Here's one example of where the difference between rational numbers and irrational numbers matters. Consider a circle of circumference $1$ (in any units you choose), and suppose we have an ant (of infinitesimal size, of course) on the circle that moves forward by $f$ instantaneously once per second. Then the ant will return to its starting point if and only if $f$ is a rational number.
Maybe that was a little contrived. How about this instead? Consider an infinite square lattice with a chosen point $O$. Choose another point $P$ and draw the line segment $O P$. Pick an angle $\theta$ and draw a line $L$ starting from $O$ so that the angle between $L$ and $O P$ is $\theta$. Then, the line $L$ passes through a lattice point other than $O$ if and only if $\tan \theta$ is rational.
In general the difference between rational and irrational becomes most apparent when you have some kind of periodicity in space or time, as in the examples above.
I think the problem is that if you simply state those definitions exactly as they are you'll fall in the problem of not having defined the notion of "a quotient of integers". So the good and cool definition of rationals that solve all of these problems is to introduce one equivalence relation in a certain set. I don't know if you are used to equivalence relations (or even relations at all), so I'll talk about that first.
If you have two sets $A$ and $B$ you can create a relation $R$ between them which is a subset of the cartesian product $R\subset A\times B$. Think for a minute, elements of $A\times B$ are pairs $(a,b)$ with $a\in A$ and $b\in B$, so if $(a,b)\in R$ we are telling that $a$ is in some way related to $b$ and in this case we write $aRb$. Now, an equivalence relation can be introduced between a set and itself to mimic equality, it is usually denoted $\sim$ and satisfies those properties:
- $a\sim a$ (Reflexivity)
- $a \sim b \Longrightarrow b \sim a$ (Symmetry)
- $a \sim b \wedge b \sim c \Longrightarrow a\sim c$ (Transitivity)
In the third one the $\wedge$ symbol means AND. Look now that equality always obeys those three properties. So when we have a set and we want to construct a notion of the objects being equivalent without being equal we use an equivalence relation. Now, given a set $A$, an equivalence relation $\sim$ in $A$ and some element $a \in A$ the set of all other elements of $A$ equivalent to $a$ by $\sim$ is called equivalence class and denoted $\left[a\right]$. The set of all equivalence classes is called the quotient set and denoted $A/\sim$ and although the elements are sets of elements of $A$ we can usually think of $A/\sim$ as just the elements of $A$ with $\sim$ imposed on them.
Now returning to your problem! Given the set $\mathbb{Z}\times (\mathbb{Z}\setminus\{0\})$, in other words, ordered pairs of integers without any pair with $0$ in the second element we introduce the following equivalence relation $\sim$ on the set:
$$(a,b)\sim(a',b') \Longleftrightarrow ab'=a'b$$
Now stop for a while and look what we did! We are almost defining the quocient of integers. If we introduce the notation:
$$\frac{a}{b}=\left\{(a',b') \in \mathbb{Z}\times (\mathbb{Z}\setminus\{0\}) : (a',b')\sim (a,b)\right\}$$
We have exactly what the quotient is: the equivalence class of all those elements $(a,b)$ with the relation imposed. Proving this is really an equivalence relation is a good exercise. Now, look what I've said before: formally we define the quotient using equivalence classes but in practice we simply think of it as the element $(a,b) \in \mathbb{Z}\times(\mathbb{Z}\setminus\{0\})$ with the relation $\sim$ imposed.
Now, we define the set of rationals by:
$$\mathbb{Q}=(\mathbb{Z}\times(\mathbb{Z}\setminus\{0\}))/\sim$$
In other words, the set of rationals is the set of all quotients of integers, recalling that we defined the quotient as that equivalence class. With this your first definition is obviously equivalent (indeed we just made it formal using this thought) and the second is the exact same thing.
I hope this helps you somehow! Good luck!
Best Answer
You are completely right in that the notion of holes depends on what you can put between your numbers. There is always a bigger set. The integers are contained in the rationals, which are contained in the real numbers, but those themselves can be viewed to have many holes, called non-standard real numbers (e.g. infinitesimals).
In order to give the ideas of “holes” and “being without holes” a meaning, you have to define what the total set of your numbers should be. If you say you want all rationals, that is fine. But there are properties which are only fulfilled by bigger sets of numbers.
For example you can easily construct a right triangle with legs of length $1$. Then the length of the hypotenuse will not be a natural number like $1$ nor even a rational number. It is $\sqrt2$, a hole in the rational numbers. The same happens if you want to measure the circumference of a circle with rational radius.
As for the statement in your book, $\sqrt2$ and $π$ are probably numbers they want on their number line. However what exactly the set of numbers on the number line is, depends on how you define it. There is not a right or wrong way. Just more and less useful ones for doing math with.
In analysis now, there are still more things you would like to describe by numbers. For example you want to find zeros of a function. For the real numbers the following property holds:
This is not true for the rational numbers. The reason is, you can always make your interval $(a, b)$ smaller and smaller, keeping $f(a) < 0 < f(b)$, thereby closing in on the zero of $f$. That way you get a sequence of (rational or real) numbers $(a_n)_{n\inℕ}$ which is Cauchy (meaning the distance $\vert a_m-a_n\vert$ gets arbitrary small for $m$ and $n$ big enough). But only in the real numbers there will always be some limit $x$ contained in all the intervals $(a_n, b_n)$. This property is called completeness.