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In summary, the conversation discusses the use of division as a way to compare two quantities, such as a ratio. It is noted that division is the inverse operation of multiplication and is useful for allocating quantities equally. The use of the notation a:b to represent a ratio is also mentioned, but it is uncommon when the second quantity is zero. The conversation also considers the possibility of comparing quantities additively or multiplicatively. Ultimately, the purpose of using division to compare quantities is to easily visualize the ratio between them.

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Don't we have that with the > and < operators?

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There are 3 oranges and no apples.

- What is the ratio of oranges to apples? "Three to zero" or 3:0 is OK, but 3/0 is not defined.

- What is the ratio of apples to oranges? "Zero to three" or 0:3 is OK, and 0/3 is defined but following through to get "0" loses information about how many oranges.

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bahamagreen said:

There are 3 oranges and no apples.

- What is the ratio of oranges to apples? "Three to zero" or 3:0 is OK, but 3/0 is not defined.

- What is the ratio of apples to oranges? "Zero to three" or 0:3 is OK, and 0/3 is defined but following through to get "0" loses information about how many oranges.

That is more along the lins of what I am asking. I'm wondering why division is used as a way to compare quantities. From your example, it would seem as though using the notation such as a:b to be a better way...

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Not really. It is very rare for someone to compare, say, 3 oranges and 0 apples.Mr Davis 97 said:That is more along the lins of what I am asking. I'm wondering why division is used as a way to compare quantities. From your example, it would seem as though using the notation such as a:b to be a better way...

If you want to know

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Mark44 said:Not really. It is very rare for someone to compare, say, 3 oranges and 0 apples.

If you want to knowhow many times larger one quantity is than another, you divide, and that's your ratio. As far as the a : b notation, I've never, ever seen it used when the second quantity (which is to be compared to) is zero.

I do see how division can be useful for expressing how two quantities compare, but my question is really why was the idea conceived in the first place? It seems that division is just the inverse operation of multiplication. So how come it turns out to be useful? This question also applies to rates to, as that is a specific type of ratio.

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I doubt that anyone knows when division first came about. An obvious use is in allocating n things amongst m people. Any time someone needs to allocate a number of things equally, they're almost certainly doing division.Mr Davis 97 said:I do see how division can be useful for expressing how two quantities compare, but my question is really why was the idea conceived in the first place?

Mr Davis 97 said:It seems that division is just the inverse operation of multiplication. So how come it turns out to be useful? This question also applies to rates to, as that is a specific type of ratio.

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Mark44 said:Not really. It is very rare for someone to compare, say, 3 oranges and 0 apples.

If you want to knowhow many times larger one quantity is than another, you divide, and that's your ratio. As far as the a : b notation, I've never, ever seen it used when the second quantity (which is to be compared to) is zero.

Zero apples is not a rare thing, it is the state wherever and whenever apples are not present... zero apples is the state of the entirety of my house right now and virtually all sub-volumes of the universe for all time...

What have you seen used when the second quantity is zero?

3:0 is still informative of the magnitude of the second compared to that of the first... it is not multiplicative information (because division by 0 is undefined), but still retains some additive information as to the magnitude of the difference in value.

Seems to me that X:Y is simply two values and the colon is not any intrinsic operation. Choosing an operation to use for comparing them is another step. One may compare these numbers additively with respect to zero, which allows either number to be zero, or one may impose a division by looking at how many times larger one is of the other, which is either undefined or results in zero if either one is zero.

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Compare the alternative type of descriptor: there are 35,448 more men than women in engineering. Without knowing the exact amount of total engineers, this number may be alarming (there are only 36,000 engineers) or marginal (there are 35,000,000 engineers). So this method of comparison is not suited to describing this particular type of information concisely.

A description of "how many times one group fits inside the other", a ratio, is more immediately descriptive in this case. As you can see, the : operator is also used to describe ratios just as efficiently as division, but division has the added benefit of being able to be reduced to a single real number descriptor.

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Mr Davis 97 said:

Ratios between two quantities give us the comparison of a quantity with a unit value of the other quantity. Now let's say a bag contains 10 oranges and 5 mangoes. the ratio is 10:2 or 10/2 = 5. That is, for every one mango we have 5 oranges in the bag or we can say 5 oranges per mango. Same applies if there are 3 oranges and 4 mangoes. For every 1 mango we would have a 0.75 of an orange.

