What does the term "undefined" actually mean?
Mia Lopez 
I have read many articles on many sites and in many books to understand what undefined means? On some sites of Maths, I read that it could be any number. and on some sites, I read that it may be some undefined thing; and there are more definitions. But they all have clashes with each other that all are defining the term "undefined" in different ways. So which concept is right for the word "undefined" in criteria of Maths among the following five?
1-A number divided by zero may be any number (real or imaginary)
2-A number divided by zero may be an entity which is not defined yet.
3-Division of a number by zero does not make sense.
4-If $x=a/0$, then no solution exists!
5-It may give a third type of number other than real or imaginary [I have not read this definition in any book or site but it's my thought.]
This query popped up into my mind while my teacher was solving a question from my book, I am showing it to you along with my teacher's work.
Q- Prove that the roots of the following equation are real.
$x^2-2x(m+\frac{1}{m})+3=0$ where, $m$ is any real number.
Teacher's attempt:
For roots to be real,
$b^2-4ac>0$
$\implies 4(m^2+\frac{1}{m^2}-1)>0$
$\implies 4(m^2+\frac{1}{m^2}-2+1)>0$
$\implies 4[ (m-\frac{1}{m})^2+1 ]>0$.
My teacher let us write that inequality is satisfied for all $m$ belongs to real number however if $m=0$, $\frac{1}{m}$ is undefined. So if "undefined" means that "a number divided by zero may be any real or imaginary number" so then I can confess only for real numbers that inequality is satisfied for all $m$ belonging to real numbers and not for imaginary numbers since we can't make sense of a statement like this $i>0$ but if the term "undefined", in Maths, is defined as "Senseless" or "something else not known" then I strongly apprehend that why my teacher let us write that Inequality is satisfied for all $m$ belonging to real numbers?
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$\begingroup$Saying that 1 divided by 0 is undefined, does not mean that you can carry out the division and that the result is some strange entity with the property “undefined”, but simply that dividing 1 by 0 has no defined meaning. That is just like when you ask whether the number 1.9 is odd or even: That is not defined. Or when you ask what colour the number 7 has.
$\endgroup$ 10 $\begingroup$To put matters straight: Division is a function $$q:\quad{\mathbb R}\times{\mathbb R}^*, \qquad(a,b)\mapsto q(a,b)=:{a\over b}\ ,$$ whereby $q(a,b)$ is the unique number $x\in{\mathbb R}$ such that $b \>x=a$.
When we say that $\displaystyle{a\over0}$ is undefined then this means no more and no less than that the pair $(a,0)$ is not in the domain of the function $q(\cdot,\cdot)$.
Now to your three ways of understanding "undefined" in the realm of division by $0$:
If $\displaystyle{a\over0}$ could be any number, say $=13$, then this would enforce $13\cdot0=a$, which is wrong when $a\ne0$.
This is even worse. Why should $\displaystyle{7\over0}$ be the Eiffel tower?
There are circumstances where division by zero makes sense, e.g. in connection with maps of the Riemann sphere, or with meromorphic functions. There one has $\infty$ as an additional point in the universe of discourse. But these circumstances require special exception handling measures, and the "usual rules of algebra" are not valid when dealing with $\infty$.
What does the term "undefined" actually mean?
In light of the already great answers provided by Carsten and Christian, I thought a more linguistic analysis of "undefined" may be in order. The following two terms are explained in the book Origins of Mathematical Words by Anthony Lo Bello:
indeterminate$\quad$ The Latin noun terminus means the end of something. From it was formed the denominative verb termino, terminare, terminavi, terminatus meaning to set bounds to. The addition of the prefix de- emphasizes that the separation is from something else and produced the compound verb determino, determinare, determinavi, determinatus meaning to fix the limits of. The addition of the negating prefix in- to the past participle of this verb resulted in the Latin adjective indeterminatus meaning undefined, unlimited.
Now to the analysis of the word originally in question:
undefined $\quad$ The Germanic negative prefix un- has been added to the word defined of Latin origin to produce the hybrid undefined. It would have been better to say indefined as we say indefinite, but it is too late now. Defined is from the verb definio, definere, definivi, definitus, which means to set the boundaries. The plural noun finus in Latin means enclosed area, territory. The force of the prefix de- is to add the sense of thoroughness to the action.
Both of these terms, especially the mentioning of "to set the boundaries" in the analysis of the term undefined, should make Christian's answer even more lucid, especially his response to your third way of understanding "undefined."
