Ever since I learned about fractions in Elementary School, I've known how to work with them. The problem is, although I remember the mathematical rules, I don't remember how I assimilated the equality between fractions and divisions when I was younger. This is really troubling to my mind because it feels like I partly understand this concept. I know that the denominator represents the amount of parts a whole was divided into and that the numerator represents the amount of parts I'm working with. Let's say, I want to divide 3 by 4. The result of this division is 3/4. But, in this case, I'm dividing 3 wholes by 4. Doesn't this come into conflict with the definition of the denominator, which is the amount of parts in which 1 whole is divided into? This confusion is really bugging me and I'd really appreciate if someone could clear it up. Thanks for the attention.
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2$\begingroup$ The question is pretty well posed, and makes great sense. I don't understand the downvote. $\endgroup$– Arnaud MortierCommented Mar 24, 2018 at 15:47
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2$\begingroup$ Possibly helpful: math.stackexchange.com/questions/1127483/… $\endgroup$– Ethan BolkerCommented Mar 24, 2018 at 15:48
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3$\begingroup$ Let's add ratios (e.g., 3:4) to this list of mathematical expressions that turn out to be equal to each other. $\endgroup$– Syntax JunkieCommented Mar 24, 2018 at 19:13
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1$\begingroup$ The really fundamental confusion here is this: say I say "a third". It's really only meaningful as a third of something. A third of a pie, a third of a house, a third of one hundred, etc etc. However, by convention, we can just write "one third" (1 over 3) on its own, and in fact by convention that means one third "of one". It's actually very confusing, very stupid, and very strange, but it makes sense once you see it that way and accept it for what it is! $\endgroup$– FattieCommented Mar 24, 2018 at 22:23
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5$\begingroup$ @Fattie This is no more or less fundamental than natural numbers. Most people have no problem with expressions such as "three" without specifying three pies, three houses, three hundred, etc etc. There is however some confusion about how fractions relate to division, or perhaps, as Henry's answer illustrates, the fact that division distributes over addition. $\endgroup$– ServaesCommented Mar 24, 2018 at 22:34
6 Answers
You are right that there are two things going on here, and it doesn't seem obvious that they are the same.
On the one hand you have $3$ units and you take a fourth of that (that's dividing $3$ by $4$).
On the other hand you have a unit, cut it in four parts and keep only three of these parts (that's the fraction "three fourths").
The reason why they are the same is that you can achieve the first operation by cutting each of your three units into four parts, and take one part from each. By doing that, you took a fourth of your three units, and at the same time what you have in your hands is three "fourths of a unit".
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6$\begingroup$ This explanation and the visual from @henry should be combined $\endgroup$– MicFinCommented Mar 24, 2018 at 21:34
It should not be too difficult to convince yourself that the sum of the pink areas in the single circle in the top row is the same as the sum of the pink areas in the three circles in the bottom row. Both are three quarters: the first as three quarters of a single circle, and the second as a one quarter of three circles
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$\begingroup$ +1 This clearly illustrates that division distributes over addition, which seems to be at the heart of the question. $\endgroup$– ServaesCommented Mar 24, 2018 at 22:38
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8$\begingroup$ @Servaes Well, if we're going to describe it that way, we might as well write out what it means in this case. $$\frac{3}{4}=\frac{1+1+1}{4}=\frac{1}{4}+\frac{1}{4}+\frac{1}{4}$$ $\endgroup$ Commented Mar 25, 2018 at 1:58
Any definition of a fraction should agree with the common ways of looking at a fraction. The three most common ways are:
- Break an interval that is three units long into four equal pieces and take one of them:
- Break a unit interval into four equal pieces and take three of them.
- Divide four into three.
Related to the fraction $\dfrac 34$ is the ratio $3:4$. Two segments are said to have a ratio of $3:4$ if there is a common measure that can be "laid out" exactly 3 times onto the first segment and exactly 4 times onto the second segment.
The idea of a common measure is more primitive than the idea of a fraction. It was once thought that any two line segments had a common measure (were commensurable). We now know that isn't true. A side of a square and a diagonal of that square are not commensurable for example.
It would take more time than I'm willing to spend to show formally how to create the set, $\mathbb Q$, of rational numbers. The important properties are
Q1. $\dfrac ab = \dfrac cd$ if and only if $ad= bc$.
Q2. $\dfrac ab + \dfrac cd = \dfrac{ad+bc}{cd}$.
Q3. $\dfrac ab \cdot \dfrac cd = \dfrac{ac}{bd}$.
