Can anybody explain what a fraction is in a way that makes sense. I will tell you what I find so confusing:
Suppose that we are in a mindset where the word "number" means "positive integer" (let's leave negative numbers and zero out out the discussion, just to keep things simpler). Then, as you rightly point out, there is no answer to the problem of dividing $1$ by $2$, since there is no "number" that you can multiply by $2$ to obtain $1$.
So if there exists some object "one half" (whatever that would be) which solves this problem, it can't be a "number" in our old sense – it must be a new kind of number. Some would say that it's intuitively clear what "one half" is: it's just the mathematical object which corresponds to our idea of cutting a pizza in half. But this argument is not completely convincing; if it were, you wouldn't be asking this question, right? ;-)
So what mathematicians have done is that they have actually constructed a larger system of numbers which visibly includes this mysterious object. For the purposes of this answer, and just for fun, I will call these new numbers "lumbers", but they are nothing but the positive fractions (also called positive rational numbers).
By definition, a "lumber" is a set of equivalent pairs of numbers, where we use the pair $(a,b)$ to represent the idea of "dividing $a$ things into $b$ pieces", and the pair $(a,b)$ is considered equivalent to the pair $(c,d)$ if "dividing $c$ things into $d$ pieces" would give the same result (i.e., if $ad=bc$; this way of expressing things is a way of saying "$a/b=c/d$" without mentioning division, which is taboo at this point, since we haven't defined division yet).
For example, if $5$ people share $3$ pizzas equally, they each get the same amount of pizza as if $10$ people share $6$ pizzas, or if $15$ people share $9$ pizzas, etc. So the pairs of numbers $(3,5)$, $(6,10)$, $(9,15)$, etc., are equivalent, and the set of all these equivalent pairs is a "lumber" which we can give the name "three fifths": $$ \text{the lumber “three fifths”} = \{ (3,5), (6,10), (9,15), \dots \}. $$ Similarly, the mysterious object "one half" is defined to be the following lumber: $$ \text{the lumber “one half”} = \{ (1,2), (2,4), (3,6), \dots \}. $$
Now when it comes to the question of dividing the number $1$ by the number $2$, there is a problem, namely that a "number" is not a "lumber", but we can fix this by saying that to each number $n$ there is a unique corresponding lumber, namely the lumber containing the pair $(n,1)$. For example, $$ \text{the number $1$} $$ corresponds to $$ \text{the lumber “one”} = \{ (1,1), (2,2), (3,3), \dots \} $$ and the $$ \text{the number $2$} $$ corresponds to $$ \text{the lumber “two”} = \{ (2,1), (4,2), (6,3), \dots \}. $$ ("If $6$ pizzas are shared among $3$ people, they get $2$ pizzas each.")
Next, we define how to multiply two lumbers $X$ and $Y$: take any pair $(a,b)$ from the set of pairs $X$ and any pair $(c,d)$ from $Y$, and let $XY$ be the lumber containing the pair $(ac,bd)$. (This is our way of saying "$\frac{a}{b} \cdot \frac{c}{d} = \frac{ac}{bd}$" without mentioning division.) Here I'm sweeping one detail under the rug; one needs to check that the definition makes sense, in that the result doesn't depend on the choice of pairs $(a,b)$ and $(c,d)$. Let's not go into that here.
Similarly, we let $X/Y$ be the lumber which contains the pair $(ad,bc)$ (which corresponds to the idea "$\frac{a/b}{c/d} = \frac{ad}{bc}$").
Now we can finally check that $$ \frac{\text{the lumber “one”}}{\text{the lumber “two”}} = \text{the lumber “one half”}, $$ which shows that in the "lumber system" we can indeed divide "one" by "two", something that we couldn't do in our old "number system". Here's how: the pair $(a,b)=(1,1)$ belongs to the lumber "one", and the pair $(c,d)=(2,1)$ belongs to the lumber "two", and the recipe for division was to form the pair $(ad,bc)=(1\cdot 1,1\cdot 2)=(1,2)$, which indeed belongs to the lumber "one half"!
