True Definition of the Real Numbers












18












$begingroup$


I've found lots of resources that say this is a real number if it's not rational, but what is a real number, really? I mean what is the definition of a real number? If nothing else, anyone know of a resource where I could find out myself?



Thanks!






real-analysis definition







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  • 4




    $begingroup$
    A real number may be defined to be an equivalence class of Cauchy sequence of rational numbers. Alternatively, it can be thought of as a Dedekind cut.
    $endgroup$
    – Eric Naslund
    Dec 17 '12 at 23:45


















18












$begingroup$


I've found lots of resources that say this is a real number if it's not rational, but what is a real number, really? I mean what is the definition of a real number? If nothing else, anyone know of a resource where I could find out myself?



Thanks!






real-analysis definition







share|cite|improve this question











$endgroup$








  • 4




    $begingroup$
    A real number may be defined to be an equivalence class of Cauchy sequence of rational numbers. Alternatively, it can be thought of as a Dedekind cut.
    $endgroup$
    – Eric Naslund
    Dec 17 '12 at 23:45
















18












18








18


9



$begingroup$


I've found lots of resources that say this is a real number if it's not rational, but what is a real number, really? I mean what is the definition of a real number? If nothing else, anyone know of a resource where I could find out myself?



Thanks!






real-analysis definition







share|cite|improve this question











$endgroup$




I've found lots of resources that say this is a real number if it's not rational, but what is a real number, really? I mean what is the definition of a real number? If nothing else, anyone know of a resource where I could find out myself?



Thanks!






real-analysis definition




real-analysis definition






share|cite|improve this question















share|cite|improve this question













share|cite|improve this question




share|cite|improve this question








edited Dec 17 '12 at 23:50









Asaf Karagila♦

303k32429760




303k32429760










asked Dec 17 '12 at 23:43









JonathonJonathon

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180126








  • 4




    $begingroup$
    A real number may be defined to be an equivalence class of Cauchy sequence of rational numbers. Alternatively, it can be thought of as a Dedekind cut.
    $endgroup$
    – Eric Naslund
    Dec 17 '12 at 23:45
















  • 4




    $begingroup$
    A real number may be defined to be an equivalence class of Cauchy sequence of rational numbers. Alternatively, it can be thought of as a Dedekind cut.
    $endgroup$
    – Eric Naslund
    Dec 17 '12 at 23:45










4




4




$begingroup$
A real number may be defined to be an equivalence class of Cauchy sequence of rational numbers. Alternatively, it can be thought of as a Dedekind cut.
$endgroup$
– Eric Naslund
Dec 17 '12 at 23:45






$begingroup$
A real number may be defined to be an equivalence class of Cauchy sequence of rational numbers. Alternatively, it can be thought of as a Dedekind cut.
$endgroup$
– Eric Naslund
Dec 17 '12 at 23:45












2 Answers
2






active

oldest

votes


















31












$begingroup$

There is no "true" definition of the real numbers because there are several ways to think of the real numbers either as mathematical notions (i.e. we don't really care what are the objects which represent the numbers, we just care about the structure) and there are concrete ways to construct the real numbers, e.g. as sets of rational numbers or equivalence classes of sequences.



The structure of the real numbers is unique. It is an order field which is order-complete. It is also the unique complete Archimedean field. This means that if we construct any other field which is ordered and order complete, then we built something which is isomorphic to the real numbers.



Generally speaking, if we accept the rational numbers as "atomic" (namely, objects whose existence we take for granted, and do not investigate further) then the real numbers can be constructed either as particular sets of rationals, called Dedekind cuts, or as equivalence classes of Cauchy sequences.



It is a nontrivial task (at least without seeing it a couple of times before) to prove that either definition gives us this structure we seek. That complete ordered field. It is even less trivial to actually prove the uniqueness of that structure. I won't go into either subjects.



In either definition we can find the rationals are embedded into the real numbers, and in most cases we think about the rationals as being part of the real numbers as much as we think about integers being rational numbers.



One final remark is that if one prefers not to accept the rational numbers as atomic then it is possible to construct them from the integers, and we can construct those from the natural numbers, and in fact we can construct those just from the empty set.




