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.

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

The number systems nested as concentric rectanglesFive concentric rounded rectangles. The outermost is labelled complex numbers C, containing i and 2 plus 3 i. The next rectangle in is labelled real numbers R, containing square root of two, pi, and e. Next is rationals Q, containing one half, minus seven over three, and zero point two five. Next is integers Z, containing minus two and minus seven. At the centre is naturals N, containing one, two, three, four, and five, with zero shown just outside the naturals but inside the integers.ℂ — Complex numbersℝ — Real numbersℚ — Rationalsℤ — Integersℕ — Naturals1 2 3 4 50 −2 −71/2 −7/3 0.25√2 π ei 2 + 3i
Each ring labels the layer that admits the examples in its band but were *not* allowed in the ring inside it. The naturals $1,2,3,4,5$ sit at the centre. Zero and the negatives live in $\mathbb{Z}$ but not $\mathbb{N}$. Fractions live in $\mathbb{Q}$ but not $\mathbb{Z}$. Irrationals live in $\mathbb{R}$ but not $\mathbb{Q}$. And $i, 2 + 3i$ live in $\mathbb{C}$ but not $\mathbb{R}$.

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.

A table showing which smallest set contains each numberA five-column table. The left column lists numbers: seven, minus four, three over five, zero point six repeating, square root of two, pi, and two plus i. The remaining columns are labelled N, Z, Q, R, C. Each row shows which set first contains the number by highlighting the cell.numberℕℤℚℝℂ7−43/50.6̄√2π2 + i
Each red dot marks the smallest set a 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 $\mathbb{C}$. But every dot also belongs to every set to the *right* of it — that is the whole point of nesting.

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.