Now, in practice it would be too tedious to keep making all these distinctions, so when one uses the symbol "$1$" it can mean either the number $1$ (a positive integer), or the lumber "one" (the set defined above), or many other things as well, all depending on the context. And instead of introducing some new special symbol for the lumber "one half", one simply says that it's the result of dividing $1$ (meaning the lumber "one") by $2$ (meaning the lumber "two"), and writes it as "$1/2$".
I like several of the other answers. I want to add that you have discovered something about what makes fractions hard, and hard for teachers to explain to kids. What we write as "1/2" has (at least) three separate (but related) meanings.
It represents "cut something in two pieces and take one of them". That's the way fractions are first taught and understood.
It's a "number" - the spot on the number line halfway between 0 and 1. (There's a prominent number line in almost every first and second grade classroom.)
It's the answer to the question "what do you multiply 2 by to get 1?" That's the meaning kids have to grasp when they begin to grapple with algebra.
Mathematicians and other people who have somehow gotten used to these three meanings can easily and subconsciously switch back and forth among them in any particular situation. But explaining just how these three different ways to think about "1/2" are related is subtle - and more than I will attempt here.
I hope this helps you.
The symbol $1/n$ is just only a "symbol" that represents the number $x$ defined by $ n \cdot x=1$, i.e. the inverse of $n$ in the group $(\mathbb{Q}, \cdot)$ and the existence of this number is guaranteed by the axioms defining $\mathbb{Q}$.
Formally you can construct $\mathbb{Q}$ as the set of equivalence classes of ordered pairs of integers $(m,n)$ with $n \ne 0$, where the equivalence relation is $(m_1,n_1) \sim (m_2,n_2)$ iff $m_1n_2 - m_2n_1 = 0$ and with suitable definitions of the $+$ and $\cdot$ operations as you can see in the Wikipedia page.
So the symbol $m/n$ stay for a representative element of the equivalence class $(m,n)$ in such quotient set.
Something that may help is changing how you think about 1, 2, 3...
You are not "Tony." Hopefully you're more than that. You're probably a person (though its possible you could be an AI). A more accurate wording would be "Your name is Tony."
Likewise, numbers could use a little freedom. Think of them as "the number called 1" "the number called 2" and so forth. The number is its own thing, we just use symbols like 1 and 2 to represent them. This actually becomes a big deal in programming: The number you call 10 is often called 0x0A by programmers (for reasons well beyond this question). Whether we call it 10 or 0x0A, it's the same number. Add 10+10 and you get the number we call 20. Programmers might call it 0x14, because we're confusing that way, but it's all just names. Nothing more!
We can do some division here also, we can divide 10/5 and get 2 (or more accurately, the number called 2). You can think of 10/5 as a way to construct the number called 2. 10/5 and 2 are representations of the same number -- we just call the number 2 because that's easier on our heads to give numbers just one good name. "One fifth of ten" is just as good of a name for that number, but its longer and harder to remember.
Now what about your fractions, like 1/2. They are also a way to construct a number, but it doesn't have an easy name. We just know the number constructed by 1/2 is "the number half way between zero and one," if we're confident such a number exists in the first place. It's the same number as we construct with 2/4, "the number one quarter of the way between zero and two." Now, as before, we want to make things easy on ourselves. 1/2 happens to be the simplest way to construct that particular number (2/4, 3/6, and many others also construct the same number). We declare the name of that number to be named "1/2".
There are two final questions left to answer. One is whether such a number exists in the first place. That really depends on what you're dividing. If you have a box, and are told "divide it into two equal halves," it might be a bit hard to do so. But if someone opens the box, and inside are a thousand tiny pebbles, and you are told to divide this into two equal halves, you can do it (500 pebbles in each pile). On other other hand, if King Solomon orders you to cut your baby in half to solve a squabble, you would righteously argue that "a baby divided into two equal halves" doesn't quite make sense.