To read more:




  1. Completion of rational numbers via Cauchy sequences

  2. question about construction of real numbers

  3. Constructing $mathbb R$

  4. Why does the Dedekind Cut work well enough to define the Reals?

  5. Construction of $Bbb R$ from $Bbb Q$

  6. In set theory, how are real numbers represented as sets?






share|cite|improve this answer











$endgroup$













  • $begingroup$
    When you say true, dude, what does that really mean?
    $endgroup$
    – Will Jagy
    Dec 18 '12 at 20:03










  • $begingroup$
    Will, in which part?
    $endgroup$
    – Asaf Karagila♦
    Dec 18 '12 at 20:04






  • 2




    $begingroup$
    What is the meaning of Christmas? As an Israeli I always thought it meant giving the non-Jewish Russians a day off midweek, and the non-observant Jews have a senseless reason to drink without feeling guilty. Hellenism is fun.
    $endgroup$
    – Asaf Karagila♦
    Dec 18 '12 at 20:20






  • 1




    $begingroup$
    And Ramanujan's Birthday the next day. If there is a next day.
    $endgroup$
    – Will Jagy
    Dec 18 '12 at 20:28






  • 2




    $begingroup$
    Will, to paraphrase Pearl Jam, we're still alive.
    $endgroup$
    – Asaf Karagila♦
    Dec 30 '12 at 20:31



















-2












$begingroup$

We start with natural numbers $$mathbb N={0,1,2,...}$$ then expand to ad them negative numbers to get the set of whole numbers
$$mathbb Z={...,-2,-1,0,1,2,...}$$. Next step is definition of division of numbers and if we divide two whole numbers result is not always whole number then we define the rationals or fractions
$$mathbb Q={frac{a}{b}:a,binmathbb Z,bneq 0}$$
Cantor proved that all three sets are equivalent or countable. Historically problem came when ancient greeks want to find diagonal of square of size 1 that is $sqrt2$ then Pitagora or Euclid proved that $sqrt2$ can not be written as ratio of two whole numbers. So $sqrt2$ is not a rational numbers. All numbers that can not be written as ratio of whole numbers we call irrational numbers. The set of irrational numbers we denote by $$mathbb I$$ Cantor proved that the set $mathbb I$ is uncountable finally the set of real numbers is $$mathbb R=mathbb Qcupmathbb I$$






share|cite|improve this answer











$endgroup$









  • 2




    $begingroup$
    I'm not sure I like this idea of giving the irrational numbers their own notation. As far as I know, the irrational numbers cannot be made into an interesting algebraic structure in their own right, and hence probably don't deserve a notation other thand $mathbb{R} setminus mathbb{Q}.$ Just my 2 cents.
    $endgroup$
    – goblin
    Jul 26 '16 at 14:34








  • 1




    $begingroup$
    @goblin For what it's worth, the Wikipedia page on Blackboard bold says that $mathbb{J}$ "occasionally represents the set of irrational numbers, $mathbf{R} setminus mathbf{Q}$ ($mathbb{R} setminus mathbb{Q}$)".
    $endgroup$
    – Calum Gilhooley
    Aug 5 '17 at 13:08












  • $begingroup$
    This does not answer the question which asked what the formal definition of a real number is. Some may think the real numbers already existed but despite that, some people invented a definition of a real number in ZF as an object satisfying a certain property which they probably constructed from the finite ordinals and proved that one object from the Von-Neumann universe contains all of those objects and only those objects and that the object with the operations of addition, multiplication, and inequality as they were defined is a complete ordered field. Another problem with this answer is why
    $endgroup$
    – Timothy
    Dec 8 '18 at 20:12










  • $begingroup$
    can you assume that there exists a set of all real numbers with the operations of addition, multiplication, and inequality as an undefined concept and it's a complete ordered field? If you make the additional assumption that all of those real numbers are rational, you can derive a contradiction but if the Pythagoreans were wrong with that additional assumption, then maybe you are also wrong without it.
    $endgroup$
    – Timothy
    Dec 8 '18 at 20:16












  • $begingroup$
    Very late comment: "all numbers that cannot be written as a ratio of whole numbers we call irrational numbers" is false. Is $1+2i$ irrational? Is $1+2iinmathbb R$ then?
    $endgroup$
    – YiFan
    Dec 9 '18 at 12:28













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2 Answers
2






active

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2 Answers
2






active

oldest

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active

oldest

votes






active

oldest

votes









31












$begingroup$

There is no "true" definition of the real numbers because there are several ways to think of the real numbers either as mathematical notions (i.e. we don't really care what are the objects which represent the numbers, we just care about the structure) and there are concrete ways to construct the real numbers, e.g. as sets of rational numbers or equivalence classes of sequences.



The structure of the real numbers is unique. It is an order field which is order-complete. It is also the unique complete Archimedean field. This means that if we construct any other field which is ordered and order complete, then we built something which is isomorphic to the real numbers.



Generally speaking, if we accept the rational numbers as "atomic" (namely, objects whose existence we take for granted, and do not investigate further) then the real numbers can be constructed either as particular sets of rationals, called Dedekind cuts, or as equivalence classes of Cauchy sequences.



