Can a number be non-imaginary and non-real?












3












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I passed across this chart on the web and got confused. The diagram implies that there are numbers that are neither real nor imaginary. Is that possible, or is it just an incorrect diagram?



Numbers Diagram



I understand the box titled "Imaginary numbers", might actually mean to say "Pure imaginary numbers"? But even then, it implies that all real numbers are imaginary (as there are examples like 4 + 2i under complex numbers, which are imaginary), which is nonsense! Kindly help me. Thank you!






number-theory complex-numbers







share|cite|improve this question











$endgroup$








  • 4




    $begingroup$
    The chart is fine.
    $endgroup$
    – Wuestenfux
    Dec 27 '18 at 15:05






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    It depends on definition. I have seen $3+4i$, for instance, called "imaginary"; when the real part is zero like $(0+)4i$, then it's "pure imaginary".
    $endgroup$
    – Oscar Lanzi
    Dec 27 '18 at 15:08








  • 1




    $begingroup$
    There are a couple things that are useless and distracting about this chart. The chain of inclusions is perhaps the most useful part. By the way, you should probably rid yourself of the ideal of a “number” as a particular thing. There is no real definition for it.
    $endgroup$
    – rschwieb
    Dec 27 '18 at 19:29








  • 1




    $begingroup$
    The chart should have $1,2,3,ldots$ only in $N$ and not also in $W$; $0$ only in $W$ and not also in $Z$; and $sin(pi/3)=sqrt3/2$ in $A$. But that seems to be unrelated to the main question.
    $endgroup$
    – Teepeemm
    Dec 27 '18 at 21:46






  • 1




    $begingroup$
    If you want to know about numbers that are not complex numbers (but still make sense in some way), you can look up p-adic numbers, those are pretty interesting
    $endgroup$
    – Yuriy S
    Dec 27 '18 at 22:53
















3












$begingroup$


I passed across this chart on the web and got confused. The diagram implies that there are numbers that are neither real nor imaginary. Is that possible, or is it just an incorrect diagram?



Numbers Diagram



I understand the box titled "Imaginary numbers", might actually mean to say "Pure imaginary numbers"? But even then, it implies that all real numbers are imaginary (as there are examples like 4 + 2i under complex numbers, which are imaginary), which is nonsense! Kindly help me. Thank you!






number-theory complex-numbers







share|cite|improve this question











$endgroup$








  • 4




    $begingroup$
    The chart is fine.
    $endgroup$
    – Wuestenfux
    Dec 27 '18 at 15:05






  • 1




    $begingroup$
    It depends on definition. I have seen $3+4i$, for instance, called "imaginary"; when the real part is zero like $(0+)4i$, then it's "pure imaginary".
    $endgroup$
    – Oscar Lanzi
    Dec 27 '18 at 15:08








  • 1




    $begingroup$
    There are a couple things that are useless and distracting about this chart. The chain of inclusions is perhaps the most useful part. By the way, you should probably rid yourself of the ideal of a “number” as a particular thing. There is no real definition for it.
    $endgroup$
    – rschwieb
    Dec 27 '18 at 19:29








  • 1




    $begingroup$
    The chart should have $1,2,3,ldots$ only in $N$ and not also in $W$; $0$ only in $W$ and not also in $Z$; and $sin(pi/3)=sqrt3/2$ in $A$. But that seems to be unrelated to the main question.
    $endgroup$
    – Teepeemm
    Dec 27 '18 at 21:46






  • 1




    $begingroup$
    If you want to know about numbers that are not complex numbers (but still make sense in some way), you can look up p-adic numbers, those are pretty interesting
    $endgroup$
    – Yuriy S
    Dec 27 '18 at 22:53














3












3








3


1



$begingroup$


I passed across this chart on the web and got confused. The diagram implies that there are numbers that are neither real nor imaginary. Is that possible, or is it just an incorrect diagram?



Numbers Diagram



I understand the box titled "Imaginary numbers", might actually mean to say "Pure imaginary numbers"? But even then, it implies that all real numbers are imaginary (as there are examples like 4 + 2i under complex numbers, which are imaginary), which is nonsense! Kindly help me. Thank you!






number-theory complex-numbers







share|cite|improve this question











$endgroup$




I passed across this chart on the web and got confused. The diagram implies that there are numbers that are neither real nor imaginary. Is that possible, or is it just an incorrect diagram?