$\endgroup$ 0 $\begingroup$From the algebraic point of view, the Real numbers form a field under multiplication and addition. If we look at the Reals as a field, there is no separate operation of "division", instead we multiply by the reciprocal. Since $0$ has no reciprocal in $\mathbb{R}$ (in fact the additive identity in a field never does), there exists no element available to multiply by to perform what would be commonly called "division by zero".
We know $0$ has no reciprocal in $\mathbb{R}$ because there is no real number $z$ that makes the following equation true: $$0z=1$$
If there were such a $z$, then we could multiply by that number and obtain an answer for "dividing by zero", but since there is no $z$, we have no way to "divide by zero".
To expand on user38858's answer, it would be like "dividing by orange". "Orange" is not a Real number, so the idea that we could perform any mathmatics with "Orange" in the context of Real numbers makes no sense. There is no real number that represent the reciprocal of $0$, so it also makes no sense to attempt to perform any mathematics on such an object.
From the algebraic standpoint, I would say that "Division by zero does not make sense" is the closest of your three concepts. Maybe I would change it slightly to say "Division by zero is not even possible".
$\endgroup$ $\begingroup$On some site of Mathematics, I read that it could be any number. and on some sites, I read that it may be anything. These are two major meanings of undefined but I think, they have a clash.
I just wanted to address this because it looked like none of the other (very good) answers have.
It looks like you're seeing two different definitions on those various websites:
Undefined = not yet defined. As in, this may have a value or meaning or definition, but we haven't gotten there yet. A lot of sites may do this with a variable X, saying it hasn't yet been defined, or we haven't found the value yet, but we can make some statements.
Undefined = not defined; we have not given an explanation for what this means. This is how I typically see undefined used. Undefined means there does not exist an answer in the way we defined it.
For example, if we define a function $f$ $|$ $\mathbb R \mapsto \mathbb R$ by $f(x) = 2x$, then $f(x)$ is "defined" for $x$ a real number. $x = 1$, $f(x) = 2$. $f(4) = 8$.
However, $f(orange)$ makes no sense. It's undefined since orange is not an element of $\mathbb R$.
We could start out the problem by saying $f$ $|$ $\mathbb R$ $U$ $(orange) \mapsto \mathbb R$ where $f(x) = 2x$ for $x$ in $\mathbb R$, and $f(orange) = 0$, then $f(orange)$ is not undefined. I've given it an answer.
Similarly, you could start out your problem by giving an output for $\frac{1}{m}$, for $m = 0$, maybe by saying $\frac{1}{m} = 0$. Whether or not you'd find that construction very useful is altogether another matter, but it is possible.
$\endgroup$ 3 $\begingroup$Something is undefined because none has defined it (yet). It is easy to define addition and multiplication of real numbers.
One can then show the theorem that for arbitrary $a,b\in\mathbb R$, the equation $a+x=b$ has exactly one solution. By virtue of this theorem, one can define $b-a$ as this unique solution.
One can also show that theorem that for $b\in \mathbb R$ and $a\in\mathbb R\setminus\{0\}$, the equation $ax=b$ has a unique solution. By virtue of this theorem, one can define $\frac ba$ as this unique solution under these circumstances. In principle, none could prevent you from defining $\frac ba$ as something completely different, for example $\frac ba:=a^2+b^2-7$. However, this makes little sense as one cannot express nice theorems with this (well, you'd have that this "division" is commutative and there'd we funny equations such as $\frac55=\frac71$). At any rate, such a definition, though feasible, would not be related to multiplication.
So what if we consider only definitions of division $\frac ab$ that coincide with our usual division (unique solution of $bx=a$) as long as we divide by something nonzero? Whatever you define $\frac 10$ to be (say, $\frac 10:=\star$), you'd lose nice properties of division (i.e., this modified notion of division is less helpful to express general statements succinctly). For example, following general rules we should have $\frac10=\frac1{2\cdot 0}=\frac12\cdot \frac10$, so $\star=\frac12\star$. So either we no longer have "$\frac a{2b}=\frac12 \frac ab$ whenever $\frac ab$ is defined"; or we no longer have "the unique solution of $x=\frac12 x$ is $x=0$"; or the symbol $\star$ is actually just $0$ and we no longer have "$\frac ab=c$ implies $a=bc$". Some of these options would be especially awful because the uniqueness of solutions of similar equations was what motivated the introduction of "division" in the first place!
