Why should a non-commutative operation even be called “multiplication”?
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As per my knowledge and what was taught in school,
$a*b$ is a times b or b times a
Obviously this is commutative as a times b and b times a are same thing.
on the other hand there are multiplications like vector multiplication and matrix multiplication that are not commutative.
What does multiplication mean in general, for these?
Or should they even be called multiplication?
number-theory terminology noncommutative-algebra binary-operations
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up vote
5
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As per my knowledge and what was taught in school,
$a*b$ is a times b or b times a
Obviously this is commutative as a times b and b times a are same thing.
on the other hand there are multiplications like vector multiplication and matrix multiplication that are not commutative.
What does multiplication mean in general, for these?
Or should they even be called multiplication?
number-theory terminology noncommutative-algebra binary-operations
New contributor
13
Much of the time it's tradition. But if you have an operation which is associative and distributes over addition (whatever "addition" means), then it's definitely worthy of the name "multiplication".
– Arthur
18 hours ago
According to inc.com/bill-murphy-jr/…, kids can learn the material from grade 1 better if they start grade when at the age of 8. Maybe by starting so late, they could avoid having damaging past knowledge and be introduced to the concept of a natural number as a finite ordinal and completely avoid teaching them any concept of cardinal numbers until they're ready for it later. That's because it's much harder to learn to add by creating two groups each of which has a certain number of objects then
– Timothy
8 hours ago
counting the objects in the union than directly from the inductive definition of natural number addition. Also, they should be taught only the bijective unary numeral system in grade 1 and how to add and multiply using it. They should also be guided on how to write a formal proof in a certain weak system of pure number theory and taught how to master that skill. They should later be tested on their ability to write a proof that natural number addition is associative and commutative and that natural number multiplication is also associative and commutative and distributes over addition and
– Timothy
8 hours ago
should not see the proof before they figure it out themself. They should later be given an inductive definition of the decimal notation of each natural number that just says to take the successor, you increase the last digit by 1 if it's not 9 or if it is 9, change it to 0 and change string before the last digit to that which represents the next natural number and be left to figure out on their own a short cut for adding or multiplying natural numbers given in decimal notation and write the answer in decimal notation. That way, when they're taught the definition of matrix multiplication, they
– Timothy
8 hours ago
won't think that operation is commutative. Also, waiting until later to teach them about cardinality of a finite set actually enables them to learn the material faster. I don't want to write this as an answer because I can't find a way to answer the question that I'm sure is not worse than no answer.
– Timothy
8 hours ago
add a comment |
up vote
5
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favorite
up vote
5
down vote
favorite
As per my knowledge and what was taught in school,
$a*b$ is a times b or b times a
Obviously this is commutative as a times b and b times a are same thing.
on the other hand there are multiplications like vector multiplication and matrix multiplication that are not commutative.
What does multiplication mean in general, for these?
Or should they even be called multiplication?
number-theory terminology noncommutative-algebra binary-operations
New contributor
As per my knowledge and what was taught in school,
$a*b$ is a times b or b times a
Obviously this is commutative as a times b and b times a are same thing.
on the other hand there are multiplications like vector multiplication and matrix multiplication that are not commutative.
What does multiplication mean in general, for these?
Or should they even be called multiplication?
number-theory terminology noncommutative-algebra binary-operations
number-theory terminology noncommutative-algebra binary-operations
New contributor
New contributor
edited 12 hours ago
Asaf Karagila♦
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299k32420750
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asked 18 hours ago
mathaholic
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13
Much of the time it's tradition. But if you have an operation which is associative and distributes over addition (whatever "addition" means), then it's definitely worthy of the name "multiplication".
– Arthur
18 hours ago
According to inc.com/bill-murphy-jr/…, kids can learn the material from grade 1 better if they start grade when at the age of 8. Maybe by starting so late, they could avoid having damaging past knowledge and be introduced to the concept of a natural number as a finite ordinal and completely avoid teaching them any concept of cardinal numbers until they're ready for it later. That's because it's much harder to learn to add by creating two groups each of which has a certain number of objects then
– Timothy
8 hours ago
counting the objects in the union than directly from the inductive definition of natural number addition. Also, they should be taught only the bijective unary numeral system in grade 1 and how to add and multiply using it. They should also be guided on how to write a formal proof in a certain weak system of pure number theory and taught how to master that skill. They should later be tested on their ability to write a proof that natural number addition is associative and commutative and that natural number multiplication is also associative and commutative and distributes over addition and
– Timothy
8 hours ago
should not see the proof before they figure it out themself. They should later be given an inductive definition of the decimal notation of each natural number that just says to take the successor, you increase the last digit by 1 if it's not 9 or if it is 9, change it to 0 and change string before the last digit to that which represents the next natural number and be left to figure out on their own a short cut for adding or multiplying natural numbers given in decimal notation and write the answer in decimal notation. That way, when they're taught the definition of matrix multiplication, they
– Timothy
8 hours ago
won't think that operation is commutative. Also, waiting until later to teach them about cardinality of a finite set actually enables them to learn the material faster. I don't want to write this as an answer because I can't find a way to answer the question that I'm sure is not worse than no answer.
– Timothy
8 hours ago
add a comment |
13
Much of the time it's tradition. But if you have an operation which is associative and distributes over addition (whatever "addition" means), then it's definitely worthy of the name "multiplication".
– Arthur
18 hours ago
According to inc.com/bill-murphy-jr/…, kids can learn the material from grade 1 better if they start grade when at the age of 8. Maybe by starting so late, they could avoid having damaging past knowledge and be introduced to the concept of a natural number as a finite ordinal and completely avoid teaching them any concept of cardinal numbers until they're ready for it later. That's because it's much harder to learn to add by creating two groups each of which has a certain number of objects then
– Timothy
8 hours ago
counting the objects in the union than directly from the inductive definition of natural number addition. Also, they should be taught only the bijective unary numeral system in grade 1 and how to add and multiply using it. They should also be guided on how to write a formal proof in a certain weak system of pure number theory and taught how to master that skill. They should later be tested on their ability to write a proof that natural number addition is associative and commutative and that natural number multiplication is also associative and commutative and distributes over addition and
– Timothy
8 hours ago
should not see the proof before they figure it out themself. They should later be given an inductive definition of the decimal notation of each natural number that just says to take the successor, you increase the last digit by 1 if it's not 9 or if it is 9, change it to 0 and change string before the last digit to that which represents the next natural number and be left to figure out on their own a short cut for adding or multiplying natural numbers given in decimal notation and write the answer in decimal notation. That way, when they're taught the definition of matrix multiplication, they
– Timothy
8 hours ago
won't think that operation is commutative. Also, waiting until later to teach them about cardinality of a finite set actually enables them to learn the material faster. I don't want to write this as an answer because I can't find a way to answer the question that I'm sure is not worse than no answer.
– Timothy
8 hours ago
13
13
Much of the time it's tradition. But if you have an operation which is associative and distributes over addition (whatever "addition" means), then it's definitely worthy of the name "multiplication".
– Arthur
18 hours ago
Much of the time it's tradition. But if you have an operation which is associative and distributes over addition (whatever "addition" means), then it's definitely worthy of the name "multiplication".
