Relation between Division and multiplication

In summary: I forget the symbol. It's the inverse of multiplication. So, ##b/a = 1##. This can be rewritten as: ##\frac{a}{b} = \frac{1}{b}##. So, if we want to make ##10## cookies, we would need to use ##10/5 = 2## cookies for the dough, and ##5/2 = 1## for the sugar and butter.
  • #1
mark2142
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TL;DR Summary
It’s says that “every division problem has a related multiplication problem”
I am trying to understand this sentence.
For example what is ##\frac {169}{13} = ?##
This says “When ##169## is divided into ##13## groups how many there are in each group?”
This can be converted into a multiplication problem like this “##13## groups of how many in each group makes ##169##?”
This is ##13 * ? = 169##. It can be solved by repeated addition of ##13##.

Am I correct? Does this makes sense?
 
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  • #2
The point is that there is actually no such thing as a division. It is a concept that is meant to be convenient. I personally doubt that it is convenient as it causes problems as soon as someone thinks about it. So it is meant to be used in the supermarket or at construction sites. However, it is not suitable to be used in mathematics. I would even change the term division algebra if I could.

To understand division we only need multiplication and an inverse. ##\dfrac{a}{b}:=a\cdot b^{-1}## Thus, division is actually a multiplication with an element that satisfies the equation ##b\cdot x=1## which is again a multiplication. So all we need is
a) Solve ##b\cdot x=1##
b) Multiply ##a\cdot x## to get ##\dfrac{a}{b}.##
Division is obsolete.
 
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  • #3
[tex]169 \times \frac{1}{13} = ?[/tex]
 
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  • #4
mark2142 said:
TL;DR Summary: It’s says that “every division problem has a related multiplication problem”
I am trying to understand this sentence.

For example what is ##\frac {169}{13} = ?##
This says “When ##169## is divided into ##13## groups how many there are in each group?”
This can be converted into a multiplication problem like this “##13## groups of how many in each group makes ##169##?”
This is ##13 * ? = 169##. It can be solved by repeated addition of ##13##.

Am I correct? Does this makes sense?
No, that's not what is meant here. Here's the division equation: ##\frac {169}{13} = 13##.
The corresponding multiplication equation is ##13 \cdot 13 = 169##.

More generally, if ##\frac a b = q##, then the related multiplication equation is ##b \cdot q = a##, with the only exception being when b = 0.

Looking at multiplication as repeated addition makes sense if the numbers involved are integers, but the analogy falls apart if you're working with real numbers that aren't rational.
 
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  • #5
fresh_42 said:
The point is that there is actually no such thing as a division. It is a concept that is meant to be convenient. I personally doubt that it is convenient as it causes problems as soon as someone thinks about it. So it is meant to be used in the supermarket or at construction sites. However, it is not suitable to be used in mathematics. I would even change the term division algebra if I could.
Ok. That is some fresh information.

So what I said is right. Yes?

(##\frac pq=n## means ##p=qn##)
 
  • #6
mark2142 said:
Ok. That is some fresh information.

So what I said is right. Yes?

(##\frac pq=n## means ##p=qn##)
Yes, but not the other way around. ##p=qn## does not mean ##\frac pq=n## as long as we haven't ruled out ##q=0.##
 
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  • #7
mark2142 said:
Ok. That is some fresh information.
I see what you did there. :wink:
 
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  • #8
fresh_42 said:
The point is that there is actually no such thing as a division.
A "fact" that no maker of calculators and most manufacturers of computer CPUs would agree with. Intel might have wished there was no such thing as division when they rolled out their first Pentium processor back in about 1992 or so. Because of a flaw in their firmware that produced some incorrect division results, they had to recall about 1 billion USD worth of processors.
 
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  • #9
Mark44 said:
A "fact" that no maker of calculators and most manufacturers of computer CPUs would agree with. Intel might have wished there was no such thing as division when they rolled out their first Pentium processor back in about 1992 or so. Because of a flaw in their firmware that produced some incorrect division results, they had to recall about 1 billion USD worth of processors.
Ok. Supermarkets, construction sites, and programmers of calculators.

Division is for schools. It has no place at universities. We invert, we do not divide.
 
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  • #10
fresh_42 said:
Ok. Supermarkets, construction sites, and programmers of calculators.
Or engineering firms, manufacturers, etc., etc.
fresh_42 said:
Division is for schools. It has no place at universities. We invert, we do not divide.
You say "tomato" I say "tomahto." Granted that universities don't teach much arithmetic, but arithmetic is a part or mathematics, and to say that "there is no such thing as division" is a silly thing to say.
 
