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What is a number in math? [closed]
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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.
soft-question arithmetic definition integers natural-numbers
edited Aug 2 at 14:46
gt6989b
30.2k22148
asked Aug 2 at 14:35
Ethan Chan
591322
closed as too broad by amWhy, José Carlos Santos, Peter, Xander Henderson, Jam Aug 2 at 15:04
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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up vote
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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.
soft-question arithmetic definition integers natural-numbers
edited Aug 2 at 14:46
gt6989b
30.2k22148
asked Aug 2 at 14:35
Ethan Chan
591322
closed as too broad by amWhy, José Carlos Santos, Peter, Xander Henderson, Jam Aug 2 at 15:04
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.
1
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 2 at 14:46
Can you link to the previous question you're referencing?
– Matthew Leingang
Aug 2 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 2 at 14:48
1
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 2 at 14:49
2
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 $xmapsto x^2$ 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 2 at 14:54
 |Â
show 8 more comments
up vote
3
down vote
favorite
up vote
3
down vote
favorite
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.
soft-question arithmetic definition integers natural-numbers
edited Aug 2 at 14:46
gt6989b
30.2k22148
asked Aug 2 at 14:35
Ethan Chan
591322
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.
soft-question arithmetic definition integers natural-numbers
edited Aug 2 at 14:46
gt6989b
30.2k22148
asked Aug 2 at 14:35
Ethan Chan
591322
edited Aug 2 at 14:46
gt6989b
30.2k22148
edited Aug 2 at 14:46
gt6989b
30.2k22148
edited Aug 2 at 14:46
gt6989b
30.2k22148
30.2k22148
asked Aug 2 at 14:35
Ethan Chan
591322
asked Aug 2 at 14:35
Ethan Chan
591322
asked Aug 2 at 14:35
Ethan Chan
591322
591322
closed as too broad by amWhy, José Carlos Santos, Peter, Xander Henderson, Jam Aug 2 at 15:04
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.
closed as too broad by amWhy, José Carlos Santos, Peter, Xander Henderson, Jam Aug 2 at 15:04
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.
1
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 2 at 14:46
Can you link to the previous question you're referencing?
– Matthew Leingang
Aug 2 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 2 at 14:48
1
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 2 at 14:49
2
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 $xmapsto x^2$ 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 2 at 14:54
 |Â
show 8 more comments
1
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 2 at 14:46
Can you link to the previous question you're referencing?
– Matthew Leingang
Aug 2 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 2 at 14:48
1
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 2 at 14:49
2
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 $xmapsto x^2$ 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 2 at 14:54
1
1
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 2 at 14:46
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 2 at 14:46
Can you link to the previous question you're referencing?
– Matthew Leingang
Aug 2 at 14:48
Can you link to the previous question you're referencing?
– Matthew Leingang
Aug 2 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 2 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 2 at 14:48
1
1
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 2 at 14:49
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 2 at 14:49
2
2
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 $xmapsto x^2$ 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 2 at 14:54
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 $xmapsto x^2$ 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 2 at 14:54
 |Â
show 8 more comments
6 Answers
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active
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1
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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...
answered Aug 2 at 14:44
gt6989b
30.2k22148
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 2 at 14:46
1
@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 2 at 14:49
@EthanChan tried to bring this down a little bit
– gt6989b
Aug 2 at 14:51
add a comment |Â
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0
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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.
answered Aug 2 at 14:54
John
21.9k32346
1
The book that you reference is likely Russell and Whitehead's Principia.
– Xander Henderson
Aug 2 at 14:54
@XanderHenderson Thank you!!
– John
Aug 2 at 15:01
1
Wikipedia has a pretty good article on it.
– Xander Henderson
Aug 2 at 15:03
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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.
answered Aug 2 at 14:58
Vasya
2,4601514
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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 $3.141ldots$) 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 $left(beginalign1\2endalignright)$ 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".
answered Aug 2 at 15:02
Jam
4,10111230
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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 $$-xne x,$$ 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 $xne 0$ 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.
answered Aug 2 at 14:49
Allawonder
1,274412
That does not follow from the field axioms. You just get $2x=0$, and there are fields of characteristic $2$. If $2neq 0$ in your particular field, then it follows from the field axioms.
– Kyle Miller
Aug 2 at 19:34
@KyleMiller I don't know why you think this needs to be pointed out, but if $xinmathbf C$ and $xne0,$ then doesn't it follow that $x+xne0$?
– Allawonder
Aug 2 at 21:12
Because it's ambiguous whether "say, the complex numbers $mathbfC$" 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 2 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 2 at 22:13
1
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 $mathbfC$ is characteristic $0$.