It is a nontrivial task (at least without seeing it a couple of times before) to prove that either definition gives us this structure we seek. That complete ordered field. It is even less trivial to actually prove the uniqueness of that structure. I won't go into either subjects.



In either definition we can find the rationals are embedded into the real numbers, and in most cases we think about the rationals as being part of the real numbers as much as we think about integers being rational numbers.



One final remark is that if one prefers not to accept the rational numbers as atomic then it is possible to construct them from the integers, and we can construct those from the natural numbers, and in fact we can construct those just from the empty set.




To read more:




  1. Completion of rational numbers via Cauchy sequences

  2. question about construction of real numbers

  3. Constructing $mathbb R$

  4. Why does the Dedekind Cut work well enough to define the Reals?

  5. Construction of $Bbb R$ from $Bbb Q$

  6. In set theory, how are real numbers represented as sets?






share|cite|improve this answer











$endgroup$













  • $begingroup$
    When you say true, dude, what does that really mean?
    $endgroup$
    – Will Jagy
    Dec 18 '12 at 20:03










  • $begingroup$
    Will, in which part?
    $endgroup$
    – Asaf Karagila♦
    Dec 18 '12 at 20:04






  • 2




    $begingroup$
    What is the meaning of Christmas? As an Israeli I always thought it meant giving the non-Jewish Russians a day off midweek, and the non-observant Jews have a senseless reason to drink without feeling guilty. Hellenism is fun.
    $endgroup$
    – Asaf Karagila♦
    Dec 18 '12 at 20:20






  • 1




    $begingroup$
    And Ramanujan's Birthday the next day. If there is a next day.
    $endgroup$
    – Will Jagy
    Dec 18 '12 at 20:28






  • 2




    $begingroup$
    Will, to paraphrase Pearl Jam, we're still alive.
    $endgroup$
    – Asaf Karagila♦
    Dec 30 '12 at 20:31
















31












$begingroup$

There is no "true" definition of the real numbers because there are several ways to think of the real numbers either as mathematical notions (i.e. we don't really care what are the objects which represent the numbers, we just care about the structure) and there are concrete ways to construct the real numbers, e.g. as sets of rational numbers or equivalence classes of sequences.



The structure of the real numbers is unique. It is an order field which is order-complete. It is also the unique complete Archimedean field. This means that if we construct any other field which is ordered and order complete, then we built something which is isomorphic to the real numbers.



Generally speaking, if we accept the rational numbers as "atomic" (namely, objects whose existence we take for granted, and do not investigate further) then the real numbers can be constructed either as particular sets of rationals, called Dedekind cuts, or as equivalence classes of Cauchy sequences.



It is a nontrivial task (at least without seeing it a couple of times before) to prove that either definition gives us this structure we seek. That complete ordered field. It is even less trivial to actually prove the uniqueness of that structure. I won't go into either subjects.



In either definition we can find the rationals are embedded into the real numbers, and in most cases we think about the rationals as being part of the real numbers as much as we think about integers being rational numbers.



One final remark is that if one prefers not to accept the rational numbers as atomic then it is possible to construct them from the integers, and we can construct those from the natural numbers, and in fact we can construct those just from the empty set.




To read more:




  1. Completion of rational numbers via Cauchy sequences

  2. question about construction of real numbers

  3. Constructing $mathbb R$

  4. Why does the Dedekind Cut work well enough to define the Reals?

  5. Construction of $Bbb R$ from $Bbb Q$

  6. In set theory, how are real numbers represented as sets?






share|cite|improve this answer











$endgroup$













  • $begingroup$
    When you say true, dude, what does that really mean?
    $endgroup$
    – Will Jagy
    Dec 18 '12 at 20:03










  • $begingroup$
    Will, in which part?
    $endgroup$
    – Asaf Karagila♦
    Dec 18 '12 at 20:04






  • 2




    $begingroup$
    What is the meaning of Christmas? As an Israeli I always thought it meant giving the non-Jewish Russians a day off midweek, and the non-observant Jews have a senseless reason to drink without feeling guilty. Hellenism is fun.
    $endgroup$
    – Asaf Karagila♦
    Dec 18 '12 at 20:20






  • 1




    $begingroup$
    And Ramanujan's Birthday the next day. If there is a next day.
    $endgroup$
    – Will Jagy
    Dec 18 '12 at 20:28






  • 2




    $begingroup$
    Will, to paraphrase Pearl Jam, we're still alive.
    $endgroup$
    – Asaf Karagila♦
    Dec 30 '12 at 20:31














31












31








31





$begingroup$

There is no "true" definition of the real numbers because there are several ways to think of the real numbers either as mathematical notions (i.e. we don't really care what are the objects which represent the numbers, we just care about the structure) and there are concrete ways to construct the real numbers, e.g. as sets of rational numbers or equivalence classes of sequences.



