Why is the Absolute value / modulus function used?












2














Why is the absolute value function or modulus function $|x|$ used ? What are its uses?



For example the square of a modulus number will always be positive, but why is it used when for example the square of any number whether positive or negative is always positive ? For example, $X^2$, will give a positive number whether negative or positive where $X$ is any number positive or negative.






absolute-value







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  • 1




    $x^2ne |x|$ so if I want the positive value of $x$ , how would I "just do" $x^2$?
    – John Douma
    46 mins ago










  • I’m just giving an example. What is the use of the modulus function ?
    – Dan
    44 mins ago










  • Obviously: Getting the absolute value of a number.
    – Henrik
    39 mins ago






  • 1




    It has many uses. Have you had Calculus? It is used in definitions where we only care about the distance between two points regardless of which one is greater. e.g. $|x-c|ltdeltaimplies |f(x)-f(c)|ltepsilon$.
    – John Douma
    38 mins ago
















2














Why is the absolute value function or modulus function $|x|$ used ? What are its uses?



For example the square of a modulus number will always be positive, but why is it used when for example the square of any number whether positive or negative is always positive ? For example, $X^2$, will give a positive number whether negative or positive where $X$ is any number positive or negative.






absolute-value







share|cite|improve this question




















  • 1




    $x^2ne |x|$ so if I want the positive value of $x$ , how would I "just do" $x^2$?
    – John Douma
    46 mins ago










  • I’m just giving an example. What is the use of the modulus function ?
    – Dan
    44 mins ago










  • Obviously: Getting the absolute value of a number.
    – Henrik
    39 mins ago






  • 1




    It has many uses. Have you had Calculus? It is used in definitions where we only care about the distance between two points regardless of which one is greater. e.g. $|x-c|ltdeltaimplies |f(x)-f(c)|ltepsilon$.
    – John Douma
    38 mins ago














2












2








2







Why is the absolute value function or modulus function $|x|$ used ? What are its uses?



For example the square of a modulus number will always be positive, but why is it used when for example the square of any number whether positive or negative is always positive ? For example, $X^2$, will give a positive number whether negative or positive where $X$ is any number positive or negative.






absolute-value







share|cite|improve this question















Why is the absolute value function or modulus function $|x|$ used ? What are its uses?



For example the square of a modulus number will always be positive, but why is it used when for example the square of any number whether positive or negative is always positive ? For example, $X^2$, will give a positive number whether negative or positive where $X$ is any number positive or negative.






absolute-value




absolute-value






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edited 41 mins ago

























asked 51 mins ago









Dan

88331128




88331128








  • 1




    $x^2ne |x|$ so if I want the positive value of $x$ , how would I "just do" $x^2$?
    – John Douma
    46 mins ago










  • I’m just giving an example. What is the use of the modulus function ?
    – Dan
    44 mins ago










  • Obviously: Getting the absolute value of a number.
    – Henrik
    39 mins ago






  • 1




    It has many uses. Have you had Calculus? It is used in definitions where we only care about the distance between two points regardless of which one is greater. e.g. $|x-c|ltdeltaimplies |f(x)-f(c)|ltepsilon$.
    – John Douma
    38 mins ago














  • 1




    $x^2ne |x|$ so if I want the positive value of $x$ , how would I "just do" $x^2$?
    – John Douma
    46 mins ago










  • I’m just giving an example. What is the use of the modulus function ?
    – Dan
    44 mins ago










  • Obviously: Getting the absolute value of a number.
    – Henrik
    39 mins ago






  • 1




    It has many uses. Have you had Calculus? It is used in definitions where we only care about the distance between two points regardless of which one is greater. e.g. $|x-c|ltdeltaimplies |f(x)-f(c)|ltepsilon$.
    – John Douma
    38 mins ago








1




1




$x^2ne |x|$ so if I want the positive value of $x$ , how would I "just do" $x^2$?
– John Douma
46 mins ago




$x^2ne |x|$ so if I want the positive value of $x$ , how would I "just do" $x^2$?
– John Douma
46 mins ago












I’m just giving an example. What is the use of the modulus function ?
– Dan
44 mins ago




I’m just giving an example. What is the use of the modulus function ?
– Dan
44 mins ago












Obviously: Getting the absolute value of a number.
– Henrik
39 mins ago




Obviously: Getting the absolute value of a number.
– Henrik
39 mins ago




1




1




It has many uses. Have you had Calculus? It is used in definitions where we only care about the distance between two points regardless of which one is greater. e.g. $|x-c|ltdeltaimplies |f(x)-f(c)|ltepsilon$.
– John Douma
38 mins ago




