Could inverse (of) multiplication operation be considered and if not, then why not?

[tylers100]

* = multiplication

i = inverse (of) multiplication

2 * 2 = 4

4 i 2 = 2

The reason I post this is because I don't see it appearing as part of arithmetic operations elsewhere apart from some references to inverse (e.g. link: https://en.wikipedia.org/wiki/Multiplicative_inverse) but probably not exactly what I am talking about.

 

[studiot]

16 minutes ago, tylers100 said:

* = multiplication

i = inverse (of) multiplication

2 * 2 = 4

4 i 2 = 2

The reason I post this is because I don't see it appearing as part of arithmetic operations elsewhere apart from some references to inverse (e.g. link: https://en.wikipedia.org/wiki/Multiplicative_inverse) but probably not exactly what I am talking about.

 

You need to delve into set theory and group theory to discuss this.

Are you ready for that ?

[tylers100]

2 hours ago, studiot said:

You need to delve into set theory and group theory to discuss this.

Are you ready for that ?

I looked up both set theory and group theory wiki, bookmarked these for later reading.

No, I'm not ready for that at this moment. It is just that while refreshing myself with math at basic level for some time, I noticed there must be a "counterpart" for every mathematical operation even at arithmetic level, maybe.. an example:

+ and -, addition and subtraction

* and i, multiplication and inverse of multiplication (I called it as I do not have a formal termed word for it as far I do not know one except similar words referenced.)

/ and \, division and its counterpart

Division and its counterpart (when I did the inverse multiplication math I realized a counterpart of division might would has inverse of multiplication and addition along with rounding up or down decimals, I called it something like, "jointion" and or denoted with a symbol like '\' while division is '/'.  It would joins numbers and round up/down numbers whereas division divides to decimals if necessary. But some math I did with it didn't checked out as I encountered some errors and confusions with it.. so perhaps I still need some work with understanding my approach and conceptualizing process or something like that.)

[studiot]

8 minutes ago, tylers100 said:

I looked up both set theory and group theory wiki, bookmarked these for later reading.

No, I'm not ready for that at this moment. It is just that while refreshing myself with math at basic level for some time, I noticed there must be a "counterpart" for every mathematical operation even at arithmetic level, maybe.. an example:

+ and -, addition and subtraction

* and i, multiplication and inverse of multiplication (I called it as I do not have a formal termed word for it as far I do not know one except similar words referenced.)

/ and \, division and its counterpart

Division and its counterpart (when I did the inverse multiplication math I realized a counterpart of division might would has inverse of multiplication and addition along with rounding up or down decimals, I called it something like, "jointion" and or denoted with a symbol like '\' while division is '/'.  It would joins numbers and round up/down numbers whereas division divides to decimals if necessary. But some math I did with it didn't checked out as I encountered some errors and confusions with it.. so perhaps I still need some work with understanding my approach and conceptualizing process or something like that.)

I wasn't suggesting you go self study those subjects, I was offering help with that.

However it is laudable that you took the interest to look these up and also to indicate where I should start. +1

I have some apple trees to prune this afternoon ,but I will make a start when I have done that.

[studiot]

Ok so a set is just an imaginary container we use to collect together itmes/objects of interest.

The simplest mathematical set is just a list of the (mathematical objects in it.

Since some sets are very large indeed (even Infinite) the list can be very long or even never ending.

The next step up (mathematically) is to collect together objects with some property to cope with this because if we can specify the property  we only need this specification to specify the set. We can do without the list.

For example { The set of all even numbers}

We write the set in between curly brackets.

But mathematicians prefer formulae to words so we write

{x : x = 2n where n  is an integer}

To read this we say to ourselves   (The colon means 'such that')

"The set of all x such that x = 2n where n is an integer"

In fact we can shorten this using the set of all integers which is given the symbol Z

{x:x=2n,∈nZ}

Note some authors use

Note some authors use a vertical line instead of a colon.

[math]\{ x|x = 2n, \in nZ\} [/math]

Edited November 10 by studiot

[studiot]

Ok so a set is just an imaginary container we use to collect together items/objects of interest.

The simplest mathematical set is just a list of the (mathematical) objects in it.

These are called the members or the elements of the set.

Since some sets are very large indeed (even Infinite) the list can be very long or even never ending.

The next step up (mathematically) is to collect together objects with some property to cope with this because if we can specify the property  we only need this specification to specify the set. We can do without the list.

For example { The set of all even numbers}

We write the set in between curly brackets.

But mathematicians prefer formulae to words so we write

{x : x = 2n where n  is an integer}

To read this we say to ourselves   (The colon means 'such that')

"The set of all x such that x = 2n where n is an integer"

In fact we can shorten this using the set of all integers which is given the symbol Z

{x:x=2n,∈nZ}

Note some authors use

Note some authors use a vertical line instead of a colon.