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  • #11
Mark44 said:
You say "tomato" I say "tomahto." Granted that universities don't teach much arithmetic, but arithmetic is a part or mathematics, and to say that "there is no such thing as division" is a silly thing to say.
Au contraire! It resolves all the silly questions and misconceptions about division. Inversion is what is left at an axiomatic level. But, I know, old habits die hard, regardless of how silly they are.

You need division to give all kids the same number of cookies, that's it. I call that partition(ing). Division ##a/b## introduces a new unit, ##b## instead of ##1##. Then it expresses ##a## according to this new unit: ##a\cdot b^{-1}.## That is what division does. It is all reduced to multiplication. The lack of associativity is a serious disadvantage.
 
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  • #12
mark2142 said:
This is ##13 * ? = 169##. It can be solved by repeated addition of ##13##.

Am I correct? Does this makes sense?
I wouldn't associate it with repeated addition but rather directly to multiplication.

In abstract algebra, a Field has a property that every nonzero element ##a \ne 0## has a multiplicative inverse, ##a^{-1}## such that ##a a^{-1} = 1##. Division by ##a## is defined as multiplication by ##a^{-1}##.
 
  • #13
fresh_42 said:
Au contraire! It resolves all the silly questions and misconceptions about division. Inversion is what is left at an axiomatic level. But, I know, old habits die hard, regardless of how silly they are.
The silliness is insisting that "there is no such thing as division," a blanket statement that is patently false, given the examples I cite below.
fresh_42 said:
You need division to give all kids the same number of cookies, that's it.
Well, that's hardly the case. Division comes into play in calculating present or future value, in performing polynomial long division, doing trigonometric calculations, and too many other types of calculations for me to list.
 
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  • #14
FactChecker said:
I wouldn't associate it with repeated addition but rather directly to multiplication.
If we are literally breaking down things into very basics then we are actually doing repeated addition. ##13## added ##13## times =##169##.
But we can save time and do direct multiplication.
 
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  • #15
mark2142 said:
If we are literally breaking down things into very basics then we are actually doing repeated addition. ##13## added ##13## times =##169##.
But we can save time and do direct multiplication.
There are problems with using repeated addition to explain division. Using repeated addition, you have to give an elaborate explanation for ##1/3 (=0.333333...) ##

On the other hand, if you know that every nonzero real number has a multiplicative inverse, you can just denote the multiplicative inverse of ##3## as ##3^{-1} = 1/3## and use it when needed. It's just a simpler approach.

PS. Certainly, repeated addition was the initial motivation, but that is not the best way to imagine it now.
 
  • #16
Mark44 said:
Granted that universities don't teach much arithmetic, but arithmetic is a part or mathematics, and to say that "there is no such thing as division" is a silly thing to say.
It’s for convenience that we teach division to kids. Actually what we are doing is inverted multiplication.
Mark44 said:
Division comes into play in calculating present or future value, in performing polynomial long division, doing trigonometric calculations, and too many other types of calculations for me to list.
That’s what he meant by “to give all kids same number of cookie”. Division is for convenience.
 
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  • #17
Mark44 said:
A "fact" that no maker of calculators and most manufacturers of computer CPUs would agree with. Intel might have wished there was no such thing as division when they rolled out their first Pentium processor back in about 1992 or so. Because of a flaw in their firmware that produced some incorrect division results, they had to recall about 1 billion USD worth of processors.
Just because division is used by companies with big names doesn’t qualify division as a concept. It’s still multiplication.
They must have used it because its convenient.
Division has made life easier.
 
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  • #18
mark2142 said:
That’s what he meant by “to give all kids same number of cookie”. Division is for convenience.
Maybe you didn't see my examples of situations where division is used that aren't as trivial as giving kids the same number of cookies.
mark2142 said:
Just because division is used by companies with big names doesn’t qualify division as a concept.
This is just nonsense. Division is used by companies, big and small, as well as individuals all around the world. The vast majority of people in the world who are at least conversant with arithmetic would differ with you on whether division is a concept.

mark2142 said:
It’s still multiplication.
Now you're confused. Division is multiplication by the reciprocal of the divisor -- you left that part out.
 