– Arthur
18 hours ago
According to inc.com/bill-murphy-jr/…, kids can learn the material from grade 1 better if they start grade when at the age of 8. Maybe by starting so late, they could avoid having damaging past knowledge and be introduced to the concept of a natural number as a finite ordinal and completely avoid teaching them any concept of cardinal numbers until they're ready for it later. That's because it's much harder to learn to add by creating two groups each of which has a certain number of objects then
– Timothy
8 hours ago
According to inc.com/bill-murphy-jr/…, kids can learn the material from grade 1 better if they start grade when at the age of 8. Maybe by starting so late, they could avoid having damaging past knowledge and be introduced to the concept of a natural number as a finite ordinal and completely avoid teaching them any concept of cardinal numbers until they're ready for it later. That's because it's much harder to learn to add by creating two groups each of which has a certain number of objects then
– Timothy
8 hours ago
counting the objects in the union than directly from the inductive definition of natural number addition. Also, they should be taught only the bijective unary numeral system in grade 1 and how to add and multiply using it. They should also be guided on how to write a formal proof in a certain weak system of pure number theory and taught how to master that skill. They should later be tested on their ability to write a proof that natural number addition is associative and commutative and that natural number multiplication is also associative and commutative and distributes over addition and
– Timothy
8 hours ago
counting the objects in the union than directly from the inductive definition of natural number addition. Also, they should be taught only the bijective unary numeral system in grade 1 and how to add and multiply using it. They should also be guided on how to write a formal proof in a certain weak system of pure number theory and taught how to master that skill. They should later be tested on their ability to write a proof that natural number addition is associative and commutative and that natural number multiplication is also associative and commutative and distributes over addition and
– Timothy
8 hours ago
should not see the proof before they figure it out themself. They should later be given an inductive definition of the decimal notation of each natural number that just says to take the successor, you increase the last digit by 1 if it's not 9 or if it is 9, change it to 0 and change string before the last digit to that which represents the next natural number and be left to figure out on their own a short cut for adding or multiplying natural numbers given in decimal notation and write the answer in decimal notation. That way, when they're taught the definition of matrix multiplication, they
– Timothy
8 hours ago
should not see the proof before they figure it out themself. They should later be given an inductive definition of the decimal notation of each natural number that just says to take the successor, you increase the last digit by 1 if it's not 9 or if it is 9, change it to 0 and change string before the last digit to that which represents the next natural number and be left to figure out on their own a short cut for adding or multiplying natural numbers given in decimal notation and write the answer in decimal notation. That way, when they're taught the definition of matrix multiplication, they
– Timothy
8 hours ago
won't think that operation is commutative. Also, waiting until later to teach them about cardinality of a finite set actually enables them to learn the material faster. I don't want to write this as an answer because I can't find a way to answer the question that I'm sure is not worse than no answer.
– Timothy
8 hours ago
won't think that operation is commutative. Also, waiting until later to teach them about cardinality of a finite set actually enables them to learn the material faster. I don't want to write this as an answer because I can't find a way to answer the question that I'm sure is not worse than no answer.
– Timothy
8 hours ago
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Terms in mathematics do not necessarily have absolute, universal definitions. The context is very important. It is common for a term to have similar but not identical meanings in multiple contexts but it is also common for the meaning to differ considerably. It can even differ from author to author in the same context. To be sure of the meaning, you need to check the author's definition.
It might be tempting to demand that multiplication be commutative and another term be used when it isn't but that would break some nice patterns such as the real numbers, to the complex numbers, to the quaternions.
In day to day life, multiplication is commutative but only because it deals only with real numbers. As you go deeper into maths, you will need to unlearn this assumption. It is very frequent that something called multiplication is not commutative.
1
(+1) It's a good answer, but my main reason for upvoting are the first two sentences.
– José Carlos Santos
18 hours ago
@JoséCarlosSantos Thanks. I considered adding some examples e.g. whether a ring has a multiplicative identity and two very different meanings of field. However, I suspected that these might not help.
– badjohn
12 hours ago
1
Matrices are probably a more commonly encountered example of non-commutative multiplication than quaternions.
– Henning Makholm
7 hours ago
@HenningMakholm, Yes and it seems that the OP first encountered non-commutative multiplication with matrices. I mentioned quaternions because the progression $mathbb{R}$ to $mathbb{C}$ to $mathbb{H}$ was neater.
– badjohn
6 hours ago
@Broman The "it" in that quote referred to maths in day to day life. Very few people use complex numbers in day to day life. Various structures in addition to the reals have commutative multiplication but are rarely encountered outside mathematics (or serious science). $mathbb{Z}_n$ is also commutative but not often used in day to day life. There are 12 and 24 hour clocks but multiplying two times is not common.
– badjohn
6 hours ago
|
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In school you might have also learned that multiplication or addition are defined for "numbers", but vectors, matrices, and functions are not numbers, why should they be allowed to be added or multiplied to begin with?
It turns out that sounds and letters form words which we use in context to convey information, usually in a concise manner.
In school, for example, most of your mathematical education is about the real numbers, maybe a bit about the complex numbers. Maybe you even learned to derive a function.
But did it ever occur to you that there are functions from $Bbb R$ to itself such that for every $a<b$, the image of the function on $(a,b)$ is the entire set of real numbers?
If all functions you dealt with in school were differentiable (or at least almost everywhere), why is that even a function? What does it even mean for something to be a function?
Well. These are questions that mathematicians dealt with a long time ago, and decided that we should stick to definitions. So in mathematics we have explicit definitions, and we give them names so we don't have to repeat the definition each time. The phrase "Let $H$ be a Hilbert space over $Bbb C$" packs in those eight words an immense amount of knowledge, that usually takes a long time to learn, for example.
Sometimes, out of convenience, and out of the sheer love of generalizations, mathematicians take a word which has a "common meaning", and decide that it is good enough to be used in a different context and to mean something else. Germ, stalk, filter, sheaf, quiver, graph, are all words that take a cue from the natural sense of the word, and give it an explicit definition in context.
(And I haven't even talked about things which have little to no relation to their real world meaning, e.g. a mouse in set theory.)
Multiplication is a word we use in context, and the context is usually an associative binary operation on some set. This set can be of functions, matrices, sets, etc. If we require the operation to have a neutral element and admit inverses, we have a group; if we adjoin another operator which is also associative, admits inverses, and commutative and posit some distributivity laws, we get a ring; or a semi-ring; or so on and so forth.
But it is convenient to talk about multiplication, because it's a word, and in most cases the pedagogical development helps us grasp why in generalizations we might want to omit commutativity from this operation.
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Terminology for mathematical structures is often built off of, and analogized to, terminology for "normal" mathematical structures such as integers, rational numbers, and real numbers. For vectors over real numbers, we have addition already defined for the coordinates. Applying this operation and adding components termwise results in a meaningful operation, and the natural terminology is to refer to that as simply "addition". Multiplying termwise result in an operation that isn't as meaningful (for one thing, this operation, unlike termwise addition, is dependent on the coordinate system). The cross product, on the other hand, is a meaningful operation, and it interacts with termwise addition in a manner similar to how real multiplication interacts with real addition. For instance, $(a+b)times c = a times c + b times c$ (distributive property).
For matrices, we again have termwise addition being a meaningful operation. Matrices represent linear operators, and the definition of linearity includes many of the properties of multiplication, such as distribution: A(u+v) = A(u) + A(v). Thus, it's natural to treat application of a linear operator as "multiplying" a vector by a matrix, and from there it's natural to define matrix multiplication as composition of the linear operators: (A*B)(v) = (A(B(v)).
2
+1, indeed the common feature of operations called "multiplication" seems to be that they are binary operations that distribute (on both sides) over "addition".
– Henning Makholm
7 hours ago
@HenningMakholm: I don't think that's quite the whole truth. For example, people talk about group "multiplication" even though there's only one operation in that setting. What does seem true is that there are lots of interesting examples of noncommutative operations which distribute over commutative operations and few (or no?) examples of commutative operations which distribute over noncommutative operations, so the tendency is to analogize commutative things with addition and noncommutative things with multiplication.
– Micah
4 hours ago
@Micah: Hmm, perhaps I'm projecting here, because the word "multiply" about arbirary group operations has never sat well me. (I'm cool with multiplicative notation, of course, and speaking about "product" and "factors" doesn't sound half as jarring as "multiply" either).
– Henning Makholm
1 hour ago
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1
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The mathematical concept below the question is that of operation.
In a very general setting, if X is a set, than an operation on X is just a function
$$Xtimes Xto X$$
Usually denoted with multiplicative notations like $cdot$ or $*$. This means that an operation takes two elements of X and give as a result an element of $X$ (exactly as when you take two numbers, say $3$ ant $5$ and the result is $5*3=15$)
Examples of oparations are the usual operations on real numbers: plus, minus, division (defined on non-zero reals), multiplications, exponentiation.
Now, if you want to use operations in mathematics you usualy requires properties that are useful in calculations. Here the most common properties that an operation can have.
