Is Zero a Number?

[guigui]

Sorry, cant help but talk about math, this is my job after all.

There is no real number that multiplied by 0 would equal 1.

^ This is absoluty truth and is proven using the distributivity of multiplication over addition. And it also implies that one can not divide by zero.
Also those "1/0 tends to infinity" and such things does not mean anything.

It is true, however, that if a function tends to 0 by positive value, then its invert tends to +infty.



Congrats on the Fanzine, long may it live.

[MintyTheCat]

Sorry, cant help but talk about math, this is my job after all.

There is no real number that multiplied by 0 would equal 1.

^ This is absoluty truth and is proven using the distributivity of multiplication over addition. And it also implies that one can not divide by zero.

This is true and works all the way until we are as infinitesimally close as we are ever going to get (actually we cannot ever exhaust an infinitesimal - so we never actually ever reach there ).

Also those "1/0 tends to infinity" and such things does not mean anything.
It is true, however, that if a function tends to 0 by positive value, then its invert tends to +infty.

Does the Frog reach the other side of the pond as he hops onto the next lilly pad, Guigui? To my mind he never does reach the other side but he gets infinitesimally close and hence the use of Limits and the entire basis for Calculus.

Of course it means something - real phenomena from the natural world does not stop as the area reduces and tends to zero as the force remains constant - eventually the area is as infinitesimally close to zero as we can get and at some point becomes infinite - if you believe that the frog crosses to the other side of the pond - I do not believe that he does.

It is open to debate of course in the Math world and thankfully it works well enough for us to use it - despite the patch ups we have to use when asking Computers to do some Maths for us.

As stated before: ZERO is not a counting number and therefore there cannot legally be a zeroth edition which is what I raised at the outset - do we agree on that math wise, Guigui?

[Obiwanshinobi]

Not saying "zeroth" is a word, but I believe "zero" is an adjective when needed (in English, that is; in Polish zerowy is a word alright). Though speaking of sequences, first element is commonly called "first", in recursive definitions I saw notation such as used on a regular basis.

[MintyTheCat]

Not saying "zeroth" is a word, but I believe "zero" is an adjective when needed (in English, that is; in Polish zerowy is a word alright). Though speaking of sequences, first element is commonly called "first", in recursive definitions I saw notation such as used on a regular basis.

Zeroth is used extensively in computing.

Computing really introduced the notion of zero being the first element as any C programmer will attest to the manner by which Strings are stored in memory with the first element being zero displacement and Byte or UNICODE Word to the next element.
Be it zero or one it is still the first element and really a matter of context and convention.

However, zero is not a counting number - which was the crux of what I raised

Thinking more about limits I concur that you guys are correct and that 1/0 is off the map. Something odd occurs very, very close either side of the zero point and leads us to these philosophical discussions - we know what happens directly before and after it but the value itself, I concur is undefined.

Cheers,

Minty.

[guigui]

So now we are completely off topic and I feel sorry for OP, mods please move or delete.

As stated before: ZERO is not a counting number and therefore there cannot legally be a zeroth edition which is what I raised at the outset - do we agree on that math wise, Guigui?

Mathwise there is an algebraic structure called unitary division ring, which R, the set of real number, is.

In such a ring, 0 and 1 exist and are not the same element, also, the distributivity of * over + stands. From there, one can quickly prove that 0*a = 0 for any element a.
This also implies that 0 is not invertible (if it were, there would exist an element such that 0*a=1).
Since division by an element a is defined by actually being the multiplication by the invert of a, there is no such thing as division by 0.

Starting from here, if absolutely wanting to divide by something close to 0, you have to add some topology to the mix and invent infinitesimal calculus. In this branch, one can define a structure where, if a function is never equals to 0, but tends (topology clearly defines what "tends" means) to 0 by positive values, then its invert tends to +infty.
But the expression 1/0 still has no sense.

Now, observe the real world for one second : the closer you are of from an object, the bigger you see it. This is what maths tried to modelize with all those "tends to infty" abstraction, and I think they did it pretty good.
Collision of the frog is also taken into account : the frog-lillypad distance may reach 0, in which case its invert does not exist.

As for the computer point of view, I dont know anything about this.

[MintyTheCat]

Thanks for the Math expertise and axiomisation and rules, Guigui - it is good to have a Math authority at hand.

The lillypad paradigm does tend to split the audience. I am in the camp that feels that the frog never reaches the other side but he gets as close to it but never exactly - these arguments are purely platonic and absolute abstractions.

[guigui]

Thanks for the Math expertise and axiomisation and rules, Guigui - it is good to have a Math authority at hand.

