Is division confused with fractions?

[ajb]

Can I consider 3/10 as an operator that carrys out the operation of dividing into 10 and discarding seven of the parts?

It is not really an operator, but a way of representing a number. Division is an operator, though really it is just multiplication by an inverse.

 

So you can say [math] 3 \div 10 := 3 \times 10^{-1} = \frac{3}{10} [/math], where [math]10 \times 10^{-1} =1[/math].

 

What is true is that [math]1 = \frac{10}{10} = \frac{3 + 7}{10} = \frac{3}{10} + \frac{7}{10}[/math].

 

Or [math]\frac{10}{10}- \frac{7}{10} = \frac{3}{10}[/math], which is a nice way to think of it.

 

So do the other seven parts exist, did they ever exist?

As this is mathematics we don't have to worry if they ever physically existed if that is what you mean? We can say "there exists a unique real number that can be represented by 7/10".

[studiot]

 

It is not really an operator

 

What about the definition prevents it being an operator?

 

Obviously it could be generalised from specifically 3 and 10.

 

edit: This is not to say that it couldn't also be something else in another context.

Edited August 6, 2013 by studiot

[gabrelov]

Why do we teach our kids that 6 ÷ 3 = 2? If I divide 6 oranges in 3, what I get is not 2 oranges, but 2+2+2, without subtracting 4 oranges. So why don't we teach 6 ÷ 3 = 2+2+2, 1÷2 = 0.5+0.5, etc etc? If I say "I would like a quarter of this cake", then I am not dividing, but I am dividing and subtracting, I am doing a fraction, 1/4. Don't you think divisions shouldn't be confused with fractions?

 

We can rephrase this as we have 6 oranges and there are three of you so we divide 6 oranges by three persons and you get 2 for each person.

 

It simply implies how many part will each person get.

Edited August 6, 2013 by gabrelov

[ajb]

What about the definition prevents it being an operator?

I would not think of "/" as an operator as such, but a way of representing rational numbers where a and b are integers. The symbol [math]\div[/math] is the operator that takes two numbers, a and b with b nonzero say, and produces another number, which can be written as a/b for integers.

 

It maybe a bit pedantic, but that is how I think you should think of it.

[studiot]

I wasn't suggesting that / is an operator.

 

I was suggesting that the expression 3/10 or more generally a/b can be considered an operator.

 

Not only on numbers, but on other objects such as circles. That is how a pie diagram is generated.

 

So I repeat my question.

 

What in the definition of operators prevents a/b being an operator?

 

[math]\frac{a}{b}(1)[/math]

[math]\frac{a}{b}(10)[/math]

[math]\frac{a}{b}(\bigcirc )[/math]

 

All have valid meanings if a/b is regarded as an operator IHMO.

[Greg H.]

The example values are all very neat, appart from 1/3. As a decimal this does not have a very nice representation.

 

You mean [math]7 = (2 \times 3) +1[/math]. Which we can then pretty much use as the definition of dividing

 

[math]7 \div 3 := \frac{7}{3} = 2 + \frac{1}{3}[/math].

 

As you have said, this leads to the notion of a remainder.

You are correct about what I meant. Copy pasta is the enemy. Thanks for the correction.

[Myuncle]

 

We can rephrase this as we have 6 oranges and there are three of you so we divide 6 oranges by three persons and you get 2 for each person.

 

It simply implies how many part will each person get.

 

 

If you imply that you divide and discard the rest, you are absolutely right, but in fact, aren't we just being lazy in writing, and still confusing divisions with fractions? Let's say we have to apply divisions and fractions to geometry. Kids are introduced to a rectangle, they are given the measurement of the area and of one side (width or length), and they have to find out the other side, without knowing any formula, making a very useful effort in finding the formula by themselves, this would be very educational. So, the length of one side of the rectangle is 4, but 4 what? 4 mm? Not necessarily, they can be 4 squared biscuits. The given area of the rectangle is 20 squared biscuits. Now the kids will easily align 4 biscuits to make one side, and they are left to wonder what to do with the rest of the biscuits (20-4=16), they will make a useful effort to align the rest of the biscuits, they don't discard or ignore any of them, until they finished the rectangle, and they will notice that the other side of the rectangle has magically only 5 biscuits. So, some of them will realize that initially they are simply dividing 20 biscuits into 4 rows, then they will have to ignore 15 biscuits to find out the length of the other side, some of them won't realize it, but that's not a problem, at least they are making a healthy effort to find out the formula, which is, divide the area by a given side, (20÷4=5x4) and discard (or ignore) 15. So are we just dividing 20 biscuits by 4 to find the other side of the rectangle? No, there is a discarding process. So, without being lazy, the formula should be written not with a division, but with a fraction, 20/4=5, and again the verb fractioning shouldn't be confused with dividing.

