Is it correct to think of a negative number as a way of showing that you have a deficit in something?












2












$begingroup$


Is it correct to think of a negative number as a way of showing that you have a deficit in something? For example,




  • Let $x$ be a real number such that $x=25$.



  • Let $y$ be a real number such that $y=27$.



    Consider this equality:
    $$x-(-2)=y$$
    The way I think of it is that $x$ is $y$ with a deficit of $2$ and in order for it to equal $y$ we need to remove that deficit and that's why:
    $$ -(-2)=+2$$
    It's like we have a deficit in something and we want to remove it by substracting it (the deficit).
    Is my intuition correct?








arithmetic real-numbers







share|cite|improve this question











$endgroup$

















    2












    $begingroup$


    Is it correct to think of a negative number as a way of showing that you have a deficit in something? For example,




    • Let $x$ be a real number such that $x=25$.



    • Let $y$ be a real number such that $y=27$.



      Consider this equality:
      $$x-(-2)=y$$
      The way I think of it is that $x$ is $y$ with a deficit of $2$ and in order for it to equal $y$ we need to remove that deficit and that's why:
      $$ -(-2)=+2$$
      It's like we have a deficit in something and we want to remove it by substracting it (the deficit).
      Is my intuition correct?








    arithmetic real-numbers







    share|cite|improve this question











    $endgroup$















      2












      2








      2





      $begingroup$


      Is it correct to think of a negative number as a way of showing that you have a deficit in something? For example,




      • Let $x$ be a real number such that $x=25$.



      • Let $y$ be a real number such that $y=27$.



        Consider this equality:
        $$x-(-2)=y$$
        The way I think of it is that $x$ is $y$ with a deficit of $2$ and in order for it to equal $y$ we need to remove that deficit and that's why:
        $$ -(-2)=+2$$
        It's like we have a deficit in something and we want to remove it by substracting it (the deficit).
        Is my intuition correct?








      arithmetic real-numbers







      share|cite|improve this question











      $endgroup$




      Is it correct to think of a negative number as a way of showing that you have a deficit in something? For example,




      • Let $x$ be a real number such that $x=25$.



      • Let $y$ be a real number such that $y=27$.



        Consider this equality:
        $$x-(-2)=y$$
        The way I think of it is that $x$ is $y$ with a deficit of $2$ and in order for it to equal $y$ we need to remove that deficit and that's why:
        $$ -(-2)=+2$$
        It's like we have a deficit in something and we want to remove it by substracting it (the deficit).
        Is my intuition correct?








      arithmetic real-numbers




      arithmetic real-numbers






      share|cite|improve this question















      share|cite|improve this question













      share|cite|improve this question




      share|cite|improve this question








      edited Dec 15 '18 at 10:27









      Blue

      48.5k870154




      48.5k870154










      asked Dec 15 '18 at 6:17









      Ashraf BenmebarekAshraf Benmebarek

      485




      485






















          1 Answer
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          active

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          1












          $begingroup$

          From the perspective of Group Theory, a negative number is simply an (the) "additive inverse" of a "positive" number. Note: This is more or less an over-simplification, because in $mathbb{R}$ (the set of all real numbers) we have an order relation $le$, and things can get more complicated.



          Let $(G,+)$ be a group (If you don't know what a group is, just think of it as a very simple algebraic structure that allows you to perform cancellation). One of the group axioms says that given any element $g$ in $G$, you can find some $tilde{g}$ satisfying
          $$g+tilde{g}=tilde{g}+g=0.$$



          Then you may ask what $0$ actually is. Again I am not going too deep, so it would be better to treat it as the ordinary $0$ so that the value of any number is unchanged after adding $0$.



          Then $tilde{g}$ is called an (the) inverse of $g$. We give it a notation $-g$. Another note that is perhaps hard for you. When you denote something (by using a notation), you have to make sure that it is unique (so you cannot have something like $tilde{g}=2$ and $tilde{g}=3$ simultaneously), and the article "the" has an implicit meaning of uniqueness.



