Level 4 – GCSE and A level maths full justification
This is an A level maths proof companion consisting of documents and videos where intuition or proofs are provided for all results in GCSE and A level maths used as black boxes, some of which you may not have even realized required justification—when that happens, I explain why justification is needed. Many steps often skipped over in textbooks are justified in this level. The proofs for statistics are included with the further maths proof companions in level 6 as those tend to be more complicated.
You would benefit from this if one of the following (or equivalent for a non‑British system) describes you:
A level maths students seeking to go beyond the curriculum
A level further maths students
A level maths students interested in better understanding the material
A level maths students interested to understand the “why” for results or formulas or methods that are not typically properly justified in A level textbooks or their exercises but still used in GCSEs or A level pure maths
Students aiming for an A or an A* in A level maths
Anyone considering pursuing maths at university
Undergraduate maths students
Maths teachers or prospective maths teachers (A level teachers in particular)
SOURCES: Cambridge notes, math stack exchange, statproofbook, wikipedia
Here is the document and videos for this level:
Table of Contents [Expand] [Note that things are subject to change when level 3 comes out due to stuff moving around]
Page 1 - Existence and uniqueness of prime factorization
Page 6 - Calculus (Rigorous proof of Product and Chain rules, Generalized power rule, existence and uniqueness of e, riemann integration, Separation of variables)
Page 10 (subject to change) - Geometric series (Reason for valid interval)
Page 11 (subject to change) - Volume of sphere and cone, surface area of sphere (GCSE but proofs involve integration)
Page 12 (subject to change) - Trigonometry addition formulae
Page 14 (subject to change) - Sin and cos derivatives
Page 16 (subject to change) - Generalized binomial theorem (deferring power series derivative assumption until later in the document)
Page 18 (subject to change) - Partial fractions justification for why we do what we do in the various cases