The structure of the real numbers is unique. It is an order field which is order-complete. It is also the unique complete Archimedean field. This means that if we construct any other field which is ordered and order complete, then we built something which is isomorphic to the real numbers.



Generally speaking, if we accept the rational numbers as "atomic" (namely, objects whose existence we take for granted, and do not investigate further) then the real numbers can be constructed either as particular sets of rationals, called Dedekind cuts, or as equivalence classes of Cauchy sequences.



It is a nontrivial task (at least without seeing it a couple of times before) to prove that either definition gives us this structure we seek. That complete ordered field. It is even less trivial to actually prove the uniqueness of that structure. I won't go into either subjects.



In either definition we can find the rationals are embedded into the real numbers, and in most cases we think about the rationals as being part of the real numbers as much as we think about integers being rational numbers.



One final remark is that if one prefers not to accept the rational numbers as atomic then it is possible to construct them from the integers, and we can construct those from the natural numbers, and in fact we can construct those just from the empty set.




To read more:




  1. Completion of rational numbers via Cauchy sequences

  2. question about construction of real numbers

  3. Constructing $mathbb R$

  4. Why does the Dedekind Cut work well enough to define the Reals?

  5. Construction of $Bbb R$ from $Bbb Q$

  6. In set theory, how are real numbers represented as sets?






share|cite|improve this answer











$endgroup$



There is no "true" definition of the real numbers because there are several ways to think of the real numbers either as mathematical notions (i.e. we don't really care what are the objects which represent the numbers, we just care about the structure) and there are concrete ways to construct the real numbers, e.g. as sets of rational numbers or equivalence classes of sequences.



The structure of the real numbers is unique. It is an order field which is order-complete. It is also the unique complete Archimedean field. This means that if we construct any other field which is ordered and order complete, then we built something which is isomorphic to the real numbers.



Generally speaking, if we accept the rational numbers as "atomic" (namely, objects whose existence we take for granted, and do not investigate further) then the real numbers can be constructed either as particular sets of rationals, called Dedekind cuts, or as equivalence classes of Cauchy sequences.



It is a nontrivial task (at least without seeing it a couple of times before) to prove that either definition gives us this structure we seek. That complete ordered field. It is even less trivial to actually prove the uniqueness of that structure. I won't go into either subjects.



In either definition we can find the rationals are embedded into the real numbers, and in most cases we think about the rationals as being part of the real numbers as much as we think about integers being rational numbers.



One final remark is that if one prefers not to accept the rational numbers as atomic then it is possible to construct them from the integers, and we can construct those from the natural numbers, and in fact we can construct those just from the empty set.




To read more:




  1. Completion of rational numbers via Cauchy sequences

  2. question about construction of real numbers

  3. Constructing $mathbb R$

  4. Why does the Dedekind Cut work well enough to define the Reals?

  5. Construction of $Bbb R$ from $Bbb Q$

  6. In set theory, how are real numbers represented as sets?







share|cite|improve this answer














share|cite|improve this answer



share|cite|improve this answer








edited Apr 13 '17 at 12:21









Community♦

1




1










answered Dec 17 '12 at 23:50









Asaf Karagila♦Asaf Karagila

303k32429760




303k32429760












  • $begingroup$
    When you say true, dude, what does that really mean?
    $endgroup$
    – Will Jagy
    Dec 18 '12 at 20:03










  • $begingroup$
    Will, in which part?
    $endgroup$
    – Asaf Karagila♦
    Dec 18 '12 at 20:04






  • 2




    $begingroup$
    What is the meaning of Christmas? As an Israeli I always thought it meant giving the non-Jewish Russians a day off midweek, and the non-observant Jews have a senseless reason to drink without feeling guilty. Hellenism is fun.
    $endgroup$
    – Asaf Karagila♦
    Dec 18 '12 at 20:20






  • 1




    $begingroup$
    And Ramanujan's Birthday the next day. If there is a next day.
    $endgroup$
    – Will Jagy
    Dec 18 '12 at 20:28






  • 2




    $begingroup$
    Will, to paraphrase Pearl Jam, we're still alive.
    $endgroup$
    – Asaf Karagila♦
    Dec 30 '12 at 20:31


















  • $begingroup$
    When you say true, dude, what does that really mean?
    $endgroup$
    – Will Jagy
    Dec 18 '12 at 20:03










