How should I think about the relationship between numerator and denominator?

Question

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:

Which value should be in the numerator? How to think about fractions? **Updated**

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!

Kind regards!

Answer 1

The Swedish word tälja has another (obsolete) meaning which has nothing to do with wood carving, namely to count (see SAOB).

And nämna means benämna, i.e., to give a name to something.

So täljaren is “the one who counts”, while nämnaren is “the one who names”. (Exactly like the English words, numerator and denominator.)

For example, in the fraction 2/5, the denominator 5 names the thing that you are counting, namely fifths (femtedelar), while the numerator 2 tells you how many of that thing you have: two fifths.

Answer 2

If one thinks of the numerator as the number of whole things you have and the denominator as the number of parts to divide them into, the quotient expresses the size of the parts relative to the size of a single thing. This meaning is more in line with the idea of $p/q$ as the solution to a division problem, with the integer and fractional part of $p/q$ having a direct analogy to the quotient and remainder in integer division.

If one thinks of the numerator as the number of parts you have and the denominator as the number of parts that makes a whole, then the quotient expresses the fraction of a whole that the parts you have gives you. This meaning is more in line with the idea of $p/q$ as its own entity, a rational number. This interpretation doesn't map as readily to the quotient and remainder of integer division, but it does help in seeing why addition and multiplication work the way they do for rational numbers.

However, while the interpretation of $p/q$ differs between the approaches, there will be no difference in the numerical values that result. If you have $p$ wholes and want to divide it into $q$ parts, the first is saying that you divide each whole into $q$ parts, then combine one part from each whole. The second is saying to combine each whole, then divide into $q$ parts. But the end result is the same--you have $q$ equal parts that combine into $p$ wholes. This idea that two different interpretations can lead to the same underlying structure is a huge part of mathematics. There are many celebrated theorems in mathematics that essentially say "These two things we thought were different are actually fundamentally the same.", and these are celebrated because things that may be nonobvious in one interpretation can become much clearer in the other.