Numbers Diagram



I understand the box titled "Imaginary numbers", might actually mean to say "Pure imaginary numbers"? But even then, it implies that all real numbers are imaginary (as there are examples like 4 + 2i under complex numbers, which are imaginary), which is nonsense! Kindly help me. Thank you!






number-theory complex-numbers




number-theory complex-numbers






share|cite|improve this question















share|cite|improve this question













share|cite|improve this question




share|cite|improve this question








edited Dec 28 '18 at 3:36









JBentley

18528




18528










asked Dec 27 '18 at 15:04









ShaanShaan

213




213








  • 4




    $begingroup$
    The chart is fine.
    $endgroup$
    – Wuestenfux
    Dec 27 '18 at 15:05






  • 1




    $begingroup$
    It depends on definition. I have seen $3+4i$, for instance, called "imaginary"; when the real part is zero like $(0+)4i$, then it's "pure imaginary".
    $endgroup$
    – Oscar Lanzi
    Dec 27 '18 at 15:08








  • 1




    $begingroup$
    There are a couple things that are useless and distracting about this chart. The chain of inclusions is perhaps the most useful part. By the way, you should probably rid yourself of the ideal of a “number” as a particular thing. There is no real definition for it.
    $endgroup$
    – rschwieb
    Dec 27 '18 at 19:29








  • 1




    $begingroup$
    The chart should have $1,2,3,ldots$ only in $N$ and not also in $W$; $0$ only in $W$ and not also in $Z$; and $sin(pi/3)=sqrt3/2$ in $A$. But that seems to be unrelated to the main question.
    $endgroup$
    – Teepeemm
    Dec 27 '18 at 21:46






  • 1




    $begingroup$
    If you want to know about numbers that are not complex numbers (but still make sense in some way), you can look up p-adic numbers, those are pretty interesting
    $endgroup$
    – Yuriy S
    Dec 27 '18 at 22:53














  • 4




    $begingroup$
    The chart is fine.
    $endgroup$
    – Wuestenfux
    Dec 27 '18 at 15:05






  • 1




    $begingroup$
    It depends on definition. I have seen $3+4i$, for instance, called "imaginary"; when the real part is zero like $(0+)4i$, then it's "pure imaginary".
    $endgroup$
    – Oscar Lanzi
    Dec 27 '18 at 15:08








  • 1




    $begingroup$
    There are a couple things that are useless and distracting about this chart. The chain of inclusions is perhaps the most useful part. By the way, you should probably rid yourself of the ideal of a “number” as a particular thing. There is no real definition for it.
    $endgroup$
    – rschwieb
    Dec 27 '18 at 19:29








  • 1




    $begingroup$
    The chart should have $1,2,3,ldots$ only in $N$ and not also in $W$; $0$ only in $W$ and not also in $Z$; and $sin(pi/3)=sqrt3/2$ in $A$. But that seems to be unrelated to the main question.
    $endgroup$
    – Teepeemm
    Dec 27 '18 at 21:46






  • 1




    $begingroup$
    If you want to know about numbers that are not complex numbers (but still make sense in some way), you can look up p-adic numbers, those are pretty interesting
    $endgroup$
    – Yuriy S
    Dec 27 '18 at 22:53








4




4




$begingroup$
The chart is fine.
$endgroup$
– Wuestenfux
Dec 27 '18 at 15:05




$begingroup$
The chart is fine.
$endgroup$
– Wuestenfux
Dec 27 '18 at 15:05




1




1




$begingroup$
It depends on definition. I have seen $3+4i$, for instance, called "imaginary"; when the real part is zero like $(0+)4i$, then it's "pure imaginary".
$endgroup$
– Oscar Lanzi
Dec 27 '18 at 15:08






$begingroup$
It depends on definition. I have seen $3+4i$, for instance, called "imaginary"; when the real part is zero like $(0+)4i$, then it's "pure imaginary".
$endgroup$
– Oscar Lanzi
Dec 27 '18 at 15:08






1




1




$begingroup$
There are a couple things that are useless and distracting about this chart. The chain of inclusions is perhaps the most useful part. By the way, you should probably rid yourself of the ideal of a “number” as a particular thing. There is no real definition for it.
$endgroup$
– rschwieb
Dec 27 '18 at 19:29






$begingroup$
There are a couple things that are useless and distracting about this chart. The chain of inclusions is perhaps the most useful part. By the way, you should probably rid yourself of the ideal of a “number” as a particular thing. There is no real definition for it.
$endgroup$
– rschwieb
Dec 27 '18 at 19:29






1




1




$begingroup$
The chart should have $1,2,3,ldots$ only in $N$ and not also in $W$; $0$ only in $W$ and not also in $Z$; and $sin(pi/3)=sqrt3/2$ in $A$. But that seems to be unrelated to the main question.
$endgroup$
– Teepeemm
Dec 27 '18 at 21:46




$begingroup$
The chart should have $1,2,3,ldots$ only in $N$ and not also in $W$; $0$ only in $W$ and not also in $Z$; and $sin(pi/3)=sqrt3/2$ in $A$. But that seems to be unrelated to the main question.
$endgroup$
– Teepeemm
Dec 27 '18 at 21:46




1




1




$begingroup$
If you want to know about numbers that are not complex numbers (but still make sense in some way), you can look up p-adic numbers, those are pretty interesting
$endgroup$
– Yuriy S
Dec 27 '18 at 22:53




$begingroup$
If you want to know about numbers that are not complex numbers (but still make sense in some way), you can look up p-adic numbers, those are pretty interesting
$endgroup$
– Yuriy S
Dec 27 '18 at 22:53










6 Answers
6






active

oldest

votes


















9












$begingroup$

A complex number is the sum of a pure imaginary number and a real number. However, there are numbers which are neither real or imaginary: Octonions.