$1)$ Associativity. That means that $(a*b)*c=a*(b*c)$ and allow you to omit parenthesis and write $a*b*c$. The usual summation and multiplications on real numbers are associative. Subtraction, division, and exponentiation are not. For example:
$$(5-2)-2=3-2=1neq 5=5-(2-2)$$
$$ (9/3)/3=3/3=1neq 9=9/1=9/(3/3)$$
$$ 2^{(3^2)}=2^9=512neq 64= 8^2=(2^3)^2$$
A famous examples of non-associative multiplication often used in mathematics is that of Cayley Octonions.
A useful example of associative operations is the composition of functions. Let $A$ be any set and $X$ be the set of all functions from $A$ to $A$, i.e. $X={f:Ato A}$. The composition of two function $f,g$ is the function $f*g$ defined by $f*g(a)=f(g(a))$. Clearly $f*(g*h)(a)=f(g(h(a)))=(f*g)*h(a)$.
$2)$ Commutativity. That means that $a*b=b*a$. Usual sum and multiplications are commutative. Matrix product is not commutative, the composition of functions is not commutative in general (a rotation composed with a translation is not the same as a translation first, and then the rotation), vector product is not commutative. Non commutativity of composition of functions is on the basis of Heisemberg uncertainty principle
. The Hamilton Quaternions are a useful structure used in math. They have a multiplication which is associative but non commutative.
$3)$ Existence of Neutral element. This means that there is an element e in $X$ so that $x*e=e*x=x$ for any $x$ of $X$. For summation the netural element is $0$, for multiplication is $1$. If you consider $X$ as the set of even integers numbers, than the usual multiplication is well-defined on $X$, but the neutral element does not exists in $X$ (it would be $1$, which is not in $X$).
$4)$ Existence of Inverse. In case there is neutral element $ein X$, this means that for any $xin X$ there exists $y$ so that $xy=yx=e$. Usually $y$ is denoted by $x^{-1}$. The inverse for usual sum is $-x$, the inverse for usual multiplication is $1/x$ (which exists only for non-zero elements). In the realm of matrices, there are many matrices that have no inverse, for instane the matrix
$begin{pmatrix} 1 & 1\1&1end{pmatrix}$.
Such properties are important because they make an operation user-friendly. For example: is it true that if $a,bneq 0$ then $a*bneq0$? This seems kind of obviuos, but it depends on the properties of the operation. For example
$begin{pmatrix} 1 & 0\3&0end{pmatrix}begin{pmatrix} 0 & 0\1&2end{pmatrix}=begin{pmatrix} 0 & 0\0&0end{pmatrix}$ but both
$begin{pmatrix} 1 & 0\3&0end{pmatrix}$ and $begin{pmatrix} 0 & 0\1&2end{pmatrix}$ are different from zero.
Concluding, I would say that when a mathematician hear the word multiplication, immediately think to an associative operation, usually (but not always) with neutral element, sometimes commutative.
Another example of non-associative multiplication is IEEE floating point numbers (where intermediate results can underflow to zero or not, depending on the order of evaluation.small * small * big
can be zero if the first two terms underflow, orsmall
if the second two terms multiply to 1.)
– Martin Bonner
5 hours ago
Not to mention the cross product in $mathbb R^3$.
– Henning Makholm
5 hours ago
@MartinBonner: An interesting example, in that it is a non-associative approximation to an associative operation. I see in Wikipedia, incidentally, that it may be subnormal rather than zero, though I do not see exactly when.
– PJTraill
5 hours ago
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4 Answers
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4 Answers
4
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active
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active
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18
down vote
Terms in mathematics do not necessarily have absolute, universal definitions. The context is very important. It is common for a term to have similar but not identical meanings in multiple contexts but it is also common for the meaning to differ considerably. It can even differ from author to author in the same context. To be sure of the meaning, you need to check the author's definition.
It might be tempting to demand that multiplication be commutative and another term be used when it isn't but that would break some nice patterns such as the real numbers, to the complex numbers, to the quaternions.
In day to day life, multiplication is commutative but only because it deals only with real numbers. As you go deeper into maths, you will need to unlearn this assumption. It is very frequent that something called multiplication is not commutative.
1
(+1) It's a good answer, but my main reason for upvoting are the first two sentences.
– José Carlos Santos
18 hours ago
@JoséCarlosSantos Thanks. I considered adding some examples e.g. whether a ring has a multiplicative identity and two very different meanings of field. However, I suspected that these might not help.
– badjohn
12 hours ago
1
Matrices are probably a more commonly encountered example of non-commutative multiplication than quaternions.
– Henning Makholm
7 hours ago
@HenningMakholm, Yes and it seems that the OP first encountered non-commutative multiplication with matrices. I mentioned quaternions because the progression $mathbb{R}$ to $mathbb{C}$ to $mathbb{H}$ was neater.
– badjohn
6 hours ago
@Broman The "it" in that quote referred to maths in day to day life. Very few people use complex numbers in day to day life. Various structures in addition to the reals have commutative multiplication but are rarely encountered outside mathematics (or serious science). $mathbb{Z}_n$ is also commutative but not often used in day to day life. There are 12 and 24 hour clocks but multiplying two times is not common.
– badjohn
6 hours ago
|
show 1 more comment
up vote
18
down vote
Terms in mathematics do not necessarily have absolute, universal definitions. The context is very important. It is common for a term to have similar but not identical meanings in multiple contexts but it is also common for the meaning to differ considerably. It can even differ from author to author in the same context. To be sure of the meaning, you need to check the author's definition.
It might be tempting to demand that multiplication be commutative and another term be used when it isn't but that would break some nice patterns such as the real numbers, to the complex numbers, to the quaternions.
In day to day life, multiplication is commutative but only because it deals only with real numbers. As you go deeper into maths, you will need to unlearn this assumption. It is very frequent that something called multiplication is not commutative.
1
(+1) It's a good answer, but my main reason for upvoting are the first two sentences.
– José Carlos Santos
18 hours ago
@JoséCarlosSantos Thanks. I considered adding some examples e.g. whether a ring has a multiplicative identity and two very different meanings of field. However, I suspected that these might not help.
– badjohn
12 hours ago
1
Matrices are probably a more commonly encountered example of non-commutative multiplication than quaternions.
– Henning Makholm
7 hours ago
@HenningMakholm, Yes and it seems that the OP first encountered non-commutative multiplication with matrices. I mentioned quaternions because the progression $mathbb{R}$ to $mathbb{C}$ to $mathbb{H}$ was neater.
– badjohn
6 hours ago
@Broman The "it" in that quote referred to maths in day to day life. Very few people use complex numbers in day to day life. Various structures in addition to the reals have commutative multiplication but are rarely encountered outside mathematics (or serious science). $mathbb{Z}_n$ is also commutative but not often used in day to day life. There are 12 and 24 hour clocks but multiplying two times is not common.
– badjohn
6 hours ago
|
show 1 more comment
up vote
18
down vote
up vote
18
down vote
Terms in mathematics do not necessarily have absolute, universal definitions. The context is very important. It is common for a term to have similar but not identical meanings in multiple contexts but it is also common for the meaning to differ considerably. It can even differ from author to author in the same context. To be sure of the meaning, you need to check the author's definition.
It might be tempting to demand that multiplication be commutative and another term be used when it isn't but that would break some nice patterns such as the real numbers, to the complex numbers, to the quaternions.
In day to day life, multiplication is commutative but only because it deals only with real numbers. As you go deeper into maths, you will need to unlearn this assumption. It is very frequent that something called multiplication is not commutative.
Terms in mathematics do not necessarily have absolute, universal definitions. The context is very important. It is common for a term to have similar but not identical meanings in multiple contexts but it is also common for the meaning to differ considerably. It can even differ from author to author in the same context. To be sure of the meaning, you need to check the author's definition.
It might be tempting to demand that multiplication be commutative and another term be used when it isn't but that would break some nice patterns such as the real numbers, to the complex numbers, to the quaternions.
