Can infinity be divided by anything? [on hold] Announcing the arrival of Valued Associate #679: Cesar Manara Planned maintenance scheduled April 17/18, 2019 at 00:00UTC (8:00pm US/Eastern)Related Rates Problem about moving shadow'Meaning' of a triple integral with $f(x,y,z)neq 1$What kinds of things are infinite series useful for?What allows us divide/multiply dx in calculus?Taylor series questionProve that $int_-infty^infty fracoperatornameAi^2(x+a_n)x^2dx = 1$Given a finite sequence, can we always find a relation that generates that sequence?if $f>g$, can it be that $int^b_af =int^b_ag$ or $int^b_af >int^b_ag ?$How to calculate the limit $lim_ntoinfty fracsqrtn+1 (1+n^2)sqrtn (2 +n^2+2n)$? Is my reasoning correct?How can I calculate this limit at infinity?
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Can infinity be divided by anything? [on hold]
Announcing the arrival of Valued Associate #679: Cesar Manara
Planned maintenance scheduled April 17/18, 2019 at 00:00UTC (8:00pm US/Eastern)Related Rates Problem about moving shadow'Meaning' of a triple integral with $f(x,y,z)neq 1$What kinds of things are infinite series useful for?What allows us divide/multiply dx in calculus?Taylor series questionProve that $int_-infty^infty fracoperatornameAi^2(x+a_n)x^2dx = 1$Given a finite sequence, can we always find a relation that generates that sequence?if $f>g$, can it be that $int^b_af =int^b_ag$ or $int^b_af >int^b_ag ?$How to calculate the limit $lim_ntoinfty fracsqrtn+1 (1+n^2)sqrtn (2 +n^2+2n)$? Is my reasoning correct?How can I calculate this limit at infinity?
$begingroup$
I'm in ninth grade and I've been thinking about this for a while. It's related to a question that came to my mind, namely which is the highest number you can divide 11 by. Is "infinite" a number and is it the highest number by which 11 can divide it?
calculus
edited Apr 12 at 20:28
Ernie060
2,940719
asked Apr 12 at 14:37
IlearnmathnotknowitIlearnmathnotknowit
412
New contributor
Ilearnmathnotknowit is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
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put on hold as unclear what you're asking by blub, José Carlos Santos, YiFan, Lord Shark the Unknown, Jam 6 hours ago
Please clarify your specific problem or add additional details to highlight exactly what you need. As it's currently written, it’s hard to tell exactly what you're asking. 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.
|
show 12 more comments
$begingroup$
I'm in ninth grade and I've been thinking about this for a while. It's related to a question that came to my mind, namely which is the highest number you can divide 11 by. Is "infinite" a number and is it the highest number by which 11 can divide it?
calculus
edited Apr 12 at 20:28
Ernie060
2,940719
asked Apr 12 at 14:37
IlearnmathnotknowitIlearnmathnotknowit
412
New contributor
Ilearnmathnotknowit is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
$endgroup$
put on hold as unclear what you're asking by blub, José Carlos Santos, YiFan, Lord Shark the Unknown, Jam 6 hours ago
Please clarify your specific problem or add additional details to highlight exactly what you need. As it's currently written, it’s hard to tell exactly what you're asking. 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.
17
$begingroup$
Infinity is not a number.
$endgroup$
– Sean Roberson
Apr 12 at 14:38
2
$begingroup$
@SeanRoberson What is a number for you to be saying that?
$endgroup$
– user647486
Apr 12 at 14:41
2
$begingroup$
@user647486 Care to justify your statement?
$endgroup$
– QuantumPenguin
Apr 12 at 14:46
6
$begingroup$
There is an arithmetic for the extended real numbers (real numbers with $pminfty.$) You can see its arithmetic here: en.wikipedia.org/wiki/….
$endgroup$
– Adrian Keister
Apr 12 at 14:50
7
$begingroup$
Your title asks about dividing infinity by a number, then in your question you start out asking about dividing eleven by something, then you go back to dividing something by eleven. Dividing something by eleven and dividing eleven by something are very different, and you need to more clearly distinguish between the two.
$endgroup$
– Acccumulation
Apr 12 at 16:05
|
show 12 more comments
$begingroup$
I'm in ninth grade and I've been thinking about this for a while. It's related to a question that came to my mind, namely which is the highest number you can divide 11 by. Is "infinite" a number and is it the highest number by which 11 can divide it?
calculus
edited Apr 12 at 20:28
Ernie060
2,940719
asked Apr 12 at 14:37
IlearnmathnotknowitIlearnmathnotknowit
412
New contributor
Ilearnmathnotknowit is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
$endgroup$
I'm in ninth grade and I've been thinking about this for a while. It's related to a question that came to my mind, namely which is the highest number you can divide 11 by. Is "infinite" a number and is it the highest number by which 11 can divide it?
calculus
calculus
edited Apr 12 at 20:28
Ernie060
2,940719
asked Apr 12 at 14:37
IlearnmathnotknowitIlearnmathnotknowit
412
New contributor
Ilearnmathnotknowit is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
edited Apr 12 at 20:28
Ernie060
2,940719
asked Apr 12 at 14:37
IlearnmathnotknowitIlearnmathnotknowit
412
New contributor
Ilearnmathnotknowit is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
edited Apr 12 at 20:28
Ernie060
2,940719
edited Apr 12 at 20:28
Ernie060
2,940719
edited Apr 12 at 20:28
Ernie060
2,940719
2,940719
asked Apr 12 at 14:37
IlearnmathnotknowitIlearnmathnotknowit
412
New contributor
Ilearnmathnotknowit is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
asked Apr 12 at 14:37
IlearnmathnotknowitIlearnmathnotknowit
412
asked Apr 12 at 14:37
IlearnmathnotknowitIlearnmathnotknowit
412
412
New contributor
Ilearnmathnotknowit is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
New contributor
Ilearnmathnotknowit is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
Ilearnmathnotknowit is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
put on hold as unclear what you're asking by blub, José Carlos Santos, YiFan, Lord Shark the Unknown, Jam 6 hours ago
Please clarify your specific problem or add additional details to highlight exactly what you need. As it's currently written, it’s hard to tell exactly what you're asking. 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.
put on hold as unclear what you're asking by blub, José Carlos Santos, YiFan, Lord Shark the Unknown, Jam 6 hours ago
Please clarify your specific problem or add additional details to highlight exactly what you need. As it's currently written, it’s hard to tell exactly what you're asking. 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.
