I've been thinking lots about the nature of fractions; of what exactly they are and what they tell you. I searched for a similar question as the one I have about fractions on this fourm, and found this thread, which I think summarizes my pondering of what value should be represented as the numerator and denominator in a fraction:
Seeing as I'm a dunce, I'm not quite satisfied with the provided answer, since I want to get a little bit more nitty gritty is how one should think about the numerator and denominator specifically.
I'm swedish, and in the swedish language the numerator is commonly called "täljare", and the denominator "nämnare". I suppose many swedish readers might disapprove, but since I was a child I always thought of "täljare" as something you "cut", since the word "täljning" simply means wood carving in english. Since I was a rather clever boy, I never quite bothered to ponder on the nature of the word "nämnare", but seeing as it's called "denominator" in english I guess it becomes more understandable, since the word "denominator" is derived from the fact that the number represented in the denominator is the one that denominates the fraction, i.e. which tells us what kind of fraction we're dealing with, or what kind of number that represents "the whole".
This is where I find myself in trouble. I've always thought of the numerator as the number you "cut" into pieces, and the number of pieces you "cut" the numerator is told in the denominator. Like woodcarving, you know.
1 ÷ 2 = 0.5, or "1 cut into 2 gives us two halves."
Now, having explored some math in my adult years once again, I find that this isn't the proper way to think of it at all.
I'm being told that the numerator represents "the parts", the denominator represents "the whole" and the quotient represents "the proportion".
When we do long division by hand, what we're doing is essentially to see how many times we can "squeese" the denominator into the numerator, like this:
Division is the opposite of multiplication, and multiplication is a more convenient and neater way of expressing repeated addition. Hence, division could be seen as a way of expressing repeated subtraction, which we can see when we do long divisions, since in the above example we can express the process of 432 ÷ 15 as "how many times can we subtract 15 from 432?".
This messes with my perception of the numerator as that which you "cut" (you know, with the täljare = täljning = woodcarving). Rather, I've found that I should think of it as me seeing how many times I can "squeeze" the denominator into the numerator. This in turn leaves me a little dumbfounded of how exactly I could explain what the quotient is. Sure, it's "the proportion", but what does that mean, exactly?
What is the numerator and denominator, exactly, and how should I think of their relationship? What would be a simplier way of explaining what "the proportion" expressed in the quotient really is?
I'm sorry for asking writing such a long message asking about something that is no doubt fiendishly simple for you guys, but this has really bothered me for some time. I'd be very thankful if you could try and help me out a little!