EDIT:



As the comments pointed out, I skipped quaternions, but octonions are so much cooler. If you are looking for even more numbers to bend your brain, there are the surreal numbers and the transfinite numbers.






share|cite|improve this answer











$endgroup$









  • 18




    $begingroup$
    Good point, but why did you skip Quaternions?
    $endgroup$
    – Tobias Kienzler
    Dec 27 '18 at 20:06












  • $begingroup$
    Because a) octonions are cooler, and b) there aren't sedecimtonions. It ends at octonions.
    $endgroup$
    – William Grannis
    Dec 28 '18 at 15:14










  • $begingroup$
    There are Sedenions ;)
    $endgroup$
    – Tobias Kienzler
    Dec 28 '18 at 15:29










  • $begingroup$
    Related, but AFAIK they don't behave the same.
    $endgroup$
    – William Grannis
    Dec 28 '18 at 15:51



















6












$begingroup$

The chart is correct. Its author calls imaginary numbers to the numbers of the form $lambda i$, with $lambdainmathbb R$. So, $1+i$ is neither real nor imaginary.






share|cite|improve this answer









$endgroup$













  • $begingroup$
    $lambda ne 0$?
    $endgroup$
    – Daniel R. Collins
    Dec 28 '18 at 2:45










  • $begingroup$
    Quite likely, but you would have to ask that question is the author of the chart to be sure.
    $endgroup$
    – José Carlos Santos
    Dec 28 '18 at 3:15



















6












$begingroup$

The number $3+4i$ is an example of a number that is neither real nor imaginary.



EDIT

What makes you say "it implies that all real numbers are imaginary"? There are real numbers (like $pi$) and there are imaginary numbers (like $ipi$). These two sets are almost disjoint (they only have number $0$ in common), but they are both subsets of complex numbers (which also contains numbers like $3+4i$).






share|cite|improve this answer











$endgroup$













  • $begingroup$
    So what's the difference between an imaginary and a pure imaginary number?
    $endgroup$
    – Shaan
    Dec 27 '18 at 15:14






  • 1




    $begingroup$
    @Shaan. I consider them as being the same, perhaps with the exception of the number $0$ that might be called imaginary but not pure imaginary.
    $endgroup$
    – md2perpe
    Dec 27 '18 at 15:16










  • $begingroup$
    They're not disjoint since both include $0$.
    $endgroup$
    – J.G.
    Dec 27 '18 at 15:21






  • 1




    $begingroup$
    @J.G. That's correct; I missed that special case. In that way the diagram is incorrect unless "Imaginary Numbers" refer to pure imaginary numbers which exclude zero.
    $endgroup$
    – md2perpe
    Dec 27 '18 at 15:24












  • $begingroup$
    So, please clarify. Real number = 1/3, Imaginary = 2i, and Complex = 1/3 + 2i
    $endgroup$
    – Shaan
    Dec 27 '18 at 15:32



















2












$begingroup$

Those are complex numbers. You might not have seen it because the letters are as dark as the background






share|cite|improve this answer









$endgroup$





















    1












    $begingroup$


    I understand the box titled "Imaginary numbers", might actually mean to say "Pure imaginary numbers"?




    At least I learned that the term "imaginary number" means what you call "pure imaginary number":



    An "imaginary number" is a number that can be written as $yi$ (or $0+yi$) with $yinmathbb R$. A number that can be written as $x+yi$ with $x,yinmathbb Rbackslash{0}$ (such as $4+2i$) is not called "imaginary number" according to this terminology.




    But even then, it implies that all real numbers are imaginary ...




    No. Why?



    In the diagram the real numbers are drawn as subset of "complex numbers" ($mathbb C$), not as subset of "imaginary numbers".



    Every real number is a "complex number".




    as there are examples like 4 + 2i under complex numbers, which are imaginary




    With exception of the integer numbers, the examples in a box show numbers that do not fit in a "sub-box":



    The examples in the "rational numbers" box do not contain an integer number; the examples in the "real numbers" box do not contain any rational number.



    This does not mean that integers numbers are not rational...






    share|cite|improve this answer









    $endgroup$





















      0












      $begingroup$

      One oddity of maths that does not fit with the typical public perception is that there are not absolute fixed definitions. Many will vary from author to author. Some definitions are agreed by pretty much everyone. For others, e.g. the exponential function, not all agree on the definition but all of the popular ones are equivalent. Sometimes, there is a significant variation e.g. rings in algebra, is a multiplicative identity required? Today, I think that most people would not say that 1 is prime but some old books might. The Bourbaki school even proposed that 0 be considered both positive and negative rather than neither.



      Back to your case, it seems that you regard imaginary as meaning not real but others restrict imaginary to real mutiples of $i$. My only issue with that chart would be the need for a tiny overlap of the imaginaires and the reals containing $0$,






      share|cite|improve this answer









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        6 Answers
        6






        active

        oldest

        votes








        6 Answers
        6






        active

        oldest

        votes









        active

        oldest

        votes






        active

        oldest

        votes









        9












        $begingroup$

        A complex number is the sum of a pure imaginary number and a real number. However, there are numbers which are neither real or imaginary: Octonions.



