Real number
How do you define "real"?

A real number is any number with an infinite decimal expansion and no imaginary component. Between any two real numbers, there are infinitely many real numbers, which is why they are often used to measure continuous quantities like distance and time. Rational and irrational numbers make up the set of reals, whose symbol is \(\mathbb{R}.\)

Real numbers are largely excluded from discrete mathematics because the focus is on non-continuous (discrete) structures such as integers, graphs, and logical statements.

Contents

Sign

The property of a real number being positive or negative is its sign. All real numbers but \(0\) have a sign.

For a real number \(x\):
If and only if \(x > 0\), \(x\) is positive.
If and only if \(x < 0\), \(x\) is negative.
If and only if \(x \ge 0\), \(x\) is non-negative.
If and only if \(x \le 0\), \(x\) is non-positive.

Absolute value

The absolute value of a real number is defined as the magnitude of its distance from \(0.\)

For a real number \(x\):
If and only if \(x\) is non-negative, \(|x|=x.\)
If and only if \(x\) is negative, \(|x|=-x.\)
Logic & Proofs
Integer • Rational number • Inequality • Real number • Theorem • Proof • Statement • Proof by exhaustion • Universal generalization • Counterexample • Existence proof • Existential instantiation • Axiom • Logic • Truth • Proposition • Compound proposition • Logical operation • Logical equivalence • Tautology • Contradiction • Logic law • Predicate • Domain • Quantifier • Argument • Rule of inference • Logical proof • Direct proof • Proof by contrapositive • Irrational number • Proof by contradiction • Proof by cases • Summation • Disjunctive normal form
Set Theory
Set • Element • Empty set • Universal set • Subset • Power set • Cartesian product • String • Binary string • Empty string • Set operation • Set identity • Set proof
Functions
Function • Floor function • Ceiling function • Inverse function
Algorithms
Algorithm • Pseudocode • Command • Asymptotic notation • Time complexity • Atomic operation • Brute-force algorithm
Relations
Relation • Reflexive relation • Symmetric relation • Transitive relation • Relation composition • Equivalence relation • Equivalence class
Number Theory
Integer division • Linear combination • Division algorithm • Modular arithmetic • Prime factorization • Greatest common divisor • Least common multiple • Primality test • Factoring algorithm • Euclid's theorem • Prime number theorem • Euclidean algorithm
Induction
Proof by induction • Fibonacci sequence • Proof by strong induction • Well-ordering principle • Sequence • Factorial • Recursive definition
Combinatorics
Rule of product • Rule of sum • Bijection rule • Permutation • Combination • Complement rule • Experiment • Outcome • Sample space • Event • Probability • Probability distribution • Uniform distribution • Multiset • Sixfold way • Inclusion-exclusion principle • Pigeonhole principle
Graph Theory
Graph • Walk • Subgraph • Regular graph • Complete graph • Empty graph • Cycle graph • Hypercube graph • Bipartite graph • Component • Eulerian circuit • Eulerian trail • Hamiltonian cycle • Hamiltonian path • Tree • Huffman tree • Substring • Forest • Path graph • Star • Spanning tree • Weighted graph • Minimum spanning tree • Greedy algorithm • Prim's algorithm
Recursion
Recursion • Recursive algorithm • Correctness proof • Divide-and-conquer algorithm • Sorting algorithm • Merge sort