In day to day life, multiplication is commutative but only because it deals only with real numbers. As you go deeper into maths, you will need to unlearn this assumption. It is very frequent that something called multiplication is not commutative.
edited 12 hours ago
answered 18 hours ago
badjohn
4,0871620
4,0871620
1
(+1) It's a good answer, but my main reason for upvoting are the first two sentences.
– José Carlos Santos
18 hours ago
@JoséCarlosSantos Thanks. I considered adding some examples e.g. whether a ring has a multiplicative identity and two very different meanings of field. However, I suspected that these might not help.
– badjohn
12 hours ago
1
Matrices are probably a more commonly encountered example of non-commutative multiplication than quaternions.
– Henning Makholm
7 hours ago
@HenningMakholm, Yes and it seems that the OP first encountered non-commutative multiplication with matrices. I mentioned quaternions because the progression $mathbb{R}$ to $mathbb{C}$ to $mathbb{H}$ was neater.
– badjohn
6 hours ago
@Broman The "it" in that quote referred to maths in day to day life. Very few people use complex numbers in day to day life. Various structures in addition to the reals have commutative multiplication but are rarely encountered outside mathematics (or serious science). $mathbb{Z}_n$ is also commutative but not often used in day to day life. There are 12 and 24 hour clocks but multiplying two times is not common.
– badjohn
6 hours ago
|
show 1 more comment
1
(+1) It's a good answer, but my main reason for upvoting are the first two sentences.
– José Carlos Santos
18 hours ago
@JoséCarlosSantos Thanks. I considered adding some examples e.g. whether a ring has a multiplicative identity and two very different meanings of field. However, I suspected that these might not help.
– badjohn
12 hours ago
1
Matrices are probably a more commonly encountered example of non-commutative multiplication than quaternions.
– Henning Makholm
7 hours ago
@HenningMakholm, Yes and it seems that the OP first encountered non-commutative multiplication with matrices. I mentioned quaternions because the progression $mathbb{R}$ to $mathbb{C}$ to $mathbb{H}$ was neater.
– badjohn
6 hours ago
@Broman The "it" in that quote referred to maths in day to day life. Very few people use complex numbers in day to day life. Various structures in addition to the reals have commutative multiplication but are rarely encountered outside mathematics (or serious science). $mathbb{Z}_n$ is also commutative but not often used in day to day life. There are 12 and 24 hour clocks but multiplying two times is not common.
– badjohn
6 hours ago
1
1
(+1) It's a good answer, but my main reason for upvoting are the first two sentences.
– José Carlos Santos
18 hours ago
(+1) It's a good answer, but my main reason for upvoting are the first two sentences.
– José Carlos Santos
18 hours ago
@JoséCarlosSantos Thanks. I considered adding some examples e.g. whether a ring has a multiplicative identity and two very different meanings of field. However, I suspected that these might not help.
– badjohn
12 hours ago
@JoséCarlosSantos Thanks. I considered adding some examples e.g. whether a ring has a multiplicative identity and two very different meanings of field. However, I suspected that these might not help.
– badjohn
12 hours ago
1
1
Matrices are probably a more commonly encountered example of non-commutative multiplication than quaternions.
– Henning Makholm
7 hours ago
Matrices are probably a more commonly encountered example of non-commutative multiplication than quaternions.
– Henning Makholm
7 hours ago
@HenningMakholm, Yes and it seems that the OP first encountered non-commutative multiplication with matrices. I mentioned quaternions because the progression $mathbb{R}$ to $mathbb{C}$ to $mathbb{H}$ was neater.
– badjohn
6 hours ago
@HenningMakholm, Yes and it seems that the OP first encountered non-commutative multiplication with matrices. I mentioned quaternions because the progression $mathbb{R}$ to $mathbb{C}$ to $mathbb{H}$ was neater.
– badjohn
6 hours ago
@Broman The "it" in that quote referred to maths in day to day life. Very few people use complex numbers in day to day life. Various structures in addition to the reals have commutative multiplication but are rarely encountered outside mathematics (or serious science). $mathbb{Z}_n$ is also commutative but not often used in day to day life. There are 12 and 24 hour clocks but multiplying two times is not common.
– badjohn
6 hours ago
@Broman The "it" in that quote referred to maths in day to day life. Very few people use complex numbers in day to day life. Various structures in addition to the reals have commutative multiplication but are rarely encountered outside mathematics (or serious science). $mathbb{Z}_n$ is also commutative but not often used in day to day life. There are 12 and 24 hour clocks but multiplying two times is not common.
– badjohn
6 hours ago
|
show 1 more comment
up vote
5
down vote
In school you might have also learned that multiplication or addition are defined for "numbers", but vectors, matrices, and functions are not numbers, why should they be allowed to be added or multiplied to begin with?
It turns out that sounds and letters form words which we use in context to convey information, usually in a concise manner.
In school, for example, most of your mathematical education is about the real numbers, maybe a bit about the complex numbers. Maybe you even learned to derive a function.
But did it ever occur to you that there are functions from $Bbb R$ to itself such that for every $a<b$, the image of the function on $(a,b)$ is the entire set of real numbers?
If all functions you dealt with in school were differentiable (or at least almost everywhere), why is that even a function? What does it even mean for something to be a function?
Well. These are questions that mathematicians dealt with a long time ago, and decided that we should stick to definitions. So in mathematics we have explicit definitions, and we give them names so we don't have to repeat the definition each time. The phrase "Let $H$ be a Hilbert space over $Bbb C$" packs in those eight words an immense amount of knowledge, that usually takes a long time to learn, for example.
Sometimes, out of convenience, and out of the sheer love of generalizations, mathematicians take a word which has a "common meaning", and decide that it is good enough to be used in a different context and to mean something else. Germ, stalk, filter, sheaf, quiver, graph, are all words that take a cue from the natural sense of the word, and give it an explicit definition in context.
(And I haven't even talked about things which have little to no relation to their real world meaning, e.g. a mouse in set theory.)
Multiplication is a word we use in context, and the context is usually an associative binary operation on some set. This set can be of functions, matrices, sets, etc. If we require the operation to have a neutral element and admit inverses, we have a group; if we adjoin another operator which is also associative, admits inverses, and commutative and posit some distributivity laws, we get a ring; or a semi-ring; or so on and so forth.
But it is convenient to talk about multiplication, because it's a word, and in most cases the pedagogical development helps us grasp why in generalizations we might want to omit commutativity from this operation.
add a comment |
up vote
5
down vote
In school you might have also learned that multiplication or addition are defined for "numbers", but vectors, matrices, and functions are not numbers, why should they be allowed to be added or multiplied to begin with?
It turns out that sounds and letters form words which we use in context to convey information, usually in a concise manner.
In school, for example, most of your mathematical education is about the real numbers, maybe a bit about the complex numbers. Maybe you even learned to derive a function.
But did it ever occur to you that there are functions from $Bbb R$ to itself such that for every $a<b$, the image of the function on $(a,b)$ is the entire set of real numbers?
If all functions you dealt with in school were differentiable (or at least almost everywhere), why is that even a function? What does it even mean for something to be a function?
Well. These are questions that mathematicians dealt with a long time ago, and decided that we should stick to definitions. So in mathematics we have explicit definitions, and we give them names so we don't have to repeat the definition each time. The phrase "Let $H$ be a Hilbert space over $Bbb C$" packs in those eight words an immense amount of knowledge, that usually takes a long time to learn, for example.
Sometimes, out of convenience, and out of the sheer love of generalizations, mathematicians take a word which has a "common meaning", and decide that it is good enough to be used in a different context and to mean something else. Germ, stalk, filter, sheaf, quiver, graph, are all words that take a cue from the natural sense of the word, and give it an explicit definition in context.
(And I haven't even talked about things which have little to no relation to their real world meaning, e.g. a mouse in set theory.)
Multiplication is a word we use in context, and the context is usually an associative binary operation on some set. This set can be of functions, matrices, sets, etc. If we require the operation to have a neutral element and admit inverses, we have a group; if we adjoin another operator which is also associative, admits inverses, and commutative and posit some distributivity laws, we get a ring; or a semi-ring; or so on and so forth.
