Further Mathematics (UK A-Level)
UK A-Level Further Mathematics (Years 12-13, ages 16-18) covering the AQA/Edexcel/OCR specification taken alongside A-Level Maths: Core Pure (complex numbers, matrices, further algebra, further calculus, further vectors, polar coordinates, hyperbolic functions, differential equations) plus optional applied modules in Further Mechanics, Further Statistics, and Decision Mathematics.
Ämne: Matematik · Nivå: Gymnasium (16–19) · 399 kort
Innehåll
- The imaginary unit i is defined by i² = −1, so i = √(−1). A complex number is written z = a + bi, where a is the real part Re(z) and b is the imaginary part Im(z).
- The complex conjugate of z = a + bi is z* = a − bi. Multiplying a complex number by its conjugate gives a real number: z·z* = a² + b².
- The modulus of z = a + bi is |z| = √(a² + b²), the distance from the origin to the point (a, b) on the Argand diagram.
- The argument of a complex number, arg(z), is the angle θ measured from the positive real axis to the line joining the origin to z, conventionally taken in the principal range −π < θ ≤ π.
- Modulus-argument (polar) form: z = r(cos θ + i sin θ), where r = |z| and θ = arg(z). This is sometimes abbreviated r cis θ.
- Exponential (Euler) form: z = r e^(iθ), where r = |z| and θ = arg(z). Euler's formula states e^(iθ) = cos θ + i sin θ.
- When multiplying complex numbers in polar form, multiply the moduli and add the arguments: |z₁z₂| = |z₁||z₂| and arg(z₁z₂) = arg(z₁) + arg(z₂).
- When dividing complex numbers in polar form, divide the moduli and subtract the arguments: |z₁/z₂| = |z₁|/|z₂| and arg(z₁/z₂) = arg(z₁) − arg(z₂).
- Euler's identity, e^(iπ) + 1 = 0, links the five fundamental constants e, i, π, 1 and 0. It is the special case of e^(iθ) = cos θ + i sin θ at θ = π.
- For a polynomial with real coefficients, non-real complex roots occur in conjugate pairs: if a + bi is a root, then a − bi is also a root. Hence such a polynomial of odd degree has at least one real root.
- de Moivre's theorem: (cos θ + i sin θ)ⁿ = cos nθ + i sin nθ for integer n. Equivalently, (r e^(iθ))ⁿ = rⁿ e^(inθ).
- de Moivre's theorem is used to derive multiple-angle identities. For example, expanding (cos θ + i sin θ)³ and equating real parts gives cos 3θ = 4cos³θ − 3cos θ.
- Using z = e^(iθ), we have z + 1/z = 2cos θ and z − 1/z = 2i sin θ. More generally zⁿ + 1/zⁿ = 2cos nθ. These relations let you express powers of cos θ and sin θ as sums of multiple angles.
- The n distinct nth roots of a complex number lie equally spaced on a circle in the Argand diagram, separated by angles of 2π/n radians.
- The nth roots of unity are the solutions of zⁿ = 1. They are 1, ω, ω², ..., ωⁿ⁻¹ where ω = e^(2πi/n). Their sum is zero for n ≥ 2, and they lie at the vertices of a regular n-gon on the unit circle.
- On an Argand diagram, |z − a| = r represents a circle of radius r centred at the point representing the complex number a.
- On an Argand diagram, |z − a| = |z − b| represents the perpendicular bisector of the line segment joining the points a and b.
- On an Argand diagram, arg(z − a) = θ represents a half-line (ray) starting from the point a (excluding a itself) making angle θ with the positive real direction.
- Squaring a complex number doubles its argument and squares its modulus: if z = r e^(iθ) then z² = r² e^(2iθ).
- The complex roots of unity are useful because they form a cyclic group under multiplication: multiplying by ω = e^(2πi/n) rotates a point by 2π/n about the origin.
- To find the nth roots of a complex number z = r e^(iθ), use z^(1/n) = r^(1/n) e^(i(θ + 2πk)/n) for k = 0, 1, ..., n−1, adding multiples of 2π to the argument before dividing.
- Adding a fixed complex number a to every point of a locus translates (shifts) the whole locus by the vector a; multiplying by a fixed complex number scales and rotates it.
- The sum of the roots of zⁿ = 1 (the nth roots of unity) is 0 for n ≥ 2 because they are the roots of zⁿ − 1 = 0, whose coefficient of z^(n−1) is 0.
