Can anyone explain the following please?

It may seem like a stupid question, and it may have been asked on here before, but why (in current leading mathematics theory) when one is divided by two, does it give '*a half'?? Surely one divided by two gives three thirds. I have attempted to exlain this with a diagram: there is one whole, that is divided by two (giving three thirds). Thanks. :)

*edit: if anything if gives two halves, not one.

"One divided by two" is equivalent to saying "1/2". It refers to the arithmetic operation of division, not dividing something by two lines, but into two equal parts.

To parallel your picture, consider a rectangle, and now divide it into two parts.

Semantic mistake - It should be "1 divided into 2" not "1 divided by two".

- KitNyx

No, I don't think he means "One divided into two" because that would just be two halfs (2/2). He's asking why "One divided by two" is one half (1/2) and not three thirds (3/3).

Dapthar pretty much hit the nail on the head. "One divided by two" is not talking about physically dividing the "one" object by "two" lines, but rather is a mathematical expression to denote a fraction.

The words "divided by" represent the mathematical operator in the expression:

1 / 2

The division operator is the inverse of multiplication. The expression

x / y ("x divided by y")

gives the answer to the question:

"How many times must y be multiplied to give x?"

Writing 12/2 = 6 means that 2 must be multipled by 6 to give 12. And writing 1/2 is the same as saying that 2 must be multiplied by one half to give 1.

There is always a problem when you put math into english (in fact, there is usually some problem with putting english into english).

Let me suggest the english interpretation:

1/2 should be said "1 part of two equal parts."


Usually, we just shorten this with a knickname: one half.




Of course, if you start with the english and then try to interpret that into mathenese, you run into even more fundamental problems, i.e. problems of meaning.

When you say "one divided by two," you have not specified the "whats." One what divided by two whats? Of course, one anything divided by two dividing boundaries will give you three pieces of your anything that, when considered as a group, return the original anything (if the boundaries are just conceptual).

I.e. One rectangle divided by two dividing lines in such a way that the dividing lines form edges of what are now three congruent rectangles, then, yes, "one divided by two," IN THIS SENSE and definition of one and two and divide, does in fact equal three thirds, but where three and thirds are in different sense than the one or the two or the meaning of divide.

I still hold that if you were to take one section of an apple divided BY two parts you would have 1/3 of the apple. If you were to take one section of an apple divided INTO two parts you would have 1/2 the apple. I still think it is a semantic debate, just a mistake in the way we convert mathematic symbols into language.

- KitNyx

Originally posted by KitNyx
I still hold that if you were to take one section of an apple divided BY two parts you would have 1/3 of the apple. If you were to take one section of an apple divided INTO two parts you would have 1/2 the apple. I still think it is a semantic debate, just a mistake in the way we convert mathematic symbols into language.

- KitNyx I see a few problems with this.

First of all, you start out with a "section of an apple" which, to me says that you don't necessarily start out with a whole apple.

Secondly, you say "divided BY two parts," but how do you define "part?"

Lastly, if the object is "divided INTO two parts," then why would you have an amount of apple any more specific than with what you started? You start with "one section of an apple." After the division you end with "1/2 the apple." Does this indicate a realization that the section was actually more specifically 1/2?

...but into two equal parts.

To parallel your picture, consider a rectangle, and now divide it into two parts. Okay, there are two halves, not a half.Semantic mistake - It should be "1 divided into 2" not "1 divided by two". Okay, again, there are two halves, not a half!The words "divided by" represent the mathematical operator in the expression:

1 / 2
Yes and 1/2 in currect mathematics is 0.5, when maybe it should be 0.3 0.3 0.3, or at least 0.5 0.5. You can't just disregard a half because it suits. Even if one divided into two is what is meant, there are still two halbves, not just one!

Hemlock, let me try to explain this in a different manner. The posters in this thread have supplied multiple answers to your question, and James R. has even given an explanation of why division is defined in its current manner, but you seem to still have difficulties understanding. To put it simply, division is defined in its current manner because it supplies a proper inverse operation to multiplication. If you "believe" in multiplication, then the way division is defined should not bother you.

The simplest way I can restate the answer to your question is by means of an example. Consider a 50% off sale. If an item is $1.00, to find out the amount one would have to pay, one computes 1.00/2 = .5. Thus one would pay $.50 for the item, not 1/3 of a dollar, or $3/3, or $2/2, but $.50. This result should appeal to one's common sense.

Finally, now that proper definitions, reasoning, motivation, and examples have been presented, any more debate on this matter becomes a concern of semantics, not Mathematics. Thus, if you continue to have a problem understanding why 1/2 = .5, then I suggest that you reread the responses in this thread, or speak with someone who is properly versed in Mathematics.

I don't want to disagree with dapthar; I just want to explicate the inverse idea:

we want to know z given that x/y = z

this is the inverse to the relationship yz = x



In terms of an operation, "dividing something by y" is the inverse of the operation of "multiplying something by y."



In terms an operator, A_y.
Let A_y(z) = x.
Let B_y(x) = z.
Therefore B_y = A_y^-1.

Consider x = 1; A_y(.) = "multiply . by 2"
We want to find z. z is the vector/function/value that satisfies the two equations above, namely, that:

"multiply z by 2" = 1

We know that z must be 0.5. Using the second equation, we can find z directly, thus division is born.

I think you're wrong Hemlock. My example shows how one divided by two equals four, and it may also be significant that you chose a rect angle (a quad(r)angle) to signify one. However you are not wrong in this because even if you chose a tri angle, or a circle, as long as the divisions create equal areas then the same answers will be given for each 'sum'.

Can you please reply to how one divided by two gives four, in the following example, because mathematics should be reason in it's purest form, and it therefore should give absolute answers. Thanks.

Mucker, well spotted and you've made some good points. The answer however is that the dividing lines cannot cross because they will therefore divided each other, i.e. in the example you have given the two dividing lines divide each other to give four lines, and a sum involving divisions of four will have to be done differently. The following example shows how a division of four will work. :)

I read 1/2 as being 1 of 2 equal parts. It is not meant to imply taking a whole and dividing it into two equal halves, which still leaves you with a whole but in two parts. It is only 1 of the two halves.