The lillypad paradigm does tend to split the audience. I am in the camp that feels that the frog never reaches the other side but he gets as close to it but never exactly - these arguments are purely platonic and absolute abstractions.

Not that much abstraction actually, the paradigm stands on the "unending" addition : 1/2 + 1/4 + 1/8 + 1/16 + ...

Mathwise, (I love the word, thanks for pointing it to me) the "unending" addition is called series, and this one tends to 1 ; the common sense says you have to add an infinite number of terms for it to reach 1, thus the impression that the frog will never land.

To lift the issue, also take into account that time is a dimension we all travel through, but that cannot be splitted.
The sentence "the frog runs half the distance, then half the remaining distance, then half the remaining distance, then ..." falls short since one cannot split time enough to let the frog actually do all those runs. The way our brains percieve it "the frog runs half the distance, then the other half of the distance, and lands."

Now imagine a being that is able to travel through time like we humans can travel through space. Then this being would be able to see the frog never reaching the pad, like I am able to see my hand close to a wall without touching it.

Thinking about it : time is a dimension we, humans, all travel through together and at the same speed ; in a way, we are stuck in time. Maybe some other being are stuck in space but not in time and sometimes see the whole world of us passing by. Hey guys, what do we look like ? Stupid I guess.

[BulletMagnet]

Split thread from here if you need to see how it began - please continue all math-related discussion here.

[HardcoreOtaku]

As the OP in the other thread I didn't really mind the math talk as it was bumping the thread and more people were finding out about the fanzine.

Also although it was kind of off topic I found it an interesting debate, BTW my stance is I believe zero is a number.

[JBC]

Zero is the Un-number, much like 7up is the Un-cola. We can understand the concept of zero, but it doesn't actually exist. Much like how we can understand the concept of nothing, but nothing doesn't exist in an infinite universe. However since in infinity every possibility must exist we are able to entertain the idea of the existence of nothing (zero). It's cool how our minds evolved to do that. Zero is the only true fiction!

So when are you guys gonna figure out antigravity for me with your maths? I want to live in the Sin & Punishment possibility.

[Ed Oscuro]

We can understand the concept of zero, but it doesn't actually exist.

You can say the same about any other number. Show me "one" somewhere. I mean, "one," not "one apple." In the same way you can show me a zero as "no difference," but not a "zero" itself.

FWIW I don't think the discussion about limits has much to do with whether zero is a number or not. I'll agree it has properties different from other numbers, but unless somebody lays down the law about things that don't exist in the way real things do, it seems just to be a definitional matter (i.e., arbitrary).

[BrianC]

Considering the name of one of their racing games, I don't think Nintendo liked zero much.

[BIL]

^THAT IS THE KIND OF JOKE I LIKE TO SEE

[copy-paster]

In logic, zero is a word and 0 is a number.

[Lord Satori]

Can we make the about the number "i" instead? I think a debate about that one would be more enjoyable.

[MintyTheCat]

Ah, that's not the question: Zero is a member of the Real set of numbers but it is not considered a counting number:

http://www.mathsisfun.com/whole-numbers.html

This is what I raised back at the start of the original post

[MintyTheCat]

Can we make the about the number "i" instead? I think a debate about that one would be more enjoyable.

That would be another Thread.

In my neck of the woods it was always "j" - "i" freaks me out

[Ed Oscuro]

This is what I raised back at the start of the original post

Yes, you guys covered that. Teach me to just post without reading everything...also I'd better not pretend to be a mathematician.

What guigui mentioned about limits is a partial expansion (mostly a restatement) of Zeno's paradoxes. I'll have to look into the things guigui mentioned, but for the moment I'm of the mind that "undefined" is a good term here, that zero's not a counting number, and that something strange happens when you approach and "cross over" the limit. Of course, one answer is (looking at a graph of multiplicative inverse values helps create the notion) that zero's multiplicative inverse is actually infinity, but "undefined" seems to work better as a definition there (getting off topic here).

Getting closer to the original question again, is this relevant or not? Does every element in the ring have to have a multiplicative inverse of 1, or is this in reference to something different?

[guigui]

Getting closer to the original question again, is this relevant or not? Does every element in the ring have to have a multiplicative inverse of 1, or is this in reference to something different?

The answer is no : in a ring, the existence of the inverse of any element is not imposed.
In a field (or division ring, depending on the country), we impose that every non-zero element has an invert.