But it is convenient to talk about multiplication, because it's a word, and in most cases the pedagogical development helps us grasp why in generalizations we might want to omit commutativity from this operation.
add a comment |
up vote
5
down vote
up vote
5
down vote
In school you might have also learned that multiplication or addition are defined for "numbers", but vectors, matrices, and functions are not numbers, why should they be allowed to be added or multiplied to begin with?
It turns out that sounds and letters form words which we use in context to convey information, usually in a concise manner.
In school, for example, most of your mathematical education is about the real numbers, maybe a bit about the complex numbers. Maybe you even learned to derive a function.
But did it ever occur to you that there are functions from $Bbb R$ to itself such that for every $a<b$, the image of the function on $(a,b)$ is the entire set of real numbers?
If all functions you dealt with in school were differentiable (or at least almost everywhere), why is that even a function? What does it even mean for something to be a function?
Well. These are questions that mathematicians dealt with a long time ago, and decided that we should stick to definitions. So in mathematics we have explicit definitions, and we give them names so we don't have to repeat the definition each time. The phrase "Let $H$ be a Hilbert space over $Bbb C$" packs in those eight words an immense amount of knowledge, that usually takes a long time to learn, for example.
Sometimes, out of convenience, and out of the sheer love of generalizations, mathematicians take a word which has a "common meaning", and decide that it is good enough to be used in a different context and to mean something else. Germ, stalk, filter, sheaf, quiver, graph, are all words that take a cue from the natural sense of the word, and give it an explicit definition in context.
(And I haven't even talked about things which have little to no relation to their real world meaning, e.g. a mouse in set theory.)
Multiplication is a word we use in context, and the context is usually an associative binary operation on some set. This set can be of functions, matrices, sets, etc. If we require the operation to have a neutral element and admit inverses, we have a group; if we adjoin another operator which is also associative, admits inverses, and commutative and posit some distributivity laws, we get a ring; or a semi-ring; or so on and so forth.
But it is convenient to talk about multiplication, because it's a word, and in most cases the pedagogical development helps us grasp why in generalizations we might want to omit commutativity from this operation.
In school you might have also learned that multiplication or addition are defined for "numbers", but vectors, matrices, and functions are not numbers, why should they be allowed to be added or multiplied to begin with?
It turns out that sounds and letters form words which we use in context to convey information, usually in a concise manner.
In school, for example, most of your mathematical education is about the real numbers, maybe a bit about the complex numbers. Maybe you even learned to derive a function.
But did it ever occur to you that there are functions from $Bbb R$ to itself such that for every $a<b$, the image of the function on $(a,b)$ is the entire set of real numbers?
If all functions you dealt with in school were differentiable (or at least almost everywhere), why is that even a function? What does it even mean for something to be a function?
Well. These are questions that mathematicians dealt with a long time ago, and decided that we should stick to definitions. So in mathematics we have explicit definitions, and we give them names so we don't have to repeat the definition each time. The phrase "Let $H$ be a Hilbert space over $Bbb C$" packs in those eight words an immense amount of knowledge, that usually takes a long time to learn, for example.
Sometimes, out of convenience, and out of the sheer love of generalizations, mathematicians take a word which has a "common meaning", and decide that it is good enough to be used in a different context and to mean something else. Germ, stalk, filter, sheaf, quiver, graph, are all words that take a cue from the natural sense of the word, and give it an explicit definition in context.
(And I haven't even talked about things which have little to no relation to their real world meaning, e.g. a mouse in set theory.)
Multiplication is a word we use in context, and the context is usually an associative binary operation on some set. This set can be of functions, matrices, sets, etc. If we require the operation to have a neutral element and admit inverses, we have a group; if we adjoin another operator which is also associative, admits inverses, and commutative and posit some distributivity laws, we get a ring; or a semi-ring; or so on and so forth.
But it is convenient to talk about multiplication, because it's a word, and in most cases the pedagogical development helps us grasp why in generalizations we might want to omit commutativity from this operation.
edited 12 hours ago
answered 12 hours ago
Asaf Karagila♦
299k32420750
299k32420750
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up vote
2
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Terminology for mathematical structures is often built off of, and analogized to, terminology for "normal" mathematical structures such as integers, rational numbers, and real numbers. For vectors over real numbers, we have addition already defined for the coordinates. Applying this operation and adding components termwise results in a meaningful operation, and the natural terminology is to refer to that as simply "addition". Multiplying termwise result in an operation that isn't as meaningful (for one thing, this operation, unlike termwise addition, is dependent on the coordinate system). The cross product, on the other hand, is a meaningful operation, and it interacts with termwise addition in a manner similar to how real multiplication interacts with real addition. For instance, $(a+b)times c = a times c + b times c$ (distributive property).
For matrices, we again have termwise addition being a meaningful operation. Matrices represent linear operators, and the definition of linearity includes many of the properties of multiplication, such as distribution: A(u+v) = A(u) + A(v). Thus, it's natural to treat application of a linear operator as "multiplying" a vector by a matrix, and from there it's natural to define matrix multiplication as composition of the linear operators: (A*B)(v) = (A(B(v)).
2
+1, indeed the common feature of operations called "multiplication" seems to be that they are binary operations that distribute (on both sides) over "addition".
– Henning Makholm
7 hours ago
@HenningMakholm: I don't think that's quite the whole truth. For example, people talk about group "multiplication" even though there's only one operation in that setting. What does seem true is that there are lots of interesting examples of noncommutative operations which distribute over commutative operations and few (or no?) examples of commutative operations which distribute over noncommutative operations, so the tendency is to analogize commutative things with addition and noncommutative things with multiplication.
– Micah
4 hours ago
@Micah: Hmm, perhaps I'm projecting here, because the word "multiply" about arbirary group operations has never sat well me. (I'm cool with multiplicative notation, of course, and speaking about "product" and "factors" doesn't sound half as jarring as "multiply" either).
– Henning Makholm
1 hour ago
add a comment |
up vote
2
down vote
Terminology for mathematical structures is often built off of, and analogized to, terminology for "normal" mathematical structures such as integers, rational numbers, and real numbers. For vectors over real numbers, we have addition already defined for the coordinates. Applying this operation and adding components termwise results in a meaningful operation, and the natural terminology is to refer to that as simply "addition". Multiplying termwise result in an operation that isn't as meaningful (for one thing, this operation, unlike termwise addition, is dependent on the coordinate system). The cross product, on the other hand, is a meaningful operation, and it interacts with termwise addition in a manner similar to how real multiplication interacts with real addition. For instance, $(a+b)times c = a times c + b times c$ (distributive property).
For matrices, we again have termwise addition being a meaningful operation. Matrices represent linear operators, and the definition of linearity includes many of the properties of multiplication, such as distribution: A(u+v) = A(u) + A(v). Thus, it's natural to treat application of a linear operator as "multiplying" a vector by a matrix, and from there it's natural to define matrix multiplication as composition of the linear operators: (A*B)(v) = (A(B(v)).
2
+1, indeed the common feature of operations called "multiplication" seems to be that they are binary operations that distribute (on both sides) over "addition".
– Henning Makholm
7 hours ago
@HenningMakholm: I don't think that's quite the whole truth. For example, people talk about group "multiplication" even though there's only one operation in that setting. What does seem true is that there are lots of interesting examples of noncommutative operations which distribute over commutative operations and few (or no?) examples of commutative operations which distribute over noncommutative operations, so the tendency is to analogize commutative things with addition and noncommutative things with multiplication.
– Micah
4 hours ago
@Micah: Hmm, perhaps I'm projecting here, because the word "multiply" about arbirary group operations has never sat well me. (I'm cool with multiplicative notation, of course, and speaking about "product" and "factors" doesn't sound half as jarring as "multiply" either).
– Henning Makholm
1 hour ago
add a comment |
up vote
2
down vote
up vote
2
down vote
Terminology for mathematical structures is often built off of, and analogized to, terminology for "normal" mathematical structures such as integers, rational numbers, and real numbers. For vectors over real numbers, we have addition already defined for the coordinates. Applying this operation and adding components termwise results in a meaningful operation, and the natural terminology is to refer to that as simply "addition". Multiplying termwise result in an operation that isn't as meaningful (for one thing, this operation, unlike termwise addition, is dependent on the coordinate system). The cross product, on the other hand, is a meaningful operation, and it interacts with termwise addition in a manner similar to how real multiplication interacts with real addition. For instance, $(a+b)times c = a times c + b times c$ (distributive property).