17
$begingroup$
Infinity is not a number.
$endgroup$
– Sean Roberson
Apr 12 at 14:38
2
$begingroup$
@SeanRoberson What is a number for you to be saying that?
$endgroup$
– user647486
Apr 12 at 14:41
2
$begingroup$
@user647486 Care to justify your statement?
$endgroup$
– QuantumPenguin
Apr 12 at 14:46
6
$begingroup$
There is an arithmetic for the extended real numbers (real numbers with $pminfty.$) You can see its arithmetic here: en.wikipedia.org/wiki/….
$endgroup$
– Adrian Keister
Apr 12 at 14:50
7
$begingroup$
Your title asks about dividing infinity by a number, then in your question you start out asking about dividing eleven by something, then you go back to dividing something by eleven. Dividing something by eleven and dividing eleven by something are very different, and you need to more clearly distinguish between the two.
$endgroup$
– Acccumulation
Apr 12 at 16:05
|
show 12 more comments
17
$begingroup$
Infinity is not a number.
$endgroup$
– Sean Roberson
Apr 12 at 14:38
2
$begingroup$
@SeanRoberson What is a number for you to be saying that?
$endgroup$
– user647486
Apr 12 at 14:41
2
$begingroup$
@user647486 Care to justify your statement?
$endgroup$
– QuantumPenguin
Apr 12 at 14:46
6
$begingroup$
There is an arithmetic for the extended real numbers (real numbers with $pminfty.$) You can see its arithmetic here: en.wikipedia.org/wiki/….
$endgroup$
– Adrian Keister
Apr 12 at 14:50
7
$begingroup$
Your title asks about dividing infinity by a number, then in your question you start out asking about dividing eleven by something, then you go back to dividing something by eleven. Dividing something by eleven and dividing eleven by something are very different, and you need to more clearly distinguish between the two.
$endgroup$
– Acccumulation
Apr 12 at 16:05
17
17
$begingroup$
Infinity is not a number.
$endgroup$
– Sean Roberson
Apr 12 at 14:38
$begingroup$
Infinity is not a number.
$endgroup$
– Sean Roberson
Apr 12 at 14:38
2
2
$begingroup$
@SeanRoberson What is a number for you to be saying that?
$endgroup$
– user647486
Apr 12 at 14:41
$begingroup$
@SeanRoberson What is a number for you to be saying that?
$endgroup$
– user647486
Apr 12 at 14:41
2
2
$begingroup$
@user647486 Care to justify your statement?
$endgroup$
– QuantumPenguin
Apr 12 at 14:46
$begingroup$
@user647486 Care to justify your statement?
$endgroup$
– QuantumPenguin
Apr 12 at 14:46
6
6
$begingroup$
There is an arithmetic for the extended real numbers (real numbers with $pminfty.$) You can see its arithmetic here: en.wikipedia.org/wiki/….
$endgroup$
– Adrian Keister
Apr 12 at 14:50
$begingroup$
There is an arithmetic for the extended real numbers (real numbers with $pminfty.$) You can see its arithmetic here: en.wikipedia.org/wiki/….
$endgroup$
– Adrian Keister
Apr 12 at 14:50
7
7
$begingroup$
Your title asks about dividing infinity by a number, then in your question you start out asking about dividing eleven by something, then you go back to dividing something by eleven. Dividing something by eleven and dividing eleven by something are very different, and you need to more clearly distinguish between the two.
$endgroup$
– Acccumulation
Apr 12 at 16:05
$begingroup$
Your title asks about dividing infinity by a number, then in your question you start out asking about dividing eleven by something, then you go back to dividing something by eleven. Dividing something by eleven and dividing eleven by something are very different, and you need to more clearly distinguish between the two.
$endgroup$
– Acccumulation
Apr 12 at 16:05
|
show 12 more comments
6 Answers
6
active
oldest
votes
$begingroup$
Thinking and playing with infinity is a popular thing to do, especially for people your age. To do so correctly however, you need to understand what we actually mean by infinity and what we mean by a "number."
Note: For the remainder of this post, by "division" and "divided by" I will be referring to the usual division operator rather than the related concepts from number theory.
Infinity is not a "real number" (that is not to say it doesn't exist in some contexts, but it does not belong to the set of numbers known as 'the real numbers') so we are not even allowed to refer to operations involving infinity. It just flat out doesn't make sense in this context. There is no largest "real number" and so there is no "highest (real) number" that $11$ can be divided by.
Since you are talking about "dividing by infinity" then you are probably working in the extended real numbers rather than the real numbers. In such a context, yes infinity is the "largest" extended real number and $11$ can indeed be divided by it. In fact, every real number when treated as an extended real number instead can be divided by infinty and the result would be zero. In that context, yes... you are correct.
The extended real numbers however is not the usual context to be working in. If the question was asked "does there exist a largest number that $11$ can be divided by" without any additional clarification about context then the answer would be no.