Exemple :
Z, the set of all integers is a ring. In this ring the element 2 has no invert, ie there exists no element a such that 2*a = 1.
R, the set of all real numbers is a field (division ring). In this field, 2 has an invert, which is noted as 2^(-1), or 1/2, and you get 2*1/2 = 1.

As for the existence of i, the imaginary number such that i^2=-1, it is precisely defined by either considering a multiplicative structure on R^2, or a quotient structure on R[X], or even a substructure of the set of 2x2 matrices.
As you see, maths has all those objects, even the non intuitive ones, very precisely defined and the DO exist, at least the second you define them correctly and write them on your paper.

[TransatlanticFoe]

This is like when someone at school tried to convince everyone black isn't a colour because it's an absence of colour. Only with more maths.

[Xyga]

Math. Not even once.

[Ed Oscuro]

Getting closer to the original question again, is this relevant or not? Does every element in the ring have to have a multiplicative inverse of 1, or is this in reference to something different?

The answer is no : in a ring, the existence of the inverse of any element is not imposed.
In a field (or division ring, depending on the country), we impose that every non-zero element has an invert.

Your answer agrees with what I've read elsewhere, but I'm still confused about what the ring having a multiplicative inverse (per the link) means. Is this what's known as the "unity element," and in what way does it appear?

Also, in your examples...

Z, the set of all integers is a ring. In this ring the element 2 has no invert, ie there exists no element a such that 2*a = 1.

This would be a field, not a psuedo-ring?

R, the set of all real numbers is a field (division ring). In this field, 2 has an invert, which is noted as 2^(-1), or 1/2, and you get 2*1/2 = 1.

This would be a unitary ring?

As for the existence of i, the imaginary number such that i^2=-1, it is precisely defined [...] As you see, maths has all those objects, even the non intuitive ones, very precisely defined and the DO exist, at least the second you define them correctly and write them on your paper.

That was in reference to division by zero, I don't know how the imaginary number would fit into Zeno's paradoxes, but everything I've seen indicates that infinity is like an "informal notion" (as said here) and any definition only holds in limited circumstances, not universally.

[HardcoreOtaku]

This is like when someone at school tried to convince everyone black isn't a colour because it's an absence of colour. Only with more maths.

Black isn't a colour though, it's a shade.

Zero is a number.

[guigui]

Ed Oscuro, you reversed guy needs some definitions, dont you ?

Ring : algebraic structure in which one can do an addition +, and a multiplication *.
- the addition should be commutative, which means that for any a,b in the ring, a+b = b+a.
- Multiplication should be distributive over the addition
- There should exist an element, usually noted 0, that is neutral for the addition ; which means that for any a in the ring, a+0 = a.
- Every element a should have an opposite, which means there exists an element, usually noted -a, such that a+(-a) = 0

Unitary ring : a ring in which there exists an unity element, usually noted 1. This element is neutral for the multiplication, which means that for any a in the unitary ring, a*1 = 1*a = a.
In an unitary ring, one defines the notion of invertible element as follows : an element a in the ring is invertible is there exists an element, usually noted a^(-1) and called the invert of a, in the ring such that a*a^(-1) = a^(-1)*a = 1.
One can prove that 0 (remember, the neutral for + ?) is never invertible, thus the impossibility to multiply by the invert of 0, ie the impossibility to divide by zero.
Z is an unitary ring, 2 has no invert in this ring.

Division ring : a unitary ring in which every non-zero element is invertible.
R is a division ring, 2 has an invert in R which is 2^(-1) = 1/2.

Field : a division ring in which the multiplication is commutative, which means that for any a,b in the field, a*b = b*a.

Math. Not even once.

Maths, everyday. This is quite fun to formulate stuffs that are either right or wrong and to mix them together to create new stuffs. Also allows to keep certains parts of the brain actives, and see how far you can push them. And it earns me big money to pay for videogames, gg.

[MintyTheCat]

Math. Not even once.

Maths, everyday. This is quite fun to formulate stuffs that are either right or wrong and to mix them together to create new stuffs. Also allows to keep certains parts of the brain actives, and see how far you can push them. And it earns me big money to pay for videogames, gg.

I wish that more people thought like this, Guigui.
If I did not need the $ I would ditch Engineering and simply do Maths, Music, Art and some electronics and programming all day.

There is a strong incidence between math education and level of earnings.

[guigui]

If I did not need the $ I would ditch Engineering and simply do Maths, Music, Art and some electronics and programming all day.

There is a strong incidence between math education and level of earnings.

Well, dont know about your country, but in France, maths do not pay even quite close to what you can get in engineering, especially if you compare the number of years required to actually start to earn decently well with maths.

