B Relation between Division and multiplication

[mark2142]

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?

 

[fresh_42]

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Division is obsolete.

 

[pasmith]

169 \times \frac{1}{13} = ?

 

[Mark44]

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.

 

[mark2142]

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 $p=qn$)

 

[fresh_42]

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 $q=0.$

 

[berkeman]

Ok. That is some fresh information.

I see what you did there.

 

[Mark44]

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.

 

[fresh_42]

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.

 

[Mark44]

Ok. Supermarkets, construction sites, and programmers of calculators.

Or engineering firms, manufacturers, etc., etc.

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.

 

[fresh_42]

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.

 

[FactChecker]

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 $a^{-1}$.

 

[Mark44]

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.

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.

 

[mark2142]

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.

 

[FactChecker]

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.

 

[mark2142]

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.

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.

 

[mark2142]

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.

 

[Mark44]

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.

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.

It’s still multiplication.

Now you're confused. Division is multiplication by the reciprocal of the divisor -- you left that part out.

 

[FactChecker]

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.

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.

 

[PeroK]

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.

 

[mark2142]

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.

 

[PeroK]

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?

 

[mark2142]

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.

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.

Now you're confused. Division is multiplication by the reciprocal of the divisor -- you left that part out.

That’s a mistake or typo.

 

[FactChecker]

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.

 

[FactChecker]

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.

 

[Frabjous]

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)

 

[Mark44]

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?

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.

 

[FactChecker]

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".

 

[fresh_42]

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.

There are division operations in Python and other programming languages.

Yep. Supermarkets.

 

[FactChecker]

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.

 

[fresh_42]

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.

 

[FactChecker]

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.

 

[fresh_42]

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".

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:

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.

 

[Frabjous]

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.

 

[FactChecker]

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.

 

[Mark44]

There are division operations in Python and other programming languages.

Yep. Supermarkets.

Yeah, like supermarkets have registers that are programmed in Python. Right...

I know that threads like "division of zero", "why is 1=0.999..." and similar nonsense are very popular.

But the OP was not asking about division by zero or why 1 = 0.999...

Why do we tend to think we know what OPs think or know?

We can make a reasonable guess by the questions that the OP is asking, both in this thread and the previous ones.

Granted that division is equivalent to multiplication by the multiplicative inverse, but at some point prior to introducing modern algebra notions (e.g., groups, rings, fields, and so on) you have to have the ability to actually do division. How else would you calculate $\frac 3 {4.71}$?

 

[FactChecker]

I consider the mention of Abstract Algebra here to be similar to the mention of General Relativity on a thread where someone proposes a simple explanation for gravity. In fact, introductory Abstract Algebra can be studied with far fewer prerequisites than General Relativity.

 

[bob012345]

I think this thread is multiplying division among our ranks.

 

[gmax137]

Is there an algorithm for finding "the inverse of b" that doesn't involve dividing 1 by b?

 

[FactChecker]

Is there an algorithm for finding "the inverse of b" that doesn't involve dividing 1 by b?

You can present it as trial and error multiplication, converging to a solution. I have dealt with students who only learned that in high school and were failing Freshman math in college. They couldn't do division problems fast enough to pass remedial math tests.
The problem becomes harder if multiplication is thought of as repeated addition. IMO, that, combined with a clumsy division method, would be unmanageable.
Those initial motivational approaches should be replaced long before a student enters High School.
In fact, my last High School tutoring involved students using the long-division algorithm on polynomials. So they were far beyond the approach of the OP.

 

[gmax137]

You can present it as trial and error multiplication, converging to a solution. I have dealt with students who only learned that in high school and were failing Freshman math in college. They couldn't do division problems fast enough to pass remedial math tests.
The problem becomes harder if multiplication is thought of as repeated addition. IMO, that, combined with a clumsy division method, would be unmanageable.
Those initial motivational approaches should be replaced long before a student enters High School.
In fact, my last High School tutoring involved students using the long-division algorithm on polynomials. So they were far beyond the approach of the OP.

That seems a horrific price to pay, to avoid sophmoric questions about 0.999999.

 

[fresh_42]

The point is that what some here call abstract algebra is indeed real algebra, namely multiplication. It is a binary operation and within this thread, the multiplication of a group; and the group properties are essential here! It is algebra per definition no matter how you paint it.

My point of view is that these basics can be learned very early in life. I strengthened the advantages of such an approach.

On the other side, I did not read a single argument of why the classical approach in our schools is better! Not a single one. I only read polemics, rhetorical onomatopoeia, silly examples where people divide numbers - as if we weren't a scientific website anymore and this wouldn't be a technical, mathematical forum, personal offenses, and a lot of hot air.

I can argue at this level but it makes no sense and I have the disadvantage that it is not in my native language. I know a priori that I cannot convince my critics here, for reasons I cannot tell without breaching the rules. So I leave it at that. However, it would have been nice if someone actually presented an argument for why divisions should not be reduced to inversions. Sad.

I apologize that I used the metaphors of supermarkets and construction sites to illustrate daily work in real life in contrast to the science of mathematics. I thought this would be clear, but it apparently wasn't.

 

[FactChecker]

The point is that there is actually no such thing as a division. ... So all we need is
a) Solve $b\cdot x=1$

I call that division. The first thing I would say is "Divide both sides by b." and I think that 99% of algebra teachers would say that. I don't understand your objection to that and why you think you have a better approach.

