Real Number Continuum: Density, Completeness, and Bounds

6 lessons

Goal

Analyze the topological structure of the real line, including Archimedean property, density of rationals, and the completeness axiom.

Set-Theoretic Foundations and Classifications of Real Numbers

6 lessons

0 of 6 done

Set-Theoretic Foundations and Classifications of Real NumbersUp next
Decimal Expansions, Periodicity, and Non-Terminating RepresentationsPremiumNot started yet
The Archimedean Property and Density of Rational NumbersNot started yet
Upper Bounds, Lower Bounds, Supremum, and InfimumNot started yet
The Completeness Axiom and the Least Upper Bound PropertyNot started yet
The Nested Intervals Theorem and Cantor's Diagonalization ArgumentNot started yet

Standards Covered

Curriculum Framework

7 lessons

Construct direct algebraic proofs, deductions from axioms, and proofs by exhaustion for finite cases and number-theoretic properties.

7 lessons

Master indirect proof methods including contradiction, contraposition, and finding minimal counterexamples to refute conjectures.

6 lessons

Analyze the topological structure of the real line, including Archimedean property, density of rationals, and the completeness axiom.

6 lessons

Manipulate complex surd expressions, apply index laws across all rational exponents, and rationalize multi-term radical denominators.

8 lessons

Perform arithmetic on complex numbers in Cartesian form, compute modulus and conjugates, and graph loci in the Argand diagram.

7 lessons

Convert between Cartesian, polar, and Euler's exponential forms, compute products and quotients, and evaluate powers via De Moivre's Theorem.

8 lessons

Apply the Fundamental Theorem of Algebra, factor polynomials completely over the complex numbers, and solve higher-degree equations.