepperberg, spontaneously and correctly used the label "none" during trials of his counting skills to describe an absence of colored blocks on a tray.

The findings, published in the current issue of The Journal of Comparative Psychology, add to a growing body of scientific evidence that the avian brain, though quite different from the brains of mammals, are capable of handling more complex processes than previously thought.

Generally, humans don't begin to understand the concept of zero until about three or four years old (longer in the case of some). Chimps and spider monkeys have been shown to have some rudimentary understanding of the concept of none. Grey parrots have brains the size of a walnut.

[bigdjiver]

Pi is irrational and so has no repeating sequences.

A normal number is one in whose decimal expansion all digits are equally likely to appear.

Is the claim about telephone numbers equivalent to being a normal number, or is irrationality sufficient?The set of of telephone numbers is far from "normal". Just think how many begin with 0.

A vacuum technologist was introduced thius: "An expert has been defined as someone who gets to know more and more about less and less, until he knows everything about nothing. Mr .... knows practically everything about practically nothing."

[LMC]

I OFFICIALY declare this thread to be the Greatest discussion about absolutely NOTHING .... EVER. :wink:

Well, that saves me asking

damn you Gus... :rofl:

[Clive Long]

And more than that, the sequences within that sequence (say you pick every other digit (or is it integer ) or every 500th or every squillionth) the resulting sequences are non-recurring.
References? I thought this was still unknown (same for fact that every digit sequence appears). Although the statistical evidence strongly implies Pi is normal, in which case both conclusions would follow.
You are very probably right - I'm really outta my league here.

All I can offer is:
"A real number is said to be irrational if it is not rational, i.e. it cannot be expressed as a quotient of integers. The decimal expansion is non-periodic for any irrational, but periodic for any rational number"

http://planetmath.org/encyclopedia/TheoryOfRationalAndIrrationalNumbers.html

Which may be equivalent to what I wrote - but I might be well off.

Enough.

Clive

[bigdjiver]

References? I thought this was still unknown (same for fact that every digit sequence appears). Although the statistical evidence strongly implies Pi is normal, in which case both conclusions would follow.There are normal, non-recurring sequences that do not contain every digit sequence, and have every sqillionth digit the same.

Take pi, and every time 34 appears replace it with 43. Swap every sqillionth digit with the nearest non-sqillionth 9.

[Feelingpink]

I OFFICIALY declare this thread to be the Greatest discussion about absolutely NOTHING .... EVER. :wink:So aren't you pleased that YOU were the one who raised the issue in the first place! :rofl:

[David Franklin]

All I can offer is:
"A real number is said to be irrational if it is not rational, i.e. it cannot be expressed as a quotient of integers. The decimal expansion is non-periodic for any irrational, but periodic for any rational number"

Which may be equivalent to what I wrote - but I might be well off.No, it's not the same; again, the obvious counterexample is Liouville's number, which is non-periodic (and therefore irrational), but only contains 0's and 1's.

There are normal, non-recurring sequences that do not contain every digit sequence, and have every sqillionth digit the same.That would be a direct contradiction of the definition of normal I'm using:

From http://mathworld.wolfram.com/NormalNumber.html:
for a normal decimal number, each digit 0-9 would be expected to occur 1/10 of the time, each pair of digits 00-99 would be expected to occur 1/100 of the time, etcIn fact every digit sequence has to occur infinitely often in a normal number according to that definition.

[bigdjiver]

Nice idea - but wrong.

CliveUnhelpful - given that it is a simplification, how is it wrong?

[bigdjiver]

....That would be a direct contradiction of the definition of normal I'm using ... In fact every digit sequence has to occur infinitely often in a normal number according to that definition.Apologies, I was using ESG's definition.

[Clive Long]

Unhelpful - given that it is a simplification, how is it wrong?
Good point.

I was being terse to the point of rudeness.

In fact I have been trying to justify what I wrote - I have go on a bit - I will PM you with my thoughts * (ramblings) rather than post them - as several people at Harrow tonight said the presence of this thread is sapping their will to live.

Clive

* and PM them to Dr Franklin for his review comments :)

[El Salsero Gringo]

Apologies, I was using ESG's definition.I was being deliberately sloppy. It's the long run in-the-limit probablity of finding any digit sequence that has to equal out.

if you want a more "mathematical" definition of a 'normal' number, it looks like this: (from wikipedia):

Suppose b is an integer greater than 1 and x is a real number. Consider the digit sequence of x in the base b positional number system. If s is a finite string of digits in base b, we write N(s,n) for the number of times the string s occurs among the first n digits of x. The number x is called normal in base b if for every string s of length k:

[killingtime]

if you want a more "mathematical" definition of a 'normal' number, it looks like this: (from wikipedia):

Do you think we can get MathML support on this board so you don't need to attach images on the math involved :D?