[math]\{ x|x = 2n, \in nZ\} [/math]

 

It is worthwhile knowing that there are some standard sets such as

N the set of Natural or counting numbers

Z the set of all integers (positive and negative)

Q the set of all fractions

R the set of all real numbers (decimal fractions)

 

OK so we have some sets.

That is all you need to know about sets to start with. Don't worry about all the stuff about Union, Intersection and so on.

Now we can do three things

1. We can consider actions from one set to another ie between sets themselves.
    
2. We can consider actions between individual members of a set.
    
3. We can increase the value of the set by creating a (mathematical) structure.
   We do this by specifying particular useful properties of the transactions between members as in 2.

 

Now if you think about school arithmetic we learn how to add, subtract, multiply and divide.

This gives us four rules, called Paeno's Rules of Arithmetic.

 

Three very useful additional rules are that

1. Every possible sum or product between members of our set are also members of our set.
2. There are no members which are not a sum or product of other members.
3. Each sum or product refers to exactly one other member.

When we have a set that conforms to not only the ordinary rules of arithmetic but also those three rules we call our set a group.

 

Now going back to actions between sets themselves.

Sometimes both the source and target set are the same or the target set is a copy of the source set.

A mapping connects elements of one set to elements of another by means of some rule or process or formula.

If every connection of the mapping connects to a unique (one and only one)  element in the target set well call it a function.

When this connection rule is satified we can also construct an inverse function.

So for instance

1 x 12 = 12

3 x4 = 12

2 x 6 = 12

So given 12 we cannot determine which of these to claim as the inverse of the multiplication.

 

That should be enough to start with.

 

[tylers100]

I looked into Set Theory and Group Theory at previously times but need to read and think a bit more about these at later time. For now, discussing about just the inverse multiplication using the multiplication concept.

Diagram Image

See the attached diagram image.

No Process / Procedure

It is starting to look like inverse multiplication is un-doable since I'm unable to find a correct process or procedure for it / that or there is none (or none yet.. ?). I mean, for example:

Multiplication
1 * 12 = 12

then

Inverse Multiplication
12 i 12 = 1

It is seemingly easy to see one can reverse the multiplication and its answer the other way to obtain answer when inversely multiplying. But.. no process or procedure.

Uniform

I inverse-multiplied using the same multiplication principle or concept as shown on the image. I should point out that when inversely-multiply (gray numbers / arrows / lines) numbers, these seems uniformly-even when divisions are slightly different despite the fact these divides evenly.

Or maybe I'm a bit over-analyzing the basics of arithmetic operations or something like that. It is just that I want to ensure that I get these basics of arithmetic operations right and explore if there is any other major arithmetic operation or not with certainty.

Also or maybe I need to step a bit back and look into the Set Theory and Group Theory a bit more to know before proceeding further?

[John Cuthber]

On 11/10/2024 at 11:06 AM, tylers100 said:

* = multiplication

i = inverse (of) multiplication

2 * 2 = 4

4 i 2 = 2

The reason I post this is because I don't see it appearing as part of arithmetic operations elsewhere apart from some references to inverse (e.g. link: https://en.wikipedia.org/wiki/Multiplicative_inverse) but probably not exactly what I am talking about.

 

Am I missing something?
 4 ÷ 2 =2

[tylers100]

3 hours ago, John Cuthber said:

Am I missing something?
 4 ÷ 2 =2

No, you are not missing something. It is just that I thought I noticed a possibly and different mathematical operation (e.g. inverse multiplication as counterpart to multiplication), but that is not apparent or actual case so.

Yes, that is correct.

[Genady]

Multiplication example. Take 13 jars. Put 19 beans in each jar. Take out all the beans and count them. The number, 247, is a result of multiplying 13 by 19.

Inverse multiplication example. Take 247 beans. Put them one-by-one in 13 jars, one bean in each jar at a time, until all the beans are in the jars. Take out beans from one of the jars and count them. The number, 19, is a result of the inverse operation (aka dividing 247 by 13.)

[studiot]

11 hours ago, tylers100 said:

No, you are not missing something. It is just that I thought I noticed a possibly and different mathematical operation (e.g. inverse multiplication as counterpart to multiplication), but that is not apparent or actual case so.

Yes, that is correct.

I think perhaps you are mixing up the three different meanings of the word inverse.

The inverse is the result of the operation of inversion or the verb to invert.

 

 

First definition:-  to invert means to turn upside down.

For example the inverse of   [math]\frac{5}{{13}}[/math]   is   [math]\frac{{13}}{5}[/math]

 

This definition has no meaning for simple numbers alone, you require a set of fractions or quotients to make it work.

This definition has more importance when we use algebra rather than just number.