(Note: infinity and negative infinity can not be divided by infinity)
answered Apr 12 at 14:55
JMoravitzJMoravitz
49.2k44091
$endgroup$
add a comment |
$begingroup$
The number $11$ can be divided by anything - real or complex. If you mean what is the greatest integer - $x$ - such that
$$11equiv 0modx$$
In other words, $11$ divided by $x$ leaves no remainder or fractional part, then the solution would be $x=11$.
answered Apr 12 at 14:41
Peter ForemanPeter Foreman
7,8921320
$endgroup$
3
$begingroup$
small suggestion : you may want to rewrite11 ≡ 0 mod x, OP may not understand that notation (at least I wouldn't on 9th grade)
$endgroup$
– Rod
Apr 12 at 17:18
add a comment |
$begingroup$
This was getting too long for a comment, but it isn't really an answer - more an encouragement to hang in there with the kind of question you are asking.
The question "what is a number?" has caused many mathematicians to reflect on the basic material they work with and how it is defined. It leads to rich and fundamental questions about the foundation of mathematics.
Combined with geometric insights we can extend familiar numbers/systems to obtain projective spaces or the Riemann Sphere, both of which accommodate notions of infinity.
Algebraically it leads to reflection on what properties we expect our numbers to have, and the kinds of mathematical objects which have those properties.
And you might want to look up Cantor's diagonal argument and reflection on how infinite sets don't always behave in accord with our intuition. (An infinite set can be put in one-to-one correspondence with a proper subset of itself - we can match the integers with the even integers by pairing $n$ with $2n$)
It is possible to create systems of numbers in which the existence of infinite numbers is consistent with arithmetical operations - JH Conway's "Surreal Numbers" are an example.
To return a bit to the question:
If you are working with non-negative integers, then there is no infinite integer, and you can't divide by something which doesn't exist.
If you were working within the rational numbers (fractions), you could divide by any rational number (including any integer) except for zero. That would give you a valid fraction as an answer.
The non-negative integers and the rational numbers are systems of numbers which are carefully constructed to obey clearly stated rules - the need for careful definition arises because of the kind of question you are asking.
answered Apr 12 at 15:03
Mark BennetMark Bennet
82k984183
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5
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+1. Your answer has the flavor of an encouragement. If you want to encourage the asker then you may also consider "liking" the question. [I am always amazed when several people give very long answers to a question that they do not "like." So far, I am the only person who has "liked" (+1d) the question.]
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– Michael
Apr 12 at 15:36
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PS: In the comment trail of the question, my son wrote his comment using my account name "Michael." I encouraged him to look at the question because I thought he would appreciate it.
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– Michael
Apr 12 at 15:56
add a comment |
$begingroup$
Julia thinks ∞ is evenly divisible by everything. Julia got ∞ by Π ℙ = ∞ (that is, the product of all prime numbers).
Anna thinks ∞ is evenly divisible by only by 1. Anna got ∞ by Π ℙ + 1 = ∞ (that is, one more than the product of all prime numbers).
∞ is not a number. Trying to treat it like one will eventually cause a headache.
(If you're having trouble getting the reference, there was this famous question I cannot find anymore that had five students giving five different answers to a "what is the next number in this sequence" question using five different fourth order polynomials.)
answered Apr 12 at 17:23
JoshuaJoshua
333210
$endgroup$
add a comment |
$begingroup$
It all depends on the defintions of $infty$, "number" and "divides".
I will assume that you're asked what is the greatest integer number dividing 11 then basically in any sensible interpretation of words $infty$, "integer number" and "divides" the answer to your question will be "11". As far as I know, $infty$ is always interpreted distinct from the usual numbers (by usual I mean real numbers) (including integers), that is, you have an axiom (or a theorem)
$$forall x in mathbbZ , x neq infty.$$ Hence $infty$ could not possibly be the greatest integer number dividing 11 since it then would be an integer contradicting this axiom (or a theorem, whatever you wanna call it)
edited Apr 12 at 16:01
Eric Towers
33.9k22371
answered Apr 12 at 14:54
famesyasdfamesyasd
349313
$endgroup$
add a comment |
$begingroup$
It is really great to be thinking of stuff like this.
You will find people all around you who will say infinity is not a number and they will stop there, but there are a set of math classes in your future (Calculus) that are all about dealing with numbers that are virtually infinite--they call them limits.
All those people who say that it's absolutely impossible to deal with infinity will find it perfectly fine to deal with "x as x approaches infinity" and you get the same results you would intuit for infinity (Division by infinity is still a bit tricky if I recall correctly)
Anyway, if you have ever noticed that math is often about describing things, calculus is how you describe strange (arbitrary) curves and such--Exactly how much water fits into your strangely shaped sink, or how long will it take to drain said water since the speed it drains is slower as there is less water in the sink.
This kind of thing is calculated by slicing up curves into infinitely (Approaching infinite anyway) thin slices and adding the slices together to get an exact result.
(PS, I haven't done much calculus in decades, if this is factually wrong please feel free to fix it rather than just blindly downvoting. Thanks)
answered Apr 12 at 18:12
Bill KBill K
1274
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$begingroup$
I don't think it's a great idea to compare limits to infinity. Limits are (usually) real numbers with fairly easily definable properties, unlike infinity.
$endgroup$
– Jam
6 hours ago
add a comment |
6 Answers
6
active
oldest
votes
6 Answers
6
active
oldest
votes
active
oldest
votes
active
oldest
votes
$begingroup$
Thinking and playing with infinity is a popular thing to do, especially for people your age. To do so correctly however, you need to understand what we actually mean by infinity and what we mean by a "number."
Note: For the remainder of this post, by "division" and "divided by" I will be referring to the usual division operator rather than the related concepts from number theory.
Infinity is not a "real number" (that is not to say it doesn't exist in some contexts, but it does not belong to the set of numbers known as 'the real numbers') so we are not even allowed to refer to operations involving infinity. It just flat out doesn't make sense in this context. There is no largest "real number" and so there is no "highest (real) number" that $11$ can be divided by.
Since you are talking about "dividing by infinity" then you are probably working in the extended real numbers rather than the real numbers. In such a context, yes infinity is the "largest" extended real number and $11$ can indeed be divided by it. In fact, every real number when treated as an extended real number instead can be divided by infinty and the result would be zero. In that context, yes... you are correct.
