Explain $\Bbb Q$ is a dense set

Question

This is approximately my explanation for 14-year-old students in saying that $\Bbb Q$ is a dense set.

In contrast to the set $\Bbb N$ and the set $\Bbb Z$, between any two rationals another rational is always included, and thus we can say that between two rationals infinite rationals are included.

For example, let us put the numbers $0$ and $1$ on the straight line. Now let us denote on the line a rational number between $0$ and $1$, for example, their half.

Now we indicate on the line a rational number between $0$ and $\frac 12$, for example, their half. I will obtain the sequence $0, \frac 14, \frac 12$. Now we indicate on the line a rational number between $0$ and $\frac 14$, for example, their half $\frac 18$.

Considering to always divide by $\dfrac{1}{2^n}$ with $n\in\Bbb N$ the fraction $\dfrac{1}{2^n}$ with the large $n$ the points in the subdivision will all accumulate toward $0$ (zero becomes an accumulation point).

For this reason we can say that the set $\Bbb Q$ is a DENSE set.

By this expression we mean an ordered set in which, given any INTERVAL, THERE IS AT LEAST ONE ELEMENT INSIDE it.

Is there another easy explanation to give my students or is the one I have given enough?

Answer 1

Definition: A set $S$ is order-dense* if and only if

$$\forall x_{\in S} \forall y_{\in S} (x < y \to \exists z_{\in S} (x < z \land z < y))$$

That is, if $x < y$, there is some $z$ between them.

Theorem: $\mathbb{Q}$ is dense.

Proof: Let $x, y \in \mathbb{Q}$. Suppose $x < y$. $\mathbb{Q}$ is closed under addition and division, so $z = \frac{x + y}{2} \in \mathbb{Q}$. Furthermore, \begin{align} x &= \frac{2x}{2} \\ &= \frac{x + x}{2} \\ &< \frac{x + y}{2} & x < y \\ &= z \end{align} and \begin{align} z &= \frac{x + y}{2} \\ &< \frac{y + y}{2} & x < y \\ &= \frac{2y}{2} \\ &= y \end{align} $\Box$

* Not to be confused with topologically dense.

Answer 2

I would slightly modify your explanation by adding one preliminary step and one final step.

Step 1: Motivate the definition of "dense" which is jargon but students already have a sense of what it is.

Let $X$ be a collection of points that can be (totally) ordered: for all $x,y$ in $X$, either $x<y$ or $y<x$ or $x=y$. For example, $X$ may look like $$x_1<x_2<x_3,\qquad X=\{x_1,x_2,x_3\}$$ or $$x_1<x_2<\cdots<x_{999}<\cdots<x_{n}<y,\qquad X=\{y,x_1,x_2,\ldots\}$$ or $$x_1<x_2<\cdots<x_{999}<\cdots<x_{n}<y<z,\qquad X=\{z,y,x_1,x_2,\ldots\}$$

Henceforth assume $X$ is an infinite collection.

Say $X$ is dense if for each $y$ in $X$, there exists an infinite sequence of points $(x_1,x_2,\ldots)$ such that the points $x_n$ never coincide with $y$ (i.e. $x_n\ne y$) and either $$x_1<x_2<\cdots<y\qquad\text{i.e.}\qquad \begin{cases}m<n\Rightarrow x_m<x_n\\n=0,1,2,\ldots\Rightarrow(x_n<y)\end{cases}$$ OR $$y<\cdots<x_2<x_1\qquad\text{i.e.}\qquad \begin{cases}m<n\Rightarrow x_n<x_m\\n=0,1,2,\ldots\Rightarrow(y<x_n)\end{cases}$$ I recommend having a chalkboard for this previous part, and I recommend saying "the 'x-sub-n'approach y either from the left or from the right." Also, for the children, please do not use logic notation such as $\wedge$ or $\exists$.

Step 2: [insert your explanation that $\mathbb{Q}$ is dense]

Step 3: Remark that there is a simpler and equivalent stricter definition of dense-ness.

Say $X$ is dense if for all $x,y$ in $X$, if $x\ne y$, then there exists $z$ in $X$ such that either $x<z<y$ or $y<z<x$. [justify the equivalence with intuition learned from the explanation in Step 2]

Let the students ponder why this definition is "stricter" (I almost missed it entirely before logging off MSE, for example)