Now why does this work? When we divide 10 by 2 we find the number that when repeatedly subtracted twice (or repeatedly added twice gives us 10) gives us 0. This means for every '1' in '2' we need to subtract a 5 from 10. Keep repeating this (twice) till we get a 0. Thus for every '1' we have a 5 just like we had 5 oranges for every mango in the above example.

I hope this was clear.

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Of course zero apples is not a rarity, but almost no one would be interested in a ratio comparison between three oranges and zero apples, and a ratio is exactly what you get with the : notation. See the wikipedia article, http://en.wikipedia.org/wiki/Ratio.bahamagreen said:Zero apples is not a rare thing, it is the state wherever and whenever apples are not present... zero apples is the state of the entirety of my house right now and virtually all sub-volumes of the universe for all time...

Subtraction, as in there are three more oranges than apples.bahamagreen said:What have you seen used when the second quantity is zero?

The colon notation IS division, just in a different form of notation. It's a ratio, and the indicated operation is division.bahamagreen said:3:0 is still informative of the magnitude of the second compared to that of the first... it is not multiplicative information (because division by 0 is undefined), but still retains some additive information as to the magnitude of the difference in value.

Seems to me that X:Y is simply two values and the colon is not any intrinsic operation.

Nope.bahamagreen said:Choosing an operation to use for comparing them is another step.

Sorry, this is nonsense.bahamagreen said:One may compare these numbers additively with respect to zero, which allows either number to be zero, or one may impose a division by looking at how many times larger one is of the other, which is either undefined or results in zero if either one is zero.

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Thanks for the remarks... let me elaborate on my thinking and maybe we can get to the bottom of what's amiss.

I looked at the Wiki page you linked; I saw nothing mentioning 0 specifically as a term in a ratio, but did notice that all the Euclidean examples when converted to modern form stipulated positive integers... that suggests 0 may be improper in ratios.

I can see how all defined divisions are ratios, but no ratios where the second place is 0 are defined divisions.

Therefore it seems improper to hold that ratios and divisions are the same unless ratios must not have 0 in the second place.

That seems consistent with your reply and Wiki.

To hold that all ratios are divisions one must restrict ratios to those for which the corresponding division must not include a division by zero.

However, this restriction seems to extend to natural examples that seem perfectly defined with a zero in the second place.

Mixtures of paint colors A and B can be a ratio A:B in which either A or B may range from 0 to N..

A bucket of paint that is all A and no B has the ratio A:B where B=0 and the color is fully saturated A..

How do you view the paint example? Does it represent a class of improper ratios?

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As has been noted, we can have mixtures of fruit where we have 5 apples for every 10 oranges and we would represent it as the ratio 5 : 10 with the understanding that it's apples to oranges. In this case, we would never consider dividing one by the other because the items are different and so we wouldn't use a fraction to represent it. However, we might use percentages like 33% apples to 66% oranges in the mixture of fruit.

With respect to ratios representing division, I would argue that it's more general than that in its common usage. Consider the case of apples, oranges and bananas, we might say that 5 : 10 : 15 to mean 5 apples to 10 oranges to 15 bananas. It wouldn't make sense to say it's

My take is that we can always use ratios or percentages for mixtures no matter how many items or how much material is used. However, when ratios show zero elements then it's meaningless to use fractions and by extension division.

Perhaps, this is why we learn about all three techniques in school to cover the case where we may have to deal with zero.

Having said that, I think the thread has run its course and it's time to close it now.

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Not "may be improper" -- division by zero is undefined. Period.bahamagreen said:I looked at the Wiki page you linked; I saw nothing mentioning 0 specifically as a term in a ratio, but did notice that all the Euclidean examples when converted to modern form stipulated positive integers... that suggests 0 may be improper in ratios.

If something is not present in a mixture, you don't describe its absence by a ratio. If you buy a gallon of paint, they don't list all the things that arebahamagreen said:I can see how all defined divisions are ratios, but no ratios where the second place is 0 are defined divisions.

Therefore it seems improper to hold that ratios and divisions are the same unless ratios must not have 0 in the second place.

That seems consistent with your reply and Wiki.

To hold that all ratios are divisions one must restrict ratios to those for which the corresponding division must not include a division by zero.

However, this restriction seems to extend to natural examples that seem perfectly defined with a zero in the second place.

Mixtures of paint colors A and B can be a ratio A:B in which either A or B may range from 0 to N..

A bucket of paint that is all A and no B has the ratio A:B where B=0 and the color is fully saturated A..

bahamagreen said:How do you view the paint example?

Does it represent a class of improper ratios?

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