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Before I begin, let me give you so background. I previously asked a question on "How to prove that −x is not equal to x just because they yield the same result when in $x^2$". This got me thinking. What is a number in math anyway?

For instance, aside from the fact that 5 looks the same as 5, how do we know that 5=5? How do we know that 2 numbers are equal, and 2 numbers are different?

A can't be because they give the same input when plugged into a function. If this definition is true, then x and -x would be the same.

So can some tell me what is the definition of a number in a math? And please, can you also use that definition to prove/show that −x is not equal to x just because they yield the same result when in $x^2$.

I'm sorry if this question sounds far fetched. But it would really help with my understanding of math if I found an answer to it. Also, can you try to give the answer at the level of a high school Pre-Calculus student? Thanks.

gt6989b
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Ethan Chan
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    The word "number" is used in various different scenarios. There is no formal definition of a number. There is a formal definition of a natural number or integer or real number or complex number or $p$-adic number or transfinite number, etc. Sometimes these things are very loosely related to what we intuitively perceive as a number. – freakish Aug 02 '18 at 14:46
  • Can you link to the previous question you're referencing? – Matthew Leingang Aug 02 '18 at 14:48
  • @freakish Okay. Last question. How do any of these definitions "prove/show that −x is not equal to x just because they yield the same result when in $x^2$? – Ethan Chan Aug 02 '18 at 14:48
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    And please, can you also use that definition to prove/show that −x is not equal to x just because they yield the same result when in $x^2$. It's not possible to assume $f(x) = f(-x)$ and from that derive $x \neq -x.$ Especially since it is not true; to see this, take $x=0$. – md2perpe Aug 02 '18 at 14:49
  • @EthanChan I don't know what you are refering to. Please provide a link. In order to prove that "$-x$ is not equal to $x$" you need to know what "$x$" is, what "$-$" is and what "equal" means. – freakish Aug 02 '18 at 14:50
  • Previous question link: https://math.stackexchange.com/questions/2828249/how-to-prove-that-x-is-not-equal-to-x-just-because-they-yield-the-same-resu – Ethan Chan Aug 02 '18 at 14:50
  • In the real numbers , $x=-x$ implies $2x=0$ and this implies $x=0$. Hence, for every real $x\ne 0$, we have $x\ne -x$. "$5=5$" is an obvious statement, who would try to prove it formally ? – Peter Aug 02 '18 at 14:51
  • @EthanChan In order to fully understand what all of that means I think you need a course or a book in the set theory (numbers are derived from it). Try this: https://books.google.pl/books?id=x6cZBQ9qtgoC&printsec=frontcover&source=gbs_ge_summary_r&redir_esc=y&hl=pl#v=onepage&q&f=false There's a deep and well defined meaning to everything in maths. – freakish Aug 02 '18 at 14:54
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    The property that you seem to want to be true is that if $f(x) = f(y)$, then $x=y$. This property, called injectivity does not hold in general. The function $x\mapsto x^2$ is an example. Both $-2$ and $2$ are sent to $4$ by this function. For a more pathological example, consider the function $f(x)=1$. All possible values of $x$ are mapped to 1. Since $f(0) = 1$ and $f(3) = 1$, should we conclude that $1=3$? – Xander Henderson Aug 02 '18 at 14:54
  • Possible duplicate of What is a number? – Jam Aug 02 '18 at 15:04
  • There are also rings, fields, and algebras, which are kinds of number systems. Some of these number systems counterintuitively have it that $x=-x$ even when $x$ isn't zero! (Different number systems can be incompatible.) Then there are cardinals for size, which generalize the natural numbers to infinity and beyond. – Kyle Miller Aug 02 '18 at 15:11
  • The question "aside from the fact that 5 looks the same as 5, how do we know that 5=5 ?" makes little sense. – Mauro ALLEGRANZA Aug 03 '18 at 13:04
  • Regarding the question : "to prove/show that $−x$ is not equal to $x$ just because they yield the same result when in $x^2$ why do you find strange that an "opeartion" can give the same result with two different inputs ? I assume that we can go from London to Cambridge by different ways: this does not implies that all the roads from London to Cambridge are the same. – Mauro ALLEGRANZA Aug 03 '18 at 13:06

6 Answers6

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Number are part of a real number line which has $2$ fixed points as a reference. These two points can be $0,1,{1\over2}$,$\sqrt3$ anything. And then infinite even divisions of the distances on the number line using reference points gives us some other points which are called numbers.
Simply the fact that $-5$ and $5$ represent two different points on number line proves that they aren't equal.