        EDIT:



        As the comments pointed out, I skipped quaternions, but octonions are so much cooler. If you are looking for even more numbers to bend your brain, there are the surreal numbers and the transfinite numbers.






        share|cite|improve this answer











        $endgroup$









        • 18




          $begingroup$
          Good point, but why did you skip Quaternions?
          $endgroup$
          – Tobias Kienzler
          Dec 27 '18 at 20:06












        • $begingroup$
          Because a) octonions are cooler, and b) there aren't sedecimtonions. It ends at octonions.
          $endgroup$
          – William Grannis
          Dec 28 '18 at 15:14










        • $begingroup$
          There are Sedenions ;)
          $endgroup$
          – Tobias Kienzler
          Dec 28 '18 at 15:29










        • $begingroup$
          Related, but AFAIK they don't behave the same.
          $endgroup$
          – William Grannis
          Dec 28 '18 at 15:51
















        9












        $begingroup$

        A complex number is the sum of a pure imaginary number and a real number. However, there are numbers which are neither real or imaginary: Octonions.



        EDIT:



        As the comments pointed out, I skipped quaternions, but octonions are so much cooler. If you are looking for even more numbers to bend your brain, there are the surreal numbers and the transfinite numbers.






        share|cite|improve this answer











        $endgroup$









        • 18




          $begingroup$
          Good point, but why did you skip Quaternions?
          $endgroup$
          – Tobias Kienzler
          Dec 27 '18 at 20:06












        • $begingroup$
          Because a) octonions are cooler, and b) there aren't sedecimtonions. It ends at octonions.
          $endgroup$
          – William Grannis
          Dec 28 '18 at 15:14










        • $begingroup$
          There are Sedenions ;)
          $endgroup$
          – Tobias Kienzler
          Dec 28 '18 at 15:29










        • $begingroup$
          Related, but AFAIK they don't behave the same.
          $endgroup$
          – William Grannis
          Dec 28 '18 at 15:51














        9












        9








        9





        $begingroup$

        A complex number is the sum of a pure imaginary number and a real number. However, there are numbers which are neither real or imaginary: Octonions.



        EDIT:



        As the comments pointed out, I skipped quaternions, but octonions are so much cooler. If you are looking for even more numbers to bend your brain, there are the surreal numbers and the transfinite numbers.






        share|cite|improve this answer











        $endgroup$



        A complex number is the sum of a pure imaginary number and a real number. However, there are numbers which are neither real or imaginary: Octonions.



        EDIT:



        As the comments pointed out, I skipped quaternions, but octonions are so much cooler. If you are looking for even more numbers to bend your brain, there are the surreal numbers and the transfinite numbers.







        share|cite|improve this answer














        share|cite|improve this answer



        share|cite|improve this answer








        edited Dec 28 '18 at 15:20

























        answered Dec 27 '18 at 15:24









        William GrannisWilliam Grannis

        991521




        991521








        • 18




          $begingroup$
          Good point, but why did you skip Quaternions?
          $endgroup$
          – Tobias Kienzler
          Dec 27 '18 at 20:06












        • $begingroup$
          Because a) octonions are cooler, and b) there aren't sedecimtonions. It ends at octonions.
          $endgroup$
          – William Grannis
          Dec 28 '18 at 15:14










        • $begingroup$
          There are Sedenions ;)
          $endgroup$
          – Tobias Kienzler
          Dec 28 '18 at 15:29










        • $begingroup$
          Related, but AFAIK they don't behave the same.
          $endgroup$
          – William Grannis
          Dec 28 '18 at 15:51














        • 18




          $begingroup$
          Good point, but why did you skip Quaternions?
          $endgroup$
          – Tobias Kienzler
          Dec 27 '18 at 20:06












        • $begingroup$
          Because a) octonions are cooler, and b) there aren't sedecimtonions. It ends at octonions.
          $endgroup$
          – William Grannis
          Dec 28 '18 at 15:14










        • $begingroup$
          There are Sedenions ;)
          $endgroup$
          – Tobias Kienzler
          Dec 28 '18 at 15:29










        • $begingroup$
          Related, but AFAIK they don't behave the same.
          $endgroup$
          – William Grannis
          Dec 28 '18 at 15:51








        18




        18




        $begingroup$
        Good point, but why did you skip Quaternions?
        $endgroup$
        – Tobias Kienzler
        Dec 27 '18 at 20:06






        $begingroup$
        Good point, but why did you skip Quaternions?
        $endgroup$
        – Tobias Kienzler
        Dec 27 '18 at 20:06














        $begingroup$
        Because a) octonions are cooler, and b) there aren't sedecimtonions. It ends at octonions.
        $endgroup$
        – William Grannis
        Dec 28 '18 at 15:14




        $begingroup$
        Because a) octonions are cooler, and b) there aren't sedecimtonions. It ends at octonions.
        $endgroup$
        – William Grannis
        Dec 28 '18 at 15:14