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  • #19
mark2142 said:
Just because division is used by companies with big names doesn’t qualify division as a concept.
A "concept" can be very elaborate and far removed from initial definitions. To say that division is not a concept is just wrong. It may be far removed from the basic definitions that you want to use, but it is as much a concept as General Relativity is even though it is far removed from Newton's basic ideas.
mark2142 said:
It’s still multiplication.
There are many things where multiplication is defined but division is not. The existence of a multiplicative identity, ##1##, and multiplicative inverses, ##a * 1/a = 1## for every nonzero ##a## is important.

If you are interested in this type of thing, you might be interested in abstract algebra.
 
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  • #20
fresh_42 said:
Au contraire! It resolves all the silly questions and misconceptions about division. Inversion is what is left at an axiomatic level. But, I know, old habits die hard, regardless of how silly they are.

You need division to give all kids the same number of cookies, that's it. I call that partition(ing). Division ##a/b## introduces a new unit, ##b## instead of ##1##. Then it expresses ##a## according to this new unit: ##a\cdot b^{-1}.## That is what division does. It is all reduced to multiplication. The lack of associativity is a serious disadvantage.
Not everything in life is abstract algebra! There are division operations in Python and other programming languages. Your mistake is to imagine that if pure mathematics goes down a certain road, then the rest of the world must follow.

In abstract algebra, there are only two binary operations. But, in science, economics and computer programming there are four.
 
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  • #21
Isn’t everything just addition?
Subtraction is addition of opposite. Multiplication is repeated addition and division is just opposite of multiplication which is also just addition.
 
  • #22
mark2142 said:
Isn’t everything just addition?
Subtraction is addition of opposite. Multiplication is repeated addition and division is just opposite of multiplication which is also just addition.
Perhaps for integers multiplication can be reduced to multiple additions, but not beyond that. It's not clear how you would define the multiplicative inverse in terms of addition only:

For every non-zero number ##a##, there exists a number ##a^{-1}## such that ##a\cdot a^{-1} = 1##.

Can you define that in terms of only addition? Without recognising multiplication as a binary operation in its own right?
 
  • #23
Mark44 said:
Maybe you didn't see my examples of situations where division is used that aren't as trivial as giving kids the same number of cookies.
I saw.
Mark44 said:
Division is used by companies, big and small, as well as individuals all around the world. The vast majority of people in the world who are at least conversant with arithmetic would differ with you on whether division is a concept.
and this proves my point. “Vast majority” always do what’s convenient. They would not understand inverse of a. Same thing with percentage. People understand 100. They are able to Compare things with 100. That’s why we use percent today.
Mark44 said:
Now you're confused. Division is multiplication by the reciprocal of the divisor -- you left that part out.
That’s a mistake or typo.
 
  • #24
mark2142 said:
Isn’t everything just addition?
Subtraction is addition of opposite. Multiplication is repeated addition
As long as you can define fractional addition. But I think that is tricky. I suspect that can't be done without already having division defined. So it would be a circular definition.

In Abstract Algebra, you start with two operations, addition and multiplication, and define how they interact. There is no attempt to define one in terms of the other. It applies to a greater variety of situations. In fact, I think it would be hard in the Real number system to give a good definition of multiplication in terms of addition.
 
  • #25
PeroK said:
Not everything in life is abstract algebra! There are division operations in Python and other programming languages. Your mistake is to imagine that if pure mathematics goes down a certain road, then the rest of the world must follow.

In abstract algebra, there are only two binary operations. But, in science, economics and computer programming there are four.
Abstract Algebra, like computer science, can be used to study mathematical systems with more operations (some basic, some derived). A Field postulates the existance of a multiplicative identity, ##1##, and a multiplicative inverse, ##a^{-1}##, for every nonzero element, ##a##. Some common things are Fields and others are not. Whether division is basic or derived from multiplication, it requires some additional assumptions.
 
  • #26
How does one calculate the decimal equivalent of a fraction without doing something “division like.”
For example how does one calculate ½=0.5 (I know that you can prove it by multiplying both sides by 2)
 
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  • #27
Frabjous said:
How does one calculate the decimal equivalent of a fraction without doing something “division like.”
For example how does one calculate ½=0.5 (I know that you can prove it by multiplying both sides by 2)
Exactly.
As another example, how do you calculate ##\frac 3 {4.73}##? One way is to use ordinary division to go at it directly. Another way is to calculate the reciprocal of 4.73 (i.e., the multiplicative inverse of 4.73. That is, calculate ##\frac 1 {4.73}##, and then multiply your result by 3. How are you going to go about getting ##\frac 1 {4.73}## if not by using the "obsolete" operation of division?

mark2142 said:
and this proves my point. “Vast majority” always do what’s convenient.
As to proving your point, which was that division doesn't qualify as a concept, well, that's laughable at best. If the vast majority find it to be convenient, it's a concept.
 