  • $begingroup$
    Will, in which part?
    $endgroup$
    – Asaf Karagila♦
    Dec 18 '12 at 20:04






  • 2




    $begingroup$
    What is the meaning of Christmas? As an Israeli I always thought it meant giving the non-Jewish Russians a day off midweek, and the non-observant Jews have a senseless reason to drink without feeling guilty. Hellenism is fun.
    $endgroup$
    – Asaf Karagila♦
    Dec 18 '12 at 20:20






  • 1




    $begingroup$
    And Ramanujan's Birthday the next day. If there is a next day.
    $endgroup$
    – Will Jagy
    Dec 18 '12 at 20:28






  • 2




    $begingroup$
    Will, to paraphrase Pearl Jam, we're still alive.
    $endgroup$
    – Asaf Karagila♦
    Dec 30 '12 at 20:31
















$begingroup$
When you say true, dude, what does that really mean?
$endgroup$
– Will Jagy
Dec 18 '12 at 20:03




$begingroup$
When you say true, dude, what does that really mean?
$endgroup$
– Will Jagy
Dec 18 '12 at 20:03












$begingroup$
Will, in which part?
$endgroup$
– Asaf Karagila♦
Dec 18 '12 at 20:04




$begingroup$
Will, in which part?
$endgroup$
– Asaf Karagila♦
Dec 18 '12 at 20:04




2




2




$begingroup$
What is the meaning of Christmas? As an Israeli I always thought it meant giving the non-Jewish Russians a day off midweek, and the non-observant Jews have a senseless reason to drink without feeling guilty. Hellenism is fun.
$endgroup$
– Asaf Karagila♦
Dec 18 '12 at 20:20




$begingroup$
What is the meaning of Christmas? As an Israeli I always thought it meant giving the non-Jewish Russians a day off midweek, and the non-observant Jews have a senseless reason to drink without feeling guilty. Hellenism is fun.
$endgroup$
– Asaf Karagila♦
Dec 18 '12 at 20:20




1




1




$begingroup$
And Ramanujan's Birthday the next day. If there is a next day.
$endgroup$
– Will Jagy
Dec 18 '12 at 20:28




$begingroup$
And Ramanujan's Birthday the next day. If there is a next day.
$endgroup$
– Will Jagy
Dec 18 '12 at 20:28




2




2




$begingroup$
Will, to paraphrase Pearl Jam, we're still alive.
$endgroup$
– Asaf Karagila♦
Dec 30 '12 at 20:31




$begingroup$
Will, to paraphrase Pearl Jam, we're still alive.
$endgroup$
– Asaf Karagila♦
Dec 30 '12 at 20:31











-2












$begingroup$

We start with natural numbers $$mathbb N={0,1,2,...}$$ then expand to ad them negative numbers to get the set of whole numbers
$$mathbb Z={...,-2,-1,0,1,2,...}$$. Next step is definition of division of numbers and if we divide two whole numbers result is not always whole number then we define the rationals or fractions
$$mathbb Q={frac{a}{b}:a,binmathbb Z,bneq 0}$$
Cantor proved that all three sets are equivalent or countable. Historically problem came when ancient greeks want to find diagonal of square of size 1 that is $sqrt2$ then Pitagora or Euclid proved that $sqrt2$ can not be written as ratio of two whole numbers. So $sqrt2$ is not a rational numbers. All numbers that can not be written as ratio of whole numbers we call irrational numbers. The set of irrational numbers we denote by $$mathbb I$$ Cantor proved that the set $mathbb I$ is uncountable finally the set of real numbers is $$mathbb R=mathbb Qcupmathbb I$$






share|cite|improve this answer











$endgroup$









  • 2




    $begingroup$
    I'm not sure I like this idea of giving the irrational numbers their own notation. As far as I know, the irrational numbers cannot be made into an interesting algebraic structure in their own right, and hence probably don't deserve a notation other thand $mathbb{R} setminus mathbb{Q}.$ Just my 2 cents.
    $endgroup$
    – goblin
    Jul 26 '16 at 14:34








  • 1




    $begingroup$
    @goblin For what it's worth, the Wikipedia page on Blackboard bold says that $mathbb{J}$ "occasionally represents the set of irrational numbers, $mathbf{R} setminus mathbf{Q}$ ($mathbb{R} setminus mathbb{Q}$)".
    $endgroup$
    – Calum Gilhooley
    Aug 5 '17 at 13:08