Accordingly we don't always allow fractions when doing math. There are some problems restricted to "integers" (..., -3, -2, -1, 0, 1, 2, 3, ...), where 1/2 actually has no value. However, lots of problems deal with things that can be split into any sized fractions, which we call "rationals". (There are also "real" numbers, which is a very loaded name, but you'll learn about them later in your career).
The final question is "is this useful." Math is just making models. I can summon up some symbols and declare "4 $ Wobble = bleep!" and not be wrong. It's just not a very useful model. I won't be able to communicate much to other mathematicians with it.
Consider if you had a pile of 1000 pebbles. You're told to multiply that number by four. You dig deep into your bag of spare pebbles and pour them out on the table until you have 4000 pebbles. Now you are told to divide it by four, so you take away pebbles until you have 1000 pebbles left.
Now its done in the other order. You start with 1000 pebbles, and you are told to divide that number by four, and remove the rest. You comply, leaving a pile of 250 pebbles. Now you are told to multiply that number by four, so you add more pebbles to do so. You end up with a pile of 1000 pebbles.
In both cases, you end up with the same number of pebbles as you started. In symbols, 1000*4/4 = 1000 and 1000/4*4 = 1000
Now lets do the same with smaller numbers, and to make it interesting, we're going to force it to be an integer problem by using babies again. Forgive me for any graphic imagery which may follow.
You start with one baby. You are told to multiply the number of babies by 4. Fortunately, lots of your friends are of the child rearing age, so you russle up a few more babies, so that you now have 4 babies. We then tell you to divide the number of babies by 4, so you give 3 of them back to their parents, leaving you with one baby.
Now you are are told to divide the number of babies by 4. You look at the guy asking the questions really oddly, and make sure he can see how many babies are on the table. He says, "don't worry, we're going to multiply by 4 next, so you'll end up at one baby again."
For some reason, the parent who gave you the first baby really wants it back, and isn't comfortable with you having it on the table. The mother understands that, with babies (integers), 1*4/4 = 1, but 1/4*4 is NOT 1. Emphatically not 1.
This only occurred because we were using integers. Lets hand the relieved baby back to their mother, and start dividing something a little more forgiving: sticks of chewing gum.
You put one stick of chewing gum on the table. The guy asking you to do really odd things now tells you, "divide the amount of chewing gum on the table by 4. Then multiply that amount by 4. I want you to give me that much gum." Without wasting time to chop it into pieces (which you could have), you simply hand the gum over. He asks why you didn't chop anything up, so you tell him "1 stick of gum divide by four is 1/4 of a stick of gum. When I multiply 1/4 of a stick of gum by 4, I get 1 stick of gum. I'm lazy, so I just gave it to you."
Thus, for rational numbers (the integers plus the fractions), 1/4*4 = 1. Things that can be described with rational numbers can be split up in any number of ways, so its like there was a very large number of pieces, and you simply divide the pieces.
This is why mathematicians like rational numbers. They have all sorts of neat properties, like "If I name a rational number Bleep and another rational number Floop, and Floop is not the same number as zero, Bleep/Floop*Floop = Bleep." Other than making sure I didn't divide by zero, I don't actually care what Bleep and Floop were. This is really convenient for mathematicians, so we keep using rational numbers.
Disclaimer: most math with fractions is done over "real" numbers, which is these rational numbers plus another oddball set of numbers called the irrational numbers (not joking). Real numbers also have these neat properties like Bleep/Floop*Floop = Bleep, but they're a little harder to explain here. You'll find they're really intuitive to use later
The idea of quantity can be extended beyond just counting. Even as little children we knew that "one half" means one piece of a thing broken into two equal pieces. So "one half" is ALSO a quantity. The problem then is how to denote it symbolically; and a wonderfully ingenious idea is to REPRESENT it as a DIVISION that previously was not sensible. The denominator of this fraction is the total number of equal pieces and the numerator is the number of equal pieces that we are representing. This gives us a notational device that is unambiguous. The next idea is, why not extend this to all ratios of integers? This way we can represent an integer itself as the fraction integer/1 as well as ratios that CAN be divided like 4/2. This way we have a way to express ALL quantities as fractions. Unlike the counting quantities, fractions are not unique since if the numerator and denominator have common factors, these can be cancelled. Since fractions are quantities we have fraction arithmetic. For a long time people thought that now we have all possible quantities. This is why they got so upset when it was discovered that the square root of 2 CANNOT be expressed as a fraction! This meant that there are quantities that cannot be expressed as fractions! But wait; there's more! There are quantities like pi and e and gamma that cannot be expressed with roots either! Quantity is fecund (if that's the word I want). There are also two dimensional quantities called "complex numbers" and three dimensional quantities that we don't talk about because they don't have nice properties, and four dimensional quantities called quaternions, and so on.