The extended real numbers however is not the usual context to be working in. If the question was asked "does there exist a largest number that $11$ can be divided by" without any additional clarification about context then the answer would be no.
(Note: infinity and negative infinity can not be divided by infinity)
answered Apr 12 at 14:55
JMoravitzJMoravitz
49.2k44091
$endgroup$
add a comment |
$begingroup$
Thinking and playing with infinity is a popular thing to do, especially for people your age. To do so correctly however, you need to understand what we actually mean by infinity and what we mean by a "number."
Note: For the remainder of this post, by "division" and "divided by" I will be referring to the usual division operator rather than the related concepts from number theory.
Infinity is not a "real number" (that is not to say it doesn't exist in some contexts, but it does not belong to the set of numbers known as 'the real numbers') so we are not even allowed to refer to operations involving infinity. It just flat out doesn't make sense in this context. There is no largest "real number" and so there is no "highest (real) number" that $11$ can be divided by.
Since you are talking about "dividing by infinity" then you are probably working in the extended real numbers rather than the real numbers. In such a context, yes infinity is the "largest" extended real number and $11$ can indeed be divided by it. In fact, every real number when treated as an extended real number instead can be divided by infinty and the result would be zero. In that context, yes... you are correct.
The extended real numbers however is not the usual context to be working in. If the question was asked "does there exist a largest number that $11$ can be divided by" without any additional clarification about context then the answer would be no.
(Note: infinity and negative infinity can not be divided by infinity)
answered Apr 12 at 14:55
JMoravitzJMoravitz
49.2k44091
$endgroup$
add a comment |
$begingroup$
Thinking and playing with infinity is a popular thing to do, especially for people your age. To do so correctly however, you need to understand what we actually mean by infinity and what we mean by a "number."
Note: For the remainder of this post, by "division" and "divided by" I will be referring to the usual division operator rather than the related concepts from number theory.
Infinity is not a "real number" (that is not to say it doesn't exist in some contexts, but it does not belong to the set of numbers known as 'the real numbers') so we are not even allowed to refer to operations involving infinity. It just flat out doesn't make sense in this context. There is no largest "real number" and so there is no "highest (real) number" that $11$ can be divided by.
Since you are talking about "dividing by infinity" then you are probably working in the extended real numbers rather than the real numbers. In such a context, yes infinity is the "largest" extended real number and $11$ can indeed be divided by it. In fact, every real number when treated as an extended real number instead can be divided by infinty and the result would be zero. In that context, yes... you are correct.
The extended real numbers however is not the usual context to be working in. If the question was asked "does there exist a largest number that $11$ can be divided by" without any additional clarification about context then the answer would be no.
(Note: infinity and negative infinity can not be divided by infinity)
answered Apr 12 at 14:55
JMoravitzJMoravitz
49.2k44091
$endgroup$
Thinking and playing with infinity is a popular thing to do, especially for people your age. To do so correctly however, you need to understand what we actually mean by infinity and what we mean by a "number."
Note: For the remainder of this post, by "division" and "divided by" I will be referring to the usual division operator rather than the related concepts from number theory.
Infinity is not a "real number" (that is not to say it doesn't exist in some contexts, but it does not belong to the set of numbers known as 'the real numbers') so we are not even allowed to refer to operations involving infinity. It just flat out doesn't make sense in this context. There is no largest "real number" and so there is no "highest (real) number" that $11$ can be divided by.
Since you are talking about "dividing by infinity" then you are probably working in the extended real numbers rather than the real numbers. In such a context, yes infinity is the "largest" extended real number and $11$ can indeed be divided by it. In fact, every real number when treated as an extended real number instead can be divided by infinty and the result would be zero. In that context, yes... you are correct.
The extended real numbers however is not the usual context to be working in. If the question was asked "does there exist a largest number that $11$ can be divided by" without any additional clarification about context then the answer would be no.
(Note: infinity and negative infinity can not be divided by infinity)
answered Apr 12 at 14:55
JMoravitzJMoravitz
49.2k44091
answered Apr 12 at 14:55
JMoravitzJMoravitz
49.2k44091
answered Apr 12 at 14:55
JMoravitzJMoravitz
49.2k44091
answered Apr 12 at 14:55
JMoravitzJMoravitz
49.2k44091
49.2k44091
add a comment |
add a comment |
$begingroup$
The number $11$ can be divided by anything - real or complex. If you mean what is the greatest integer - $x$ - such that
$$11equiv 0modx$$
In other words, $11$ divided by $x$ leaves no remainder or fractional part, then the solution would be $x=11$.
answered Apr 12 at 14:41
Peter ForemanPeter Foreman
7,8921320
$endgroup$
3
$begingroup$
small suggestion : you may want to rewrite11 ≡ 0 mod x, OP may not understand that notation (at least I wouldn't on 9th grade)
$endgroup$
– Rod
Apr 12 at 17:18
add a comment |
$begingroup$
The number $11$ can be divided by anything - real or complex. If you mean what is the greatest integer - $x$ - such that
$$11equiv 0modx$$
In other words, $11$ divided by $x$ leaves no remainder or fractional part, then the solution would be $x=11$.
answered Apr 12 at 14:41
Peter ForemanPeter Foreman
7,8921320
$endgroup$
3
$begingroup$
small suggestion : you may want to rewrite11 ≡ 0 mod x, OP may not understand that notation (at least I wouldn't on 9th grade)
$endgroup$
– Rod
Apr 12 at 17:18
add a comment |
$begingroup$
The number $11$ can be divided by anything - real or complex. If you mean what is the greatest integer - $x$ - such that
$$11equiv 0modx$$
In other words, $11$ divided by $x$ leaves no remainder or fractional part, then the solution would be $x=11$.