        $begingroup$
        There are Sedenions ;)
        $endgroup$
        – Tobias Kienzler
        Dec 28 '18 at 15:29




        $begingroup$
        There are Sedenions ;)
        $endgroup$
        – Tobias Kienzler
        Dec 28 '18 at 15:29












        $begingroup$
        Related, but AFAIK they don't behave the same.
        $endgroup$
        – William Grannis
        Dec 28 '18 at 15:51




        $begingroup$
        Related, but AFAIK they don't behave the same.
        $endgroup$
        – William Grannis
        Dec 28 '18 at 15:51











        6












        $begingroup$

        The chart is correct. Its author calls imaginary numbers to the numbers of the form $lambda i$, with $lambdainmathbb R$. So, $1+i$ is neither real nor imaginary.






        share|cite|improve this answer









        $endgroup$













        • $begingroup$
          $lambda ne 0$?
          $endgroup$
          – Daniel R. Collins
          Dec 28 '18 at 2:45










        • $begingroup$
          Quite likely, but you would have to ask that question is the author of the chart to be sure.
          $endgroup$
          – José Carlos Santos
          Dec 28 '18 at 3:15
















        6












        $begingroup$

        The chart is correct. Its author calls imaginary numbers to the numbers of the form $lambda i$, with $lambdainmathbb R$. So, $1+i$ is neither real nor imaginary.






        share|cite|improve this answer









        $endgroup$













        • $begingroup$
          $lambda ne 0$?
          $endgroup$
          – Daniel R. Collins
          Dec 28 '18 at 2:45










        • $begingroup$
          Quite likely, but you would have to ask that question is the author of the chart to be sure.
          $endgroup$
          – José Carlos Santos
          Dec 28 '18 at 3:15














        6












        6








        6





        $begingroup$

        The chart is correct. Its author calls imaginary numbers to the numbers of the form $lambda i$, with $lambdainmathbb R$. So, $1+i$ is neither real nor imaginary.






        share|cite|improve this answer









        $endgroup$



        The chart is correct. Its author calls imaginary numbers to the numbers of the form $lambda i$, with $lambdainmathbb R$. So, $1+i$ is neither real nor imaginary.







        share|cite|improve this answer












        share|cite|improve this answer



        share|cite|improve this answer










        answered Dec 27 '18 at 15:07









        José Carlos SantosJosé Carlos Santos

        162k22128232




        162k22128232












        • $begingroup$
          $lambda ne 0$?
          $endgroup$
          – Daniel R. Collins
          Dec 28 '18 at 2:45










        • $begingroup$
          Quite likely, but you would have to ask that question is the author of the chart to be sure.
          $endgroup$
          – José Carlos Santos
          Dec 28 '18 at 3:15


















        • $begingroup$
          $lambda ne 0$?
          $endgroup$
          – Daniel R. Collins
          Dec 28 '18 at 2:45










        • $begingroup$
          Quite likely, but you would have to ask that question is the author of the chart to be sure.
          $endgroup$
          – José Carlos Santos
          Dec 28 '18 at 3:15
















        $begingroup$
        $lambda ne 0$?
        $endgroup$
        – Daniel R. Collins
        Dec 28 '18 at 2:45




        $begingroup$
        $lambda ne 0$?
        $endgroup$
        – Daniel R. Collins
        Dec 28 '18 at 2:45












        $begingroup$
        Quite likely, but you would have to ask that question is the author of the chart to be sure.
        $endgroup$
        – José Carlos Santos
        Dec 28 '18 at 3:15




        $begingroup$
        Quite likely, but you would have to ask that question is the author of the chart to be sure.
        $endgroup$
        – José Carlos Santos
        Dec 28 '18 at 3:15











        6












        $begingroup$

        The number $3+4i$ is an example of a number that is neither real nor imaginary.



        EDIT

        What makes you say "it implies that all real numbers are imaginary"? There are real numbers (like $pi$) and there are imaginary numbers (like $ipi$). These two sets are almost disjoint (they only have number $0$ in common), but they are both subsets of complex numbers (which also contains numbers like $3+4i$).






        share|cite|improve this answer











        $endgroup$













        • $begingroup$
          So what's the difference between an imaginary and a pure imaginary number?
          $endgroup$
          – Shaan
          Dec 27 '18 at 15:14






        • 1




          $begingroup$
          @Shaan. I consider them as being the same, perhaps with the exception of the number $0$ that might be called imaginary but not pure imaginary.
          $endgroup$
          – md2perpe
          Dec 27 '18 at 15:16










        • $begingroup$
          They're not disjoint since both include $0$.
          $endgroup$
          – J.G.
          Dec 27 '18 at 15:21






        • 1




          $begingroup$
          @J.G. That's correct; I missed that special case. In that way the diagram is incorrect unless "Imaginary Numbers" refer to pure imaginary numbers which exclude zero.
          $endgroup$
          – md2perpe
          Dec 27 '18 at 15:24












        • $begingroup$
          So, please clarify. Real number = 1/3, Imaginary = 2i, and Complex = 1/3 + 2i
          $endgroup$
          – Shaan
          Dec 27 '18 at 15:32
















        6












        $begingroup$

        The number $3+4i$ is an example of a number that is neither real nor imaginary.