Division is obsolete.

What?

 

[fresh_42]

I call that division. The first thing I would say is "Divide both sides by b." and I think that 99% of algebra teachers would say that. I don't understand your objection to that and why you think you have a better approach.

If you call the multiplication by an inverse a division then I don't have objections. But it means to say good-bye to equations like $a\, : \,b=c$ or even $\dfrac{a}{b}=c.$ And these are at the same time my objections. They create trouble: $a\, : \,b\, : \,c$ means what? Or double quotients where you need longer lines to note the main quotient. Division requires exception after exception; dozens of rules which are completely unnecessary. Or the naive question about the division by zero. All gone if we used $ab^{-1}$ and $b^{-1}$ as the solution of $bx=1$ right from the start as it should be in my opinion.

What?

We need inversions, no divisions. That doesn't mean that we won't use long divisions anymore, but as a consequence of the Euclidean algorithm and not as an operation in its own right. Will I still use $15:3=5?$ Yes, of course, after I learned it right and I know what I do. It means to accept $15:3=15\cdot 3^{-1}=5\cdot (3\cdot3^{-1})=5,$ i.e. reversing the multiple of $3$ to get $5.$ Divisions are not necessary. Inversions will do.

My opinion can be stated as:
Start teaching mathematics, not calculating. We have calculators for that.

Nobody questions that biology, chemistry, and physics are taught as close to actual science as possible. Only mathematics is taught like the kids were all dull.

And if it is even impossible to say it right on a scientific website without being shouted out for doing so, then things are really bad.

Not everything in life is abstract algebra!

... is an oath of disclosure. I looked it up. This thread is still in a mathematical forum, or only on my screen?

 

[Mark44]

Division requires exception after exception; dozens of rules which are completely unnecessary. Or the naive question about the division by zero.

What exactly are these "dozens of rules"? The only one I can think of is that division by zero is undefined. Instead of "dozens of rules" I'd be happy to hear, say, a half dozen.

That doesn't mean that we won't use long divisions anymore, but as a consequence of the Euclidean algorithm and not as an operation in its own right. Will I still use 15:3=5? Yes, of course, after I learned it right and I know what I do.

I don't think anyone is arguing with you that 15:3 (or as it would be more commonly presented,$\frac {15} 3$) is the same as $15 \cdot 3^{-1}$. What we're arguing against is all the hyperbole you've previously written in this thread (direct quotes):

• "Division is obsolete."
• "There is actually no such thing as a division."
• "You need division to give all kids the same number of cookies, that's it."

However, it would have been nice if someone actually presented an argument for why divisions should not be reduced to inversions. Sad.

Arguments were presented multiple times, if you had bothered to read them -- e.g., find decimal representations for $\frac 1 3$ and $\frac 3 {4.76}$. Expressions such as the one you gave, $\frac{15} 3$ are too trivial to bother commenting on.

My opinion can be stated as:
Start teaching mathematics, not calculating. We have calculators for that.

Nobody questions that biology, chemistry, and physics are taught as close to actual science as possible. Only mathematics is taught like the kids were all dull.

I don't know whether you have ever taught a class in mathematics at any level. If so, I'm fairly certain that you have never taught a class in a grade school or high school. No teacher in his or her right mind would start teaching 4th graders (about 9 years old) arithmetic using abstract algebra constructs such as groups and group properties, $\mathbb Z$, $\mathbb Q$, and quotient fields (all terms you used), especially if said kids were still a bit weak on adding and multiplying single digit integers. A child needs to crawl before she walks and walk before she runs. Tossing her a calculator before she is able to add, subtract, multiply, and divide one- and two-digit numbers borders on the criminal, IMO.

 

[mark2142]

@fresh_42 This totally is the case. I agree with you and Feynman. People here don’t explain in simple language and pour lots of information as a reply. This makes hard to understand and then they say we have pointed this or that out many times but you ignored. I don’t ignore. It gets lost into the sea of lots of facts. In the end it gets hard to follow and make sense out of anything.

 

[fresh_42]

@fresh_42 This totally is the case. I agree with you and Feynman. People here don’t explain in simple language and pour lots of information as a reply. This makes hard to understand and then they say we have pointed this or that out many times but you ignored. I don’t ignore. It gets lost into the sea of lots of facts. In the end it gets hard to follow and make sense out of anything.

I often think what would have happened to Ramanujan's talent if he was forced into our Western education system. The kids I have met in my life were so much more gifted than what they were asked for at school. Mathematics can be exciting instead of a synonym for horror at school where one algorithm hunts the other without ever explaining the background. Most kids over here meet mathematics for the first time at university - ok, maybe with the exception of geometry.

 

[PeroK]

No teacher in his or her right mind would start teaching 4th graders (about 9 years old) arithmetic using abstract algebra constructs such as groups and group properties, $\mathbb Z$, $\mathbb Q$, and quotient fields (all terms you used), especially if said kids were still a bit weak on adding and multiplying single digit integers.

That's the way mathematics is taught in cloud cuckoo land!

 

[fresh_42]

That's the way mathematics is taught in cloud cuckoo land!

This discussion is getting more and more profound by the minute. Exactly my argument: personal offenses replace rational arguments - very professional, Sirs.

Can we close this now?

 

[Drakkith]

Thread locked for moderation.