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<h1>Cambridge Pure Mathematics (Syllabus)</h1>
<div class="keepws">
(<span class="bold">notes</span>
(date 2025-07-11)
(category <a href="../../notes.html">psychology</a>)
(status <span class="status-rough">rough</span>)
(confidence certain)
(importance 9)
(tags methodology learning syllabus))
</div>
<p class="preview">A comprehensive syllabus for ultralearning pure mathematics, based on Cambridge's curriculum. This includes core topics, recommended textbooks, and resources for self-study.</p>
<div class="prose">
<p>This is a curriculum designed for ultralearning. For more info on my modified methodology for ultralearning, see the post
<a href=""></a>. To learn about it's foundation read the book <a href="https://www.amazon.com/Ultralearning-Master-Hard-Skills-Fast/dp/006285268X">Ultralearning</a> by Scott Young.</p>
<h2>Introduction</h2>
<p>This is a curriculum designed for those who are interested in pursuing a advanced understanding of pure mathematics, particularly one based on the
Cambride Mathematical Tripos. It is designed as a accelerated learning path, allowing you to cover the material in a much shorter time frame than tradtional methods.
If done right it is intended to be a full replacement for a university degree in pure mathematics as far as the core material is concerned.</p>
<h3>Core Philosophy</h3>
<h3>Prerequisites</h3>
<h4>Core Pure Mathematics</h4>
<ul>
<li><strong>Proof Writing</strong>: Understanding and constructing mathematical proofs (direct proof, proof by contradiction, proof by induction).</li>
<li><strong>Algebra</strong>: Polynomials, algebraic division, factor theorem, inequalities, functions (domain, range, inverse, composite), graph sketching.</li>
<li><strong>Coordinate Geometry</strong>: Equations of lines and circles, parametric equations.</li>
<li><strong>Sequences and Series</strong>: Arithmetic and geometric series, binomial expansion.</li>
<li><strong>Trigonometry</strong>: Identities, equations, inverse trigonometric functions, sum and product formulae.</li>
<li><strong>Exponentials and Logarithms</strong>: Properties and graphs.</li>
<li><strong>Calculus</strong>: Differentiation (from first principles, rules for differentiation, applications to gradients, tangents, normals, stationary points), Integration (as a limit of a sum, fundamental theorem, techniques of integration, applications to areas and volumes).</li>
<li><strong>Numerical Methods</strong>: Iteration, locating roots.</li>
</ul>
<h4>Further Pure Mathematics</h4>
<ul>
<li><strong>Complex Numbers</strong>:</li>
<li><strong>Matrices</strong>:</li>
<li><strong>Further Algebra & Functions</strong>:</li>
<li><strong>Further Calculus</strong>:</li>
<li><strong>Vectors</strong>:</li>
<li><strong>Hyperbolic Functions</strong>:</li>
<li><strong>Differential Equations</strong>:</li>
<li><strong>Polar Coordinates</strong>:</li>
</ul>
<h3>General Resources to Gather</h3>
</div>
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