        EDIT

        What makes you say "it implies that all real numbers are imaginary"? There are real numbers (like $pi$) and there are imaginary numbers (like $ipi$). These two sets are almost disjoint (they only have number $0$ in common), but they are both subsets of complex numbers (which also contains numbers like $3+4i$).






        share|cite|improve this answer











        $endgroup$













        • $begingroup$
          So what's the difference between an imaginary and a pure imaginary number?
          $endgroup$
          – Shaan
          Dec 27 '18 at 15:14






        • 1




          $begingroup$
          @Shaan. I consider them as being the same, perhaps with the exception of the number $0$ that might be called imaginary but not pure imaginary.
          $endgroup$
          – md2perpe
          Dec 27 '18 at 15:16










        • $begingroup$
          They're not disjoint since both include $0$.
          $endgroup$
          – J.G.
          Dec 27 '18 at 15:21






        • 1




          $begingroup$
          @J.G. That's correct; I missed that special case. In that way the diagram is incorrect unless "Imaginary Numbers" refer to pure imaginary numbers which exclude zero.
          $endgroup$
          – md2perpe
          Dec 27 '18 at 15:24












        • $begingroup$
          So, please clarify. Real number = 1/3, Imaginary = 2i, and Complex = 1/3 + 2i
          $endgroup$
          – Shaan
          Dec 27 '18 at 15:32














        6












        6








        6





        $begingroup$

        The number $3+4i$ is an example of a number that is neither real nor imaginary.



        EDIT

        What makes you say "it implies that all real numbers are imaginary"? There are real numbers (like $pi$) and there are imaginary numbers (like $ipi$). These two sets are almost disjoint (they only have number $0$ in common), but they are both subsets of complex numbers (which also contains numbers like $3+4i$).






        share|cite|improve this answer











        $endgroup$



        The number $3+4i$ is an example of a number that is neither real nor imaginary.



        EDIT

        What makes you say "it implies that all real numbers are imaginary"? There are real numbers (like $pi$) and there are imaginary numbers (like $ipi$). These two sets are almost disjoint (they only have number $0$ in common), but they are both subsets of complex numbers (which also contains numbers like $3+4i$).







        share|cite|improve this answer














        share|cite|improve this answer



        share|cite|improve this answer








        edited Dec 27 '18 at 15:45

























        answered Dec 27 '18 at 15:07









        md2perpemd2perpe

        8,00611028




        8,00611028












        • $begingroup$
          So what's the difference between an imaginary and a pure imaginary number?
          $endgroup$
          – Shaan
          Dec 27 '18 at 15:14






        • 1




          $begingroup$
          @Shaan. I consider them as being the same, perhaps with the exception of the number $0$ that might be called imaginary but not pure imaginary.
          $endgroup$
          – md2perpe
          Dec 27 '18 at 15:16










        • $begingroup$
          They're not disjoint since both include $0$.
          $endgroup$
          – J.G.
          Dec 27 '18 at 15:21






        • 1




          $begingroup$
          @J.G. That's correct; I missed that special case. In that way the diagram is incorrect unless "Imaginary Numbers" refer to pure imaginary numbers which exclude zero.
          $endgroup$
          – md2perpe
          Dec 27 '18 at 15:24












        • $begingroup$
          So, please clarify. Real number = 1/3, Imaginary = 2i, and Complex = 1/3 + 2i
          $endgroup$
          – Shaan
          Dec 27 '18 at 15:32


















        • $begingroup$
          So what's the difference between an imaginary and a pure imaginary number?
          $endgroup$
          – Shaan
          Dec 27 '18 at 15:14






        • 1




          $begingroup$
          @Shaan. I consider them as being the same, perhaps with the exception of the number $0$ that might be called imaginary but not pure imaginary.
          $endgroup$
          – md2perpe
          Dec 27 '18 at 15:16










        • $begingroup$
          They're not disjoint since both include $0$.
          $endgroup$
          – J.G.
          Dec 27 '18 at 15:21






        • 1




          $begingroup$
          @J.G. That's correct; I missed that special case. In that way the diagram is incorrect unless "Imaginary Numbers" refer to pure imaginary numbers which exclude zero.
          $endgroup$
          – md2perpe
          Dec 27 '18 at 15:24












        • $begingroup$
          So, please clarify. Real number = 1/3, Imaginary = 2i, and Complex = 1/3 + 2i
          $endgroup$
          – Shaan
          Dec 27 '18 at 15:32
















        $begingroup$
        So what's the difference between an imaginary and a pure imaginary number?
        $endgroup$
        – Shaan
        Dec 27 '18 at 15:14




        $begingroup$
        So what's the difference between an imaginary and a pure imaginary number?
        $endgroup$
        – Shaan
        Dec 27 '18 at 15:14