– Kyle Miller
Aug 2 at 23:29
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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 $0,1,1over2$,$sqrt3$ 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.
answered Aug 2 at 14:41
Love Invariants
77715
1
This exclude complex numbers.
– md2perpe
Aug 2 at 14:42
Yes, I was obviously telling about real numbers. Complex numbers are defined entirely different.
– Love Invariants
Aug 2 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 2 at 14:44
1
$-x$ 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 2 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 2 at 14:55
 |Â
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6 Answers
6
active
oldest
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6 Answers
6
active
oldest
votes
active
oldest
votes
active
oldest
votes
up vote
1
down vote
accepted
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...
answered Aug 2 at 14:44
gt6989b
30.2k22148
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 2 at 14:46
1
@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 2 at 14:49
@EthanChan tried to bring this down a little bit
– gt6989b
Aug 2 at 14:51
add a comment |Â
up vote
1
down vote
accepted
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...
answered Aug 2 at 14:44
gt6989b
30.2k22148
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 2 at 14:46
1
@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 2 at 14:49
@EthanChan tried to bring this down a little bit
– gt6989b
Aug 2 at 14:51
add a comment |Â
up vote
1
down vote
accepted
up vote
1
down vote
accepted
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...
answered Aug 2 at 14:44
gt6989b
30.2k22148
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...
answered Aug 2 at 14:44
gt6989b
30.2k22148
edited Aug 2 at 14:51
answered Aug 2 at 14:44
gt6989b
30.2k22148
answered Aug 2 at 14:44
gt6989b
30.2k22148
answered Aug 2 at 14:44
gt6989b
30.2k22148
30.2k22148
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 2 at 14:46
1
@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 2 at 14:49
@EthanChan tried to bring this down a little bit
– gt6989b
Aug 2 at 14:51
add a comment |Â
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 2 at 14:46
1
@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 2 at 14:49
@EthanChan tried to bring this down a little bit
– gt6989b
Aug 2 at 14:51
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 2 at 14:46
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 2 at 14:46
1
1
@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 2 at 14:49
@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 2 at 14:49
@EthanChan tried to bring this down a little bit
– gt6989b
Aug 2 at 14:51
@EthanChan tried to bring this down a little bit
– gt6989b
Aug 2 at 14:51
add a comment |Â
up vote
0
down vote
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.
answered Aug 2 at 14:54
John
21.9k32346
1
The book that you reference is likely Russell and Whitehead's Principia.
– Xander Henderson
Aug 2 at 14:54
@XanderHenderson Thank you!!
– John
Aug 2 at 15:01
1
Wikipedia has a pretty good article on it.
– Xander Henderson
Aug 2 at 15:03
add a comment |Â
up vote
0
down vote
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.
answered Aug 2 at 14:54
John
21.9k32346
1
The book that you reference is likely Russell and Whitehead's Principia.
– Xander Henderson
Aug 2 at 14:54
@XanderHenderson Thank you!!
– John
Aug 2 at 15:01
1
Wikipedia has a pretty good article on it.
– Xander Henderson
Aug 2 at 15:03
add a comment |Â
up vote
0
down vote
up vote
0
down vote
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.
answered Aug 2 at 14:54
John
21.9k32346
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.
answered Aug 2 at 14:54
John
21.9k32346
edited Aug 2 at 14:57
answered Aug 2 at 14:54
John
21.9k32346
answered Aug 2 at 14:54
John
21.9k32346
answered Aug 2 at 14:54
John
21.9k32346
21.9k32346
1
The book that you reference is likely Russell and Whitehead's Principia.
– Xander Henderson
Aug 2 at 14:54
@XanderHenderson Thank you!!
– John
Aug 2 at 15:01
1
Wikipedia has a pretty good article on it.
– Xander Henderson
Aug 2 at 15:03
add a comment |Â
1
The book that you reference is likely Russell and Whitehead's Principia.
– Xander Henderson
Aug 2 at 14:54
@XanderHenderson Thank you!!
– John
Aug 2 at 15:01
1
Wikipedia has a pretty good article on it.
– Xander Henderson
Aug 2 at 15:03
1
1
The book that you reference is likely Russell and Whitehead's Principia.
– Xander Henderson
Aug 2 at 14:54
The book that you reference is likely Russell and Whitehead's Principia.
– Xander Henderson
Aug 2 at 14:54
@XanderHenderson Thank you!!
– John
Aug 2 at 15:01
@XanderHenderson Thank you!!
– John
Aug 2 at 15:01
1
1
Wikipedia has a pretty good article on it.
– Xander Henderson
Aug 2 at 15:03
Wikipedia has a pretty good article on it.