$1/2$ is a number in the same way that $1+2$ is a number. $2+1$ can be simplified to $3$, but it's still the same number. $1/2$ cannot be simplified any more in this way, but you can choose to write it in decimal form, but this is optional. Some fractions can, like $4/2 = 2$ and $5/5 = 1$. Knowing this you can convert any fraction into a decimal by making sure the denominator is $10$: $\dfrac{1}{2}\times1 = \dfrac{1}{2}\times\dfrac{5}{5} = \dfrac{1\times5}{2\times5}=\dfrac{5}{10} = 0.5$.
Fractions are incomplete integers, and thus cannot be written as an integer. (Except in cases like $4/2$)
In that case addition $1+2=3$ doesn't either give an answer, because $3$ is just an other sign and $\displaystyle 3=\frac{3}{1}$ is also a division problem.
We got ten symbols for figures as a sort of alphabet to write the names of numbers. The naming of the fractional part may seems to be awkward but is as consequent as the integer part.
Yes $\displaystyle \frac{1}{2}=0.5=\frac{5}{10}$ and those are some ways human represent fractional numbers with figures and dot or figures and slash.
If I understand your confusion right, the answer is that $\frac{a}{b}$ represent what it should be if multiplying it by $b$ gives $a$. What is half a pizza? Some might say it is impossible to divide it into exactly half, but it shows that we do have a certain expectation of what it ought to be to qualify as "exactly half". In particular, it should be a certain part of the pizza such that doubling it perfectly gives exactly the original pizza. That concept we call "$\frac{1}{2}$", regardless of whether it is achievable in real life or not. Likewise for all other fractions.
Can you measure exactly $\frac{1}{10} \text{cm}$? No, because your ruler markings are not perfectly fine. But we can talk about $\frac{1}{10}$ anyway, and we can use our ruler and pencil to approximate it. Mathematics tells us how the exact quantity $\frac{1}{10}$ behaves, and if our approximation is close enough we expect it to have similar behaviour. For example we know from Mathematics that $\frac{1}{10} + \frac{2}{5} = \frac{1}{2}$. If we use a precise enough ruler and sharp enough pencil, we can verify for instance that $\frac{1}{10} \text{cm} + \frac{2}{5} \text{cm} \approx \frac{1}{2} \text{cm}$, so our Mathematical fractions seem to model distance measurements sufficiently well.
Now what about negative numbers like $-\frac{1}{2}$? In fact positive fractions were so sufficient for daily life in history that it took a long time for negative numbers to get accepted as real or even just useful. However, there is a natural way to understand both positive and negative fractions together.
Instead of thinking of a number as something that describes a quantity, one can think of a number as describing a process done to a certain extent, such as "$+1$" representing "move forward $1$ meter", and "$+n$" representing "do $+1$ for $n$ times", which ends up being "move forward $n$ meters".
When the process can be undone, we can then think of negative numbers as describing how much is undone. We can let "$-n$" represent "undo $+1$ for $n$ times", which ends up being "move backward $n$ meters".