- The triangle inequality for complex numbers states |z₁ + z₂| ≤ |z₁| + |z₂|, with equality when z₁ and z₂ have the same argument.
- The Fundamental Theorem of Algebra guarantees that every polynomial of degree n ≥ 1 with complex coefficients has exactly n roots in the complex numbers, counted with multiplicity.
- Matrix multiplication is associative (AB)C = A(BC) and distributive over addition, but it is NOT commutative in general: AB ≠ BA for most matrices.
- Two matrices can be multiplied only if the number of columns of the first equals the number of rows of the second. An m×n matrix times an n×p matrix gives an m×p matrix.
- For a 2×2 matrix [[a, b], [c, d]], the determinant is ad − bc. A matrix is singular (non-invertible) if and only if its determinant is zero.
- The inverse of the 2×2 matrix [[a, b], [c, d]] is (1/det)[[d, −b], [−c, a]], where det = ad − bc. Swap the leading diagonal, negate the other diagonal, divide by the determinant.
- The determinant of a product equals the product of the determinants: det(AB) = det(A)·det(B). It follows that det(A⁻¹) = 1/det(A).
- The inverse of a product reverses the order: (AB)⁻¹ = B⁻¹A⁻¹. The order matters because matrix multiplication is not commutative.
- The determinant of a 2×2 transformation matrix gives the area scale factor of the transformation. A negative determinant means the transformation reverses orientation (includes a reflection).
- For a 3×3 transformation matrix, the determinant gives the volume scale factor of the transformation in three dimensions.
- The matrix for an anticlockwise rotation by angle θ about the origin is [[cos θ, −sin θ], [sin θ, cos θ]].
- To find the matrix of a composite transformation 'first A then B', multiply B×A (the later transformation goes on the left, because it acts on the result of A).
- An eigenvector of a square matrix M is a non-zero vector v such that Mv = λv for some scalar λ. The scalar λ is the corresponding eigenvalue; the matrix stretches the eigenvector without changing its direction (or reverses it if λ < 0).
- Eigenvalues are found by solving the characteristic equation det(M − λI) = 0, where I is the identity matrix. For a 2×2 matrix this is a quadratic in λ.
- Diagonalising a matrix: if M has eigenvalues forming diagonal matrix D and eigenvectors forming the columns of P, then M = PDP⁻¹ and Mⁿ = PDⁿP⁻¹, making high powers easy to compute.
- A system of linear equations can be written as a matrix equation Ax = b. If A is invertible, the unique solution is x = A⁻¹b.
- If the determinant of the coefficient matrix of a linear system is zero, the system is singular: it has either no solutions (inconsistent equations) or infinitely many solutions (dependent equations).
- Geometrically, three linear equations in three unknowns represent three planes. A unique solution is a single point of intersection; no solution can form a 'sheaf' or 'prism'; infinitely many solutions form a common line of intersection.
- The identity matrix I has 1s on the leading diagonal and 0s elsewhere. It acts as the multiplicative identity: MI = IM = M for any compatible matrix M.
- The transpose of a matrix, written Aᵀ or A', is formed by swapping rows and columns. (AB)ᵀ = BᵀAᵀ.
- An invariant point of a transformation maps to itself (Mv = v); an invariant line maps onto itself as a set, though points on it may move along it. Eigenvectors give the directions of invariant lines through the origin.
- For a 3×3 matrix the determinant can be found by expansion along a row or column using cofactors, with the checkerboard sign pattern +−+ / −+− / +−+.
- For a quadratic ax² + bx + c = 0 with roots α, β: the sum of roots α + β = −b/a and the product αβ = c/a (Vieta's formulas).
- For a cubic ax³ + bx² + cx + d = 0 with roots α, β, γ: Σα = −b/a, Σαβ = c/a, and αβγ = −d/a.
- For a quartic with roots α, β, γ, δ and leading coefficient a: Σα = −b/a, Σαβ = c/a, Σαβγ = −d/a, and αβγδ = e/a. The signs alternate as the symmetric functions increase in order.
- The identity α² + β² = (α + β)² − 2αβ lets you find the sum of squares of roots from the elementary symmetric functions without finding the roots themselves.
- To form a polynomial whose roots are a transformation of the original roots (e.g. each root +2, or each root doubled), substitute the inverse transformation w = f(x) into the original equation, e.g. let x = w − 2.