For matrices, we again have termwise addition being a meaningful operation. Matrices represent linear operators, and the definition of linearity includes many of the properties of multiplication, such as distribution: A(u+v) = A(u) + A(v). Thus, it's natural to treat application of a linear operator as "multiplying" a vector by a matrix, and from there it's natural to define matrix multiplication as composition of the linear operators: (A*B)(v) = (A(B(v)).
Terminology for mathematical structures is often built off of, and analogized to, terminology for "normal" mathematical structures such as integers, rational numbers, and real numbers. For vectors over real numbers, we have addition already defined for the coordinates. Applying this operation and adding components termwise results in a meaningful operation, and the natural terminology is to refer to that as simply "addition". Multiplying termwise result in an operation that isn't as meaningful (for one thing, this operation, unlike termwise addition, is dependent on the coordinate system). The cross product, on the other hand, is a meaningful operation, and it interacts with termwise addition in a manner similar to how real multiplication interacts with real addition. For instance, $(a+b)times c = a times c + b times c$ (distributive property).
For matrices, we again have termwise addition being a meaningful operation. Matrices represent linear operators, and the definition of linearity includes many of the properties of multiplication, such as distribution: A(u+v) = A(u) + A(v). Thus, it's natural to treat application of a linear operator as "multiplying" a vector by a matrix, and from there it's natural to define matrix multiplication as composition of the linear operators: (A*B)(v) = (A(B(v)).
answered 9 hours ago
Acccumulation
6,4752616
6,4752616
2
+1, indeed the common feature of operations called "multiplication" seems to be that they are binary operations that distribute (on both sides) over "addition".
– Henning Makholm
7 hours ago
@HenningMakholm: I don't think that's quite the whole truth. For example, people talk about group "multiplication" even though there's only one operation in that setting. What does seem true is that there are lots of interesting examples of noncommutative operations which distribute over commutative operations and few (or no?) examples of commutative operations which distribute over noncommutative operations, so the tendency is to analogize commutative things with addition and noncommutative things with multiplication.
– Micah
4 hours ago
@Micah: Hmm, perhaps I'm projecting here, because the word "multiply" about arbirary group operations has never sat well me. (I'm cool with multiplicative notation, of course, and speaking about "product" and "factors" doesn't sound half as jarring as "multiply" either).
– Henning Makholm
1 hour ago
add a comment |
2
+1, indeed the common feature of operations called "multiplication" seems to be that they are binary operations that distribute (on both sides) over "addition".
– Henning Makholm
7 hours ago
@HenningMakholm: I don't think that's quite the whole truth. For example, people talk about group "multiplication" even though there's only one operation in that setting. What does seem true is that there are lots of interesting examples of noncommutative operations which distribute over commutative operations and few (or no?) examples of commutative operations which distribute over noncommutative operations, so the tendency is to analogize commutative things with addition and noncommutative things with multiplication.
– Micah
4 hours ago
@Micah: Hmm, perhaps I'm projecting here, because the word "multiply" about arbirary group operations has never sat well me. (I'm cool with multiplicative notation, of course, and speaking about "product" and "factors" doesn't sound half as jarring as "multiply" either).
– Henning Makholm
1 hour ago
2
2
+1, indeed the common feature of operations called "multiplication" seems to be that they are binary operations that distribute (on both sides) over "addition".
– Henning Makholm
7 hours ago
+1, indeed the common feature of operations called "multiplication" seems to be that they are binary operations that distribute (on both sides) over "addition".
– Henning Makholm
7 hours ago
@HenningMakholm: I don't think that's quite the whole truth. For example, people talk about group "multiplication" even though there's only one operation in that setting. What does seem true is that there are lots of interesting examples of noncommutative operations which distribute over commutative operations and few (or no?) examples of commutative operations which distribute over noncommutative operations, so the tendency is to analogize commutative things with addition and noncommutative things with multiplication.
– Micah
4 hours ago
@HenningMakholm: I don't think that's quite the whole truth. For example, people talk about group "multiplication" even though there's only one operation in that setting. What does seem true is that there are lots of interesting examples of noncommutative operations which distribute over commutative operations and few (or no?) examples of commutative operations which distribute over noncommutative operations, so the tendency is to analogize commutative things with addition and noncommutative things with multiplication.
– Micah
4 hours ago
@Micah: Hmm, perhaps I'm projecting here, because the word "multiply" about arbirary group operations has never sat well me. (I'm cool with multiplicative notation, of course, and speaking about "product" and "factors" doesn't sound half as jarring as "multiply" either).
– Henning Makholm
1 hour ago
@Micah: Hmm, perhaps I'm projecting here, because the word "multiply" about arbirary group operations has never sat well me. (I'm cool with multiplicative notation, of course, and speaking about "product" and "factors" doesn't sound half as jarring as "multiply" either).
– Henning Makholm
1 hour ago
add a comment |
up vote
1
down vote
The mathematical concept below the question is that of operation.
In a very general setting, if X is a set, than an operation on X is just a function
$$Xtimes Xto X$$
Usually denoted with multiplicative notations like $cdot$ or $*$. This means that an operation takes two elements of X and give as a result an element of $X$ (exactly as when you take two numbers, say $3$ ant $5$ and the result is $5*3=15$)
Examples of oparations are the usual operations on real numbers: plus, minus, division (defined on non-zero reals), multiplications, exponentiation.
Now, if you want to use operations in mathematics you usualy requires properties that are useful in calculations. Here the most common properties that an operation can have.
$1)$ Associativity. That means that $(a*b)*c=a*(b*c)$ and allow you to omit parenthesis and write $a*b*c$. The usual summation and multiplications on real numbers are associative. Subtraction, division, and exponentiation are not. For example:
$$(5-2)-2=3-2=1neq 5=5-(2-2)$$
$$ (9/3)/3=3/3=1neq 9=9/1=9/(3/3)$$
$$ 2^{(3^2)}=2^9=512neq 64= 8^2=(2^3)^2$$
A famous examples of non-associative multiplication often used in mathematics is that of Cayley Octonions.
A useful example of associative operations is the composition of functions. Let $A$ be any set and $X$ be the set of all functions from $A$ to $A$, i.e. $X={f:Ato A}$. The composition of two function $f,g$ is the function $f*g$ defined by $f*g(a)=f(g(a))$. Clearly $f*(g*h)(a)=f(g(h(a)))=(f*g)*h(a)$.
$2)$ Commutativity. That means that $a*b=b*a$. Usual sum and multiplications are commutative. Matrix product is not commutative, the composition of functions is not commutative in general (a rotation composed with a translation is not the same as a translation first, and then the rotation), vector product is not commutative. Non commutativity of composition of functions is on the basis of Heisemberg uncertainty principle
. The Hamilton Quaternions are a useful structure used in math. They have a multiplication which is associative but non commutative.
$3)$ Existence of Neutral element. This means that there is an element e in $X$ so that $x*e=e*x=x$ for any $x$ of $X$. For summation the netural element is $0$, for multiplication is $1$. If you consider $X$ as the set of even integers numbers, than the usual multiplication is well-defined on $X$, but the neutral element does not exists in $X$ (it would be $1$, which is not in $X$).
$4)$ Existence of Inverse. In case there is neutral element $ein X$, this means that for any $xin X$ there exists $y$ so that $xy=yx=e$. Usually $y$ is denoted by $x^{-1}$. The inverse for usual sum is $-x$, the inverse for usual multiplication is $1/x$ (which exists only for non-zero elements). In the realm of matrices, there are many matrices that have no inverse, for instane the matrix
$begin{pmatrix} 1 & 1\1&1end{pmatrix}$.