          Hence, what is $-2$? $-2$ is just the additive inverse of $2$. So if you have a $2$, after adding $-2$, you get back $0$. This is just something like a deficit of something: suppose you have $2$ apples, and after you have a deficit of $2$ apples, you have no apple left. Alternatively, if your friend got $2$ apples from you, then you have a deficit of $2$ apples. Hence if I add back $2$ apples to you, you have no deficit.






          share|cite|improve this answer











          $endgroup$













          • $begingroup$
            Isn't there a real-world example that shows why -(-2)=2 ?
            $endgroup$
            – Ashraf Benmebarek
            Dec 28 '18 at 15:41











          Your Answer





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          active

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          1












          $begingroup$

          From the perspective of Group Theory, a negative number is simply an (the) "additive inverse" of a "positive" number. Note: This is more or less an over-simplification, because in $mathbb{R}$ (the set of all real numbers) we have an order relation $le$, and things can get more complicated.



          Let $(G,+)$ be a group (If you don't know what a group is, just think of it as a very simple algebraic structure that allows you to perform cancellation). One of the group axioms says that given any element $g$ in $G$, you can find some $tilde{g}$ satisfying
          $$g+tilde{g}=tilde{g}+g=0.$$



          Then you may ask what $0$ actually is. Again I am not going too deep, so it would be better to treat it as the ordinary $0$ so that the value of any number is unchanged after adding $0$.



          Then $tilde{g}$ is called an (the) inverse of $g$. We give it a notation $-g$. Another note that is perhaps hard for you. When you denote something (by using a notation), you have to make sure that it is unique (so you cannot have something like $tilde{g}=2$ and $tilde{g}=3$ simultaneously), and the article "the" has an implicit meaning of uniqueness.



          Hence, what is $-2$? $-2$ is just the additive inverse of $2$. So if you have a $2$, after adding $-2$, you get back $0$. This is just something like a deficit of something: suppose you have $2$ apples, and after you have a deficit of $2$ apples, you have no apple left. Alternatively, if your friend got $2$ apples from you, then you have a deficit of $2$ apples. Hence if I add back $2$ apples to you, you have no deficit.






          share|cite|improve this answer











          $endgroup$













          • $begingroup$
            Isn't there a real-world example that shows why -(-2)=2 ?
            $endgroup$
            – Ashraf Benmebarek
            Dec 28 '18 at 15:41
















          1












          $begingroup$

          From the perspective of Group Theory, a negative number is simply an (the) "additive inverse" of a "positive" number. Note: This is more or less an over-simplification, because in $mathbb{R}$ (the set of all real numbers) we have an order relation $le$, and things can get more complicated.



          Let $(G,+)$ be a group (If you don't know what a group is, just think of it as a very simple algebraic structure that allows you to perform cancellation). One of the group axioms says that given any element $g$ in $G$, you can find some $tilde{g}$ satisfying
          $$g+tilde{g}=tilde{g}+g=0.$$



          Then you may ask what $0$ actually is. Again I am not going too deep, so it would be better to treat it as the ordinary $0$ so that the value of any number is unchanged after adding $0$.



          Then $tilde{g}$ is called an (the) inverse of $g$. We give it a notation $-g$. Another note that is perhaps hard for you. When you denote something (by using a notation), you have to make sure that it is unique (so you cannot have something like $tilde{g}=2$ and $tilde{g}=3$ simultaneously), and the article "the" has an implicit meaning of uniqueness.



          Hence, what is $-2$? $-2$ is just the additive inverse of $2$. So if you have a $2$, after adding $-2$, you get back $0$. This is just something like a deficit of something: suppose you have $2$ apples, and after you have a deficit of $2$ apples, you have no apple left. Alternatively, if your friend got $2$ apples from you, then you have a deficit of $2$ apples. Hence if I add back $2$ apples to you, you have no deficit.






          share|cite|improve this answer











          $endgroup$













          • $begingroup$
            Isn't there a real-world example that shows why -(-2)=2 ?
            $endgroup$
            – Ashraf Benmebarek
            Dec 28 '18 at 15:41














          1












          1








          1





          $begingroup$

          From the perspective of Group Theory, a negative number is simply an (the) "additive inverse" of a "positive" number. Note: This is more or less an over-simplification, because in $mathbb{R}$ (the set of all real numbers) we have an order relation $le$, and things can get more complicated.