– Xander Henderson
Aug 2 at 15:03
add a comment |Â
up vote
0
down vote
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.
answered Aug 2 at 14:58
Vasya
2,4601514
add a comment |Â
up vote
0
down vote
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.
answered Aug 2 at 14:58
Vasya
2,4601514
add a comment |Â
up vote
0
down vote
up vote
0
down vote
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.
answered Aug 2 at 14:58
Vasya
2,4601514
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.
answered Aug 2 at 14:58
Vasya
2,4601514
answered Aug 2 at 14:58
Vasya
2,4601514
answered Aug 2 at 14:58
Vasya
2,4601514
answered Aug 2 at 14:58
Vasya
2,4601514
2,4601514
add a comment |Â
add a comment |Â
up vote
0
down vote
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 $3.141ldots$) 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 $left(beginalign1\2endalignright)$ 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".
answered Aug 2 at 15:02
Jam
4,10111230
add a comment |Â
up vote
0
down vote
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 $3.141ldots$) 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 $left(beginalign1\2endalignright)$ 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".
answered Aug 2 at 15:02
Jam
4,10111230
add a comment |Â
up vote
0
down vote
up vote
0
down vote
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 $3.141ldots$) 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 $left(beginalign1\2endalignright)$ 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".
answered Aug 2 at 15:02
Jam
4,10111230
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 $3.141ldots$) 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 $left(beginalign1\2endalignright)$ 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".
answered Aug 2 at 15:02
Jam
4,10111230
answered Aug 2 at 15:02
Jam
4,10111230
answered Aug 2 at 15:02
Jam
4,10111230
answered Aug 2 at 15:02
Jam
4,10111230
4,10111230
add a comment |Â
add a comment |Â
up vote
0
down vote
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 $$-xne x,$$ 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 $xne 0$ 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.
answered Aug 2 at 14:49
Allawonder
1,274412
That does not follow from the field axioms. You just get $2x=0$, and there are fields of characteristic $2$. If $2neq 0$ in your particular field, then it follows from the field axioms.
– Kyle Miller
Aug 2 at 19:34
@KyleMiller I don't know why you think this needs to be pointed out, but if $xinmathbf C$ and $xne0,$ then doesn't it follow that $x+xne0$?
– Allawonder
Aug 2 at 21:12
Because it's ambiguous whether "say, the complex numbers $mathbfC$" 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 2 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 2 at 22:13
1
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 $mathbfC$ is characteristic $0$.
– Kyle Miller
Aug 2 at 23:29
 |Â
show 1 more comment
up vote
0
down vote
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 $$-xne x,$$ 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 $xne 0$ 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.
answered Aug 2 at 14:49
Allawonder
1,274412
That does not follow from the field axioms. You just get $2x=0$, and there are fields of characteristic $2$. If $2neq 0$ in your particular field, then it follows from the field axioms.
– Kyle Miller
Aug 2 at 19:34
@KyleMiller I don't know why you think this needs to be pointed out, but if $xinmathbf C$ and $xne0,$ then doesn't it follow that $x+xne0$?
– Allawonder
Aug 2 at 21:12
Because it's ambiguous whether "say, the complex numbers $mathbfC$" 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 2 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 2 at 22:13
1
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 $mathbfC$ is characteristic $0$.
– Kyle Miller
Aug 2 at 23:29
 |Â
show 1 more comment
up vote
0
down vote
up vote
0
down vote
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 $$-xne x,$$ 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 $xne 0$ 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.
answered Aug 2 at 14:49
Allawonder
1,274412
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 $$-xne x,$$ 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 $xne 0$ 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.
answered Aug 2 at 14:49
Allawonder
1,274412
edited Aug 3 at 13:50
answered Aug 2 at 14:49
Allawonder
1,274412
answered Aug 2 at 14:49
Allawonder
1,274412
answered Aug 2 at 14:49
Allawonder
1,274412
1,274412
That does not follow from the field axioms. You just get $2x=0$, and there are fields of characteristic $2$. If $2neq 0$ in your particular field, then it follows from the field axioms.
– Kyle Miller
Aug 2 at 19:34
@KyleMiller I don't know why you think this needs to be pointed out, but if $xinmathbf C$ and $xne0,$ then doesn't it follow that $x+xne0$?
– Allawonder
Aug 2 at 21:12
Because it's ambiguous whether "say, the complex numbers $mathbfC$" 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 2 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 2 at 22:13
1
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 $mathbfC$ is characteristic $0$.
– Kyle Miller
Aug 2 at 23:29
 |Â
show 1 more comment
That does not follow from the field axioms. You just get $2x=0$, and there are fields of characteristic $2$. If $2neq 0$ in your particular field, then it follows from the field axioms.
– Kyle Miller
Aug 2 at 19:34
@KyleMiller I don't know why you think this needs to be pointed out, but if $xinmathbf C$ and $xne0,$ then doesn't it follow that $x+xne0$?