Such properties are important because they make an operation user-friendly. For example: is it true that if $a,bneq 0$ then $a*bneq0$? This seems kind of obviuos, but it depends on the properties of the operation. For example
$begin{pmatrix} 1 & 0\3&0end{pmatrix}begin{pmatrix} 0 & 0\1&2end{pmatrix}=begin{pmatrix} 0 & 0\0&0end{pmatrix}$ but both
$begin{pmatrix} 1 & 0\3&0end{pmatrix}$ and $begin{pmatrix} 0 & 0\1&2end{pmatrix}$ are different from zero.
Concluding, I would say that when a mathematician hear the word multiplication, immediately think to an associative operation, usually (but not always) with neutral element, sometimes commutative.
Another example of non-associative multiplication is IEEE floating point numbers (where intermediate results can underflow to zero or not, depending on the order of evaluation.small * small * big
can be zero if the first two terms underflow, orsmall
if the second two terms multiply to 1.)
– Martin Bonner
5 hours ago
Not to mention the cross product in $mathbb R^3$.
– Henning Makholm
5 hours ago
@MartinBonner: An interesting example, in that it is a non-associative approximation to an associative operation. I see in Wikipedia, incidentally, that it may be subnormal rather than zero, though I do not see exactly when.
– PJTraill
5 hours ago
add a comment |
up vote
1
down vote
The mathematical concept below the question is that of operation.
In a very general setting, if X is a set, than an operation on X is just a function
$$Xtimes Xto X$$
Usually denoted with multiplicative notations like $cdot$ or $*$. This means that an operation takes two elements of X and give as a result an element of $X$ (exactly as when you take two numbers, say $3$ ant $5$ and the result is $5*3=15$)
Examples of oparations are the usual operations on real numbers: plus, minus, division (defined on non-zero reals), multiplications, exponentiation.
Now, if you want to use operations in mathematics you usualy requires properties that are useful in calculations. Here the most common properties that an operation can have.
$1)$ Associativity. That means that $(a*b)*c=a*(b*c)$ and allow you to omit parenthesis and write $a*b*c$. The usual summation and multiplications on real numbers are associative. Subtraction, division, and exponentiation are not. For example:
$$(5-2)-2=3-2=1neq 5=5-(2-2)$$
$$ (9/3)/3=3/3=1neq 9=9/1=9/(3/3)$$
$$ 2^{(3^2)}=2^9=512neq 64= 8^2=(2^3)^2$$
A famous examples of non-associative multiplication often used in mathematics is that of Cayley Octonions.
A useful example of associative operations is the composition of functions. Let $A$ be any set and $X$ be the set of all functions from $A$ to $A$, i.e. $X={f:Ato A}$. The composition of two function $f,g$ is the function $f*g$ defined by $f*g(a)=f(g(a))$. Clearly $f*(g*h)(a)=f(g(h(a)))=(f*g)*h(a)$.
$2)$ Commutativity. That means that $a*b=b*a$. Usual sum and multiplications are commutative. Matrix product is not commutative, the composition of functions is not commutative in general (a rotation composed with a translation is not the same as a translation first, and then the rotation), vector product is not commutative. Non commutativity of composition of functions is on the basis of Heisemberg uncertainty principle
. The Hamilton Quaternions are a useful structure used in math. They have a multiplication which is associative but non commutative.
$3)$ Existence of Neutral element. This means that there is an element e in $X$ so that $x*e=e*x=x$ for any $x$ of $X$. For summation the netural element is $0$, for multiplication is $1$. If you consider $X$ as the set of even integers numbers, than the usual multiplication is well-defined on $X$, but the neutral element does not exists in $X$ (it would be $1$, which is not in $X$).
$4)$ Existence of Inverse. In case there is neutral element $ein X$, this means that for any $xin X$ there exists $y$ so that $xy=yx=e$. Usually $y$ is denoted by $x^{-1}$. The inverse for usual sum is $-x$, the inverse for usual multiplication is $1/x$ (which exists only for non-zero elements). In the realm of matrices, there are many matrices that have no inverse, for instane the matrix
$begin{pmatrix} 1 & 1\1&1end{pmatrix}$.
Such properties are important because they make an operation user-friendly. For example: is it true that if $a,bneq 0$ then $a*bneq0$? This seems kind of obviuos, but it depends on the properties of the operation. For example
$begin{pmatrix} 1 & 0\3&0end{pmatrix}begin{pmatrix} 0 & 0\1&2end{pmatrix}=begin{pmatrix} 0 & 0\0&0end{pmatrix}$ but both
$begin{pmatrix} 1 & 0\3&0end{pmatrix}$ and $begin{pmatrix} 0 & 0\1&2end{pmatrix}$ are different from zero.
Concluding, I would say that when a mathematician hear the word multiplication, immediately think to an associative operation, usually (but not always) with neutral element, sometimes commutative.
Another example of non-associative multiplication is IEEE floating point numbers (where intermediate results can underflow to zero or not, depending on the order of evaluation.small * small * big
can be zero if the first two terms underflow, orsmall
if the second two terms multiply to 1.)
– Martin Bonner
5 hours ago
Not to mention the cross product in $mathbb R^3$.
– Henning Makholm
5 hours ago
@MartinBonner: An interesting example, in that it is a non-associative approximation to an associative operation. I see in Wikipedia, incidentally, that it may be subnormal rather than zero, though I do not see exactly when.
– PJTraill
5 hours ago
add a comment |
up vote
1
down vote
up vote
1
down vote
The mathematical concept below the question is that of operation.
In a very general setting, if X is a set, than an operation on X is just a function
$$Xtimes Xto X$$
Usually denoted with multiplicative notations like $cdot$ or $*$. This means that an operation takes two elements of X and give as a result an element of $X$ (exactly as when you take two numbers, say $3$ ant $5$ and the result is $5*3=15$)
Examples of oparations are the usual operations on real numbers: plus, minus, division (defined on non-zero reals), multiplications, exponentiation.
Now, if you want to use operations in mathematics you usualy requires properties that are useful in calculations. Here the most common properties that an operation can have.
$1)$ Associativity. That means that $(a*b)*c=a*(b*c)$ and allow you to omit parenthesis and write $a*b*c$. The usual summation and multiplications on real numbers are associative. Subtraction, division, and exponentiation are not. For example:
$$(5-2)-2=3-2=1neq 5=5-(2-2)$$
$$ (9/3)/3=3/3=1neq 9=9/1=9/(3/3)$$
$$ 2^{(3^2)}=2^9=512neq 64= 8^2=(2^3)^2$$
A famous examples of non-associative multiplication often used in mathematics is that of Cayley Octonions.
A useful example of associative operations is the composition of functions. Let $A$ be any set and $X$ be the set of all functions from $A$ to $A$, i.e. $X={f:Ato A}$. The composition of two function $f,g$ is the function $f*g$ defined by $f*g(a)=f(g(a))$. Clearly $f*(g*h)(a)=f(g(h(a)))=(f*g)*h(a)$.
$2)$ Commutativity. That means that $a*b=b*a$. Usual sum and multiplications are commutative. Matrix product is not commutative, the composition of functions is not commutative in general (a rotation composed with a translation is not the same as a translation first, and then the rotation), vector product is not commutative. Non commutativity of composition of functions is on the basis of Heisemberg uncertainty principle
. The Hamilton Quaternions are a useful structure used in math. They have a multiplication which is associative but non commutative.
$3)$ Existence of Neutral element. This means that there is an element e in $X$ so that $x*e=e*x=x$ for any $x$ of $X$. For summation the netural element is $0$, for multiplication is $1$. If you consider $X$ as the set of even integers numbers, than the usual multiplication is well-defined on $X$, but the neutral element does not exists in $X$ (it would be $1$, which is not in $X$).
$4)$ Existence of Inverse. In case there is neutral element $ein X$, this means that for any $xin X$ there exists $y$ so that $xy=yx=e$. Usually $y$ is denoted by $x^{-1}$. The inverse for usual sum is $-x$, the inverse for usual multiplication is $1/x$ (which exists only for non-zero elements). In the realm of matrices, there are many matrices that have no inverse, for instane the matrix
$begin{pmatrix} 1 & 1\1&1end{pmatrix}$.