[studiot]

No I don't think I am being lazy, I am agreeing with you that either the word division (noun) or divide(verb) or the ratio symbol a/b have more than one meaning in mathematics, and I have been trying to demonstrate some of these.

 

eg " A line divides the plane into two regions"

 

However I am also suggesting that the particular meaning is normally very clear from the context.

 

This is unlike many of the other words in my mathematical dictionary that benefit from multiple meanings and give rise to far worse confusion IHMO.

[gabrelov]

 

 

If you imply that you divide and discard the rest, you are absolutely right, but in fact, aren't we just being lazy in writing, and still confusing divisions with fractions? Let's say we have to apply divisions and fractions to geometry. Kids are introduced to a rectangle, they are given the measurement of the area and of one side (width or length), and they have to find out the other side, without knowing any formula, making a very useful effort in finding the formula by themselves, this would be very educational. So, the length of one side of the rectangle is 4, but 4 what? 4 mm? Not necessarily, they can be 4 squared biscuits. The given area of the rectangle is 20 squared biscuits. Now the kids will easily align 4 biscuits to make one side, and they are left to wonder what to do with the rest of the biscuits (20-4=16), they will make a useful effort to align the rest of the biscuits, they don't discard or ignore any of them, until they finished the rectangle, and they will notice that the other side of the rectangle has magically only 5 biscuits. So, some of them will realize that initially they are simply dividing 20 biscuits into 4 rows, then they will have to ignore 15 biscuits to find out the length of the other side, some of them won't realize it, but that's not a problem, at least they are making a healthy effort to find out the formula, which is, divide the area by a given side, (20÷4=5x4) and discard (or ignore) 15. So are we just dividing 20 biscuits by 4 to find the other side of the rectangle? No, there is a discarding process. So, without being lazy, the formula should be written not with a division, but with a fraction, 20/4=5, and again the verb fractioning shouldn't be confused with dividing.

 

I get your idea that something is lost in the process of division but heres the thing. We surely know the rules of equality right, if we equate and use your equation, 20/4 = 5x4 it is not follwing the rules since if we balance the equation on both side the statement is not true, but if we remove 4 on either one side, it becomes true.

The problem here is that as children we are taught of to follow these basic rules and neglect to discover for ourselves how this things come up. You were right to say we became lazy and that is why many students fails in math because they don't understand the concept behind the equation and they just apply it without knowing its limitations.

 

Well it depends on the learning process we impart to them and math is only one way, there are others such music and playing instruments which will enhance our skills but we cannot force someone to learn something the way it should be, each one has its own way of learning, mabe for others they like shortcut method and so on. We cannot blame those who hate math since maybe they have other talents not related to math.

 

The quality of education I say is not good in some since advent of technology we rely on our calculators for computing unlike on old days where most of the formula we use today came from, they aren't lazy as before.

 

Another thing about a/b, in engineering this is very useful, suppose a/b is a nontermnating non repeating decimal, we dont want to get such number with so many digits. So we express it as fraction instead, such examples of it are pi and e which is really bad to express as decimal notation especially when doing accurate calculations. To lessen the error we let it stay as fraction until the we reach the end of the computation and then there we can express it in decimals.

 

For example: 1/3 = .33333..... adding 1/3 + 1/3 +1/3 = 1, but adding a rounded off 0.3333 + 0.3333 + 0.3333 = .9999 which is close but not equal to 1

 

Expression of a/b is very important specially in structural computation wherein a minute error can be catastrophic.

Edited August 6, 2013 by gabrelov

[Myuncle]

20/4 = 5x4 it is not follwing the rules

But the equations I wrote are 20÷4=5x4, and 20/4=5. Using "÷" only for division, and "/" only for fractions.

 

Edited August 6, 2013 by Myuncle

[John Cuthber]

 

 

Pity you have started mud slinging again.

 

I am quite well aware of the difference between dividing in half and by half.

 

That is the whole crux of the question, that even in mathematics, it (edit : dividing ) has at least two meanings.

 

I notice, incidentally, that you have avoided my questions about dividing pies or integrals.