PS: This is my understanding of numbers. Not from any book.

Love Invariants
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    This exclude complex numbers. – md2perpe Aug 02 '18 at 14:42
  • Yes, I was obviously telling about real numbers. Complex numbers are defined entirely different. – Love Invariants Aug 02 '18 at 14:42
  • @LoveInvariants Thanks. Okay. Last question. How does your definition prove/show that −x is not equal to x just because they yield the same result when in $x^2$? Thanks. – Ethan Chan Aug 02 '18 at 14:44
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    $-x$ isn't equal to $x$ because both the numbers represent different point on number line. We know that a unique point defines a unique numbers. Negative numbers were discovered to make number line a line instead of a ray starting from $0$ – Love Invariants Aug 02 '18 at 14:46
  • Or: when you take the negative, it corresponds to rotating the number line 180 degrees about the point $0$. If $x$ isn't $0$, then $-x$ is on the opposite side of zero. $x^2$ has nothing to do with it. – Kyle Miller Aug 02 '18 at 14:55
  • By the way, the real numbers consist of all the points of a line, with a $0$ and $1$ marked out so that you can do geometric constructions to get products and quotients. Any sort of scheme to evenly divide the line to name the numbers will miss almost all of them, if I'm understanding what you're meaning. – Kyle Miller Aug 02 '18 at 15:03
  • @KyleMiller- Its a hypothetical concept. You know we aren't dividing the number line when adding, subtracting etc. We just perform the calculations. – Love Invariants Aug 02 '18 at 15:06
  • @LoveInvariants This was a big point of debate in mathematical philosophy, and a reason Dedekind introduced his cuts, as I understand it. If you're not careful, you end up with only the constructible real numbers. When you say "evenly dividing the distances on the number line using reference points gives us some other points which are called numbers," it seems like you're allowing for some points to not be numbers. It is unclear what you mean. – Kyle Miller Aug 02 '18 at 19:32
  • I should have added infinitely dividing distances. Btw by evenly dividing I meant the distance between 1 and 2 should be equal to that of 0 and 1, etc. – Love Invariants Aug 03 '18 at 19:08
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You can find the set-theoretic construction of natural numbers in this Wiki article. The obvious operation of $+$ (addition) can be defined on this set, as the construction includes how to find the next number (i.e. add 1 to it).

Then you can define subtraction as an inverse of addition, and quickly notice that the natural numbers are closed under addition, but not closed under subtraction (e.g., 3-5 is not a natural number).

So you extend the natural numbers to the integers to close the set under subtraction, defining $-x$ as the unique integer you need to add to $x$ to get $0$. This way one can show that $-x = x$ if and only if $x=0$, so unless $x=0$, yuo always have $x=-x$.

UPDATE

You are asking a deep question, which requires basic foundations to understand the answers completely. But on some very simple level, you can consider the natural numbers defining the basic count of objects in a group. This way, for example, $0$ is defined as having no objects, $1$ as a unique object, $2$ as a unique object and another object (i.e. $2=1+1$) and any group with $n$ objects can be this extended to $n+1$ by adding another object. This is very simplistic but will intuitively work, which seems what you are asking for.

Now as above, note addition is defined for this group but subtraction is not, since $3-5$ is not a number of objects in a collection. Then apply the discussion above...

gt6989b
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  • I'm sorry, set-theoretic constructions are way to complex for me; I don't even know what set-theoretic means. Can you please try to explain all this at the level of a high school Pre-calc student? – Ethan Chan Aug 02 '18 at 14:46
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    @EthanChan The question "What is a number?" is a hard problem. The high school answer is "It is a thing that can be added to or multiplied by other similar things." A better answer is going to require more advanced ideas. Perhaps those ideas can be distilled down to something that a high school student can understand, but that is a heavy lift. – Xander Henderson Aug 02 '18 at 14:49
  • @EthanChan tried to bring this down a little bit – gt6989b Aug 02 '18 at 14:51
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The word number, like the words set, line, plane, point, etc., is called a primitive; that is, it cannot be usefully defined a priori. Therefore, its meaning, if we could call it meaning, is dependent on context; and in each context, whatever is regarded as a number is usually specified by a set of properties so as to delineate to what object the theory refers. Apart from this rather rarefied notion of a definition -- which is admittedly better than nothing -- there can be no definition (in the classical sense) of the word number, given that it applies to so many objects as to be uselessly general or peter out in a triviality were one to attempt such a definition.