– Allawonder
Aug 2 at 21:12
Because it's ambiguous whether "say, the complex numbers $mathbfC$" 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 2 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 2 at 22:13
1
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 $mathbfC$ is characteristic $0$.
– Kyle Miller
Aug 2 at 23:29
That does not follow from the field axioms. You just get $2x=0$, and there are fields of characteristic $2$. If $2neq 0$ in your particular field, then it follows from the field axioms.
– Kyle Miller
Aug 2 at 19:34
That does not follow from the field axioms. You just get $2x=0$, and there are fields of characteristic $2$. If $2neq 0$ in your particular field, then it follows from the field axioms.
– Kyle Miller
Aug 2 at 19:34
@KyleMiller I don't know why you think this needs to be pointed out, but if $xinmathbf C$ and $xne0,$ then doesn't it follow that $x+xne0$?
– Allawonder
Aug 2 at 21:12
@KyleMiller I don't know why you think this needs to be pointed out, but if $xinmathbf C$ and $xne0,$ then doesn't it follow that $x+xne0$?
– Allawonder
Aug 2 at 21:12
Because it's ambiguous whether "say, the complex numbers $mathbfC$" 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 2 at 21:50
Because it's ambiguous whether "say, the complex numbers $mathbfC$" 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 2 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 2 at 22:13
@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 2 at 22:13
1
1
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 $mathbfC$ is characteristic $0$.
– Kyle Miller
Aug 2 at 23:29
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 $mathbfC$ is characteristic $0$.
– Kyle Miller
Aug 2 at 23:29
 |Â
show 1 more comment
up vote
0
down vote
Number are part of a real number line which has $2$ fixed points as a reference. These two points can be $0,1,1over2$,$sqrt3$ 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.
answered Aug 2 at 14:41
Love Invariants
77715
1
This exclude complex numbers.
– md2perpe
Aug 2 at 14:42
Yes, I was obviously telling about real numbers. Complex numbers are defined entirely different.
– Love Invariants
Aug 2 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 2 at 14:44
1
$-x$ 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 2 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 2 at 14:55
 |Â
show 4 more comments
up vote
0
down vote
Number are part of a real number line which has $2$ fixed points as a reference. These two points can be $0,1,1over2$,$sqrt3$ 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.
answered Aug 2 at 14:41
Love Invariants
77715
1
This exclude complex numbers.
– md2perpe
Aug 2 at 14:42
Yes, I was obviously telling about real numbers. Complex numbers are defined entirely different.
– Love Invariants
Aug 2 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 2 at 14:44
1
$-x$ 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 2 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 2 at 14:55
 |Â
show 4 more comments
up vote
0
down vote
up vote
0
down vote
Number are part of a real number line which has $2$ fixed points as a reference. These two points can be $0,1,1over2$,$sqrt3$ 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.
answered Aug 2 at 14:41
Love Invariants
77715
Number are part of a real number line which has $2$ fixed points as a reference. These two points can be $0,1,1over2$,$sqrt3$ 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.
answered Aug 2 at 14:41
Love Invariants
77715
edited Aug 3 at 19:09
answered Aug 2 at 14:41
Love Invariants
77715
answered Aug 2 at 14:41
Love Invariants
77715
answered Aug 2 at 14:41
Love Invariants
77715
77715
1
This exclude complex numbers.
– md2perpe
Aug 2 at 14:42
Yes, I was obviously telling about real numbers. Complex numbers are defined entirely different.
– Love Invariants
Aug 2 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 2 at 14:44
1
$-x$ 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 2 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 2 at 14:55
 |Â
show 4 more comments
1
This exclude complex numbers.
– md2perpe
Aug 2 at 14:42
Yes, I was obviously telling about real numbers. Complex numbers are defined entirely different.
– Love Invariants
Aug 2 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 2 at 14:44
1
$-x$ 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 2 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 2 at 14:55
1
1
This exclude complex numbers.
– md2perpe
Aug 2 at 14:42
This exclude complex numbers.
– md2perpe
Aug 2 at 14:42
Yes, I was obviously telling about real numbers. Complex numbers are defined entirely different.
– Love Invariants
Aug 2 at 14:42
Yes, I was obviously telling about real numbers. Complex numbers are defined entirely different.
– Love Invariants
Aug 2 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 2 at 14:44
@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 2 at 14:44
1
1
$-x$ 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 2 at 14:46
$-x$ 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 2 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 2 at 14:55
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 2 at 14:55
 |Â
show 4 more comments
1
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 2 at 14:46
Can you link to the previous question you're referencing?
– Matthew Leingang
Aug 2 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 2 at 14:48
1
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 2 at 14:49
2
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 $xmapsto x^2$ 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 2 at 14:54