Now what if we want to divide up our movement into smaller equal parts? In other words, if we move halfway, then move halfway again, we would move the original amount. Indeed we can think of "$+\frac{m}{n}$" as representing "do something that if repeated $n$ times will be the same as doing $+m$". So "$+\frac{2}{3}$" would mean "move a certain amount such that if repeated $3$ times would result in moving forward $2$ meters". It may not be possible to move exactly that amount in the real world, but all our machines are built based on the assumption that if we have a good enough approximation, we can get close enough to the desired behaviour. We don't need our wheels to be perfectly circular; being able to roll smoothly enough is good enough.
If we compare this with the pizza analogy, we can see that pizza cannot be used to explain negative numbers because pizza is not a process that can be undone! In history, $a-b$ was defined only for positive $a,b$ and only when $a$ was bigger than $b$. That works for pizza since we can't take away more pizza than there originally was, but negative numbers make it easy to describe other things like position (with respect to some landmark) or velocity (positive meaning forward and negative meaning backward).
I guess everyday examples are absolutely clear for you ($\frac 12$ apple is just a half apple), so let's focus on symbols instead.
Symbols are brilliant inventions in mathematics. Using symbols we can write down operations without actually performing them. See the following example. I couldn't write it down without the symbols +, = and $x$. (For now, let's forget the fact that digits are also symbols.)
$5+3=x$
The solution is obvious: just add up $5$ and $3$ to get $8$. But what if I change it a bit?
$x-3=5$
You can't subtract $3$ from $x$. Here comes the power of symbols: known some rules that you learnt in school you can rearrange the equotion to match the first example, where two integers stand on the two sides of a + symbol. You have also learnt how to make this pattern into the number $8$. You have learnt it so early that you take it granted.
I think you get the idea. Mathematics is about manipulating symbols and patterns of symbols. A fraction is just another pattern using the horizontal line symbol. Here is a useful rearrangement pattern involving fractions:
$\frac ab \cdot \frac cd = \frac {a \cdot c}{b \cdot d}$
Using rearrangement rules you can even eliminate a variable without knowing it's exact value. (I've been using two other rules here which I assume you will spot.)
$\frac 2y \cdot \frac y3 = \frac {2 \cdot y}{y \cdot 3} = \frac 23 \cdot \frac yy = \frac 23$
For the sake of clarity I have to note that the mentioned rules are not just conventions. New rules should be built on top of existing rules, but there is always a set of basic rules and arrangements you should accept in order to speak the language of math. (Think of the decimal numeral system and the four basic operations.)
Finally, to make use of fractions in real life situations, you need some mental model. For example before cutting an apple into halves, you will esteem where to start the cut, using your models on spheres, circles and symmetry.
It is often convenient to convert a fraction into decimal form that well aligns with our concept of scale. For example, if you have 1 kg of wheat flour and you need to take one quarter of it, you will probably use a kitchen scale to measure 0.25 kg.
[disclaimer: I wrote this answer to another question (Why are fractions the same as divisions?) but it was markes as a duplicate with this one and therefor locked, before I posted it. I haven't read all the answers in this one though, cause now I spent a lot of time and just want to post it (mostly to @Gugabd who asked the above mentioned. I think his question is more interesting and disagree with the duplicate-stamp).]
I will try to formulate something useful in words. It's just as much for my own sake, as I'm going to be a teacher, and therefor it may sound as I'm teaching for children. I found the question surprisingly tricky and very interesting. So, here we go.
"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?"
I suspect that the definition of the denominator you give here, is something used in educational purpose only, when slicing pies or pizzas. I'm wondering how you know that you actually dividing three WHOLES by 4. And what is a whole? This could be three whole fifth's, or anything.
But the definition may be useful anyway. If I interpret your question correctly, you wonder why, for example, dividing 3 whole pies by 4, gives the same result as taking the fraction 3/4, and multiplying it with 1 whole pie?
What if we instead consider the denominator being the whole? So that if you're dividing 3 by 4, then 4 is "the whole".
When you got 3 pies (sorry if the pie example feels a bit childish, I'll upgrade to choclate bars later), and dividing it by 4, then 3 is not "the whole", 4 is! If you and three friend is splitting on 3 pies, and everyone wants a whole, 3 is just enough for three of you. You split 3 pies on 4 persons. Here "the whole" is the whole gang! 3/4 of you got a pie. This is one way of seeing it.
