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There are several things that I don't understand. I know that zero is a real number, but I'm confused on the how and why aspects. What defines a real number, compared to a number that's considered non-real? For what reason is zero real, and how is it that a value such as infinity isn't when zero is?

Also, what purpose does zero even have? It's obvious that it's used as a place holder, but what value does it have alone? What makes it any different from other numbers? I realize that it is typically interchanged with nothing, but that would technically mean that zero is nothing—and that's not true. People have told me already so many times that zero does not necessarily mean nothing, and I figured it out a long while ago anyway. So, what I want to know is the actual value that can be associated with zero.

However, my question brings up another topic that has been argued and disputed about and disagreed on for years. Zero divided by zero, or 0/0. There are a lot of varying answers about this, but what's been constant throughout everything is that people claim 0/0 is not a real number. So, if I may ask, why is it that zero is considered a real number, and not 0/0? And why is it considered non-real in the first place? True, there is no defined value, but the expression is anything but meaningless; if it was, then there wouldn't be so many people trying to understand it. I guess what I'm trying to figure out is the reason behind the expression's classification as non-real—especially when the sole number of zero is real, despite not having an evident value.

This covers a few different things, so I was uncertain about what to make the title of these questions. I apologize for that. If anyone can answer, though, I'd be extremely grateful!

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closed as too broad by Matthew Towers, JMP, 6005, Watson, user91500 Jun 24 '16 at 14:50

Please edit the question to limit it to a specific problem with enough detail to identify an adequate answer. Avoid asking multiple distinct questions at once. See the How to Ask page for help clarifying this question. If this question can be reworded to fit the rules in the help center, please edit the question.

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    $\begingroup$ How comfortable are you with the idea of $0$ being an integer? $\endgroup$ – Ken Duna Jun 23 '16 at 21:22
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    $\begingroup$ Well, for one thing, zero is the additive identity of the reals. $\endgroup$ – Solomon Slow Jun 23 '16 at 21:29
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    $\begingroup$ If you want your arithmetical operations to make any sort of intuitive sense, for instance that $a+0 = a$ and $1a = a$ for any number $a$, then it follows relatively immediately that $\frac00$ cannot be given a value in any consistent way. $\endgroup$ – Arthur Jun 23 '16 at 21:29
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    $\begingroup$ You kinda mixed around the order of things: First there was the property, the idea of there being something like a 0 - Element. The naming process of calling "0" null/zero/nil, came later. The same happened with the real numbers, first there was the idea of a system with certain properties - one property was there being a "0", then everything was given a name. $\endgroup$ – Imago Jun 23 '16 at 21:35
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    $\begingroup$ Don't confuse yourself with wordplay when contemplating the phrase "is nothing". $\endgroup$ – Excluded and Offended Jun 24 '16 at 3:22
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The term "real" in "real number" has a precise mathematical meaning. This is a set of numbers constructed in one of several very specific ways. The name is confusing, as it has various connotations in English. If they were called Dedekind numbers, for example, then the confusion would be less. $0$ is a Dedekind number because that's how they are defined. $\infty$ is not a Dedekind number, because of the way they are defined. $0/0$ is not a Dedekind number, because Dedekind numbers do not allow division by zero. All these questions can be answered by looking at exactly how Dedekind numbers are defined, and what operations are and are not allowed.

Although Dedekind numbers (aka real numbers) are very commonly used in mathematics, and very important, their definition is rather complicated. Hence students encounter them, called $\mathbb{R}$, early on in their studies, without really knowing what these numbers are.

It's easier for learners of mathematics to first make sense of $\mathbb{Q}$, the set of rational numbers. This is the set $\{\frac{a}{b}:a,b\in\mathbb{Z}, b\neq 0\}$. Basic operations on fractions are familiar and well-understood, so it's not too difficult to answer questions about $\mathbb{Q}$. The reason $\infty$ is not in $\mathbb{Q}$ is because you can't express $\infty$ as a fraction of two integers. The reason $0$ is in $\mathbb{Q}$ is that $0=\frac{0}{1}$, the ratio of two numbers. And so on.