answered Apr 12 at 14:41
Peter ForemanPeter Foreman
7,8921320
$endgroup$
The number $11$ can be divided by anything - real or complex. If you mean what is the greatest integer - $x$ - such that
$$11equiv 0modx$$
In other words, $11$ divided by $x$ leaves no remainder or fractional part, then the solution would be $x=11$.
answered Apr 12 at 14:41
Peter ForemanPeter Foreman
7,8921320
edited Apr 12 at 18:20
answered Apr 12 at 14:41
Peter ForemanPeter Foreman
7,8921320
answered Apr 12 at 14:41
Peter ForemanPeter Foreman
7,8921320
answered Apr 12 at 14:41
Peter ForemanPeter Foreman
7,8921320
7,8921320
3
$begingroup$
small suggestion : you may want to rewrite11 ≡ 0 mod x, OP may not understand that notation (at least I wouldn't on 9th grade)
$endgroup$
– Rod
Apr 12 at 17:18
add a comment |
3
$begingroup$
small suggestion : you may want to rewrite11 ≡ 0 mod x, OP may not understand that notation (at least I wouldn't on 9th grade)
$endgroup$
– Rod
Apr 12 at 17:18
3
3
$begingroup$
small suggestion : you may want to rewrite
11 ≡ 0 mod x, OP may not understand that notation (at least I wouldn't on 9th grade)$endgroup$
– Rod
Apr 12 at 17:18
$begingroup$
small suggestion : you may want to rewrite
11 ≡ 0 mod x, OP may not understand that notation (at least I wouldn't on 9th grade)$endgroup$
– Rod
Apr 12 at 17:18
add a comment |
$begingroup$
This was getting too long for a comment, but it isn't really an answer - more an encouragement to hang in there with the kind of question you are asking.
The question "what is a number?" has caused many mathematicians to reflect on the basic material they work with and how it is defined. It leads to rich and fundamental questions about the foundation of mathematics.
Combined with geometric insights we can extend familiar numbers/systems to obtain projective spaces or the Riemann Sphere, both of which accommodate notions of infinity.
Algebraically it leads to reflection on what properties we expect our numbers to have, and the kinds of mathematical objects which have those properties.
And you might want to look up Cantor's diagonal argument and reflection on how infinite sets don't always behave in accord with our intuition. (An infinite set can be put in one-to-one correspondence with a proper subset of itself - we can match the integers with the even integers by pairing $n$ with $2n$)
It is possible to create systems of numbers in which the existence of infinite numbers is consistent with arithmetical operations - JH Conway's "Surreal Numbers" are an example.
To return a bit to the question:
If you are working with non-negative integers, then there is no infinite integer, and you can't divide by something which doesn't exist.
If you were working within the rational numbers (fractions), you could divide by any rational number (including any integer) except for zero. That would give you a valid fraction as an answer.
The non-negative integers and the rational numbers are systems of numbers which are carefully constructed to obey clearly stated rules - the need for careful definition arises because of the kind of question you are asking.
answered Apr 12 at 15:03
Mark BennetMark Bennet
82k984183
$endgroup$
5
$begingroup$
+1. Your answer has the flavor of an encouragement. If you want to encourage the asker then you may also consider "liking" the question. [I am always amazed when several people give very long answers to a question that they do not "like." So far, I am the only person who has "liked" (+1d) the question.]
$endgroup$
– Michael
Apr 12 at 15:36
$begingroup$
PS: In the comment trail of the question, my son wrote his comment using my account name "Michael." I encouraged him to look at the question because I thought he would appreciate it.
$endgroup$
– Michael
Apr 12 at 15:56
add a comment |
$begingroup$
This was getting too long for a comment, but it isn't really an answer - more an encouragement to hang in there with the kind of question you are asking.
The question "what is a number?" has caused many mathematicians to reflect on the basic material they work with and how it is defined. It leads to rich and fundamental questions about the foundation of mathematics.
Combined with geometric insights we can extend familiar numbers/systems to obtain projective spaces or the Riemann Sphere, both of which accommodate notions of infinity.
Algebraically it leads to reflection on what properties we expect our numbers to have, and the kinds of mathematical objects which have those properties.
And you might want to look up Cantor's diagonal argument and reflection on how infinite sets don't always behave in accord with our intuition. (An infinite set can be put in one-to-one correspondence with a proper subset of itself - we can match the integers with the even integers by pairing $n$ with $2n$)
It is possible to create systems of numbers in which the existence of infinite numbers is consistent with arithmetical operations - JH Conway's "Surreal Numbers" are an example.
To return a bit to the question:
If you are working with non-negative integers, then there is no infinite integer, and you can't divide by something which doesn't exist.
If you were working within the rational numbers (fractions), you could divide by any rational number (including any integer) except for zero. That would give you a valid fraction as an answer.
The non-negative integers and the rational numbers are systems of numbers which are carefully constructed to obey clearly stated rules - the need for careful definition arises because of the kind of question you are asking.
answered Apr 12 at 15:03
Mark BennetMark Bennet
82k984183
$endgroup$
5
$begingroup$
+1. Your answer has the flavor of an encouragement. If you want to encourage the asker then you may also consider "liking" the question. [I am always amazed when several people give very long answers to a question that they do not "like." So far, I am the only person who has "liked" (+1d) the question.]
$endgroup$
– Michael
Apr 12 at 15:36
$begingroup$
PS: In the comment trail of the question, my son wrote his comment using my account name "Michael." I encouraged him to look at the question because I thought he would appreciate it.
$endgroup$
– Michael
Apr 12 at 15:56
add a comment |
$begingroup$
This was getting too long for a comment, but it isn't really an answer - more an encouragement to hang in there with the kind of question you are asking.
The question "what is a number?" has caused many mathematicians to reflect on the basic material they work with and how it is defined. It leads to rich and fundamental questions about the foundation of mathematics.
Combined with geometric insights we can extend familiar numbers/systems to obtain projective spaces or the Riemann Sphere, both of which accommodate notions of infinity.
