France Terminale Expert Mathematics

Chapters: 3

1. Complex numbers

Algebraic form, conjugate, equations · Modulus, argument, exponential form, roots of unity

Complex Numbers and Quadratic Equations – Some equations like x² + 1 = 0 have no real answer, because no real number has a negative square. So we add a new number i with i² = −1. A complex number is z = a + ib: a is the real part and b is the imaginary part. We add, subtract and multiply complex numbers like algebra, and replace i² by −1. To divide, we multiply top and bottom by the conjugate. On the Argand plane, z = a + ib is the point (a, b). The conjugate a − ib is its mirror image in the real axis, and the modulus √(a² + b²) is its distance from the origin. Every quadratic ax² + bx + c = 0 with D = b² − 4ac < 0 has two complex roots that are conjugates of each other.

Divisibility, Euclidean division, congruences · GCD, Bezout and Gauss theorems · Prime numbers

Modular Arithmetic: Remainders and Congruences – Euclidean division writes any integer a as a = n × q + r with 0 ≤ r < n. The remainder r is "a mod n". Two numbers are congruent modulo n (a ≡ b mod n) when they leave the same remainder, which means n divides a − b. Congruences can be added, subtracted, multiplied and raised to powers, so we can find remainders of huge numbers using small ones. Remainders of powers repeat in cycles.
Divisibility: Multiples, Divisors, GCD and LCM – a is divisible by b when a = b × k for a whole number k: b is a divisor (factor) of a and a is a multiple of b. Every whole number a can be written as a = b × q + r with 0 ≤ r < b (division with remainder); b divides a exactly when r = 0. Quick tests tell divisibility by 2, 3, 4, 5, 6, 8, 9, 10 and 11 from the digits. The GCD is the greatest common divisor, found by prime factors or by Euclid's algorithm; LCM is the least common multiple, and GCD × LCM = a × b. Two numbers are coprime when their GCD is 1.
Fundamental Theorem of Arithmetic – Every whole number bigger than 1 is either a prime or can be written as a product of primes in exactly one way (only the order can change). This is the Fundamental Theorem of Arithmetic. Using these prime "building blocks": HCF = product of the smallest powers of the common primes; LCM = product of the greatest powers of all primes. For two numbers, HCF × LCM = a × b.

Matrix calculations · Graphs, random walks, Markov chains

Matrices: Order, Types, Transpose, Operations and Inverse – A matrix is a box of numbers set in rows and columns. Its order is rows × columns. Special matrices include zero, identity, diagonal, scalar, row, column and square matrices. The transpose swaps rows and columns. Symmetric means Aᵀ = A and skew-symmetric means Aᵀ = −A. We add matrices place by place, and multiply them row × column. Matrix multiplication is not commutative (AB is usually not BA). A square matrix A is invertible if some B gives AB = BA = I, and that B is unique.
Graph Theory: Dots, Lines and Networks – A graph is a set of vertices (dots) joined by edges (lines). The degree of a vertex is how many edges touch it, and the sum of all degrees is twice the number of edges. An Euler trail uses every edge once and exists only when 0 or 2 vertices have odd degree. A tree is a connected graph with no cycles and n − 1 edges. Weighted graphs model roads and networks; Kruskal’s and Prim’s algorithms find a minimum spanning tree.