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    $\begingroup$ The terminology goes back to Descartes, who named quantités réelles the roots of equations that can be represented on a line and quantités imaginaires those which cannot. Among the quantités réelles, he distinguished between fausses (false, negative) and vraies (true, positive). $\endgroup$ – egreg Jun 23 '16 at 21:38
  • $\begingroup$ Indeed, with the Dedekind construction it's quite obvious that it is pure definition that $\pm\infty$ are not "Dedekind numbers": The empty set and full set of rationals are explicitly excluded from the cuts, and if they were not, they'd naturally give the two infinities (at least concerning the order; maybe the addition/multiplication formulas would give results not compatible with what we expect from $\pm\infty$). $\endgroup$ – celtschk Jun 24 '16 at 7:24
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    $\begingroup$ You address why $\infty$ is not in $\mathbf{Q}$, but only by shifting the question to why $\infty$ is not in $\mathbf{Z}$ which you do not address. $\endgroup$ – hkBst Jun 24 '16 at 11:58
  • $\begingroup$ @hkBst, that's right, we kick the can until we get to a set of numbers whose definition we understand well, such as $\mathbb{N}$. $\endgroup$ – vadim123 Jun 24 '16 at 13:50
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This is very old question. The Greeks wondered if zero was a number. The understood the concept of nothingness, but they couldn't agree if it had a quantity. We give credit to the Indians for the invention of zero. And treating 0 as a real number.

It is not entirely dissimilar to some discussions about infinity. Sometimes we talk about infinity like it is a number, and then we say but infinity isn't really a number it is a concept. It is a stand-in for boundless.

Why do we need zero? The problem with the natural numbers {1,2,3...} is that the natural numbers aren't closed under subtraction. If we can add and subtract natural numbers, we should come up with rules of arithmetic that are consitent. Enter zero. And, enter the negative numbers. Great. The integers are closed under addition and subtraction. What about infinity? We don't need it. No two numbers ever sum to infinity. Furthermore, infinity brings in a bunch of stupid rules that we don't need. Such as, infinity - 3 = infinity. Better to leave infinity out of the real numbers.

One philosophical problem from the ancients. Are fractions numbers? The ancients talked about ratios. They loved their ratios. But they viewed them as a relationship between two whole numbers rather than some sort of intermediate value between whole numbers. Eventually that became unwieldy. Especially after they realized that there were some quantities that could not be expressed as the ratio of whole numbers. So, now we agree that fractions are real numbers. It is just easier that way. And fractions allow us to close the operations of multiplication and division.

We haven't actually arrived at a definition of real numbers. And I don't think I am going to do so. Not here, at least.

What about 0/0? It is undefined, because as soon as we try to write a definition, we will find that we have created a contradiction. And if the definitions lead to contradictions, they are not very good definitions. Eventually, we come to a place in mathematics when we are forced to confront 0/0, and we engineer a solution that fits the need, for example the limit. The limit does not allow us to define 0/0 in any general sort of way, but it does give us an non-contradictory approach to address a certain set of problems in the neighborhood of 0.

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You are confusing a few things. The real numbers, we we denote $\mathbb{R}$, are usually called that to compare them to, say the rational numbers ($\mathbb{Q}$), or the complex numbers ($\mathbb{C}$).

$0$ is a real number, because it happens to be in the collection of things we call "real numbers". If we instead called them "queal numbers" then we would call $0$ a "queal number". I think you are confusing the common usage of the word "real" with its mathematical usage, which is different. Calling $0$ a real number, is not the same as saying it is an object which exists, like tables and chairs. We like to talk about $0$ because it has the convenient property that $x + 0 = 0 + x=x$.