Algebraically it leads to reflection on what properties we expect our numbers to have, and the kinds of mathematical objects which have those properties.
And you might want to look up Cantor's diagonal argument and reflection on how infinite sets don't always behave in accord with our intuition. (An infinite set can be put in one-to-one correspondence with a proper subset of itself - we can match the integers with the even integers by pairing $n$ with $2n$)
It is possible to create systems of numbers in which the existence of infinite numbers is consistent with arithmetical operations - JH Conway's "Surreal Numbers" are an example.
To return a bit to the question:
If you are working with non-negative integers, then there is no infinite integer, and you can't divide by something which doesn't exist.
If you were working within the rational numbers (fractions), you could divide by any rational number (including any integer) except for zero. That would give you a valid fraction as an answer.
The non-negative integers and the rational numbers are systems of numbers which are carefully constructed to obey clearly stated rules - the need for careful definition arises because of the kind of question you are asking.
answered Apr 12 at 15:03
Mark BennetMark Bennet
82k984183
$endgroup$
This was getting too long for a comment, but it isn't really an answer - more an encouragement to hang in there with the kind of question you are asking.
The question "what is a number?" has caused many mathematicians to reflect on the basic material they work with and how it is defined. It leads to rich and fundamental questions about the foundation of mathematics.
Combined with geometric insights we can extend familiar numbers/systems to obtain projective spaces or the Riemann Sphere, both of which accommodate notions of infinity.
Algebraically it leads to reflection on what properties we expect our numbers to have, and the kinds of mathematical objects which have those properties.
And you might want to look up Cantor's diagonal argument and reflection on how infinite sets don't always behave in accord with our intuition. (An infinite set can be put in one-to-one correspondence with a proper subset of itself - we can match the integers with the even integers by pairing $n$ with $2n$)
It is possible to create systems of numbers in which the existence of infinite numbers is consistent with arithmetical operations - JH Conway's "Surreal Numbers" are an example.
To return a bit to the question:
If you are working with non-negative integers, then there is no infinite integer, and you can't divide by something which doesn't exist.
If you were working within the rational numbers (fractions), you could divide by any rational number (including any integer) except for zero. That would give you a valid fraction as an answer.
The non-negative integers and the rational numbers are systems of numbers which are carefully constructed to obey clearly stated rules - the need for careful definition arises because of the kind of question you are asking.
answered Apr 12 at 15:03
Mark BennetMark Bennet
82k984183
answered Apr 12 at 15:03
Mark BennetMark Bennet
82k984183
answered Apr 12 at 15:03
Mark BennetMark Bennet
82k984183
answered Apr 12 at 15:03
Mark BennetMark Bennet
82k984183
82k984183
5
$begingroup$
+1. Your answer has the flavor of an encouragement. If you want to encourage the asker then you may also consider "liking" the question. [I am always amazed when several people give very long answers to a question that they do not "like." So far, I am the only person who has "liked" (+1d) the question.]
$endgroup$
– Michael
Apr 12 at 15:36
$begingroup$
PS: In the comment trail of the question, my son wrote his comment using my account name "Michael." I encouraged him to look at the question because I thought he would appreciate it.
$endgroup$
– Michael
Apr 12 at 15:56
add a comment |
5
$begingroup$
+1. Your answer has the flavor of an encouragement. If you want to encourage the asker then you may also consider "liking" the question. [I am always amazed when several people give very long answers to a question that they do not "like." So far, I am the only person who has "liked" (+1d) the question.]
$endgroup$
– Michael
Apr 12 at 15:36
$begingroup$
PS: In the comment trail of the question, my son wrote his comment using my account name "Michael." I encouraged him to look at the question because I thought he would appreciate it.
$endgroup$
– Michael
Apr 12 at 15:56
5
5
$begingroup$
+1. Your answer has the flavor of an encouragement. If you want to encourage the asker then you may also consider "liking" the question. [I am always amazed when several people give very long answers to a question that they do not "like." So far, I am the only person who has "liked" (+1d) the question.]
$endgroup$
– Michael
Apr 12 at 15:36
$begingroup$
+1. Your answer has the flavor of an encouragement. If you want to encourage the asker then you may also consider "liking" the question. [I am always amazed when several people give very long answers to a question that they do not "like." So far, I am the only person who has "liked" (+1d) the question.]
$endgroup$
– Michael
Apr 12 at 15:36
$begingroup$
PS: In the comment trail of the question, my son wrote his comment using my account name "Michael." I encouraged him to look at the question because I thought he would appreciate it.
$endgroup$
– Michael
Apr 12 at 15:56
$begingroup$
PS: In the comment trail of the question, my son wrote his comment using my account name "Michael." I encouraged him to look at the question because I thought he would appreciate it.
$endgroup$
– Michael
Apr 12 at 15:56
add a comment |
$begingroup$
Julia thinks ∞ is evenly divisible by everything. Julia got ∞ by Π ℙ = ∞ (that is, the product of all prime numbers).
Anna thinks ∞ is evenly divisible by only by 1. Anna got ∞ by Π ℙ + 1 = ∞ (that is, one more than the product of all prime numbers).
∞ is not a number. Trying to treat it like one will eventually cause a headache.
(If you're having trouble getting the reference, there was this famous question I cannot find anymore that had five students giving five different answers to a "what is the next number in this sequence" question using five different fourth order polynomials.)
answered Apr 12 at 17:23
JoshuaJoshua
333210
$endgroup$
add a comment |
$begingroup$
Julia thinks ∞ is evenly divisible by everything. Julia got ∞ by Π ℙ = ∞ (that is, the product of all prime numbers).
Anna thinks ∞ is evenly divisible by only by 1. Anna got ∞ by Π ℙ + 1 = ∞ (that is, one more than the product of all prime numbers).
∞ is not a number. Trying to treat it like one will eventually cause a headache.