So the more complete answer is: Division by zero is undefined, and is left undefined on purpose because whatever custom extended definition someone might want to introduce, it would not be useful (let alone meaningful) in the sense of an inverse of multiplication.
$\endgroup$ 0 $\begingroup$"a number divided by zero may be any real number" is a wrong statement and is never said.
A number divided by zero is just not a real number.
"zero divided by zero may be any real number" is an informal way to express that certain indeterminate expressions have a limit.
$\endgroup$ 2 $\begingroup$When referring to the result of a mathematical operation, undefined means there is no meaningful result.
When referring to a mathematical object which satisfies some mathematical relationship, undefined means no object satisfies that relationship.
The operation $f$(🍊) has no meaningful result.
There is no $x$ that satisfies the relationship $x=\frac{a}{0}$; equivalently, there is no $x$ that satisfies the relationship $x \cdot 0 = a$.
$\endgroup$ 5 $\begingroup$In this case, undefined means "not in the domain." The division operator, $div(numerator, denominator)$ is defined over the domain of real numbers except that y=0 is not in the domain.
Now, that very specific wording aside, I think you might get an intuitive grasp of the meaning if I can draw from a non-mathematics topic, programming languages. Programming language specifications have two terms which are highly related to your query: unspecified behavior and undefined behavior
- Unspecified behavior - the exact result of an operation is not specified, but the general behavior is bounded to some reasonable results. If you're writing software that uses unspecified behavior, you cannot rely on its exact behavior to be the same from compiler to compiler (example: there's a feature in C++ called a "null pointer," which is often represented with the symbol
0. Its actual value is unspecified. On most machines, its value is 0, but on some microcontrollers, a value of 65535 turns out to be surprisingly more useful, so those compilers use that.) - Undefined behavior - The behavior can be absolutely anything. The compiler is allowed to give you an error. You may get an error at runtime. You may get an error before you even execute the command that has undefined behavior. You may silently get really weird and obtuse behavior, or it may even work exactly like you think it should do. Undefined behavior is truly undefined in every way shape and form. The presence of undefined behavior invalidates the entire program, just like using an undefined value in mathematics invalidates your entire proof.
Now there is some room for undefined behavior to be defined. As a classic example, taking the square root of a negative number is undefined behavior when you're considering the set of real numbers. The square root of -1 simply has no definition if you're only looking at reals. However, if you extend your interest to not only real numbers, but complex numbers, the square root of -1 becomes defined (its value is $i$), but only as long as your concerned with complex numbers.
You could define division by 0 to result in an unspecified value. In fact, there are schools of thought that do. However, it turns out that doing so causes all sorts of problems with the rules of arithmetic that we are so used to. For example math on a Riemann sphere allows the statement $1/0=\infty$. However, they pay a dark price in doing so: the normal rules of arithmetic you are used to do not apply to arithmetic done on Riemann spheres. In particular, you lose a guarantee of a multiplicative inverse for equations that result in $\infty$. $0/0$ and $\infty/\infty$ remain undefined.
Generally speaking though, people do not use such number systems unless they explicitly call them out. For normal every day use, undefined is simply undefined.
$\endgroup$ $\begingroup$The answer by user38858 is very much to the point:
It looks like you're seeing two different definitions on those various websites:
Undefined = not yet defined. As in, this may have a value or meaning or definition, but we haven't gotten there yet. A lot of sites may do this with a variable X, saying it hasn't yet been defined, or we haven't found the value yet, but we can make some statements.
Undefined = not defined; we have not given an explanation for what this means. This is how I typically see undefined used. Undefined means there does not exist an answer in the way we defined it.
I want to stress here that this is really relevant. After reading the answer by Christian Blatter, I can't help but wonder Why is mathematics fond of infinity, but dismissive towards partially (un)defined operations? Here are contrast answers for Christian Blatter comments on your three ways of understanding "undefined" for division by zero
1-A number divided by zero may be any number (real or imaginary)
In Meadows and the equational specification of division, Jan Bergstra et al rediscover among others the fact that one of the most useful way to define division by zero is that the result is zero. Now that I write this answer, I begin to appreciate why he introduced the new name zero totalized field for a field where division by zero is defined as zero. I'm still not convinced that introducing the name meadows for commutative inverse rings (which are also known as strongly von Neuman regular rings) was a good idea.
2-A number divided by zero may be any thing.