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  • #28
mark2142 said:
If we are literally breaking down things into very basics then we are actually doing repeated addition. ##13## added ##13## times =##169##.
Several people, including myself, have pointed out the problem of fractions like 1/3 not being the result of repeated addition. You seem to have ignored that. In science and mathematics, rather than ignoring the problems with a theory, you have to pay special attention to them. Otherwise, you run the risk of becoming just like the "flat earthers".
 
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  • #29
PeroK said:
Not everything in life is abstract algebra!
And that is why we get so many threads about division. They all learned it wrong and were left insecure. To call ##a/b=ab^{-1}## abstract algebra is ridiculous. The transition from ##\mathbb{Z}## to ##\mathbb{Q}## is the example of a quotient field. It should be done right.

PeroK said:
There are division operations in Python and other programming languages.
Yep. Supermarkets.
 
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  • #30
fresh_42 said:
And that is why we get so many threads about division. They all learned it wrong and were left insecure. To call ##a/b=ab^{-1}## abstract algebra is ridiculous.
Too harsh.
 
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  • #31
FactChecker said:
Too harsh.
Echo.

I know that threads like "division of zero", "why is 1=0.999..." and similar nonsense are very popular. Fact is, they would all be meaningless if people had learned it correctly: 0 is not part of the multiplicative group, and 0.999... is a limit and a limit is a number, end of debate.

I see that some of you do not agree with me. So? That doesn't make me wrong, especially if you consider who is on that list. Too harsh? Is that a polite way to say "shut up"? Why? Because different positions are inconvenient? Because someone says "we have always done it like this" is no valid argument. Instead, it means: "I am too lazy to even consider your point of view."

Sorry, but harsh had been started by those who claimed that ##a/b=ab^{-1}## was abstract algebra.
 
  • #32
fresh_42 said:
The transition from ##\mathbb{Z}## to ##\mathbb{Q}## is the example of a quotient field.
Yes, but that is still a long way from ##1/\pi##. I think that the OP was contending that everything can be logically derived from repeated addition of the denominator, which I think is a misconception. Other assumptions and postulates are needed. There are better ways to approach it.
 
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  • #33
FactChecker said:
Yes, but that is still a long way from ##1/\pi##.
Not really. One only needs to show that ##x \cdot \pi -1 =0## has a solution, and that is a function graph crossing the x-axis. That's it. It could be taught even in primary schools if the established system wasn't aimed to create "secret knowledge", hiding behind a "too complicated for you" position and similar obstacles for kids to learn things right the first time.
"If you can't explain something to a first year student, then you haven't really understood." ~ Richard P. Feynman
That is my position. I am willing to defend it, preferably on a reasonable level instead of a rhetorical one like platitudes such as "not everything is abstract algebra".
FactChecker said:
I hink that the OP ...
Why do we tend to think we know what OPs think or know? They can speak for themselves. If something is "too complicated" then let's explain it instead of assuming it cannot be understood. By the way, you did exactly this:
FactChecker said:
In Abstract Algebra, you start with two operations, addition and multiplication, and define how they interact. There is no attempt to define one in terms of the other. It applies to a greater variety of situations.
I am only demanding that we should always proceed like this rather than hiding behind unspoken assumptions, laziness, and secret knowledge.
 
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  • #34
fresh_42 said:
One only needs to show that x⋅π−1=0 has a solution, and that is a function graph crossing the x-axis. That's it. It could be taught even in primary schools if the established system wasn't aimed to create "secret knowledge", hiding behind a "too complicated for you" position and similar obstacles for kids to learn things right the first time.
No. An existence proof is not sufficient. People need to be taught how to calculate.
 
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I agree with the OP as far as motivation goes, but not as far as rigorous, logical development goes.
The OP gave no information about his background or mathematical level and there is nothing to be seen in his profile. So we have to guess what level of answer is appropriate. I think that if he is interested in this subject matter, than mentioning Abstract Algebra is appropriate for a great deal of further study. But it is far too much to get into in a thread.
 

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