But we could also talk about wholes in the numerator, as you did in your question. But why then should the whole be 1 pie? Isn't 3 pies the whole supply? The whole = 3. So you actually got ONE whole that should be divided, here by something larger than the whole (4>3). Which means the result is something that's smaller than the whole. But if 3 pies is 1 whole, what is 4 then?
Here is the core of the question I think. It is important here to have in mind that the value of the fraction is the same whatever "scale" it is. If you make 6 more pies you have tripled the amount of pies. But what good does that do, when 8 more hungry friend pour into your kitchen, also tripling the number of people? You get the same amount of pie per person, of course. 9/12 = 3/4. The best thing about this is that you can delete the denumerator entirely, by dividing it by itself (I'm sure the mathematics here aren't new to you, just trying to make an explanation). So when your 11 friend got their pie, they all disappeared for some reason, leaving you with your fraction. You got this when 12 persons shared 9 pies equally (or 4 persons and 3 pies) but now you're just one person left, making the denominator into a 1. Of course this doesn't chance the amount of pie on your plate, but it change the value of "the whole", from 12 into 1. So the amount of pie on your plate is obviously 9/12, but A MORE ADEQUATE WAY OF EXPRESSING THE SAME THING, would be to say that the amount of pie is (your unknown fraction of 1 pie) / (1 person). So to not magically change the amount of pie you got, you have to adapt the change of the numerator so the factor, with which it's changing, it's the same as in the denominator. So you got (9/12 pie) / (1 person). Or (3/4 pie)/1 or just 3/4 pie or three quarters of a pie. That is, you got a fraction of a pie! "3/4 pie" is by no means an inferior way of expressing it, compared to 0.75 pie. (For example 2/3 is a more exact answer than 0.666... unless you're manage to write down an infinite number of sixes.) Hope this help at least a little bit!
I don't know the exact definition of a fraction so I will instead use the term rational number. By my standards, this answer doesn't help everyone understand what a rational number really is so I'm writing this answer to give my description of a rational number.
Most mathematicians use the real number system and in it, not all numbers are rational. For any integer $p$ and nonzero integer $q$, $\frac{p}{q}$ is another way of saying $(1 \div q) \times p$. A real number is said to be rational if for some integer $p$ and nonzero integer $q$, $\frac{p}{q}$ is that number. Only the rational numbers have a fraction notation with an integer numerator and denominator that people do arithmetic on. Just because real number addition, subtraction, multiplication, and division restricted to the rational numbers is closed to the rational numbers doesn't mean irrational numbers don't exist.
You could construct only the rational numbers from the integers and use the systems that contains only rational numbers. Then you would have to accept the fact that in that system, there is no solution to $x^2 = 2$. Instead, I will construct all the real numbers first then take the subset of all the real numbers that are rational. I will construct the numbers with a terminating binary notation, the dyadic rationals then construct the real numbers from them because it's easier to construct the real numbers from the dyadic rationals than from the rational numbers. The real numbers with +, $\times$, and $\leq$ have been defined in ZF and been shown to be a complete ordered field which is unique up to isomorphism. A rational number is a real number such that for some integer $p$ and nonzero integer $q$, $p \div q$ is that number, and it can be shown that not all real numbers are rational.
To construct the real numbers, first we construct the natural numbers as finite ordinal numbers and define +, $\times$, and $\leq$ on them. Next we construct the integers and redefine +, $\times$, and $\leq$ on them. Next we construct the dyadic rationals, the numbers with a terminating notation in base 2 as follows. Each odd number $x$ is not a solution to $2 \times y = x$ in $\mathbb{Z}$ so we invent a solution to it. Let's call each newly invented number a half integer. Each half integer $y$ is still not a solution to $2 \times z = y$ so we invent a solution for it. We keep going for ever and then again redefine +, $\times$, and $\leq$ on them.
