Foundations — Real number system — ℕ ⊂ ℤ ⊂ ℚ ⊂ ℝ

Everything in the parent topic leans on a handful of tiny symbols. Below, each one gets: plain words → the picture → why the topic needs it, ordered so every symbol is earned before it is used.

1. A dot on a line — the number line

Before any symbol, the picture. Draw one straight, endless line. Pick a spot, call it $0$. Step right some fixed distance, call that $1$. Keep stepping. Every number we will ever meet is a specific point on this line.

Why the topic needs it: the whole parent note is about which points on this line each number-set covers. "Gaps", "dense", "no holes" — all of these are statements about dots on this one picture.

2. The counting numbers and the symbol $\mathbb{N}$

Start where humans started: counting objects. $1,2,3,4,\dots$ — the marks you make when you tally sheep.

Picture: a row of dots on the number line, starting at $1$, marching right forever, skipping $0$ and everything to its left.

Why the topic needs it: $\mathbb{N}$ is the smallest bag — the seed. Every later number-set is built by adding new points to fix a problem $\mathbb{N}$ cannot solve.

3. "Is inside" — the symbol $\in$

We need a word for "this dot is one of the dots in that bag."

Picture: point at a single dot, then point at the bag — $\in$ is the arrow saying "that dot belongs here."

Why the topic needs it: every classification sentence ("$0\in \mathbb{Z}$", "$\sqrt{2}\notin \mathbb{Q}$") is built from this one symbol.

4. Bags inside bags — subset $\subseteq$ and proper subset $\subset$

Now compare whole bags, not single dots.

Picture: nested loops, like Russian dolls. $\mathbb{N}$ is the innermost loop, $\mathbb{Z}$ a bigger loop around it, then $\mathbb{Q}$, then $\mathbb{R}$ — each loop containing everything inside it and more.

Why the topic needs it: the entire spine of the chapter is the single line $\mathbb{N}\subset \mathbb{Z}\subset \mathbb{Q}\subset \mathbb{R}$. It says "four nested dolls, each strictly bigger than the last."

5. Zero, negatives, and the symbol $\mathbb{Z}$

Counting can't answer "I owe you 3 sheep." So we grow the bag leftward past $0$.

Picture: the number line with dots at every whole step in both directions — a mirror placed at $0$.

Why the topic needs it: $\mathbb{Z}$ is the first growth. It exists precisely because $3-5$ has no answer inside $\mathbb{N}$. And clearly $\mathbb{N}\subset \mathbb{Z}$: every counting dot is still there, but $\mathbb{Z}$ adds $0$ and the negatives.

6. Ratios, division, and the symbol $\mathbb{Q}$

Integers can't split one pizza among three people. So we fill in the spaces between the whole steps.

Picture: the number line now speckled with dots between the integers — halves, thirds, quarters — densely sprinkled.

Why the topic needs it: $\mathbb{Q}$ is the third doll, and $\mathbb{Z}\subset \mathbb{Q}$ because any integer $n$ is the fraction $\frac{n}{1}$ — but $\frac{1}{2}$ shows $\mathbb{Q}$ has strictly more.

7. The whole line, holes, and the symbols $\mathbb{R}$ and $\mathbb{R}\setminus \mathbb{Q}$

Even packed with fractions, the line still has pinprick holes. The length of a square's diagonal (which the parent proves is $\sqrt{2}$) lands on one of these holes — no fraction sits exactly there.

First we need a name for the holes themselves.

Picture: the number line drawn as a solid, unbroken line — every conceivable point filled.

Why the topic needs it: $\mathbb{R}$ is the outermost doll. $\mathbb{Q}\subset \mathbb{R}$ is proper because $\sqrt{2}\in \mathbb{R}\setminus \mathbb{Q}$, i.e. $\sqrt{2}\in \mathbb{R}$ yet $\sqrt{2}\notin \mathbb{Q}$. See Decimal Representations and Dedekind Cuts for two ways to build these missing points.

8. Small operator symbols you'll meet

Why the topic needs it: the equality rule $\frac{p}{q}=\frac{r}{s}⟺ps=qr$, the density step $a<\frac{a+b}{2}<b$, the irrationals $\mathbb{R}\setminus \mathbb{Q}$, and the $\sqrt{2}$ proof (which uses "lowest terms", i.e. $\gcd (p,q)=1$) all rely on these.

9. Two sizes of infinity — ${\aleph }_0$ and $2^{{\aleph }_0}$

The parent claims some infinite bags are "the same size" and others "bigger". We need the vocabulary.

Picture: ${\aleph }_0$ = an endless numbered queue; $2^{{\aleph }_0}$ = a crowd so dense no queue can enumerate it.

Why the topic needs it: these are the punchlines of Cardinality and Infinity and Cantor's Diagonal Argument. The subtlety $2^{{\aleph }_0}={\aleph }_1$? — is the Continuum Hypothesis, deliberately left open.

Prerequisite map

How the foundations feed the topic, from the bottom up:

• The number line (a dot = a number) is the ground everything stands on.
• On it, $\mathbb{N}$ (counting numbers) is the seed set; $\in$ (is-inside) and $\subseteq /\subset$ (bag-inside-bag) are the two comparison tools.
• Adding $0$ and negatives grows $\mathbb{N}$ into $\mathbb{Z}$; allowing ratios $\frac{p}{q}$ grows $\mathbb{Z}$ into $\mathbb{Q}$; filling the holes grows $\mathbb{Q}$ into $\mathbb{R}$, with $\mathbb{R}\setminus \mathbb{Q}$ naming those holes.
• The small operators ($<,=,⟺,\setminus ,\gcd$) and the two infinity sizes (${\aleph }_0,2^{{\aleph }_0}$) are the extra tools the parent's proofs and size-claims lean on.
• All of these together support the spine: $\mathbb{N}\subset \mathbb{Z}\subset \mathbb{Q}\subset \mathbb{R}$.

Equipment checklist

Cover the right side; can you answer each before revealing?