  • $begingroup$
    This does not answer the question which asked what the formal definition of a real number is. Some may think the real numbers already existed but despite that, some people invented a definition of a real number in ZF as an object satisfying a certain property which they probably constructed from the finite ordinals and proved that one object from the Von-Neumann universe contains all of those objects and only those objects and that the object with the operations of addition, multiplication, and inequality as they were defined is a complete ordered field. Another problem with this answer is why
    $endgroup$
    – Timothy
    Dec 8 '18 at 20:12










  • $begingroup$
    can you assume that there exists a set of all real numbers with the operations of addition, multiplication, and inequality as an undefined concept and it's a complete ordered field? If you make the additional assumption that all of those real numbers are rational, you can derive a contradiction but if the Pythagoreans were wrong with that additional assumption, then maybe you are also wrong without it.
    $endgroup$
    – Timothy
    Dec 8 '18 at 20:16












  • $begingroup$
    Very late comment: "all numbers that cannot be written as a ratio of whole numbers we call irrational numbers" is false. Is $1+2i$ irrational? Is $1+2iinmathbb R$ then?
    $endgroup$
    – YiFan
    Dec 9 '18 at 12:28


















-2












$begingroup$

We start with natural numbers $$mathbb N={0,1,2,...}$$ then expand to ad them negative numbers to get the set of whole numbers
$$mathbb Z={...,-2,-1,0,1,2,...}$$. Next step is definition of division of numbers and if we divide two whole numbers result is not always whole number then we define the rationals or fractions
$$mathbb Q={frac{a}{b}:a,binmathbb Z,bneq 0}$$
Cantor proved that all three sets are equivalent or countable. Historically problem came when ancient greeks want to find diagonal of square of size 1 that is $sqrt2$ then Pitagora or Euclid proved that $sqrt2$ can not be written as ratio of two whole numbers. So $sqrt2$ is not a rational numbers. All numbers that can not be written as ratio of whole numbers we call irrational numbers. The set of irrational numbers we denote by $$mathbb I$$ Cantor proved that the set $mathbb I$ is uncountable finally the set of real numbers is $$mathbb R=mathbb Qcupmathbb I$$






share|cite|improve this answer











$endgroup$









  • 2




    $begingroup$
    I'm not sure I like this idea of giving the irrational numbers their own notation. As far as I know, the irrational numbers cannot be made into an interesting algebraic structure in their own right, and hence probably don't deserve a notation other thand $mathbb{R} setminus mathbb{Q}.$ Just my 2 cents.
    $endgroup$
    – goblin
    Jul 26 '16 at 14:34








  • 1




    $begingroup$
    @goblin For what it's worth, the Wikipedia page on Blackboard bold says that $mathbb{J}$ "occasionally represents the set of irrational numbers, $mathbf{R} setminus mathbf{Q}$ ($mathbb{R} setminus mathbb{Q}$)".
    $endgroup$
    – Calum Gilhooley
    Aug 5 '17 at 13:08












  • $begingroup$
    This does not answer the question which asked what the formal definition of a real number is. Some may think the real numbers already existed but despite that, some people invented a definition of a real number in ZF as an object satisfying a certain property which they probably constructed from the finite ordinals and proved that one object from the Von-Neumann universe contains all of those objects and only those objects and that the object with the operations of addition, multiplication, and inequality as they were defined is a complete ordered field. Another problem with this answer is why
    $endgroup$
    – Timothy
    Dec 8 '18 at 20:12










  • $begingroup$
    can you assume that there exists a set of all real numbers with the operations of addition, multiplication, and inequality as an undefined concept and it's a complete ordered field? If you make the additional assumption that all of those real numbers are rational, you can derive a contradiction but if the Pythagoreans were wrong with that additional assumption, then maybe you are also wrong without it.
    $endgroup$
    – Timothy
    Dec 8 '18 at 20:16












  • $begingroup$
    Very late comment: "all numbers that cannot be written as a ratio of whole numbers we call irrational numbers" is false. Is $1+2i$ irrational? Is $1+2iinmathbb R$ then?
    $endgroup$
    – YiFan
    Dec 9 '18 at 12:28
