I know this is not the definition of a Dedekind cut but I will define a Dedekind cut of the dyadic rationals to be a subset of the dyadic rationals such that it is nonempty and its complement is nonempty and for every dyadic rational in the subset, all smaller dyadic rationals are in the subset. Some Dedekind cuts have a maximal element. Some Dedekind cuts have a complement with a minimal element. Some Dedekind cuts have no maximal element nor does their complement have a minimal element. For each Dedekind cut that has no maximal element nor does its complement have a minimal element, we can invent a number that's larger than all elements of the Dedekind cut and smaller than all elements of its complement to get the real numbers. Again, we can redefine +, $\times$, and $\leq$ on them in the intuitive way and show that it is a complete ordered field which is unique up to isomorphism and therefore satisfies the following properties, and I'm pretty sure that also everything that satisfies the following properties is a complete ordered field:
Let's define a real number $x$ to be rational if $\exists p \in \mathbb{Z}\exists q \in \mathbb{Z} - \text{{0}}$ such that $\frac{p}{q} = x$. It can be shown that for any integer $p$ and nonzero integer $q$, $\frac{p}{q} = p \div q$ as follows. $\frac{p}{q} = (1 \div q) \times p = ((1 \div q) \times p) \times (q \div q) = (1 \div q) \times (p \times (q \div q)) = (1 \div q) \times ((p \times q) \div q) = (1 \div q) \times ((q \times p) \div q) = ((1 \div q) \times (q \times p)) \div q = (((1 \div q) \times q) \times p) \div q = ((1 \div (q \times q)) \times p) \div q = (1 \times p) \div q = p \div q$. For any real numbers $x$ and $y$, we can define $x - y$ to mean $x + (-y)$ and if $y$ is nonzero, we can also define $x \div y$ to mean $x \times y^{-1}$. It can be shown that the rational numbers with +, $\times$, and $\leq$ satisfy all the properties but the last one which also shows that any rational number divided by any nonzero rational number is a rational number. However, given two numbers written in the form $\frac{p}{q}$ where $p$ is an integer and $q$ is a nonzero integer, to express their sum, difference, product, quotient in the same form or compute their inequality, all you need to know is how to perform those operations with integers and laws 4, 5, 6, 10, 11, 12, 13, and 16. Adding is done as follows $\frac{a}{b} + \frac{c}{d} = (a \div b) + (c \div d) = ((a \times d) \div (b \times d)) + ((c \times b) \div (b \times d)) = ((a \times d) + (c \times b)) \div (b \times d) = \frac{ad + cb}{bd}$ when $\frac{a}{b}$ and $\frac{c}{d}$ are defined. Multiplying is done as follows $\frac{a}{b} \times \frac{c}{d} = (a \div b) \times (c \div d) = (a \times c) \div (b \times d) = \frac{ac}{bd}$ when $\frac{a}{b}$ and $\frac{c}{d}$ are defined. Division is done as follows $\frac{a}{b} \div \frac{c}{d} = (a \div b) \div (c \div d) = (a \times d) \div (b \times c) = \frac{ad}{bc}$ when $\frac{a}{b} \div \frac{c}{d}$ is defined and for some integers $a$, $b$, $c$, and $d$, $\frac{ad}{bc}$ is defined even when $\frac{a}{b} \div \frac{c}{d}$ is not defined. For inequality, this is how it is done when both denominators are positive $\frac{a}{b} \leq \frac{c}{d} \equiv (a \div b) \leq (c \div d) \equiv (a \times d) \div (b \times d) \leq (b \times c) \div (b \times d) \equiv a \times d \leq b \times c$. When some of the denominators are negative and neither of the denominators are zero, all you have to do is negate both the numerator and denominator of each fraction notation with a negative denominator then compute the inequality the way you do for positive denominators.