Who is mud slinging?

Not really, Dividing in has a different meaning from dividing by.

Standing in has a different meaning from standing by.

Screwing up isn't the opposite of screwing down.

It's pretty common for a word's meaning to change depending on context.

Confusing them in maths is no different from confusing them in English.

 

 

 

It's not clear to me that an operator like integration can be divided meaningfully.

How do you stop half way through measuring the area under a curve?

On the other hand, dividing the outcomes of integrals is easy enough- it's just arithmetic.

 

I missed the one about pies, probably because they weren't mentioned in the thread before. Pie charts were, but that's another thing altogether.

[overtone]

So why don't we teach 6 ÷ 3 = 2+2+2, 1÷2 = 0.5+0.5, etc etc?

 

20÷4=5x4

Seriously: quit writing equal signs between unequal entities.

 

Never do that. Please.

 

Especially, do not do that when teaching children mathematics - you are screwing up other people, then, not just yourself. You are punishing the student for paying attention too closely, for being alert and focused.

 

If you quit writing falsehoods, and write using only correct factual statements, many of your questions will be answered automatically or otherwise clarified.

 

The feedback boost between what you write and what you think -

 

the use of notation to do as much of your thinking as possible on the paper where you can monitor and check and remember more easily and accurately -

 

the entrainment of the mental areas devoted to hand manipulation, physical feel, proprioception, one's "blind" sense of space and arrangement -

 

the entrainment of the mental areas devoted to sound and rhythm as you "read" what you've written or drawn -

 

increases one's powers of mathematical reasoning and comprehension and memory. But you can't take advantage of that if you are habituated to writing garbage and muddle.

Edited August 6, 2013 by overtone

[studiot]

 

Not really, Dividing in has a different meaning from dividing by.

 

So you actually agreed with me all along.

 

Thank you for confirming that.

[Myuncle]

 

Seriously: quit writing equal signs between unequal entities.

 

Never do that. Please.

 

Especially, do not do that when teaching children mathematics - you are screwing up other people, then, not just yourself. You are punishing the student for paying attention too closely, for being alert and focused.

 

If you quit writing falsehoods, and write using only correct factual statements, many of your questions will be answered automatically or otherwise clarified.

 

The feedback boost between what you write and what you think -

 

the use of notation to do as much of your thinking as possible on the paper where you can monitor and check and remember more easily and accurately -

 

the entrainment of the mental areas devoted to hand manipulation, physical feel, proprioception, one's "blind" sense of space and arrangement -

 

the entrainment of the mental areas devoted to sound and rhythm as you "read" what you've written or drawn -

 

increases one's powers of mathematical reasoning and comprehension and memory. But you can't take advantage of that if you are habituated to writing garbage and muddle.

What's your take on this? Does division imply any subtraction? Is a division the same thing as a fraction? Does the sign "÷" means exactly the same thing as "/"? I don't teach any children.

Edited August 6, 2013 by Myuncle

[studiot]

Good evening Myuncle,

 

 

Is a division the same thing as a fraction

 

No it isn't, but a fraction is always the same thing as a division.

 

That is because 'a division' applies to a wider class of things than just numbers.

 

For instance my earlier comment

 

A line divides a plane into two regions.

 

It makes no sense to attempt to write this geometric statement as a fraction.

 

But all fractions imply that you have to divide the bottom into the top, or divide the top by the bottom if you prefer.

 

Of course you should distinguish between proper fractions that are numerically less than or equal to 1, and improper fractions that are greater than one.

 

You should also note that there are numbers that cannot be expressed as a fraction (if you count fractions as being only of the form a/b where a and b are single numbers) unless the bottom number is one. These are called irrational numbers, and their existence greatly troubled the ancient Greeks.

Edited August 6, 2013 by studiot

[ajb]

What in the definition of operators prevents a/b being an operator?

 

[math]\frac{a}{b}(1)[/math]

[math]\frac{a}{b}(10)[/math]

 

[math]\frac{a}{b}(\bigcirc )[/math]

 

All have valid meanings if a/b is regarded as an operator IHMO.

All you have done is multiply two numbers together. You can think of any number as a map from the real line to the real line by multiplication.

For example: 1/3 = .33333..... adding 1/3 + 1/3 +1/3 = 1, but adding a rounded off 0.3333 + 0.3333 + 0.3333 = .9999 which is close but not equal to 1

Let me just stress that one has made a truncation here. That is after some finite number of decimal places we throw away all decimal places higher than this. In this particular case 1/3 is approximated by 0.3333.