Such properties are important because they make an operation user-friendly. For example: is it true that if $a,bneq 0$ then $a*bneq0$? This seems kind of obviuos, but it depends on the properties of the operation. For example
$begin{pmatrix} 1 & 0\3&0end{pmatrix}begin{pmatrix} 0 & 0\1&2end{pmatrix}=begin{pmatrix} 0 & 0\0&0end{pmatrix}$ but both
$begin{pmatrix} 1 & 0\3&0end{pmatrix}$ and $begin{pmatrix} 0 & 0\1&2end{pmatrix}$ are different from zero.
Concluding, I would say that when a mathematician hear the word multiplication, immediately think to an associative operation, usually (but not always) with neutral element, sometimes commutative.
The mathematical concept below the question is that of operation.
In a very general setting, if X is a set, than an operation on X is just a function
$$Xtimes Xto X$$
Usually denoted with multiplicative notations like $cdot$ or $*$. This means that an operation takes two elements of X and give as a result an element of $X$ (exactly as when you take two numbers, say $3$ ant $5$ and the result is $5*3=15$)
Examples of oparations are the usual operations on real numbers: plus, minus, division (defined on non-zero reals), multiplications, exponentiation.
Now, if you want to use operations in mathematics you usualy requires properties that are useful in calculations. Here the most common properties that an operation can have.
$1)$ Associativity. That means that $(a*b)*c=a*(b*c)$ and allow you to omit parenthesis and write $a*b*c$. The usual summation and multiplications on real numbers are associative. Subtraction, division, and exponentiation are not. For example:
$$(5-2)-2=3-2=1neq 5=5-(2-2)$$
$$ (9/3)/3=3/3=1neq 9=9/1=9/(3/3)$$
$$ 2^{(3^2)}=2^9=512neq 64= 8^2=(2^3)^2$$
A famous examples of non-associative multiplication often used in mathematics is that of Cayley Octonions.
A useful example of associative operations is the composition of functions. Let $A$ be any set and $X$ be the set of all functions from $A$ to $A$, i.e. $X={f:Ato A}$. The composition of two function $f,g$ is the function $f*g$ defined by $f*g(a)=f(g(a))$. Clearly $f*(g*h)(a)=f(g(h(a)))=(f*g)*h(a)$.
$2)$ Commutativity. That means that $a*b=b*a$. Usual sum and multiplications are commutative. Matrix product is not commutative, the composition of functions is not commutative in general (a rotation composed with a translation is not the same as a translation first, and then the rotation), vector product is not commutative. Non commutativity of composition of functions is on the basis of Heisemberg uncertainty principle
. The Hamilton Quaternions are a useful structure used in math. They have a multiplication which is associative but non commutative.
$3)$ Existence of Neutral element. This means that there is an element e in $X$ so that $x*e=e*x=x$ for any $x$ of $X$. For summation the netural element is $0$, for multiplication is $1$. If you consider $X$ as the set of even integers numbers, than the usual multiplication is well-defined on $X$, but the neutral element does not exists in $X$ (it would be $1$, which is not in $X$).
$4)$ Existence of Inverse. In case there is neutral element $ein X$, this means that for any $xin X$ there exists $y$ so that $xy=yx=e$. Usually $y$ is denoted by $x^{-1}$. The inverse for usual sum is $-x$, the inverse for usual multiplication is $1/x$ (which exists only for non-zero elements). In the realm of matrices, there are many matrices that have no inverse, for instane the matrix
$begin{pmatrix} 1 & 1\1&1end{pmatrix}$.
Such properties are important because they make an operation user-friendly. For example: is it true that if $a,bneq 0$ then $a*bneq0$? This seems kind of obviuos, but it depends on the properties of the operation. For example
$begin{pmatrix} 1 & 0\3&0end{pmatrix}begin{pmatrix} 0 & 0\1&2end{pmatrix}=begin{pmatrix} 0 & 0\0&0end{pmatrix}$ but both
$begin{pmatrix} 1 & 0\3&0end{pmatrix}$ and $begin{pmatrix} 0 & 0\1&2end{pmatrix}$ are different from zero.
Concluding, I would say that when a mathematician hear the word multiplication, immediately think to an associative operation, usually (but not always) with neutral element, sometimes commutative.
answered 17 hours ago
user126154
5,280716
5,280716
Another example of non-associative multiplication is IEEE floating point numbers (where intermediate results can underflow to zero or not, depending on the order of evaluation.small * small * big
can be zero if the first two terms underflow, orsmall
if the second two terms multiply to 1.)
– Martin Bonner
5 hours ago
Not to mention the cross product in $mathbb R^3$.
– Henning Makholm
5 hours ago
@MartinBonner: An interesting example, in that it is a non-associative approximation to an associative operation. I see in Wikipedia, incidentally, that it may be subnormal rather than zero, though I do not see exactly when.
– PJTraill
5 hours ago
add a comment |
Another example of non-associative multiplication is IEEE floating point numbers (where intermediate results can underflow to zero or not, depending on the order of evaluation.small * small * big
can be zero if the first two terms underflow, orsmall
if the second two terms multiply to 1.)
– Martin Bonner
5 hours ago
Not to mention the cross product in $mathbb R^3$.
– Henning Makholm
5 hours ago
@MartinBonner: An interesting example, in that it is a non-associative approximation to an associative operation. I see in Wikipedia, incidentally, that it may be subnormal rather than zero, though I do not see exactly when.
– PJTraill
5 hours ago
Another example of non-associative multiplication is IEEE floating point numbers (where intermediate results can underflow to zero or not, depending on the order of evaluation.
small * small * big
can be zero if the first two terms underflow, or small
if the second two terms multiply to 1.)– Martin Bonner
5 hours ago
Another example of non-associative multiplication is IEEE floating point numbers (where intermediate results can underflow to zero or not, depending on the order of evaluation.
small * small * big
can be zero if the first two terms underflow, or small
if the second two terms multiply to 1.)– Martin Bonner
5 hours ago
Not to mention the cross product in $mathbb R^3$.
– Henning Makholm
5 hours ago
Not to mention the cross product in $mathbb R^3$.
– Henning Makholm
5 hours ago
@MartinBonner: An interesting example, in that it is a non-associative approximation to an associative operation. I see in Wikipedia, incidentally, that it may be subnormal rather than zero, though I do not see exactly when.
– PJTraill
5 hours ago
@MartinBonner: An interesting example, in that it is a non-associative approximation to an associative operation. I see in Wikipedia, incidentally, that it may be subnormal rather than zero, though I do not see exactly when.
– PJTraill
5 hours ago
add a comment |
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13
Much of the time it's tradition. But if you have an operation which is associative and distributes over addition (whatever "addition" means), then it's definitely worthy of the name "multiplication".
– Arthur
18 hours ago
According to inc.com/bill-murphy-jr/…, kids can learn the material from grade 1 better if they start grade when at the age of 8. Maybe by starting so late, they could avoid having damaging past knowledge and be introduced to the concept of a natural number as a finite ordinal and completely avoid teaching them any concept of cardinal numbers until they're ready for it later. That's because it's much harder to learn to add by creating two groups each of which has a certain number of objects then
– Timothy
8 hours ago
counting the objects in the union than directly from the inductive definition of natural number addition. Also, they should be taught only the bijective unary numeral system in grade 1 and how to add and multiply using it. They should also be guided on how to write a formal proof in a certain weak system of pure number theory and taught how to master that skill. They should later be tested on their ability to write a proof that natural number addition is associative and commutative and that natural number multiplication is also associative and commutative and distributes over addition and
– Timothy
8 hours ago
should not see the proof before they figure it out themself. They should later be given an inductive definition of the decimal notation of each natural number that just says to take the successor, you increase the last digit by 1 if it's not 9 or if it is 9, change it to 0 and change string before the last digit to that which represents the next natural number and be left to figure out on their own a short cut for adding or multiplying natural numbers given in decimal notation and write the answer in decimal notation. That way, when they're taught the definition of matrix multiplication, they
– Timothy
8 hours ago
won't think that operation is commutative. Also, waiting until later to teach them about cardinality of a finite set actually enables them to learn the material faster. I don't want to write this as an answer because I can't find a way to answer the question that I'm sure is not worse than no answer.
– Timothy
8 hours ago