Real Number
A real number is any value on the continuous number line, including all rational and irrational numbers.
Definition
A real number is any number you can place on a number line, including whole numbers, fractions, decimals, and numbers like $\pi$. Formally, the real numbers $\mathbb{R}$ are the union of all rational numbers and all irrational numbers, representable as decimal expansions (terminating, repeating, or infinite non-repeating), and $\mathbb{R}$ is a complete ordered field: every bounded set of real numbers has a least upper bound (supremum), a property absent in $\mathbb{Q}$ that makes calculus and analysis work. $\mathbb{R}$ is the unique complete ordered field up to isomorphism, constructible from $\mathbb{Q}$ via Dedekind cuts or Cauchy sequences, with cardinality $c = 2^{\aleph_0}$, strictly greater than the countable infinity of $\mathbb{Q}$.
Example
$-5$, $0$, $3/4$, $1.7$, and $\pi$ are all real numbers, and every point on the number line represents exactly one of them; the set of reals in $[0, 1]$ alone contains $1/2$ (rational), $\sqrt{2}/2$ (irrational algebraic), and $\pi/4$ (irrational transcendental). The intermediate value theorem holds for $\mathbb{R}$: if $f$ is continuous on $[a,b]$ and $f(a) < 0 < f(b)$, there exists $c$ in $(a,b)$ with $f(c) = 0$, a result that fails for $\mathbb{Q}$ since $f(x) = x^2 - 2$ has no root in $\mathbb{Q}$ on $[1, 2]$.
Key Insight
The word "real" distinguishes these numbers from "imaginary" numbers (like $\sqrt{-1}$), which cannot be placed on the regular number line. Completeness, the key property that every bounded set has a supremum, is what makes analysis possible, and it shows up structurally too: every open cover of a closed bounded interval has a finite subcover (Heine-Borel), a cornerstone of analysis. Yet $\mathbb{R}$ is not algebraically closed, since $x^2 + 1 = 0$ has no real solution; the complex numbers $\mathbb{C}$ extend $\mathbb{R}$ to achieve algebraic closure, as guaranteed by the Fundamental Theorem of Algebra.