It has many uses. Have you had Calculus? It is used in definitions where we only care about the distance between two points regardless of which one is greater. e.g. $|x-c|ltdeltaimplies |f(x)-f(c)|ltepsilon$.
– John Douma
38 mins ago










3 Answers
3






active

oldest

votes


















3














One use of it is to define the distance between numbers. For example, im Calculus you may want to say "the distance between $x$ and $y$ is less than $1$". The way to write that mathematically is $|x-y|<1$. And you want to write it methematically so you can work with it mathematiclly.






share|cite|improve this answer





























    1














    In the context of real numbers the absolute value of a number is used in many ways but perhaps very elementarily it is used to write numbers in a canonical form. Every real number $ane 0$ is uniquely equal to $pm left |aright|$. So if we define the sign function $scolon mathbb Rsetminus{0}to {+,-}$ given by $s(a)=+$ if $a>0$ and $s(a)=-$ if $a<0$, then: for all $ane 0$ in $mathbb R$ we have $a=sign(a)cdot left | a right |$. In a sense this is a way to build all the reals from the positive ones. This is all just a special case of the polar representation of complex numbers, a representation of utmost importance.






    share|cite|improve this answer





























      1














      Origin of absolute value is from complex analysis:




      Absolute Value: In 1841, Weierstrass called $a + b$ the absolute value of the complex number $a + bi$ and represented it by $|a+bi|$.
      Absolute if from the Latin absoluere, "to free from"; hence, to free from its sign.




      Has the quote explains, its main role is to free a number from its sign.



      (caveat: note that, with modern notation, $|a+bi|$ is now the modulus of the complex number $sqrt{a^2+b^2}$, which is not the same as Weierstrauss "absolute value")




      Reference: A Brief Historical Dictionary of Mathematical Terms



      More solid reference:

      The notation $vert xvert$ for absolute value of $x$ was introduced by Weierstrass in 1841 [Vol 2,page 123] from:



      Florian Cajori.
      A History of Mathematical Notations(Two volumes bound as one).
      Dover Publications, 1993.






      share|cite|improve this answer























      • jeff560.tripod.com/a.html lists several earlier uses. I'm automatically sceptical towards a source that spells “Weierstrass” wrong, and I would be very surprised if Weierstrass actually used $a+b$ instead of $sqrt{a^2+b^2}$.
        – Hans Lundmark
        22 mins ago










      • @HansLundmark me too, that was the reason why I have another reference, let me write it down. Done.
        – Picaud Vincent
        21 mins ago













      Your Answer





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      3 Answers
      3






      active

      oldest

      votes








      3 Answers
      3






      active

      oldest

      votes









      active

      oldest

      votes






      active

      oldest

      votes









      3














      One use of it is to define the distance between numbers. For example, im Calculus you may want to say "the distance between $x$ and $y$ is less than $1$". The way to write that mathematically is $|x-y|<1$. And you want to write it methematically so you can work with it mathematiclly.






      share|cite|improve this answer


























        3














        One use of it is to define the distance between numbers. For example, im Calculus you may want to say "the distance between $x$ and $y$ is less than $1$". The way to write that mathematically is $|x-y|<1$. And you want to write it methematically so you can work with it mathematiclly.






        share|cite|improve this answer
























          3












          3








          3






          One use of it is to define the distance between numbers. For example, im Calculus you may want to say "the distance between $x$ and $y$ is less than $1$". The way to write that mathematically is $|x-y|<1$. And you want to write it methematically so you can work with it mathematiclly.






          share|cite|improve this answer












          One use of it is to define the distance between numbers. For example, im Calculus you may want to say "the distance between $x$ and $y$ is less than $1$". The way to write that mathematically is $|x-y|<1$. And you want to write it methematically so you can work with it mathematiclly.







          share|cite|improve this answer












          share|cite|improve this answer



          share|cite|improve this answer










          answered 37 mins ago









          Ovi

          12.2k1038109




          12.2k1038109























              1














              In the context of real numbers the absolute value of a number is used in many ways but perhaps very elementarily it is used to write numbers in a canonical form. Every real number $ane 0$ is uniquely equal to $pm left |aright|$. So if we define the sign function $scolon mathbb Rsetminus{0}to {+,-}$ given by $s(a)=+$ if $a>0$ and $s(a)=-$ if $a<0$, then: for all $ane 0$ in $mathbb R$ we have $a=sign(a)cdot left | a right |$. In a sense this is a way to build all the reals from the positive ones. This is all just a special case of the polar representation of complex numbers, a representation of utmost importance.






