How I Learn Math

2025-08-01   blogpage sketch howilearn

Venn diagrams to choose math books/articles

Venn diagram of great mathematicians

Venn diagram of great math explainers

Read books from those who are at the intersection of above two venn diagrams

Those books that I know of are Davenport's Higher Arithmetics, Gower's Princeton Companion to Mathematics, ?

Tree/graph of proofs

Math dependency graph

e.g.

[THEOREM] Euler’s Totient Theorem
└── If a ∈ ℤ, gcd(a, n) = 1 ⇒ a^φ(n) ≡ 1 mod n
    ├── [DEFINITION] Euler’s Totient Function φ(n)
    │   └── φ(n) = count of integers 1 ≤ k < n such that gcd(k, n) = 1
    │       ├── [DEFINITION] Coprimality: gcd(a, b) = 1
    │       │   └── [ALGORITHM] Euclidean Algorithm (to compute gcd)
    │       │       └── [AXIOM] Division Algorithm
    │       └── [ALGORITHM] Compute φ(n) using prime factorization
    │           └── φ(n) = n × ∏(1 - 1/p), for p | n
    │               └── [DEFINITION] Prime Factorization
    │                   └── [THEOREM] Fundamental Theorem of Arithmetic
    │                       └── Every integer > 1 is uniquely a product of primes
    │
    ├── [LEMMA] Multiplicative Property of φ(n)
    │   └── If gcd(m, n) = 1 ⇒ φ(mn) = φ(m)·φ(n)
    │       └── [PROOF] Uses Chinese Remainder Theorem
    │           └── [THEOREM] Chinese Remainder Theorem (CRT)
    │               └── [CONSTRUCTION] Isomorphism of ℤ/mnℤ ≅ ℤ/mℤ × ℤ/nℤ
    │
    ├── [LEMMA] Units Modulo n form a Group under Multiplication
    │   └── [CONSTRUCTION] (ℤ/nℤ)* = {a < n | gcd(a, n) = 1}
    │       ├── [DEFINITION] Group (Algebraic structure)
    │       └── [AXIOM] Associativity, identity, inverse, closure
    │
    ├── [THEOREM] Lagrange’s Theorem (Group Theory)
    │   └── In finite group G, order of any element divides |G|
    │       └── [PROOF] Coset partitioning argument
    │
    ├── [THEOREM] Fermat’s Little Theorem (special case of Euler’s)
    │   └── If p prime, a not divisible by p ⇒ a^(p−1) ≡ 1 mod p
    │       ├── [DEFINITION] Prime Number
    │       ├── [PROOF] Uses multiplicative group modulo p
    │       │   └── [LEMMA] (ℤ/pℤ)* is cyclic for prime p
    │       └── [COROLLARY] Euler’s Theorem generalizes Fermat’s
    │
    ├── [DEFINITION] Modular Arithmetic
    │   ├── [DEFINITION] Congruence: a ≡ b mod n ⇔ n | (a − b)
    │   ├── [DEFINITION] Modular Exponentiation: a^k mod n
    │   ├── [THEOREM] Properties of Congruences
    │   │   └── e.g., a ≡ b ⇒ a+c ≡ b+c and ac ≡ bc
    │   └── [ALGORITHM] Fast Modular Exponentiation
    │
    └── [EXAMPLE] Euler’s Theorem with n = 9, a = 2
        ├── φ(9) = 6 ⇒ 2^6 ≡ 1 mod 9
        └── (2^6 = 64; 64 mod 9 = 1 ✔️)

See how I generate dependency graphs at https://chatgpt.com/share/688c4b20-9050-8010-8070-52c2f85b0fac

Looking at this graph, one can divide the problem into sub-pieces. If you can prove/understand each node, then you can construct the whole proof leading to the theorem at hand. An ambitious project would be to generate an enormous map for the whole math where each node has its own document of proof.

• Relevantly, Timothy Gowers works on an automatic theorem proving project.
  

Take multiple perspectives

Learn to Think in Multiple Ways — The Anthology of Balaji

• verbal, visual, algebraic, numerical, computational, historical
  

Visual perspective is especially helpful to me.

• e.g.
  

  • group--abelian-group--ring--commutative-field--vector-space--ideal-visualizations

    Open: Pasted image 20250731210522.png
    

    

    • https://library.fiveable.me/algebraic-number-theory/unit-1/basic-algebraic-structures-groups-rings-fields/study-guide/0Gb7TWoXSam9Ng7c
      
    
    

  • group--abelian-group--ring--commutative-field--vector-space--ideal-visualizations

    

    • ref
      
    ]
    

Another very helpful is code representation, as in Python.

• e.g. RSA algorithm