(If you're having trouble getting the reference, there was this famous question I cannot find anymore that had five students giving five different answers to a "what is the next number in this sequence" question using five different fourth order polynomials.)
answered Apr 12 at 17:23
JoshuaJoshua
333210
$endgroup$
add a comment |
$begingroup$
Julia thinks ∞ is evenly divisible by everything. Julia got ∞ by Π ℙ = ∞ (that is, the product of all prime numbers).
Anna thinks ∞ is evenly divisible by only by 1. Anna got ∞ by Π ℙ + 1 = ∞ (that is, one more than the product of all prime numbers).
∞ is not a number. Trying to treat it like one will eventually cause a headache.
(If you're having trouble getting the reference, there was this famous question I cannot find anymore that had five students giving five different answers to a "what is the next number in this sequence" question using five different fourth order polynomials.)
answered Apr 12 at 17:23
JoshuaJoshua
333210
$endgroup$
Julia thinks ∞ is evenly divisible by everything. Julia got ∞ by Π ℙ = ∞ (that is, the product of all prime numbers).
Anna thinks ∞ is evenly divisible by only by 1. Anna got ∞ by Π ℙ + 1 = ∞ (that is, one more than the product of all prime numbers).
∞ is not a number. Trying to treat it like one will eventually cause a headache.
(If you're having trouble getting the reference, there was this famous question I cannot find anymore that had five students giving five different answers to a "what is the next number in this sequence" question using five different fourth order polynomials.)
answered Apr 12 at 17:23
JoshuaJoshua
333210
edited Apr 12 at 17:30
answered Apr 12 at 17:23
JoshuaJoshua
333210
answered Apr 12 at 17:23
JoshuaJoshua
333210
answered Apr 12 at 17:23
JoshuaJoshua
333210
333210
add a comment |
add a comment |
$begingroup$
It all depends on the defintions of $infty$, "number" and "divides".
I will assume that you're asked what is the greatest integer number dividing 11 then basically in any sensible interpretation of words $infty$, "integer number" and "divides" the answer to your question will be "11". As far as I know, $infty$ is always interpreted distinct from the usual numbers (by usual I mean real numbers) (including integers), that is, you have an axiom (or a theorem)
$$forall x in mathbbZ , x neq infty.$$ Hence $infty$ could not possibly be the greatest integer number dividing 11 since it then would be an integer contradicting this axiom (or a theorem, whatever you wanna call it)
edited Apr 12 at 16:01
Eric Towers
33.9k22371
answered Apr 12 at 14:54
famesyasdfamesyasd
349313
$endgroup$
add a comment |
$begingroup$
It all depends on the defintions of $infty$, "number" and "divides".
I will assume that you're asked what is the greatest integer number dividing 11 then basically in any sensible interpretation of words $infty$, "integer number" and "divides" the answer to your question will be "11". As far as I know, $infty$ is always interpreted distinct from the usual numbers (by usual I mean real numbers) (including integers), that is, you have an axiom (or a theorem)
$$forall x in mathbbZ , x neq infty.$$ Hence $infty$ could not possibly be the greatest integer number dividing 11 since it then would be an integer contradicting this axiom (or a theorem, whatever you wanna call it)
edited Apr 12 at 16:01
Eric Towers
33.9k22371
answered Apr 12 at 14:54
famesyasdfamesyasd
349313
$endgroup$
add a comment |
$begingroup$
It all depends on the defintions of $infty$, "number" and "divides".
I will assume that you're asked what is the greatest integer number dividing 11 then basically in any sensible interpretation of words $infty$, "integer number" and "divides" the answer to your question will be "11". As far as I know, $infty$ is always interpreted distinct from the usual numbers (by usual I mean real numbers) (including integers), that is, you have an axiom (or a theorem)
$$forall x in mathbbZ , x neq infty.$$ Hence $infty$ could not possibly be the greatest integer number dividing 11 since it then would be an integer contradicting this axiom (or a theorem, whatever you wanna call it)
edited Apr 12 at 16:01
Eric Towers
33.9k22371
answered Apr 12 at 14:54
famesyasdfamesyasd
349313
$endgroup$
It all depends on the defintions of $infty$, "number" and "divides".
I will assume that you're asked what is the greatest integer number dividing 11 then basically in any sensible interpretation of words $infty$, "integer number" and "divides" the answer to your question will be "11". As far as I know, $infty$ is always interpreted distinct from the usual numbers (by usual I mean real numbers) (including integers), that is, you have an axiom (or a theorem)
$$forall x in mathbbZ , x neq infty.$$ Hence $infty$ could not possibly be the greatest integer number dividing 11 since it then would be an integer contradicting this axiom (or a theorem, whatever you wanna call it)
edited Apr 12 at 16:01
Eric Towers
33.9k22371
answered Apr 12 at 14:54
famesyasdfamesyasd
349313
edited Apr 12 at 16:01
Eric Towers
33.9k22371
edited Apr 12 at 16:01
Eric Towers
33.9k22371
edited Apr 12 at 16:01
Eric Towers
33.9k22371
33.9k22371
answered Apr 12 at 14:54
famesyasdfamesyasd
349313
answered Apr 12 at 14:54
famesyasdfamesyasd
349313
answered Apr 12 at 14:54
famesyasdfamesyasd
349313
349313
add a comment |
add a comment |
$begingroup$
It is really great to be thinking of stuff like this.
You will find people all around you who will say infinity is not a number and they will stop there, but there are a set of math classes in your future (Calculus) that are all about dealing with numbers that are virtually infinite--they call them limits.
All those people who say that it's absolutely impossible to deal with infinity will find it perfectly fine to deal with "x as x approaches infinity" and you get the same results you would intuit for infinity (Division by infinity is still a bit tricky if I recall correctly)
Anyway, if you have ever noticed that math is often about describing things, calculus is how you describe strange (arbitrary) curves and such--Exactly how much water fits into your strangely shaped sink, or how long will it take to drain said water since the speed it drains is slower as there is less water in the sink.