-2












-2








-2





$begingroup$

We start with natural numbers $$mathbb N={0,1,2,...}$$ then expand to ad them negative numbers to get the set of whole numbers
$$mathbb Z={...,-2,-1,0,1,2,...}$$. Next step is definition of division of numbers and if we divide two whole numbers result is not always whole number then we define the rationals or fractions
$$mathbb Q={frac{a}{b}:a,binmathbb Z,bneq 0}$$
Cantor proved that all three sets are equivalent or countable. Historically problem came when ancient greeks want to find diagonal of square of size 1 that is $sqrt2$ then Pitagora or Euclid proved that $sqrt2$ can not be written as ratio of two whole numbers. So $sqrt2$ is not a rational numbers. All numbers that can not be written as ratio of whole numbers we call irrational numbers. The set of irrational numbers we denote by $$mathbb I$$ Cantor proved that the set $mathbb I$ is uncountable finally the set of real numbers is $$mathbb R=mathbb Qcupmathbb I$$






share|cite|improve this answer











$endgroup$



We start with natural numbers $$mathbb N={0,1,2,...}$$ then expand to ad them negative numbers to get the set of whole numbers
$$mathbb Z={...,-2,-1,0,1,2,...}$$. Next step is definition of division of numbers and if we divide two whole numbers result is not always whole number then we define the rationals or fractions
$$mathbb Q={frac{a}{b}:a,binmathbb Z,bneq 0}$$
Cantor proved that all three sets are equivalent or countable. Historically problem came when ancient greeks want to find diagonal of square of size 1 that is $sqrt2$ then Pitagora or Euclid proved that $sqrt2$ can not be written as ratio of two whole numbers. So $sqrt2$ is not a rational numbers. All numbers that can not be written as ratio of whole numbers we call irrational numbers. The set of irrational numbers we denote by $$mathbb I$$ Cantor proved that the set $mathbb I$ is uncountable finally the set of real numbers is $$mathbb R=mathbb Qcupmathbb I$$







share|cite|improve this answer














share|cite|improve this answer



share|cite|improve this answer








edited May 5 '15 at 13:29

























answered Dec 18 '12 at 0:38









Adi DaniAdi Dani

15.3k32246




15.3k32246








  • 2




    $begingroup$
    I'm not sure I like this idea of giving the irrational numbers their own notation. As far as I know, the irrational numbers cannot be made into an interesting algebraic structure in their own right, and hence probably don't deserve a notation other thand $mathbb{R} setminus mathbb{Q}.$ Just my 2 cents.
    $endgroup$
    – goblin
    Jul 26 '16 at 14:34








  • 1




    $begingroup$
    @goblin For what it's worth, the Wikipedia page on Blackboard bold says that $mathbb{J}$ "occasionally represents the set of irrational numbers, $mathbf{R} setminus mathbf{Q}$ ($mathbb{R} setminus mathbb{Q}$)".
    $endgroup$
    – Calum Gilhooley
    Aug 5 '17 at 13:08












  • $begingroup$
    This does not answer the question which asked what the formal definition of a real number is. Some may think the real numbers already existed but despite that, some people invented a definition of a real number in ZF as an object satisfying a certain property which they probably constructed from the finite ordinals and proved that one object from the Von-Neumann universe contains all of those objects and only those objects and that the object with the operations of addition, multiplication, and inequality as they were defined is a complete ordered field. Another problem with this answer is why
    $endgroup$
    – Timothy
    Dec 8 '18 at 20:12










  • $begingroup$
    can you assume that there exists a set of all real numbers with the operations of addition, multiplication, and inequality as an undefined concept and it's a complete ordered field? If you make the additional assumption that all of those real numbers are rational, you can derive a contradiction but if the Pythagoreans were wrong with that additional assumption, then maybe you are also wrong without it.
    $endgroup$
    – Timothy
    Dec 8 '18 at 20:16












  • $begingroup$
    Very late comment: "all numbers that cannot be written as a ratio of whole numbers we call irrational numbers" is false. Is $1+2i$ irrational? Is $1+2iinmathbb R$ then?
    $endgroup$
    – YiFan
    Dec 9 '18 at 12:28
















  • 2




    $begingroup$
    I'm not sure I like this idea of giving the irrational numbers their own notation. As far as I know, the irrational numbers cannot be made into an interesting algebraic structure in their own right, and hence probably don't deserve a notation other thand $mathbb{R} setminus mathbb{Q}.$ Just my 2 cents.
    $endgroup$
    – goblin
    Jul 26 '16 at 14:34








  • 1




    $begingroup$
    @goblin For what it's worth, the Wikipedia page on Blackboard bold says that $mathbb{J}$ "occasionally represents the set of irrational numbers, $mathbf{R} setminus mathbf{Q}$ ($mathbb{R} setminus mathbb{Q}$)".
    $endgroup$
    – Calum Gilhooley
    Aug 5 '17 at 13:08












  • $begingroup$
    This does not answer the question which asked what the formal definition of a real number is. Some may think the real numbers already existed but despite that, some people invented a definition of a real number in ZF as an object satisfying a certain property which they probably constructed from the finite ordinals and proved that one object from the Von-Neumann universe contains all of those objects and only those objects and that the object with the operations of addition, multiplication, and inequality as they were defined is a complete ordered field. Another problem with this answer is why
    $endgroup$
    – Timothy
    Dec 8 '18 at 20:12










