01. A roadmap for advanced mathematics
Advanced mathematics is not one subject. It is a collection of connected fields, each with its own questions, language, and tools. Build a solid foundation before moving into proofs and abstract structures.
Suggested learning order
- Strengthen algebra, functions, and trigonometry.
- Learn limits, derivatives, integrals, and series.
- Study vectors, matrices, and systems of equations.
- Practise logic, sets, and mathematical proofs.
- Choose a field and explore its deeper questions.
02. Linear algebra
Linear algebra studies vectors, matrices, linear transformations, and systems of linear equations. It is central to physics, computer graphics, data science, and machine learning.
Vectors
A vector can represent a quantity with components and direction.
Matrix multiplication
Matrices can represent transformations that rotate, stretch, reflect, or otherwise map vectors to new vectors.
Consider the system:
\[ x+y=5,\qquad x-y=1 \]
Adding the equations gives \(2x=6\), so \(x=3\). Substituting gives \(y=2\).
03. Differential equations
A differential equation relates an unknown function to one or more of its derivatives. These equations model change in mechanics, electrical circuits, population models, and many other systems.
A simple example
This equation says that the rate of change of \(y\) is proportional to its current value. Its general solution is:
Here \(C\) is a constant determined by an initial condition, and \(k\) controls the growth or decay behaviour.
04. Number theory
Number theory explores integers and their properties. It asks questions about divisibility, prime numbers, remainders, and patterns in whole numbers.
Modular arithmetic
Modular arithmetic studies remainders after division. For example, \(17\) leaves remainder \(5\) when divided by \(6\).
This notation is useful for clock arithmetic, cyclic patterns, and some areas of cryptography.
05. Mathematical logic and proofs
Mathematical logic studies statements and the rules used to reason about them. Proofs establish why a claim must be true under specified assumptions.
Implication
This means “if \(P\), then \(Q\).” A proof must establish the conclusion from the assumptions, not merely show that a few examples work.
Common proof methods
- Direct proof: derive the conclusion from the assumptions.
- Contrapositive: prove that not-\(Q\) implies not-\(P\).
- Contradiction: show that denying the claim leads to an inconsistency.
- Induction: prove a base case and then a step connecting successive cases.
06. Sequences and infinite series
A sequence is an ordered list of terms. An infinite series adds the terms of a sequence, and its behaviour depends on whether the partial sums approach a finite limit.
Geometric series
For an infinite geometric series, the sum is finite when \(|r|<1\):
07. Choose your next field
Once the foundations are strong, choose a field based on the questions you want to investigate.
For physics and the universe
Explore multivariable calculus, differential equations, vector calculus, and linear algebra.
For probability and prediction
Explore combinatorics, probability distributions, statistical inference, and stochastic processes.
For abstract patterns
Explore number theory, abstract algebra, mathematical logic, and set theory.
For computation
Explore numerical methods, discrete mathematics, algorithms, and optimisation.
08. Quick practice
Question 1
If \(x+y=9\) and \(x-y=3\), find \(x\) and \(y\).
Reveal solution
Add the equations: \(2x=12\), so \(x=6\). Then \(y=3\).
Question 2
What is the remainder when \(23\) is divided by \(7\)?
Reveal solution
\(23=3(7)+2\), so the remainder is \(2\). In modular notation, \(23\equiv2\pmod7\).
Question 3
Find the infinite sum \(1+\frac12+\frac14+\frac18+\cdots\).
Reveal solution
This is a geometric series with \(r=\frac12\). Its sum is \(\frac{1}{1-\frac12}=2\).