        1




        1




        $begingroup$
        @Shaan. I consider them as being the same, perhaps with the exception of the number $0$ that might be called imaginary but not pure imaginary.
        $endgroup$
        – md2perpe
        Dec 27 '18 at 15:16




        $begingroup$
        @Shaan. I consider them as being the same, perhaps with the exception of the number $0$ that might be called imaginary but not pure imaginary.
        $endgroup$
        – md2perpe
        Dec 27 '18 at 15:16












        $begingroup$
        They're not disjoint since both include $0$.
        $endgroup$
        – J.G.
        Dec 27 '18 at 15:21




        $begingroup$
        They're not disjoint since both include $0$.
        $endgroup$
        – J.G.
        Dec 27 '18 at 15:21




        1




        1




        $begingroup$
        @J.G. That's correct; I missed that special case. In that way the diagram is incorrect unless "Imaginary Numbers" refer to pure imaginary numbers which exclude zero.
        $endgroup$
        – md2perpe
        Dec 27 '18 at 15:24






        $begingroup$
        @J.G. That's correct; I missed that special case. In that way the diagram is incorrect unless "Imaginary Numbers" refer to pure imaginary numbers which exclude zero.
        $endgroup$
        – md2perpe
        Dec 27 '18 at 15:24














        $begingroup$
        So, please clarify. Real number = 1/3, Imaginary = 2i, and Complex = 1/3 + 2i
        $endgroup$
        – Shaan
        Dec 27 '18 at 15:32




        $begingroup$
        So, please clarify. Real number = 1/3, Imaginary = 2i, and Complex = 1/3 + 2i
        $endgroup$
        – Shaan
        Dec 27 '18 at 15:32











        2












        $begingroup$

        Those are complex numbers. You might not have seen it because the letters are as dark as the background






        share|cite|improve this answer









        $endgroup$


















          2












          $begingroup$

          Those are complex numbers. You might not have seen it because the letters are as dark as the background






          share|cite|improve this answer









          $endgroup$
















            2












            2








            2





            $begingroup$

            Those are complex numbers. You might not have seen it because the letters are as dark as the background






            share|cite|improve this answer









            $endgroup$



            Those are complex numbers. You might not have seen it because the letters are as dark as the background







            share|cite|improve this answer












            share|cite|improve this answer



            share|cite|improve this answer










            answered Dec 27 '18 at 15:23









            Milan StojanovicMilan Stojanovic

            401313




            401313























                1












                $begingroup$


                I understand the box titled "Imaginary numbers", might actually mean to say "Pure imaginary numbers"?




                At least I learned that the term "imaginary number" means what you call "pure imaginary number":



                An "imaginary number" is a number that can be written as $yi$ (or $0+yi$) with $yinmathbb R$. A number that can be written as $x+yi$ with $x,yinmathbb Rbackslash{0}$ (such as $4+2i$) is not called "imaginary number" according to this terminology.




                But even then, it implies that all real numbers are imaginary ...




                No. Why?



                In the diagram the real numbers are drawn as subset of "complex numbers" ($mathbb C$), not as subset of "imaginary numbers".



                Every real number is a "complex number".




                as there are examples like 4 + 2i under complex numbers, which are imaginary




                With exception of the integer numbers, the examples in a box show numbers that do not fit in a "sub-box":



                The examples in the "rational numbers" box do not contain an integer number; the examples in the "real numbers" box do not contain any rational number.



                This does not mean that integers numbers are not rational...






                share|cite|improve this answer









                $endgroup$


















                  1












                  $begingroup$


                  I understand the box titled "Imaginary numbers", might actually mean to say "Pure imaginary numbers"?




                  At least I learned that the term "imaginary number" means what you call "pure imaginary number":



                  An "imaginary number" is a number that can be written as $yi$ (or $0+yi$) with $yinmathbb R$. A number that can be written as $x+yi$ with $x,yinmathbb Rbackslash{0}$ (such as $4+2i$) is not called "imaginary number" according to this terminology.




                  But even then, it implies that all real numbers are imaginary ...




                  No. Why?



                  In the diagram the real numbers are drawn as subset of "complex numbers" ($mathbb C$), not as subset of "imaginary numbers".



                  Every real number is a "complex number".




                  as there are examples like 4 + 2i under complex numbers, which are imaginary




                  With exception of the integer numbers, the examples in a box show numbers that do not fit in a "sub-box":



                  The examples in the "rational numbers" box do not contain an integer number; the examples in the "real numbers" box do not contain any rational number.



                  This does not mean that integers numbers are not rational...






                  share|cite|improve this answer









                  $endgroup$
















                    1












                    1








                    1





                    $begingroup$


                    I understand the box titled "Imaginary numbers", might actually mean to say "Pure imaginary numbers"?




                    At least I learned that the term "imaginary number" means what you call "pure imaginary number":



                    An "imaginary number" is a number that can be written as $yi$ (or $0+yi$) with $yinmathbb R$. A number that can be written as $x+yi$ with $x,yinmathbb Rbackslash{0}$ (such as $4+2i$) is not called "imaginary number" according to this terminology.




                    But even then, it implies that all real numbers are imaginary ...




