MATH1025
Preparatory mathematics
Develop the algebraic and geometric tools used in university mathematics: proof and inequalities, complex numbers, sequences, polynomials, vectors and conic sections. Start with precise algebra, then learn to choose and justify a method.
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In the MATH1025 notes
- Theorem cards
- 89
- Definitions
- 56
- Counterexamples
- 10
- Concept lenses
- 15
- Histories of ideas
- 3
- Proof X-rays
- 1
Sections with a history of an idea
- Binomial coefficients and expansions4.1
- Complex numbers, polar form, and geometry6.1
- Divisibility, gcd, and integer equations7.1
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No concept lens from this course is recorded in the web yet.
Counts cover the 21 public sections of this course and what is written in them, not recorded connections. 0 of these sections are indexed into the web in detail so far.
In the webTheorem cards 89 · Definitions 56 · Counterexamples 10 · Concept lenses 15 · Histories of ideas 3 · Proof X-rays 1
None of these is on the timeline yet.
No concept lens from this course is recorded in the web yet.
Counts cover the 21 public sections of this course and what is written in them, not recorded connections. 0 of these sections are indexed into the web in detail so far.
Chapter 0-1 · 2 sections
Foundations and early methods
Foundational symbolic language and core transformations used across the course.
Theorems 14 · Definitions 12
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- 0.1
Course foundations and notation
Set up notation, proof habits, and algebraic prerequisites that support later trigonometry, sequences, and vectors.
- 1.1
Equation structure and trigonometric identities
Reinforce equation-solving structure and trigonometric identities with proof-aware algebraic transformations.
Chapter 2-3 · 2 sections
Proof and inequalities
Induction, order reasoning, rational inequalities, absolute value, and first classical inequalities.
Theorems 10 · Definitions 7Counterexamples 1
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- 2.1
Mathematical induction
Use base cases, induction steps, and strong induction to prove statements indexed by positive integers.
- 3.1
Inequalities and absolute value
Solve inequalities by preserving order, tracking domains, and using absolute value as distance.
Chapter 4 · 1 section
Binomial theorem
Factorials, permutations, combinations, Pascal's identity, and coefficient extraction from binomial expansions.
Theorems 2 · Definitions 3Lenses 1History 1
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- 4.1
Binomial coefficients and expansions
Connect permutations, combinations, Pascal's identity, and the binomial theorem.
Chapter 5 · 1 section
Sequences
Sequences as functions, recursive construction, arithmetic and geometric progressions, finite sums, and first applied recurrences.
Theorems 2 · Definitions 4Lenses 1
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- 5.1
Sequences, recursion, and series
Read sequences as functions on positive integers, compare explicit and recursive definitions, and derive arithmetic and geometric sum formulas.
Chapter 6 · 1 section
Complex numbers
Complex arithmetic, conjugates, modulus, polar and exponential forms, roots of unity, and complex-plane geometry.
Theorems 5 · Definitions 3Lenses 1History 1
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- 6.1
Complex numbers, polar form, and geometry
Construct complex numbers from ordered pairs, use conjugates and polar form, and connect rotation, roots of unity, and complex-plane geometry.
Chapter 7 · 2 sections
Integer methods
Divisibility, primes, gcd computations, Bezout identities, and integer linear equations.
Theorems 18 · Definitions 7Counterexamples 1Lenses 1History 1
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- 7.1
Divisibility, gcd, and integer equations
Develop divisibility, primes, the division algorithm, the Euclidean algorithm, Bezout's identity, prime factorization, and first linear Diophantine equations.
- 7.2
Rational and irrational numbers
Use rational-number closure, nth-root notation, and Euclid's lemma to prove standard irrationality results.
Chapter 8 · 3 sections
Polynomial methods
Polynomial arithmetic, division with remainder, polynomial gcds, irreducibility, rational functions, partial fractions, and Vieta formulas.
Theorems 12 · Definitions 6Counterexamples 2Lenses 2Proof X-rays 1
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- 8.1
Polynomial arithmetic and division
Define polynomials as finite formal sums, control degree, prove division with remainder, and use the remainder and factor theorems.
- 8.2
Polynomial gcds and irreducibility
Use polynomial Euclidean algorithms, Bezout identities, irreducibility, and field-dependent factorization.
- 8.3
Rational functions, partial fractions, and Vieta formulas
Decompose rational functions, use partial fractions in finite sums, and relate roots to coefficients with Vieta's formulas.
Chapter 9 · 2 sections
Vectors and geometry
Vectors, norms, inner products, projections, cross products, and area and volume geometry.
Theorems 8 · Definitions 5Counterexamples 2Lenses 2
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- 9.1
Vectors, norm, dot product, and projection
Connect coordinates with geometry through norms, dot products, orthogonal projection, and the Cauchy–Schwarz and triangle inequalities.
- 9.2
Cross product, scalar triple product, and vector geometry
Develop the cross product in three dimensions, interpret scalar triple products as signed volumes, and solve vector geometry problems.
Chapter 10 · 2 sections
Lines, planes, and curves
Lines and planes, distances and projections, and parametrized curves with velocity and speed.
Theorems 5 · Definitions 5Counterexamples 2Lenses 2
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- 10.1
Lines, planes, projections, and distances
Describe lines and planes with directions and normals, then derive intersections, angles, projections, and distances.
- 10.2
Parametric curves and vector functions
Separate a parametrized curve from its trace, differentiate vector functions, and interpret orientation, velocity, and speed.
Chapter 11 · 5 sections
Conic sections
Focal definitions, tangents and reflection, optional coordinate classification, and exact conic loci and parametric applications.
Theorems 13 · Definitions 4Counterexamples 2Lenses 5
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- 11.1
Parabolas: focus, directrix, tangents, and normals
Derive the parabola from its focus and directrix, then study tangents, normals, reflection, and parameter-based loci.
- 11.2
Ellipses: foci, reflection, and tangent geometry
Relate the focus-directrix and constant-sum definitions of an ellipse, and prove its tangent, reflection, and locus properties.
- 11.3
Hyperbolas: foci, asymptotes, and tangent geometry
Develop both hyperbola branches from focal distance conditions, and analyze asymptotes, tangents, reflection, and the pedal locus.
- 11.4
General conics, coordinate rotation, and classification
Study the optional coordinate-rotation method, quadratic-form classification, degenerate conics, and regular tangent equations.
- 11.5
Conic loci and parametric applications
Use chord and tangent parameters to determine exact loci, then connect rolling-circle motion with speed and rational parametrization.