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My example is 1/2 divide by 8.

The correct answer given : 1/16

However, in my mind this answer is 1 divide by 16, which to me is not what the question is asking.

When I construct the argument, it seems it should equal 1/8 as the question is asking "What is the division of 1/2 into 8 pieces?"

Not

"What is the total division of the whole when 1/2 is divided into 8 pieces?"

Why give the answer to the whole, when it is the fraction that is in question?

If I have a box, and I cut it into half, and then I ask "What is the division of this box into 8?" Why would I give a division of the whole box?

Edit

Thank you for the many answers, I believe it boils down to absolute quantity vs relative measurement, and the mass of the object.

So, 1/3 of half a tea spoon is an absurd but correct measurement.

but, say you have a house divided into 2 apartments, both having 8 rooms.

You would say : "An apartment has 8 rooms" not "The house has 16 rooms"

Edit

Absolute measurement, as a method of reaching an answer that is limited in complexity, is preferable to an infinite number of sets of fractions.

1/18 = Absolute Measurement

1/8 = Sets of fractions into infinity

UPDATE EDIT Just an update for visualisation.

Imagine one whole cube, then cut that cube 8 times, now divide that cube into half and you have 16.

I believe this is what the sum is describing and the problem with my visualisation.

This has really helped me appreciate set theory, thank you for your time.

Joseph
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    the question is asking "What is the division of 1/2 into 8 pieces?" Right: $\cfrac{;\cfrac{1}{2};}{8} = \cfrac{1}{16},$. – dxiv Apr 24 '17 at 02:07
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    The answer is not just $1/8$, it is $1/8$ of $1/2$. If you would want an answer consisting of a single fraction (instead of two of them), then you could assume that your answer would always be of the form a certain fraction of the whole, i.e. of $1$. That is, $1/8$ of $1/2$ is the same as $1/16$ of $1$. If we make a convention that "of $1$" is understood by default, then we could not just omit the "of $1/2$", in the answer "$1/8$ of $1/2$", since "$1/8$ of $1/2$" is not the same as "$1/8$ of $1$", but is the same as "$1/16$ of $1$". – Mirko Apr 24 '17 at 02:17
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    Why on earth would anyone ask what is the number of pieces is when that is what is given? we want to know what the result is. Your argument makes as much sense as "why is 5+7=12? Shouldn't it it be 7 because isn't the question what is being added?" To which the answer is of course not, and that is NOT what is being asked. Why in the heck would anyone ask that? – fleablood Apr 24 '17 at 02:23
  • I appreciate the conclusion. 1/8 is the same as 1/16, However, the problem I have is the phrasing of the question, If we want the precise answer 1/16, would we not ask, what is 16th of 1 whole.

    However if we are asked, what is 1/8th of 1/2? Why should we need the whole for the answer. Thank you for your reply.

     – Joseph Apr 24 '17 at 02:31
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    Written as given, $1/8$ is not the same as $1/16.$ Such a statement would necessitate nonequivalent units. – zahbaz Apr 24 '17 at 07:25

6 Answers6

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If I have a box, and I cut it into half, and then I ask "What is the division of this box into 8?" Why would I give a division of the whole box?

If I'm understanding your question correctly, you cut a box in half, giving you two boxes, each with half the size of the original. You then take one of those smaller boxes, and cut it into eight equal pieces. The question is "What is the size of a piece?" The answer is $\frac{1}{16}$ (by which people mean, of course, one sixteeth the size of the original box).

And what you are asking is, "Why don't we say that the answer is $\frac{1}{8}$ (by which you mean, of course, one eighth the size of the half box).

You are saying, "Well we already cut the original box in half. Why are we still comparing things to the original box? Why not to the half box?"

Is this what you are saying?

Answer: It's just different ways of saying the same thing. You are correct that the size is one eighth of the half box, and it is also correct that the size is one sixteenth of the original box. If you accept this, the question becomes why everyone except you wants to say that the answer is $\frac{1}{16}$ (and why you should too).

I would say it's because half boxes are stupid. You already have a perfectly good box, which you originally started with, and to which you can compare anything else that you encounter. You are right that the thing you are looking at is one eighth of a half box, but we already have a perfectly good standard of measurement: the box. Not the half box, but the box.

Imagine insisting that a $300$ lbs person acktuallee weighs $600$ half pounds.

D_S
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  • Yes, it's difficult for me to see why the whole box is in question, when we are only talking about 1/2 of the box.

    Thank you for your reply.