                    No. Why?



                    In the diagram the real numbers are drawn as subset of "complex numbers" ($mathbb C$), not as subset of "imaginary numbers".



                    Every real number is a "complex number".




                    as there are examples like 4 + 2i under complex numbers, which are imaginary




                    With exception of the integer numbers, the examples in a box show numbers that do not fit in a "sub-box":



                    The examples in the "rational numbers" box do not contain an integer number; the examples in the "real numbers" box do not contain any rational number.



                    This does not mean that integers numbers are not rational...






                    share|cite|improve this answer









                    $endgroup$




                    I understand the box titled "Imaginary numbers", might actually mean to say "Pure imaginary numbers"?




                    At least I learned that the term "imaginary number" means what you call "pure imaginary number":



                    An "imaginary number" is a number that can be written as $yi$ (or $0+yi$) with $yinmathbb R$. A number that can be written as $x+yi$ with $x,yinmathbb Rbackslash{0}$ (such as $4+2i$) is not called "imaginary number" according to this terminology.




                    But even then, it implies that all real numbers are imaginary ...




                    No. Why?



                    In the diagram the real numbers are drawn as subset of "complex numbers" ($mathbb C$), not as subset of "imaginary numbers".



                    Every real number is a "complex number".




                    as there are examples like 4 + 2i under complex numbers, which are imaginary




                    With exception of the integer numbers, the examples in a box show numbers that do not fit in a "sub-box":



                    The examples in the "rational numbers" box do not contain an integer number; the examples in the "real numbers" box do not contain any rational number.



                    This does not mean that integers numbers are not rational...







                    share|cite|improve this answer












                    share|cite|improve this answer



                    share|cite|improve this answer










                    answered Dec 28 '18 at 9:14









                    Martin RosenauMartin Rosenau

                    1,156139




                    1,156139























                        0












                        $begingroup$

                        One oddity of maths that does not fit with the typical public perception is that there are not absolute fixed definitions. Many will vary from author to author. Some definitions are agreed by pretty much everyone. For others, e.g. the exponential function, not all agree on the definition but all of the popular ones are equivalent. Sometimes, there is a significant variation e.g. rings in algebra, is a multiplicative identity required? Today, I think that most people would not say that 1 is prime but some old books might. The Bourbaki school even proposed that 0 be considered both positive and negative rather than neither.



                        Back to your case, it seems that you regard imaginary as meaning not real but others restrict imaginary to real mutiples of $i$. My only issue with that chart would be the need for a tiny overlap of the imaginaires and the reals containing $0$,






                        share|cite|improve this answer









                        $endgroup$


















                          0












                          $begingroup$

                          One oddity of maths that does not fit with the typical public perception is that there are not absolute fixed definitions. Many will vary from author to author. Some definitions are agreed by pretty much everyone. For others, e.g. the exponential function, not all agree on the definition but all of the popular ones are equivalent. Sometimes, there is a significant variation e.g. rings in algebra, is a multiplicative identity required? Today, I think that most people would not say that 1 is prime but some old books might. The Bourbaki school even proposed that 0 be considered both positive and negative rather than neither.



                          Back to your case, it seems that you regard imaginary as meaning not real but others restrict imaginary to real mutiples of $i$. My only issue with that chart would be the need for a tiny overlap of the imaginaires and the reals containing $0$,






                          share|cite|improve this answer









                          $endgroup$
















                            0












                            0








                            0





                            $begingroup$

                            One oddity of maths that does not fit with the typical public perception is that there are not absolute fixed definitions. Many will vary from author to author. Some definitions are agreed by pretty much everyone. For others, e.g. the exponential function, not all agree on the definition but all of the popular ones are equivalent. Sometimes, there is a significant variation e.g. rings in algebra, is a multiplicative identity required? Today, I think that most people would not say that 1 is prime but some old books might. The Bourbaki school even proposed that 0 be considered both positive and negative rather than neither.



                            Back to your case, it seems that you regard imaginary as meaning not real but others restrict imaginary to real mutiples of $i$. My only issue with that chart would be the need for a tiny overlap of the imaginaires and the reals containing $0$,






                            share|cite|improve this answer









                            $endgroup$



                            One oddity of maths that does not fit with the typical public perception is that there are not absolute fixed definitions. Many will vary from author to author. Some definitions are agreed by pretty much everyone. For others, e.g. the exponential function, not all agree on the definition but all of the popular ones are equivalent. Sometimes, there is a significant variation e.g. rings in algebra, is a multiplicative identity required? Today, I think that most people would not say that 1 is prime but some old books might. The Bourbaki school even proposed that 0 be considered both positive and negative rather than neither.



                            Back to your case, it seems that you regard imaginary as meaning not real but others restrict imaginary to real mutiples of $i$. My only issue with that chart would be the need for a tiny overlap of the imaginaires and the reals containing $0$,







                            share|cite|improve this answer












                            share|cite|improve this answer



                            share|cite|improve this answer










                            answered Dec 28 '18 at 9:30









                            badjohnbadjohn

                            4,3021620




                            4,3021620






























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