Why is the Absolute value / modulus function used?
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
add a comment |
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
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
add a comment |
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
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
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
add a comment |
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
add a comment |
3 Answers
3
active
oldest
votes
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.
add a comment |
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.
add a comment |
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.
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
add a comment |
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3 Answers
3
active
oldest
votes
3 Answers
3
active
oldest
votes
active
oldest
votes
active
oldest
votes
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.
add a comment |
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.
add a comment |
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.
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.
answered 37 mins ago
Ovi
12.2k1038109
12.2k1038109
add a comment |
add a comment |
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.
add a comment |
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.
add a comment |
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.
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.
answered 23 mins ago
Ittay Weiss
63.3k6101183
63.3k6101183
add a comment |
add a comment |
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.
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
add a comment |
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.
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
add a comment |
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.
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.
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
add a comment |
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
add a comment |
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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