  • $begingroup$
    can you assume that there exists a set of all real numbers with the operations of addition, multiplication, and inequality as an undefined concept and it's a complete ordered field? If you make the additional assumption that all of those real numbers are rational, you can derive a contradiction but if the Pythagoreans were wrong with that additional assumption, then maybe you are also wrong without it.
    $endgroup$
    – Timothy
    Dec 8 '18 at 20:16












  • $begingroup$
    Very late comment: "all numbers that cannot be written as a ratio of whole numbers we call irrational numbers" is false. Is $1+2i$ irrational? Is $1+2iinmathbb R$ then?
    $endgroup$
    – YiFan
    Dec 9 '18 at 12:28










2




2




$begingroup$
I'm not sure I like this idea of giving the irrational numbers their own notation. As far as I know, the irrational numbers cannot be made into an interesting algebraic structure in their own right, and hence probably don't deserve a notation other thand $mathbb{R} setminus mathbb{Q}.$ Just my 2 cents.
$endgroup$
– goblin
Jul 26 '16 at 14:34






$begingroup$
I'm not sure I like this idea of giving the irrational numbers their own notation. As far as I know, the irrational numbers cannot be made into an interesting algebraic structure in their own right, and hence probably don't deserve a notation other thand $mathbb{R} setminus mathbb{Q}.$ Just my 2 cents.
$endgroup$
– goblin
Jul 26 '16 at 14:34






1




1




$begingroup$
@goblin For what it's worth, the Wikipedia page on Blackboard bold says that $mathbb{J}$ "occasionally represents the set of irrational numbers, $mathbf{R} setminus mathbf{Q}$ ($mathbb{R} setminus mathbb{Q}$)".
$endgroup$
– Calum Gilhooley
Aug 5 '17 at 13:08






$begingroup$
@goblin For what it's worth, the Wikipedia page on Blackboard bold says that $mathbb{J}$ "occasionally represents the set of irrational numbers, $mathbf{R} setminus mathbf{Q}$ ($mathbb{R} setminus mathbb{Q}$)".
$endgroup$
– Calum Gilhooley
Aug 5 '17 at 13:08














$begingroup$
This does not answer the question which asked what the formal definition of a real number is. Some may think the real numbers already existed but despite that, some people invented a definition of a real number in ZF as an object satisfying a certain property which they probably constructed from the finite ordinals and proved that one object from the Von-Neumann universe contains all of those objects and only those objects and that the object with the operations of addition, multiplication, and inequality as they were defined is a complete ordered field. Another problem with this answer is why
$endgroup$
– Timothy
Dec 8 '18 at 20:12




$begingroup$
This does not answer the question which asked what the formal definition of a real number is. Some may think the real numbers already existed but despite that, some people invented a definition of a real number in ZF as an object satisfying a certain property which they probably constructed from the finite ordinals and proved that one object from the Von-Neumann universe contains all of those objects and only those objects and that the object with the operations of addition, multiplication, and inequality as they were defined is a complete ordered field. Another problem with this answer is why
$endgroup$
– Timothy
Dec 8 '18 at 20:12












$begingroup$
can you assume that there exists a set of all real numbers with the operations of addition, multiplication, and inequality as an undefined concept and it's a complete ordered field? If you make the additional assumption that all of those real numbers are rational, you can derive a contradiction but if the Pythagoreans were wrong with that additional assumption, then maybe you are also wrong without it.
$endgroup$
– Timothy
Dec 8 '18 at 20:16






$begingroup$
can you assume that there exists a set of all real numbers with the operations of addition, multiplication, and inequality as an undefined concept and it's a complete ordered field? If you make the additional assumption that all of those real numbers are rational, you can derive a contradiction but if the Pythagoreans were wrong with that additional assumption, then maybe you are also wrong without it.
$endgroup$
– Timothy
Dec 8 '18 at 20:16














$begingroup$
Very late comment: "all numbers that cannot be written as a ratio of whole numbers we call irrational numbers" is false. Is $1+2i$ irrational? Is $1+2iinmathbb R$ then?
$endgroup$
– YiFan
Dec 9 '18 at 12:28






$begingroup$
Very late comment: "all numbers that cannot be written as a ratio of whole numbers we call irrational numbers" is false. Is $1+2i$ irrational? Is $1+2iinmathbb R$ then?
$endgroup$
– YiFan
Dec 9 '18 at 12:28




















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