For example the inverse of  [math]\frac{{2a + b}}{{{b^2}}}[/math]   is  [math]\frac{{{b^2}}}{{2a + b}}[/math]

 

Every fraction or quotient has an inverse in this sense.

 

Second definition:-  For any number or expression  a  the inverse is another number  or expression  a^-1  from the same set such that  a x a^-1  = e

Where e is the identity element of the set and a^-1 is the multiplicative inverse of a.

For example  [math]\frac{13}{{5}}[/math]   is the multiplicative inverse of [math]\frac{{13}}{5}[/math]   and e = [math]\frac{{1}}{1}[/math]

 

 

The inclusion of an identity element is important in defining a group, ring or other algebraic structure.

But it does not guarantee the existence of a multiplicative inverse for all the elements.

For example consider the set of 2 by 2 matrices.

[math]\left[ {\begin{array}{*{20}{c}}
   1 & 2  \\
   3 & 4  \\
\end{array}} \right][/math]

Has a multiplicative inverse since

 

 

[math]\left[ {\begin{array}{*{20}{c}}
   1 & 2  \\
   3 & 4  \\
\end{array}} \right]\left[ {\begin{array}{*{20}{c}}
   { - 2} & 1  \\
   {\frac{3}{2}} & { - \frac{1}{2}}  \\
\end{array}} \right] = \left[ {\begin{array}{*{20}{c}}
   1 & 0  \\
   0 & 1  \\
\end{array}} \right][/math]

 

So the multiplicative inverse is the matrix

[math]\left[ {\begin{array}{*{20}{c}}
   { - 2} & 1  \\
   {\frac{3}{2}} & { - \frac{1}{2}}  \\
\end{array}} \right][/math]

and the identity element is the matrix

[math]\left[ {\begin{array}{*{20}{c}}
   1 & 0  \\
   0 & 1  \\
\end{array}} \right][/math]

 

 

However not all sets have an identity element.
And the identity element may not work with all members.

 

The matrix

[math]\left[ {\begin{array}{*{20}{c}}
   1 & 2  \\
   3 & 6  \\
\end{array}} \right][/math]

Has no multiplicative inverse.

 

All the examples so far have been unary operations. That is operating on one single element of the set.
We must now extend our  definition to binary operations.

Definition:- A binary operation takes two elements of a set and combines them to produce a single element of the set.

Third definition :-   The inverse of a binary operation means  the reverse or undoing of the 'forward' operation.

 

 

The example already given of 12 being the product of several different pairs of numbers that combine multiplicatively to yield the single element 12.

for convenience

1 x 12  = 2 x 6 = 3 x 4 = 12

Now it can immediately be seen that this operation cannot be undone since there are three possible pairs that satisfy the forward operation so given the number 12 we cannot determine which pair was involved in the original operation.

This lack of uniquness in the reverse direction means that 12 does not have a binary multiplicative inverse.

 

This has importantance in prime number and factorisation theory.

 

11 hours ago, Genady said:

Multiplication example. Take 13 jars. Put 19 beans in each jar. Take out all the beans and count them. The number, 247, is a result of multiplying 13 by 19.

Inverse multiplication example. Take 247 beans. Put them one-by-one in 13 jars, one bean in each jar at a time, until all the beans are in the jars. Take out beans from one of the jars and count them. The number, 19, is a result of the inverse operation (aka dividing 247 by 13.)

Genady's example only works backwards if we exclude 247 x 1 = 247.
This is often done in elementary treatments.

 

Edited November 14 by studiot

[tylers100]

On 11/14/2024 at 8:10 AM, studiot said:

The example already given of 12 being the product of several different pairs of numbers that combine multiplicatively to yield the single element 12.

for convenience

1 x 12  = 2 x 6 = 3 x 4 = 12

Now it can immediately be seen that this operation cannot be undone since there are three possible pairs that satisfy the forward operation so given the number 12 we cannot determine which pair was involved in the original operation.

This lack of uniquness in the reverse direction means that 12 does not have a binary multiplicative inverse.

I think I'm beginning to understand a bit.

I noticed something interesting about the 12 number, see attached image.

Linear 12 is a beginning and end.

The circle 12 is a bit amusing, a continuous 12 number. Clockwise manner as depicted on the image, but if arrow go in counter-clockwise manner.. same 12 result hence continuous 12 number. Heh, a bit interesting to point out.

[studiot]

8 minutes ago, tylers100 said:

I think I'm beginning to understand a bit.

I noticed something interesting about the 12 number, see attached image.

Linear 12 is a beginning and end.

The circle 12 is a bit amusing, a continuous 12 number. Clockwise manner as depicted on the image, but if arrow go in counter-clockwise manner.. same 12 result hence continuous 12 number. Heh, a bit interesting to point out.

+1 for spotting that.

You have just discovered 'modular numbers'.

Also called 'clock numbers'

https://en.wikipedia.org/wiki/Modular_arithmetic