As for dividing by $0$, we do not say it is an un-real number, we don't say it is anything. We say the expression "$0/0$" does not define anything at all. Division, roughly speaking, is undoing multiplication: So a claim that $0/0 = x$ is the claim that $x$ is the unique object so that $0 = 0 \times x$, but such an $x$ is not unique, so the expression does not define anything at all.

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  • $\begingroup$ "Real" is meant to be compared to "imaginary", which makes sense in English too. "Complex" is the combination of the two, not the comparison point of "real". $\endgroup$ – Mehrdad Jun 24 '16 at 0:57
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There are several things that I don't understand. I know that zero is a real number, but I'm confused on the how and why aspects. What defines a real number, compared to a number that's considered non-real? For what reason is zero real, and how is it that a value such as infinity isn't when zero is?

  • Draw a line in a Euclidean space. Mark one point on that line $0$ and mark a different point on that line $1$. Every point on that line now corresponds to a real number. You could also label one point $-2$ and another point $\pi$ and, still, every point on that line would correspond to a real number, including $0$. In that respect $0$ is just another real number.

  • In Euclidean geometry, there is no point on that line that corresponds to infinity. Using different geoemetries, there are ways to add one, two, or infinitely many infinities to a number line. But those are not real numbers. They are something else.

Also, what purpose does zero even have? It's obvious that it's used as a place holder, but what value does it have alone?

  • I think you are using the word "value" in more than one way here.

What makes it any different from other numbers?

  • These are some of those things. For all real numbers $x$.
    (1.) $\quad x + 0 = 0 + x = x$.
    (2.) $\quad$ There is a real number $(-x)$ such that $x+(-x)=(-x)+x=0$.
    (3.) $\quad$ If $x \ne 0$, then there is a real number $\dfrac 1x$ such that $\dfrac 1x \cdot x = x \cdot \dfrac 1x = 1$.

  • A consequence of (1.), (2.), and (3.), is the following theorem.
    (4.) $\quad x \cdot 0 = 0 \cdot x = 0$.

I realize that it is typically interchanged with nothing, but that would technically mean that zero is nothing—and that's not true.

  • You are right. Anyone who does that is wrong.

People have told me already so many times that zero does not necessarily mean nothing, and I figured it out a long while ago anyway. So, what I want to know is the actual value that can be associated with zero.

  • I don't think you understand what "value" means. I always thought that the value of $0$ is $0$.

However, my question brings up another topic that has been argued and disputed about and disagreed on for years. Zero divided by zero, or 0/0. There are a lot of varying answers about this, but what's been constant throughout everything is that people claim 0/0 is not a real number. So, if I may ask, why is it that zero is considered a real number, and not 0/0? And why is it considered non-real in the first place? True, there is no defined value, but the expression is anything but meaningless; if it was, then there wouldn't be so many people trying to understand it. I guess what I'm trying to figure out is the reason behind the expression's classification as non-real—especially when the sole number of zero is real, despite not having an evident value.

  • First, all fundamental properties of the real numbers are stated in terms of addition and multiplication. Subtraction and division are defined in terms of addition and multiplication. Subtraction, $x - y$, is defined as $x + (-y)$; $x$ plus the additive inverse of $y$. Division, $\dfrac xy$, is defined as $x \cdot \dfrac1y$; $x$ times the multiplicative inverse of $y$. By theorem $(4.)$ listed above, $0$ cannot have a multiplicative inverse; that is, the fact that $\dfrac 10$ does not exists, is a consequence of the way real numbers are defined. hence $\dfrac 00 = 0 \cdot \dfrac 10$ cannot have a value without contradicting the properties of the real numbers. It's not that the value of $\dfrac 00$ is not a real number. The value of $\dfrac 00$ simply cannot exists without creating a mathemtical paradox.
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  • $\begingroup$ Sorry, I don't particularly like formatting established users' posts, but it seemed sensible here -- you are intermittently quoting the OP, and this just seemed... less confusing :) $\endgroup$ – pjs36 Jun 24 '16 at 6:42
  • $\begingroup$ It's completely fine, and yes—it is less confusing. Thanks for the clarification as well! $\endgroup$ – EeveeKitty Jun 24 '16 at 7:07
  • $\begingroup$ Your point (2) isn't special for $0$ in the real numbers: It's also true that for every real number $x$ there's a number $x'$ such that $x+x'=\pi$. $\endgroup$ – celtschk Jun 24 '16 at 7:49
  • $\begingroup$ @celtschk - Yes. But I needed it to prove (4). $\endgroup$ – steven gregory Jun 24 '16 at 7:58
  • $\begingroup$ I guess you also needed the distributive law to prove (4), and yet you didn't put it in the list of "things that are special to $0$". $\endgroup$ – celtschk Jun 24 '16 at 8:00
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people claim 0/0 is not a real number.