How do we justify property Q1? Assume for the moment that rational numbers behave the way we would like them to. If $n$ if a non zero integer, then we want $\dfrac nn$ to act like the number $1$. If that is the case, then we must have $\dfrac ab = \dfrac ab \dfrac nn = \dfrac{an}{bn}$. If we let $c=an$ and $d=bn$ then we find that $ad = bc$. Also we want $\dfrac 0n$ to act like $0$. In which case
\begin{align} \dfrac ab = \dfrac cd &\implies \dfrac{ad}{cd} - \dfrac{bc}{bd} = \dfrac{0}{cd} \\ &\implies \dfrac{ad-bc}{cd} = \dfrac{0}{cd} \\ &\implies ad-bc = 0 \\ &\implies ad = bc \end{align}
Finally, we want to embed the rational numbers $\mathbb Q$ into the set of real numbers $\mathbb R$. The simplest way is to just accept what we want to be true.
R1. For $x,y \in \mathbb R$ with $y \ne 0$ we define $\dfrac xy = x \cdot y^{-1}$.
Note that a lot of books define $y^{-1} = \dfrac 1y$. It isn't too hard to check that Q1, Q2, and Q3 are still true. So R1 is a logically consistent way to embed the rational number into the real numbers.
(9/24/2020) In response to @user599310's question.
You can "see" this for yourself with a sheet of lined paper and another sheet of paper (thin enough to see the lines on the other sheet of paper through it) with a segment drawn on it.
To break that segment into, say, $5$ equal pieces, select any $6$ consecutive lines on the lined sheet of paper and think of them as being numbered $0, 1,2,3,4$ and $5$.
Now place the sheet of paper with the segment on it so that one end of the segment touches line number $0$ and the other end of the segment touches line number $5$. (Such a thing can be constructed using a straight edge and a compass.)
Now the lines $1$ through $4$ break the segment into $5$ equal pieces.
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$\begingroup$ How can we prove that we can break any interval in an arbitrary amount of equal pieces such as fifths, thirds etc? $\endgroup$– ado sarCommented Sep 24, 2020 at 16:48
Several good answers already, but there is another way to look at it.
The actual, formal definition of a concept like this is going to be based on axioms. Those basically just define equality ($\frac{a}{b} = \frac{c}{d}$ if $ad = bc$), addition, subtraction, multiplication and division in terms of the natural numbers. Then we prove that we haven't made arithmetic contradict itself just now, if it hadn't already, by extending it from whole numbers to fractions. We also want to prove that our operations have the properties that most theorems rely on, such as the order in which we add numbers not mattering.
But we teach kids about fractions (or rational numbers) and consider the octonions an interesting curiosity that you'll learn about if you get a degree in math because the fractions are useful in daily life. And complex numbers and quaternions are in between because we need them to solve certain problems in engineering.
So, on one level, the answer might be something like: $3 \div 4$ is $\frac{3}{1} \times \frac{1}{4} = \frac{3}{4}$. But you knew that already. If some set of things corresponds to fractions, that relationship will hold true. Otherwise, you'd have shown that the things that contradict our theorems about fractions are not like fractions. If you're really asking why that corresponds to a couple different things in the real world, a geometric proof like Henry's is a great approach.
A fraction is a single number. A division a binary operation (an operation with two input numbers) that yields a single number; this result can be represented as fraction. So, basically division and fraction are related to each other, but they are not the same.
Three over four is literally another way of writing "thre divided by four", which is 0.75.
When you see three over four "on its own" that means "0.75 'of one' ", which is of course 0.75.
Three over four, times four is of course three. Three over four times a hundred is of course 75. Three over four times a New York pizza is of course six slices of pizza. Three over four of your income is what the tax man takes. And so on.
If you see three over four "on it's own", it's just a convention that it means "times one".
" I'm dividing 3 wholes by 4. Doesn't this come into conflict with the definition of the denominator, which is the amount of parts in which 1 whole is divided into?..."
Where you wrote "1 whole" just above - near the end - you meant "the whole thing"...
That is to say "the whole thing in question"...
This is completely non-mysterious: in English it's normal to say "one whole thing" where the "one whole thing" might be "a swimming pool", "the USA", "1", "88", "the color blue," or whatever.
(The English phrase "one whole thing" happens to have a the word "one" in it. Much as one may say, "One must carefully brush one's hair before meeting one's grandmother." It has no connection to "1.0".)
The "whole thing" here is indeed ..... "3". That's all there is to it.
Just as you say, the denominator ("4.0" here) is what you divide in to the "whole thing". Here the "whole thing" is 3.
That's all you're seeing there!
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3$\begingroup$ The two uses of "one" in your parenthesised remark are fundamentally different, and the first is very much connected to the number one. $\endgroup$– ServaesCommented Mar 24, 2018 at 22:39