The number systems nest cleanly, like Russian dolls: \mathbb{N} sits inside \mathbb{Z}, which sits inside \mathbb{Q}, which sits inside \mathbb{R}, which sits inside \mathbb{C}. When you meet a new number, the question is not "is it in \mathbb{R} or in \mathbb{Q}?" — usually it is in both. The real question is: what is the smallest doll it fits in? That smallest doll tells you what the number really is.

The nesting, precisely

\mathbb{N} \;\subset\; \mathbb{Z} \;\subset\; \mathbb{Q} \;\subset\; \mathbb{R} \;\subset\; \mathbb{C}

Each arrow adds exactly what the previous layer was missing.

  • \mathbb{N} = \{1, 2, 3, \dots\} — counting.
  • \mathbb{Z} adds zero and the negatives, so you can subtract freely.
  • \mathbb{Q} adds fractions, so you can divide (except by zero).
  • \mathbb{R} adds the irrationals, so every length on a continuous scale has a name.
  • \mathbb{C} adds i = \sqrt{-1}, so every polynomial equation has a root.

Every integer can be written as a fraction (n = n/1), so \mathbb{Z} \subset \mathbb{Q}. Every rational is a real (it just has a terminating or repeating decimal). Every real is a complex number with imaginary part zero. The containments are genuine — no element of an inner set is missing from any outer set. Why: this is what the symbol \subset means. An element of the inner set is automatically an element of each outer set, by construction, not by coincidence.

The picture

pick a number:
Click any number. The ring with the bold border is the *smallest doll* that contains it, and the shaded rings are every set the number also belongs to. The naturals $1,2,3,4,5$ sit at the centre; $0$ and $-4$ live in $\mathbb{Z}$ but not $\mathbb{N}$; $3/5$ and $0.\overline{6}$ live in $\mathbb{Q}$ but not $\mathbb{Z}$; $\sqrt{2}$ and $\pi$ live in $\mathbb{R}$ but not $\mathbb{Q}$; and $2 + i$ needs $\mathbb{C}$. Note that $\sqrt{9} = 3$ lands in $\mathbb{N}$ — radicals are computations, not classifications.

Slot a number into its smallest set

The grid below lists several numbers. For each one, walk down the chain \mathbb{N} \to \mathbb{Z} \to \mathbb{Q} \to \mathbb{R} \to \mathbb{C} and stop at the first set that contains it. That is the smallest doll for that number.

The ringed dot marks the smallest set each number belongs to. $7$ is already in $\mathbb{N}$; $-4$ first appears in $\mathbb{Z}$; $\tfrac{3}{5}$ and $0.\overline{6} = \tfrac{2}{3}$ first appear in $\mathbb{Q}$ (yes — a repeating decimal is rational); $\sqrt{2}$ and $\pi$ appear only in $\mathbb{R}$; $2 + i$ needs $\mathbb{C}$. Faded dots to the right show that each number also belongs to every larger set.

Where does $\sqrt{9}$ go?

\sqrt{9} = 3. Not "some irrational under a radical" — just 3, which is a natural number. The square root sign does not change which doll a number lives in; only its numeric value does. So \sqrt{9} \in \mathbb{N}.

Compare this with \sqrt{3}, which is irrational, lives in \mathbb{R}, and is not in \mathbb{Q}. The radical symbol is a computation, not a classification. Always compute first, classify second.

"But every rational is also real, right?"

Yes. That is the whole point of \mathbb{Q} \subset \mathbb{R}. If a JEE question asks "is \tfrac{1}{2} a real number?" the answer is yes — fractions are a special kind of real number. The question that has only one right answer is "what is the smallest standard set that contains it?" and the answer there is \mathbb{Q}.

This distinction is what the Russian-doll picture trains you to see. A number has a type (its smallest doll) and a list of memberships (every doll containing its smallest one). Confusing those is the most common classification mistake in early algebra.

This satellite sits inside Number Systems.