OK, suppose that $\frac{a}{b}=c$.

Multiply both sides of that equation by $b$, and you get $a=bc$.

So, if $a$ and $b$ both equal zero, which unique real number, $c$, makes the equation true?

$0 = 0c$


...a value such as infinity isn't [real]...

First, you have to explain what you mean by "a value such as infinity." Lots of things in mathematics are infinite, but that's not the same thing as having a number called "infinity."

The set of reals is infinite because if you tried to construct a list of all of them, your task would never end.

OK, so how many reals would be in that list if you did finish it? Trick question! There is no "if you did finish it" because you can't finish it. No matter how many numbers you added to your list, there would always be at least one more that you haven't added yet. And, by induction, that means there would always be infinitely many more that you haven't added yet.

If you want to define a set of numbers, then it's more interesting to define a set of numbers that all obey the same rules. If there was a real number called "infinity", then you'd want it to obey the same rules as all of the other real numbers. Well, one thing that's true of every real number is if you add 1 to it, you get a different, greater real number. What do you call infinity+1? And if infinity+1 is greater than infinity, and so is infinity+2, and so is infinity+3, and so on.

So, how do you explain what "infinity" is supposed to mean if there are an infinite number of numbers that are greater than infinity?

There are number systems that have meaningful infinities, but the Reals aren't one of them. For example, Georg Cantor gave names to numbers that describe the size of various infinite sets, and he invented consistent, interesting rules that they all obey. Google for "transfinite number" to learn more about those.

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There multiple ways to define the real numbers, I will describe one of them. Start with the integers. That is, the numbers $\{\dots,-4,-3,-2,-1,0,1,2,3,4,\dots\}$. We form the rational numbers by taking fractions using the integers as numerator and denominator ($0$ is not allowed in the denominator, I might get around to explaining why as you asked about this as well).

This is where things get more complicated. Imagine making an infinite list of numbers such as $(1,\frac{1}{2},\frac{1}{3},\frac{1}{4},\dots)$. We call this a sequence. In the sequence I just listed, as you go out further in the list, the numbers get closer and closer to zero. You will never get zero in this list, but you get as close as is possible. We say that this sequence "converges to $0$".

Now at this point, we only have rational numbers to work with. The problem is that we can create sequences that "converge" but not to any rational number. For instance, $(1,1.4,1.41,1.414,1.4142,1.41421,\dots)$ is a sequence of rational numbers where each time we add one more decimal in the decimal expansion of $\sqrt{2}$. This sequence "converges to $\sqrt{2}$", but the problem is that $\sqrt{2}$ is not a rational number! So we set about to create it.

There is a precise mathematical way to explain how we create the "non-rational" numbers for which rational sequences converge to, but it is much too complicated for this post.

The point is that when we add these "non-rational" numbers that rational sequences converge to, we get the real numbers.

So in answer to a small part of your question, $0$ is a real number because it is a rational number and there is a rational sequence converging to it (in fact there are infinitely many such sequences).

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