Even so Jan Bergstra should know that a number divided by zero is zero, he introduced common meadows where $0^{-1}=a$.
3-Division of a number by zero does not make sense.
Yes, in the most common context, division of a number by zero simply does not make sense.
$\endgroup$ $\begingroup$Suppose you have a function $f:A \to C$, but you regard $A$ as a subset of some other set $i:A \hookrightarrow B$, and you do all your work from $B$. It is natural to ask if the function $f$ extends to a function $\tilde f:B \to C$ such that $f = \tilde f \circ i$.
Now, when dealing with subsets, we like to abuse notation and talk about $x \in i(A)$ and talk about $f(x)$. But $f$ is not a function on $B$, so in a rigorous sense, this is nonsense. A type error in the first place, and this is where our errors are coming from. This undefined notion comes in when we try to take $x \not \in i(A)$ and ask what $f(x)$ is. We can't pull $x$ back to an element on $x' \in A$ and then compute $f(x')$, so we say that $f(x)$ is undefined.
For example, let $A = \mathbb R^*$, $C = \mathbb R$, and $f: \mathbb R^* \to \mathbb R$ with $f(x) = 1/x$, and $B = \mathbb R$. In this case, we can't extend $f$ to a continuous function $\tilde f$ on $B = \mathbb R$. If $x \ne 0$, then we can pull $x$ back to A and compute $f(x)$, but if $x=0$, then we can't, so we say $f(x)$ is "undefined".
$\endgroup$ 0 $\begingroup$In general, an expression $x$ is undefined iff there is no $y$ such that $y=x$, or
$$x\text{ is undefined}\Longleftrightarrow\neg\exists y:y=x$$
For example, let $f$ be a function with the set of natural numbers as its domain and codomain. $f$ consists of a set of ordered pairs representing the inputs and outputs, as follows:
$$f=\{(1,0),(2,1),(3,2),(4,3),(5,4),(6,5),\ldots\}$$
Hence $f(1)=0$, $f(2)=1$, $f(3)=2$, $f(4)=3$, etc.
You may recognize this as the predecessor function on the natural numbers. The expression is undefined for $0$ in the sense that there is no ordered pair in $f$ such that the first element of the pair is $0$, hence (trivially) there is no $n$ in the codomain of $f$ such that there is an ordered pair in $f$ such that the first element of the pair is $0$ and the second element of the pair is $n$.
One could, of course, naturally extend the definition of $f$ to include $0$ (introducing the negative integers). One could do the same with $\frac{a}{0}$ for nonzero $a$, introducing a point at infinity. In doing so you may lose certain algebraic properties, such as $0\cdot a=a\cdot0=0$ for all $a$.
$\endgroup$ $\begingroup$I think what you need to understand is that not everything that your teacher says should be taken ex cathedra. 0/0 not only can but has been defined in various ways in various contexts. The limit of x/y as x and y go to zero could be defined as infinity if, say, x converges linearly, while y converges exponentially. If the converse is the case, then we might choose (Note that it is a choice. It is not written in stone that 0/0 is undefined!) to define 0/0 as zero. If x and y both converge factorially, we might for some purpose at hand define 0/0 as being equal to 1. Sometimes the limit can even oscillate between two values and in such a case it has been found convenient to define the limit as being half-way between them! It all depends on what works for you in the case at hand.
$\endgroup$ $\begingroup$I'd like to suggest another approach to explaining what an "undefined term" is in the context of term-rewriting systems.
Given an alphabet $A$ (i.e. $A$ is a non-empty, finite set), denote by $A^*$ the set consisting of all finite strings over $A$. Suppose you are given a set $T \subseteq A^*$ of terms over $A$, and a relation $R \subseteq T\times T$, which determines whether a term $s$ can be reduced to a term $t$ in a single step. Denote by $R^*$ the reflexive-transitive closure of $R$, i.e. $s\ R^*\ t$ iff $s = t$, or $s\ R\ t$, or there is a sequence of terms $r_1, \dots, r_n \in T$ ($n \in \{2, 3, \dots\}$) with $r_1 = s$, $r_n = t$, and for every $i \in {1, \dots, n-1}$ $r_i\ R\ r_{i+1}$. Suppose, additionally, you are given a subset of values (or terminals) $V \subseteq T$. Then the set of undefined terms, $U$, is the set of all terms that cannot be reduced to a value, i.e. $U := \{t \in T: \forall v\in V.\neg(t\ R^*\ v)\}$.
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