              share|cite|improve this answer


























                1














                In the context of real numbers the absolute value of a number is used in many ways but perhaps very elementarily it is used to write numbers in a canonical form. Every real number $ane 0$ is uniquely equal to $pm left |aright|$. So if we define the sign function $scolon mathbb Rsetminus{0}to {+,-}$ given by $s(a)=+$ if $a>0$ and $s(a)=-$ if $a<0$, then: for all $ane 0$ in $mathbb R$ we have $a=sign(a)cdot left | a right |$. In a sense this is a way to build all the reals from the positive ones. This is all just a special case of the polar representation of complex numbers, a representation of utmost importance.






                share|cite|improve this answer
























                  1












                  1








                  1






                  In the context of real numbers the absolute value of a number is used in many ways but perhaps very elementarily it is used to write numbers in a canonical form. Every real number $ane 0$ is uniquely equal to $pm left |aright|$. So if we define the sign function $scolon mathbb Rsetminus{0}to {+,-}$ given by $s(a)=+$ if $a>0$ and $s(a)=-$ if $a<0$, then: for all $ane 0$ in $mathbb R$ we have $a=sign(a)cdot left | a right |$. In a sense this is a way to build all the reals from the positive ones. This is all just a special case of the polar representation of complex numbers, a representation of utmost importance.






                  share|cite|improve this answer












                  In the context of real numbers the absolute value of a number is used in many ways but perhaps very elementarily it is used to write numbers in a canonical form. Every real number $ane 0$ is uniquely equal to $pm left |aright|$. So if we define the sign function $scolon mathbb Rsetminus{0}to {+,-}$ given by $s(a)=+$ if $a>0$ and $s(a)=-$ if $a<0$, then: for all $ane 0$ in $mathbb R$ we have $a=sign(a)cdot left | a right |$. In a sense this is a way to build all the reals from the positive ones. This is all just a special case of the polar representation of complex numbers, a representation of utmost importance.







                  share|cite|improve this answer












                  share|cite|improve this answer



                  share|cite|improve this answer










                  answered 23 mins ago









                  Ittay Weiss

                  63.3k6101183




                  63.3k6101183























                      1














                      Origin of absolute value is from complex analysis:




                      Absolute Value: In 1841, Weierstrass called $a + b$ the absolute value of the complex number $a + bi$ and represented it by $|a+bi|$.
                      Absolute if from the Latin absoluere, "to free from"; hence, to free from its sign.




                      Has the quote explains, its main role is to free a number from its sign.



                      (caveat: note that, with modern notation, $|a+bi|$ is now the modulus of the complex number $sqrt{a^2+b^2}$, which is not the same as Weierstrauss "absolute value")




                      Reference: A Brief Historical Dictionary of Mathematical Terms



                      More solid reference:

                      The notation $vert xvert$ for absolute value of $x$ was introduced by Weierstrass in 1841 [Vol 2,page 123] from:



                      Florian Cajori.
                      A History of Mathematical Notations(Two volumes bound as one).
                      Dover Publications, 1993.






                      share|cite|improve this answer























                      • jeff560.tripod.com/a.html lists several earlier uses. I'm automatically sceptical towards a source that spells “Weierstrass” wrong, and I would be very surprised if Weierstrass actually used $a+b$ instead of $sqrt{a^2+b^2}$.
                        – Hans Lundmark
                        22 mins ago










                      • @HansLundmark me too, that was the reason why I have another reference, let me write it down. Done.
                        – Picaud Vincent
                        21 mins ago


















                      1














                      Origin of absolute value is from complex analysis:




                      Absolute Value: In 1841, Weierstrass called $a + b$ the absolute value of the complex number $a + bi$ and represented it by $|a+bi|$.
                      Absolute if from the Latin absoluere, "to free from"; hence, to free from its sign.




                      Has the quote explains, its main role is to free a number from its sign.