          Let $(G,+)$ be a group (If you don't know what a group is, just think of it as a very simple algebraic structure that allows you to perform cancellation). One of the group axioms says that given any element $g$ in $G$, you can find some $tilde{g}$ satisfying
          $$g+tilde{g}=tilde{g}+g=0.$$



          Then you may ask what $0$ actually is. Again I am not going too deep, so it would be better to treat it as the ordinary $0$ so that the value of any number is unchanged after adding $0$.



          Then $tilde{g}$ is called an (the) inverse of $g$. We give it a notation $-g$. Another note that is perhaps hard for you. When you denote something (by using a notation), you have to make sure that it is unique (so you cannot have something like $tilde{g}=2$ and $tilde{g}=3$ simultaneously), and the article "the" has an implicit meaning of uniqueness.



          Hence, what is $-2$? $-2$ is just the additive inverse of $2$. So if you have a $2$, after adding $-2$, you get back $0$. This is just something like a deficit of something: suppose you have $2$ apples, and after you have a deficit of $2$ apples, you have no apple left. Alternatively, if your friend got $2$ apples from you, then you have a deficit of $2$ apples. Hence if I add back $2$ apples to you, you have no deficit.






          share|cite|improve this answer











          $endgroup$



          From the perspective of Group Theory, a negative number is simply an (the) "additive inverse" of a "positive" number. Note: This is more or less an over-simplification, because in $mathbb{R}$ (the set of all real numbers) we have an order relation $le$, and things can get more complicated.



          Let $(G,+)$ be a group (If you don't know what a group is, just think of it as a very simple algebraic structure that allows you to perform cancellation). One of the group axioms says that given any element $g$ in $G$, you can find some $tilde{g}$ satisfying
          $$g+tilde{g}=tilde{g}+g=0.$$



          Then you may ask what $0$ actually is. Again I am not going too deep, so it would be better to treat it as the ordinary $0$ so that the value of any number is unchanged after adding $0$.



          Then $tilde{g}$ is called an (the) inverse of $g$. We give it a notation $-g$. Another note that is perhaps hard for you. When you denote something (by using a notation), you have to make sure that it is unique (so you cannot have something like $tilde{g}=2$ and $tilde{g}=3$ simultaneously), and the article "the" has an implicit meaning of uniqueness.



          Hence, what is $-2$? $-2$ is just the additive inverse of $2$. So if you have a $2$, after adding $-2$, you get back $0$. This is just something like a deficit of something: suppose you have $2$ apples, and after you have a deficit of $2$ apples, you have no apple left. Alternatively, if your friend got $2$ apples from you, then you have a deficit of $2$ apples. Hence if I add back $2$ apples to you, you have no deficit.







          share|cite|improve this answer














          share|cite|improve this answer



          share|cite|improve this answer








          edited Dec 15 '18 at 9:47

























          answered Dec 15 '18 at 9:39









          tonychow0929tonychow0929

          29825




          29825












          • $begingroup$
            Isn't there a real-world example that shows why -(-2)=2 ?
            $endgroup$
            – Ashraf Benmebarek
            Dec 28 '18 at 15:41


















          • $begingroup$
            Isn't there a real-world example that shows why -(-2)=2 ?
            $endgroup$
            – Ashraf Benmebarek
            Dec 28 '18 at 15:41
















          $begingroup$
          Isn't there a real-world example that shows why -(-2)=2 ?
          $endgroup$
          – Ashraf Benmebarek
          Dec 28 '18 at 15:41




          $begingroup$
          Isn't there a real-world example that shows why -(-2)=2 ?
          $endgroup$
          – Ashraf Benmebarek
          Dec 28 '18 at 15:41


















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