 

It is however true that [math]0.\bar{3} + 0.\bar{3} + 0.\bar{3} =1[/math] (here I use bar to denote a repeating decimal.)

[gabrelov]

All you have done is multiply two numbers together. You can think of any number as a map from the real line to the real line by multiplication.

 

Let me just stress that one has made a truncation here. That is after some finite number of decimal places we throw away all decimal places higher than this. In this particular case 1/3 is approximated by 0.3333.

 

It is however true that [math]0.\bar{3} + 0.\bar{3} + 0.\bar{3} =1[/math] (here I use bar to denote a repeating decimal.)

 

 

Yupz but if you are just a student and having an exam and the teacher tells you to show solution it is not practical to express something in decimal.

 

By the way many things are expressed in fraction instead of decimals in measurements such as the units of inches and so on.

[studiot]

Quote ajb

All you have done is multiply two numbers together. You can think of any number as a map from the real line to the real line by multiplication.

 

 

I'm sorry you are not understanding what I wrote. All the examples are written using operator notation.

 

Yes indeed the first two examples boil down to the multiplication of two numbers, but the third example is not multiplication in any sense.

Are multiplication or division not operations?

 

In the third example I am operating on a circle with the ratio or fraction a/b. This would give me a sector and is purely geometric.

 

 

[ajb]

In the third example I am operating on a circle with the ratio or fraction a/b. This would give me a sector and is purely geometric.

Right okay.

 

How do you define such an operation?

[Are multiplication or division not operations?[/font][/color][/size]

Standard multiplication of real numbers is a binary operation. You take two numbers and get a third one. Division is really the same as multiplication.

 

You can think of any number as an operator or map on the real line by just declaring its action via multiplication. This is, in a more general setting, quite a natural thing to do as you can view an algebra as a sitting inside its automorphisms.

[studiot]

quote ajb "Right okay"

 

Hopefully this makes now sense of my other posts for you.

 

I'm sure you recognise the integrals fraction/division I posted that confused john cuthber as the x coordinate for the planar centre of gravity.

[ajb]

Hopefully this makes now sense of my other posts for you.

So how do you define the action of a fraction on the circle?

[studiot]

Personally I would say that division is a process and a fraction is one presentation of that process and that Myuncle is correct in observing that they are different, even though his arithmetic is a bit rambling.

 

You can apply division (ie operate the verb to divide) to more objects than just numbers.

Indeed engineers perfected 'dividing engines' for just that process many years ago.

 

Dividing a circle is exactly what is involved in creating pie diagrams.

 

So yes I agree with Myuncle that even at early stages in teaching we should distinguish between fractions and division as the latter can and will be applied more widely in later teaching.

 

It is an old saw that, however simplified our treatment, we should never teach something that we later have to say "that was actually false".

[overtone]

If you are too shallow in your presentation of fractions as divisions, rather than numbers, you may create confusion when it comes time to diviede fractions and by fractions.

[studiot]

 

If you are too shallow in your presentation of fractions as divisions, rather than numbers

 

 

But fractions too are more than just numbers.

 

What number, for instance, is represented by this fraction?

 

[math]\frac{{{x^2} - 2x}}{{x + 6}}[/math]

 

And what about the many types of fraction

Algebraic fractions , continued fractions, partial fractions to name but a few more.

Continued fractions are good fun becasue they represent an alternative way of thinking about recurring decimals or irrationals that have been discussed in this thread.

 

[math]1 + \frac{1}{{1 + \frac{1}{{1 + \frac{1}{{1 + \frac{1}{{1 + ..}}}}}}}} = ?[/math]

 

Is a definite (irrational) number, but is it a division?

 

Edited August 8, 2013 by studiot

[Myuncle]

If division implies a subtraction for me it's fine, I would like to read it explicitly everywhere. On wikipedia or in the dictionaries, normally they never talk about any subtraction, they just say it's the inverse of multiplication, so in theory, yes, there is a shrinking involved, and I am just being anal about it...It's always good to see a consensus in definitions. Even multiplication it's nothing but a shortcut for addition. I can calculate 3x76556 with additions, but it would take me a year to write it down...So I suppose that, in the past, the origin of multiplication was the need to create a shortcut for long tedious additions, and the origin of division was to create a shortcut for long tedious subtractions.