Year 2 — Term 1
17 study notes · 67 practice questions · Aligned with NIE syllabus
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Lessons in This Term
The complex number system
The complex number system extends real numbers to provide solutions to polynomial equations with negative discriminants, introducing the imaginary unit i defined by i² = -1 (or i = √(-1)).
Algebra of complex numbers
Fundamental algebraic operations on complex numbers follow standard field axioms with i² = -1: (1) Addition: (x₁ + iy₁) + (x₂ + iy₂) = (x₁ + x₂) + i(y₁ + y₂); (2) Subtraction: (x₁ + iy₁) - (x₂ + iy₂) = (x₁ - x₂) + i(y₁ - y₂); (3) Multiplication: (x₁ + iy₁)(x₂ + iy₂) = (x₁x₂ - y₁y₂) + i(x₁y₂ + x₂y₁); (4) Division: The quotient z₁ / z₂ (for z₂ ≠ 0) is evaluated by rationalizing the denominator, multiplying numerator and denominator by the complex conjugate z̄₂ = x₂ - iy₂: z₁/z₂ = [(x₁ + iy₁)(x₂ - iy₂)] / (x₂² + y₂²).
Complex conjugate
The complex conjugate of a complex number z = x + iy is defined as z̄ = x - iy, formed by reflecting z across the real axis on the Argand plane.
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The modulus of a complex number z = x + iy is defined as the non-negative real distance from the origin to z on the Argand plane: |z| = √(x² + y²) = √(z z̄).
The Argand diagram
The Argand diagram represents complex numbers geometrically, mapping z = x + iy to the point (x, y) or the position vector from origin O to P.
De Moivre's theorem
De Moivre's Theorem states that for any real number θ and any integer n ∈ ℤ: (cos θ + i sin θ)^n = cos(nθ) + i sin(nθ).
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Create Free AccountLoci and regions of a complex number
Loci and regions on the Argand plane describe the geometric paths and areas traced by a variable complex number z = x + iy satisfying specified equations or inequalities: (1) Circle: |z - z₀| = r represents a circle with centre z₀ and radius r; |z - z₀| ≤ r represents the closed circular disc; (2) Perpendicular Bisector: |z - z₁| = |z - z₂| represents the straight line perpendicularly bisecting the segment connecting z₁ and z₂; |z - z₁| < |z - z₂| denotes the half-plane closer to z₁; (3) Half-Line / Ray: Arg(z - z₀) = α represents a ray originating at z₀ (excluding z₀) inclined at angle α to the positive real axis; (4) Sector / Annulus: Combined bounds such as r₁ ≤ |z - z₀| ≤ r₂ and α ≤ Arg(z - z₀) ≤ β represent annular sectors.
Binomial expansion
The Binomial Theorem for a positive integer index n ∈ ℤ⁺ states that: (a + b)^n = ∑_{r=0}^n ⁿC_r a^{n-r} b^r = ⁿC₀ a^n + ⁿC₁ a^{n-1} b + ...
Binomial theorem
Practical applications of the Binomial Theorem encompass finding specific terms, determining the term independent of x, identifying greatest coefficients, and calculating numerical approximations.
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The factorial of a non-negative integer n, denoted by n!, represents the product of all positive integers less than or equal to n: n!
Fundamental principle of counting
The Fundamental Principle of Counting encompasses two core logical rules that govern combinatorial mathematics: (1) The Multiplication Principle: If an operation can be performed in m distinct ways, and following this a second independent operation can be performed in n distinct ways, the composite sequence of both operations can be executed in m × n distinct ways.
Permutations
A permutation is an arrangement of objects in a specific order. Explanation: When we talk about permutations, the order in which items are arranged is crucial.
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A combination is an unordered selection of r distinct items chosen from a set of n distinct objects, denoted by ⁿC_r or (n \atop r) = n!
Properties of matrices
A matrix is a rectangular array of numbers arranged in m rows and n columns, said to have order m × n.
Special square matrices
A square matrix has an equal number of rows and columns (n × n). Key special square matrices include: (1) Identity Matrix I: Main diagonal entries are 1 and all off-diagonal entries are 0, satisfying AI = IA = A; (2) Zero Matrix O: All entries are 0, satisfying A + O = A and AO = OA = O; (3) Diagonal Matrix: All off-diagonal entries are zero; (4) Symmetric Matrix: Satisfies A^T = A, where a_{ij} = a_{ji} for all i, j; (5) Skew-Symmetric Matrix: Satisfies A^T = -A, requiring a_{ij} = -a_{ji}, which strictly mandates that all principal diagonal entries are zero (a_{ii} = 0).
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The transpose of an m × n matrix A, denoted by A^T, is the n × m matrix formed by interchanging the rows and columns of A.
Simultaneous equations by matrices
A system of two linear simultaneous equations in two variables: a₁x + b₁y = d₁ and a₂x + b₂y = d₂ can be structured in matrix form as AX = B, where A = [[a₁, b₁], [a₂, b₂]] is the coefficient matrix, X = [[x], [y]] is the variable vector, and B = [[d₁], [d₂]] is the constant vector.