Number sets N, Z, Q, R

The numbers we work with are not one big family. Counting items needs only the natural numbers, frost and debts need the integers, splitting a pizza needs the rationals, and the diagonal of a square escapes all three. Each family has a name, a symbol and a place in a chain where every set contains the one before it.

The four sets

The natural numbers $\mathbb{N}$ are the numbers you count items with — no sign, no fractional part:

The integers $\mathbb{Z}$ add their opposites, which is the whole left half of the line:

The rational numbers $\mathbb{Q}$ are everything writable as a fraction with an integer numerator and a non-zero integer denominator:

More numbers are rational than one expects: every integer ($5 = \tfrac{5}{1}$), every terminating decimal ($0.25 = \tfrac{1}{4}$) and every repeating one ($0.(3) = \tfrac{1}{3}$).

The real numbers $\mathbb{R}$ are every number sitting on the number line — the rationals plus the ones no fraction can write, the irrationals: $\sqrt{2}$, $\sqrt{3}$, $\pi$.

The families do not sit side by side — they nest:

So the number $3$ belongs to all four sets at once. When a question asks which set a number belongs to, it usually means the smallest one — because that is the one that says the most about it.

The symbols ∈, ∉, ⊂ and ∅

Four signs are enough to write every statement about membership:

• $x \in A$ — the number $x$ belongs to the set $A$: $-5 \in \mathbb{Z}$,
• $x \notin A$ — the number $x$ does not belong to $A$: $-5 \notin \mathbb{N}$,
• $A \subset B$ — the set $A$ is contained in the set $B$: $\mathbb{Z} \subset \mathbb{Q}$,
• $\varnothing$ — the empty set, a set with no element at all.

The empty set is not decoration: the equation $x^2 = -1$ has no solutions among the real numbers, so its solution set is $\varnothing$. Mind the difference: $\{0\}$ has one element, $\varnothing$ has none.

Writing a number in binary or hexadecimal changes none of this: $1011_{(2)}$ and $11$ are the same integer in two notations — the number-system calculator converts the notation, not the value, which is exactly why it takes integers only.

Why √2 is not a fraction

That $\sqrt{2} \approx 1.41$ follows from arithmetic. That no fraction gives that number exactly has to be proved — and the proof is short. It goes by contradiction: we assume the opposite of the claim and run it into an impossibility.

Suppose $\sqrt{2}$ is rational, that is

where the fraction $\tfrac{p}{q}$ is in lowest terms (every fraction reduces to such a form). Square both sides:

Since $p^2$ is even, $p$ must be even too — the square of an odd number is odd, which we show below. So write $p = 2k$:

By the same reasoning $q$ is even as well. But a fraction with both numbers even reduces by $2$ — and we assumed it was already in lowest terms. Contradiction. The assumption was therefore false:

The same proof works for $\sqrt{3}$, $\sqrt{5}$ and the root of any number that is not a perfect square.

Divisibility proofs

The proof above rested on one move that recurs across all of mathematics: a number of known divisibility gets written as a formula. If $k \in \mathbb{Z}$, then

so every integer is either even or odd — there is no third option. In the same way, a multiple of $3$ is $3k$, and a number leaving remainder $1$ on division by $3$ is $3k + 1$.

With that notation, proofs are done by calculation rather than by examples:

The same move shows that the sum of two odd numbers is even: $(2k + 1) + (2m + 1) = 2(k + m + 1)$, which is divisible by $2$.

Common mistakes

• Treating every root as irrational — $\sqrt{9} = 3$ is a natural number. What is irrational is the root of a number that is not a perfect square.
• Excluding integers from the rationals — $5$ is $\tfrac{5}{1}$, so $5 \in \mathbb{Q}$. The sets nest; they do not exclude one another.
• Confusing $\varnothing$ with zero — the empty set has no elements, while $\{0\}$ has one.
• Calling $-3$ a natural number — natural numbers carry no minus sign; $-3$ starts only at $\mathbb{Z}$.
• Proving by example — checking a claim on five numbers is not a proof. A proof speaks about every number at once, which is why it writes them as $2k$ or $2k + 1$.