[MintyTheCat]

If I did not need the $ I would ditch Engineering and simply do Maths, Music, Art and some electronics and programming all day.

There is a strong incidence between math education and level of earnings.

Well, dont know about your country, but in France, maths do not pay even quite close to what you can get in engineering, especially if you compare the number of years required to actually start to earn decently well with maths.

Yes, Maths in itself does not pay but some job that uses a fair amount of maths tends to pay well - e.g. finance.
I find this with many pure subjects though; very few who study literature can make millions out of it alone.
I suppose this is part of the sacrifice people make when they work on the edges.

[Weak Boson]

Maths woo!

Zero is definitely a number, don't be put off by the fact that not all numbers are alike.

One of our lecturers made a point of including zero in the counting numbers. "It's perfectly natural," he would say, "What if I asked you to count the number of elephants in this room?!"

To which the obvious response was "One - whether or not zero is a counting number!"

[Ed Oscuro]

@ guigui: That's helpful, as a quick condensation of what I've seen. It is interesting how math often goes "here's the definition, now figure it out" instead of giving Yes or No answers, but I can deal with it

On that note I linked earlier, it does look as if you're following Noether's convention, calling rings that allow a multiplicative identity of 1 unitary rings. Is that right?

Today in the US listeners to NPR's long-running Science Friday show got to hear a panel discussion - among other things (like Phineas Gage) they talked about math a bit. Math is surprisingly fun if you can see the use. I try to mentally multiply things whenever I can, and do percentage comparisons (though I do that by calculator). Even though things like infinities and "is zero a number" don't seem practical, these questions have philosophical uses at least, so I enjoy them.

Long tangent on color:

Wikipedia says that black is a color. That's probably using one common-sense definition, but if you think about two commonly-mentioned scientific ways the presence of "color" is talked about - as photon wavelengths, or as a material reflecting or absorbing light - you only have part of the story. Arguably the most important part happens in that "black box" of the mind. Black paint or ink absorbs all visible colors, rather than passing on specific wavelengths like other colored paints - in either form, black is just like what we call a "color" except that it doesn't pass on any wavelengths we could see, but at the eye the perception is the same ("no signal present" on the part of the eye focused on the black color). With no stimulation of that part of the eye, that leaves the job of defining what "black" looks like to the mind. Although it's just a blank space, and our electrically noisy eyes often make it hard to even see what it should look like even in a perfectly dark room, we know what it should look like, not in outline, not in contrast with other colors - but the phenomena of black apparently only exists in contrast to other colors. After all, we have a word for it, and we have black pencils - despite blackness being "no signal," we have converted that "nothing" into a phenomenon of note. This is an issue which "colors" all scientific measurements, since there is at the end of the day a measurement being made, rather than a direct apprehension of the phenomena itself.



It is interesting how we perceive the absence of stimuli in a certain way - and it always seems to be the same - but what if one person's color perception actually inverted colors so that one person's black was actually another person's perceptual white? For reasons of simplicity we generally talk as if that is the end of the story and we assume that there's no difference from my "black" to a dog's, which is just color blind, but does that apply to insects with the ability to see things we simply can't, so that what looks blank to us might actually be brightly patterned with an ultraviolet target circle, like the center of many flowers which look featureless to us? The most interesting thing is that I don't think our perception of "black" would really change if suddenly we could see more wavelengths - the mind might "right" itself, like the documented ability of some people to flip inverted images after wearing upside-down glasses so that their mind corrected it to look normal again. It's hard to understand what kind of internal language minds use in coming to agreement on these issues, but it's clearly based in part on connections of like things into definitions that then come close to being universal. The process should be mostly the same for color as it is for math, or language, or understanding economics; we just have constant and more complete understanding of color phenomena or our native tongues than we do of economic concepts.

[guigui]

@ guigui: That's helpful, as a quick condensation of what I've seen. It is interesting how math often goes "here's the definition, now figure it out" instead of giving Yes or No answers, but I can deal with it

You're right, this is because defining things correctly is certainly the most important things in maths.
Also, it happens that maths are the domain in which it is the easiest to define things correctly. Now look at philosophy for instance, those guys need entire books, lives sometimes, to define correctly what is "nature", "humanity", "thoughts", "freedom" and such. In this regard, maths are way too easy : define, experiment, prove, do it again.

On that note I linked earlier, it does look as if you're following Noether's convention, calling rings that allow a multiplicative identity of 1 unitary rings. Is that right?

Yes.
Was that a short enough answer ? Actually, unitary rings are the ones in which we work the most, the non-unitary ones seem to be a complete mess and we cant do anything in here.