     – Joseph Apr 24 '17 at 02:23
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    See my edit. I hope what I've written will satisfy you. – D_S Apr 24 '17 at 02:35
  • A problem comes to mind, say I have a house that I have divided into 2 apartments. 1 apartment is divided into 8 rooms. (1/2 divide by 8 again) Would you say "Each apartment has 8 rooms" or "The house has 16 rooms" Wouldn't you say "1 apartment has 8 rooms" ? – Joseph Apr 24 '17 at 03:10
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    Yes, in this particular case, a "half house" would be a preferable unit of measure to a house. But in general, if you start with some unit of measure, the convention in mathematics is that you keep it as the standard to which everything else is compared. – D_S Apr 24 '17 at 03:25
  • Absolute measurement as convention, as a method of reaching an answer with limited complexity, is something I can appreciate. Thank you very much for your time. – Joseph Apr 24 '17 at 03:42
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    To echo @D_S. While sometimes it can be useful to change units of measurement (house -> apartment), most of the time -- for simplicity and to avoid errors -- you want to keep using the same units throughout. A common source of errors by beginners (of applied maths or physics) is to get units mixed-up, which is why (at least when I was at school) you used to get problems that mixed units (m/s, km/hour etc.) to train you to spot such things and convert everything to common units. – TripeHound Apr 24 '17 at 06:47
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Suppose you have a square which is $1$ unit by $1$ unit, for an area of $1$ square unit. Now you take half of the square, which is $1/2$ square units.

Next you are told to divide this half square into $8$ pieces. Each will have $1/16$ square unit of area. So $1/2$ divided by $8$ is $1/16$. Sure they all have $1/8$ the size of the half square, but that is a relative measure. $1/2$ divided by $8$ is an absolute quantity.

Also, if you claimed $1/2$ divided by $8$ should be $1/8$, then $1/3$ divided by $8$ should also be $1/8$. But then you would have $\frac{1/2}{8}=\frac{1/3}{8}$, so $\frac{1}{2}=\frac{1}{3}$.

kccu
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  • If I understand, we agree on there being two possible answers, the matter is on relative measure vs absolute quantity.

    Is one more correct than the other? Or is this a case of completeness? It's nicer to have the full whole number than the answer to a part of a whole.

     – Joseph Apr 24 '17 at 02:42
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    I don't agree that there are two possible answers to $1/2$ divided by $8$ mathematically, but I do see that one can choose to view eight pieces of $1/2$ as parts of $1/2$ or parts of $1$. If you choose to view them as parts of $1/2$, then you are treating $1/2$ as a whole, not as $1/2$, which is why you do not get the mathematical answer of $\frac{1/2}{8}=\frac{1}{16}$. – kccu Apr 24 '17 at 02:52
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Interpret the question as follows: we want to find $x$ such that "$8$ lots of $x$ equals a half", which we can write as: $$8x=\frac{1}{2}$$

This gives us $$x=\frac{1}{2}\cdot \frac{1}{8}=\frac{1}{16}$$

mrnovice
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When using real numbers, we typically call the unit $1$. It convenient to compare things to $1$.

Half of $1$ for example is $\frac12(1)=\frac12$.

Now if we consider a quantity, $a$, where $a\ne1$, then half of $a$ is $\frac12a$.

Clearly $\frac12$ differs from $\frac12a$, both in value and in symbolic representation. As it should! We have been working in a single system of units.

You seem to be arguing that $\frac12a$ could or should be referred to as $\frac12$. But doing so redefines the value of your unit to $a$. That is, you're setting $a\to1$ in your new system! This is a transformation of the space you're working in.

zahbaz
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Why give the answer to the whole, when it is the fraction that is in question?

The final answer to the division question is "What is the value each part gets". In your example, each part of the 8 parts gets $0.0625$ (as Algebraically explained by other comments and answers in this post), which may be written as $1/16$.

NoChance
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Yep I had the same doubt so probably my answer would fit see if we take an example is 1 is divided by 2 to get 1/2 in each , now our doubt is why don’t we take the part which is 1/2 and we’ll divide only this part leaving other part of the whole but according to my perception if I take this part which is 1/2 ; now leaving other 1/2 part would give me a new whole which is 1/2 becomes 1 but of smaller size so probably in order to avoid confusion we understood fraction as number of parts we have to number of parts divided of whole in this case first we divided into 2 later again this 2 into 2 which is 2*2 =4 so the total parts we divided into is 4 we get when we divide any of 1/2 is 1/4

Pvs M
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