This kind of thing is calculated by slicing up curves into infinitely (Approaching infinite anyway) thin slices and adding the slices together to get an exact result.
(PS, I haven't done much calculus in decades, if this is factually wrong please feel free to fix it rather than just blindly downvoting. Thanks)
answered Apr 12 at 18:12
Bill KBill K
1274
$endgroup$
$begingroup$
I don't think it's a great idea to compare limits to infinity. Limits are (usually) real numbers with fairly easily definable properties, unlike infinity.
$endgroup$
– Jam
6 hours ago
add a comment |
$begingroup$
It is really great to be thinking of stuff like this.
You will find people all around you who will say infinity is not a number and they will stop there, but there are a set of math classes in your future (Calculus) that are all about dealing with numbers that are virtually infinite--they call them limits.
All those people who say that it's absolutely impossible to deal with infinity will find it perfectly fine to deal with "x as x approaches infinity" and you get the same results you would intuit for infinity (Division by infinity is still a bit tricky if I recall correctly)
Anyway, if you have ever noticed that math is often about describing things, calculus is how you describe strange (arbitrary) curves and such--Exactly how much water fits into your strangely shaped sink, or how long will it take to drain said water since the speed it drains is slower as there is less water in the sink.
This kind of thing is calculated by slicing up curves into infinitely (Approaching infinite anyway) thin slices and adding the slices together to get an exact result.
(PS, I haven't done much calculus in decades, if this is factually wrong please feel free to fix it rather than just blindly downvoting. Thanks)
answered Apr 12 at 18:12
Bill KBill K
1274
$endgroup$
$begingroup$
I don't think it's a great idea to compare limits to infinity. Limits are (usually) real numbers with fairly easily definable properties, unlike infinity.
$endgroup$
– Jam
6 hours ago
add a comment |
$begingroup$
It is really great to be thinking of stuff like this.
You will find people all around you who will say infinity is not a number and they will stop there, but there are a set of math classes in your future (Calculus) that are all about dealing with numbers that are virtually infinite--they call them limits.
All those people who say that it's absolutely impossible to deal with infinity will find it perfectly fine to deal with "x as x approaches infinity" and you get the same results you would intuit for infinity (Division by infinity is still a bit tricky if I recall correctly)
Anyway, if you have ever noticed that math is often about describing things, calculus is how you describe strange (arbitrary) curves and such--Exactly how much water fits into your strangely shaped sink, or how long will it take to drain said water since the speed it drains is slower as there is less water in the sink.
This kind of thing is calculated by slicing up curves into infinitely (Approaching infinite anyway) thin slices and adding the slices together to get an exact result.
(PS, I haven't done much calculus in decades, if this is factually wrong please feel free to fix it rather than just blindly downvoting. Thanks)
answered Apr 12 at 18:12
Bill KBill K
1274
$endgroup$
It is really great to be thinking of stuff like this.
You will find people all around you who will say infinity is not a number and they will stop there, but there are a set of math classes in your future (Calculus) that are all about dealing with numbers that are virtually infinite--they call them limits.
All those people who say that it's absolutely impossible to deal with infinity will find it perfectly fine to deal with "x as x approaches infinity" and you get the same results you would intuit for infinity (Division by infinity is still a bit tricky if I recall correctly)
Anyway, if you have ever noticed that math is often about describing things, calculus is how you describe strange (arbitrary) curves and such--Exactly how much water fits into your strangely shaped sink, or how long will it take to drain said water since the speed it drains is slower as there is less water in the sink.
This kind of thing is calculated by slicing up curves into infinitely (Approaching infinite anyway) thin slices and adding the slices together to get an exact result.
(PS, I haven't done much calculus in decades, if this is factually wrong please feel free to fix it rather than just blindly downvoting. Thanks)
answered Apr 12 at 18:12
Bill KBill K
1274
answered Apr 12 at 18:12
Bill KBill K
1274
answered Apr 12 at 18:12
Bill KBill K
1274
answered Apr 12 at 18:12
Bill KBill K
1274
1274
$begingroup$
I don't think it's a great idea to compare limits to infinity. Limits are (usually) real numbers with fairly easily definable properties, unlike infinity.
$endgroup$
– Jam
6 hours ago
add a comment |
$begingroup$
I don't think it's a great idea to compare limits to infinity. Limits are (usually) real numbers with fairly easily definable properties, unlike infinity.
$endgroup$
– Jam
6 hours ago
$begingroup$
I don't think it's a great idea to compare limits to infinity. Limits are (usually) real numbers with fairly easily definable properties, unlike infinity.
$endgroup$
– Jam
6 hours ago
$begingroup$
I don't think it's a great idea to compare limits to infinity. Limits are (usually) real numbers with fairly easily definable properties, unlike infinity.
$endgroup$
– Jam
6 hours ago
add a comment |
17
$begingroup$
Infinity is not a number.
$endgroup$
– Sean Roberson
Apr 12 at 14:38
2
$begingroup$
@SeanRoberson What is a number for you to be saying that?
$endgroup$
– user647486
Apr 12 at 14:41
2
$begingroup$
@user647486 Care to justify your statement?
$endgroup$
– QuantumPenguin
Apr 12 at 14:46
6
$begingroup$
There is an arithmetic for the extended real numbers (real numbers with $pminfty.$) You can see its arithmetic here: en.wikipedia.org/wiki/….
$endgroup$
– Adrian Keister
Apr 12 at 14:50
7
$begingroup$
Your title asks about dividing infinity by a number, then in your question you start out asking about dividing eleven by something, then you go back to dividing something by eleven. Dividing something by eleven and dividing eleven by something are very different, and you need to more clearly distinguish between the two.
$endgroup$
– Acccumulation
Apr 12 at 16:05