Source: https://en.wikipedia.org/wiki/Construction_of_the_real_numbers
For any ordered pair of integers that are not both zero, let's define the greatest common factor to be the smallest positive integer such that both of those numbers are an integral multiple of it. There is an easy way to compute the greatest common factor of any ordered pair of integers that are not both zero. For any positive integers x and y, if y > x, use the fact gcf(x, y) = gcf(x, y mod x). If x > y, use the fact gcf(x, y) = gcf(x mod y, y). Keep going until one of the coordinates is 0. For any ordered pair of integers whose second coordinate is nonzero, there is exactly one ordered triplet of integers x, y, and z such that gcf(x, y) = 1; y is positive; and (xz, yz) is that ordered pair. For that reason, we can define for any integer a, nonzero integer b, integer c, and nonzero integer d, $\frac{a}{b}$ and $\frac{c}{d}$ to represent the same rational number when the way to express (a, b) in that way and the way to expresses (c, d) in that way have the exact same ordered pair in that way of expressing it. Not only that but when you find integers x, y, and z such that gcf(x, y) = 1; y is positive; and (xz, yz) is that ordered pair, it turns out that z = gcf(xz, yz). That gives an easy way to find the way to express any ordered pair of integers whose second coordinate is nonzero in that way. We can define addition of rational numbers as follows. $\frac{a}{b} + \frac{c}{d} = \frac{ad + cb}{bd}$ when b and d are both nonzero. That's because when ever you change each of the pairs in the expression to one in the same class, the result always ends up being in the same class. Yes, doing it that way also works and is much easier to learn than the method of using the common denominator. We can also define multiplication of rational numbers as follows $\frac{a}{b} \times \frac{c}{d} = \frac{ac}{bd}$ when b and d are both nonzero. That's because when ever you change each of the terms to one in the same class, the result will always end up in the same class. According to this definition, an integer is not actually a rational number. However, we can decide that for any rational number that can be expressed as $\frac{x}{1}$, that rational number is equivalent to the integer x.
It can be shown that using this definition, the following statements all hold:
Subtraction can then be defined as addition of the additive inverse and division can be defined as multiplication of the multiplicative inverse. It can also be shown that for any integer a and nonzero integer b, $\frac{a}{b} = a \div b$. Be careful to realize that it is not correct to say that $\frac{a}{b} \div \frac{c}{d} = \frac{ad}{bc}$ even when bc is nonzero if d is zero because $\frac{c}{d}$ is still undefined.
Alternatively, you can construct the dyadic rational numbers, the numbers with a terminating notation in binary, from the integers then construct the real numbers from the Dedekind cuts of the dyadic rational numbers and then define +, $\times$, and $\leq$ in the intuitive way and show that that system is a complete ordered field. Again we can define - and $\div$ in terms of + and $\times$ the same way. Then you can define that for any integer a and nonzero integer b, $\frac{a}{b}$ is another way of saying $a \div b$ and say that a real number is rational when it can be expressed as $\frac{a}{b}$ for some integer a and nonzero integer b. It can be shown that addition and multiplication of rational numbers given in this notation will still get you the right answer if you do it the same way.
Conversely, if you take for granted that the real number system is a complete ordered field and decide that by definition, for any integer a and nonzero integer b, $\frac{a}{b} = (1 \div b) \times a$, and a rational number is a real number that can be expressed as $\frac{a}{b}$ for some integer a and nonzero integer b, you can show that addition and multiplication of rational numbers is closed to the rational numbers and adding and multiplying rational numbers given in that notation can be done the exact same way as I already described.
Some authors define the fractional notation differently and define it as meaning the same thing as division, instead of meaning division just restricted to when the numerator is an integer and the denominator is a nonzero integer. For example for the rule for differentiating a quotient of functions, they write $\frac{d}{dx}(\frac{f}{g}) = \frac{f'g - fg'}{g^2}$ and not $\frac{d}{dx}(f \div g) = (f'g - fg') \div g^2$. $\frac{d}{dx}$ actually has the meaning of being an operation on functions rather than a number or a function of $x$ so $\frac{d}{dx}(f \div g)$ does not mean $(\frac{d}{dx}) \times (f \div g)$. This is actually a third meaning of the fractional notation.
