Real Number
ArithmeticA 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. Real numbers include whole numbers, fractions, decimals, and numbers like $\pi$.
Example
$-5$, $0$, $3/4$, $1.7$, and $\pi$ are all real numbers. Every point on the number line represents exactly one real number.
Key Insight
The word "real" distinguishes these numbers from "imaginary" numbers (like $\sqrt{-1}$), which cannot be placed on the regular number line.
Definition
The real numbers $\mathbb{R}$ include all rational numbers and all irrational numbers. $\mathbb{R}$ is the union of $\mathbb{Q}$ and its complement. Real numbers can be represented as decimal expansions (terminating, repeating, or infinite non-repeating). $\mathbb{R}$ is a complete ordered field.
Example
The set of reals in $[0, 1]$ contains $1/2$ (rational), $\sqrt{2}/2$ (irrational algebraic), and $\pi/4$ (irrational transcendental), showing the diversity within $\mathbb{R}$.
Key Insight
Completeness is the key property: every bounded set of real numbers has a least upper bound (supremum). This property, absent in $\mathbb{Q}$, is what makes calculus and analysis work.
Definition
$\mathbb{R}$ is the unique complete ordered field up to isomorphism. It can be constructed from $\mathbb{Q}$ via Dedekind cuts or Cauchy sequences. $\mathbb{R}$ has cardinality $c = 2^{\aleph_0}$, which is strictly greater than the countable infinity of $\mathbb{Q}$. Every open cover of a closed bounded interval has a finite subcover (Heine-Borel), a cornerstone of analysis.
Example
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 $f(c) = 0$. This fails for $\mathbb{Q}$: $f(x) = x^2 - 2$ has no root in $\mathbb{Q}$ on $[1, 2]$.
Key Insight
$\mathbb{R}$ is not algebraically closed: $x^2 + 1 = 0$ has no real solution. The complex numbers $\mathbb{C}$ extend $\mathbb{R}$ to achieve algebraic closure, as guaranteed by the Fundamental Theorem of Algebra.