In response to the other part of your question, about why $$-x\ne x,$$ well, if we assume that $x$ is a member of some field of characteristic $0,$ (say, the complex numbers $\mathbf C$), then it follows trivially from the field axioms. In particular, if our field is $\mathbf C,$ then any choice of $x\ne 0$ would immediately show why that equality is not an identity -- it leads to an inconsistency -- and we don't want that since it makes everything boring.

Allawonder
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  • That does not follow from the field axioms. You just get $2x=0$, and there are fields of characteristic $2$. If $2\neq 0$ in your particular field, then it follows from the field axioms. – Kyle Miller Aug 02 '18 at 19:34
  • @KyleMiller I don't know why you think this needs to be pointed out, but if $x\in\mathbf C$ and $x\ne0,$ then doesn't it follow that $x+x\ne0$? – Allawonder Aug 02 '18 at 21:12
  • Because it's ambiguous whether "say, the complex numbers $\mathbf{C}$" means you're giving an example of a field or if you're talking about that field in particular. It follows from that field's axioms, but not the field axioms. – Kyle Miller Aug 02 '18 at 21:50
  • @KyleMiller Are you saying that there is some field where an element (apart from the zero element) is equal to its additive inverse? – Allawonder Aug 02 '18 at 22:13
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    Yes, and the simplest example is the integers modulo $2$, where $1+1=0$. In some fields, if you add $1$ to itself enough times you get $0$. The "characteristic" of such a field is the minimal number of times that is -- the characteristic $2$ fields are the only cases where anything apart from zero is its own additive inverse (in which case everything is its own additive inverse). We say $\mathbf{C}$ is characteristic $0$. – Kyle Miller Aug 02 '18 at 23:29
  • @KyleMiller I can confirm that what you say is true. On meditating on this too, I can see that my claim was false. I accordingly edit. – Allawonder Aug 03 '18 at 13:47
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You probably learned most of what know now about what a number is by the age of four.

You counted: "One, two, three, ..."

Zero came after that. Then eventually you learned about decimals, fractions, negative numbers, and irrational numbers.

I remember seeing a program (I can't remember the reference, it was probably Nova) that showed a book that took hundreds of pages to prove that $1+1=2$. You can get that deep into numbers if you like, but I'm not qualified to get you there.

Probably the next piece of interesting information you might look into is whether something is countable. (It all circles around to childhood, doesn't it?) It basically means, can you index all of the members of a set with the natural numbers ($1,2,3, ...$)? Some sets (like the integers) you can; others (like the reals), you can't.

John
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Simply put, a number was invented to represent a particular quantity. People needed something to define quantity and number gives us such abstraction. Positive numbers can be associated with obtaining items and negative numbers can be associated with losing items.

Vasili
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A "number" is an abstract object with a magnitude. There are lots of different types of numbers, which each have different qualities. There's no unifying definition of which abstract object is called "a number", it's just a handy concept we use.

Fundamentally, the natural numbers ($1,2,3,\ldots$) are numbers. But we also say that numbers between the naturals (such as $3.141\ldots$) are also numbers. Also, we can even call complex numbers (such as $1+2i$) "numbers", despite them having real and imaginary parts. Now, you may ask "how can a number have $2$ parts?". What about a different object with $2$ parts; is the set $\{1,2\}$ a number? The answer is that it doesn't really matter whether you call $\{1,2\}$ or $1+2i$ or $\left(\begin{align}1\\2\end{align}\right)$ a "number", it's a fairly arbitrary designation. What matters is how you apply its specific properties.

There are plenty of weird mathematical objects that can be called numbers (including the $p$-adic numbers, the surreal numbers, etc.). But also, there are different ways of constructing the natural numbers, such as with sets. So I think it's irrelevant to be too concerned with what we call "a number". Plus, imagine how boring math would be if we were only concerned with "numbers".

Jam
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