# Relation between Division and multiplication

• B
• mark2142
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.
PeroK said:
There are division operations in Python and other programming languages.
fresh_42 said:
Yep. Supermarkets.
Yeah, like supermarkets have registers that are programmed in Python. Right...

fresh_42 said:
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...

fresh_42 said:
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 and PeroK
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.

fresh_42
I think this thread is multiplying division among our ranks.

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

weirdoguy, FactChecker and Mark44
gmax137 said:
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.

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FactChecker said:
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.

FactChecker
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.

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romsofia, weirdoguy and PeroK
fresh_42 said:
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.
fresh_42 said:
Division is obsolete.
What?

Mark44
FactChecker said:
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.

FactChecker said:
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.
PeroK said:
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?

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fresh_42 said:
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.

fresh_42 said:
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."
fresh_42 said:
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.

fresh_42 said:
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.

DeBangis21, romsofia, PeroK and 1 other person
@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.

weirdoguy, PeroK and fresh_42
mark2142 said:
@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.

Mark44 said:
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
PeroK said:
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?

romsofia and weirdoguy

fresh_42 said:
Can we close this now?

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