                      (caveat: note that, with modern notation, $|a+bi|$ is now the modulus of the complex number $sqrt{a^2+b^2}$, which is not the same as Weierstrauss "absolute value")




                      Reference: A Brief Historical Dictionary of Mathematical Terms



                      More solid reference:

                      The notation $vert xvert$ for absolute value of $x$ was introduced by Weierstrass in 1841 [Vol 2,page 123] from:



                      Florian Cajori.
                      A History of Mathematical Notations(Two volumes bound as one).
                      Dover Publications, 1993.






                      share|cite|improve this answer























                      • jeff560.tripod.com/a.html lists several earlier uses. I'm automatically sceptical towards a source that spells “Weierstrass” wrong, and I would be very surprised if Weierstrass actually used $a+b$ instead of $sqrt{a^2+b^2}$.
                        – Hans Lundmark
                        22 mins ago










                      • @HansLundmark me too, that was the reason why I have another reference, let me write it down. Done.
                        – Picaud Vincent
                        21 mins ago
















                      1












                      1








                      1






                      Origin of absolute value is from complex analysis:




                      Absolute Value: In 1841, Weierstrass called $a + b$ the absolute value of the complex number $a + bi$ and represented it by $|a+bi|$.
                      Absolute if from the Latin absoluere, "to free from"; hence, to free from its sign.




                      Has the quote explains, its main role is to free a number from its sign.



                      (caveat: note that, with modern notation, $|a+bi|$ is now the modulus of the complex number $sqrt{a^2+b^2}$, which is not the same as Weierstrauss "absolute value")




                      Reference: A Brief Historical Dictionary of Mathematical Terms



                      More solid reference:

                      The notation $vert xvert$ for absolute value of $x$ was introduced by Weierstrass in 1841 [Vol 2,page 123] from:



                      Florian Cajori.
                      A History of Mathematical Notations(Two volumes bound as one).
                      Dover Publications, 1993.






                      share|cite|improve this answer














                      Origin of absolute value is from complex analysis:




                      Absolute Value: In 1841, Weierstrass called $a + b$ the absolute value of the complex number $a + bi$ and represented it by $|a+bi|$.
                      Absolute if from the Latin absoluere, "to free from"; hence, to free from its sign.




                      Has the quote explains, its main role is to free a number from its sign.



                      (caveat: note that, with modern notation, $|a+bi|$ is now the modulus of the complex number $sqrt{a^2+b^2}$, which is not the same as Weierstrauss "absolute value")




                      Reference: A Brief Historical Dictionary of Mathematical Terms



                      More solid reference:

                      The notation $vert xvert$ for absolute value of $x$ was introduced by Weierstrass in 1841 [Vol 2,page 123] from:



                      Florian Cajori.
                      A History of Mathematical Notations(Two volumes bound as one).
                      Dover Publications, 1993.







                      share|cite|improve this answer














                      share|cite|improve this answer



                      share|cite|improve this answer








                      edited 19 mins ago

























                      answered 31 mins ago









                      Picaud Vincent

                      1,11836




                      1,11836












                      • jeff560.tripod.com/a.html lists several earlier uses. I'm automatically sceptical towards a source that spells “Weierstrass” wrong, and I would be very surprised if Weierstrass actually used $a+b$ instead of $sqrt{a^2+b^2}$.
                        – Hans Lundmark
                        22 mins ago










                      • @HansLundmark me too, that was the reason why I have another reference, let me write it down. Done.
                        – Picaud Vincent
                        21 mins ago




















                      • jeff560.tripod.com/a.html lists several earlier uses. I'm automatically sceptical towards a source that spells “Weierstrass” wrong, and I would be very surprised if Weierstrass actually used $a+b$ instead of $sqrt{a^2+b^2}$.
                        – Hans Lundmark
                        22 mins ago










                      • @HansLundmark me too, that was the reason why I have another reference, let me write it down. Done.
                        – Picaud Vincent
                        21 mins ago


















                      jeff560.tripod.com/a.html lists several earlier uses. I'm automatically sceptical towards a source that spells “Weierstrass” wrong, and I would be very surprised if Weierstrass actually used $a+b$ instead of $sqrt{a^2+b^2}$.
                      – Hans Lundmark
                      22 mins ago




                      jeff560.tripod.com/a.html lists several earlier uses. I'm automatically sceptical towards a source that spells “Weierstrass” wrong, and I would be very surprised if Weierstrass actually used $a+b$ instead of $sqrt{a^2+b^2}$.
                      – Hans Lundmark
                      22 mins ago












                      @HansLundmark me too, that was the reason why I have another reference, let me write it down. Done.
                      – Picaud Vincent
                      21 mins ago






                      @HansLundmark me too, that was the reason why I have another reference, let me write it down. Done.
                      – Picaud Vincent
                      21 mins ago




















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