41 · Chalk Notes I

Slight revision notes, Core Pure Year 1 and 2. Roots of unity: the complex solutions form the vertices of a regular polygon on the Argand diagram; you move between roots by multiplying. Loci of a modulus: |z| equals a real number gives a circle, since the modulus is the scalar distance between two points on the Argand diagram — there are circles, perpendicular bisectors, polynomial curves. The half line: arg(z − z₁) = θ is a line extending from the fixed point z₁ at angle θ measured anticlockwise from the direction parallel to the positive real axis — and it excludes z₁ itself, since at z = z₁ the left-hand side does not exist. Dashed lines when an inequality is strict. Regions: just colour them. The modulus is the umbrella term for absolute value; taking the sign away is just the heuristic — the true meaning is that the distance from origin is quantified as a scalar. With the Taylor series — the Maclaurin series specifically — every function circumambulates back into polynomial expressions, inclusive of the infinity concept.

From the Maclaurin series we saw that exponential and trigonometric functions can be swapped into polynomials; morph them together and you get Euler's relation. So Euler's relation and the Taylor series are the same thing. And sin(nπ) = 0 — substitute π for theta and the imaginary part of the right side vanishes, giving e^(iπ) = −1: the beautiful Euler identity. Notice how the verbal terms weave together — lay them out and evoke them all when you see any one of them. E: exponential functions. i: complex numbers, Argand diagrams — a pure imaginary number stays locked gliding on the vertical axis. π: circles, bases of cylinders and cones, the trig angles where values hit the extremes of ±1 or the medium of 0. De Moivre's theorem takes the multiple of an angle outside — useful for trig functions inside integrands, the IMC-style analysis questions. The number inside a log is also called an argument; the shared terminology exists because when you apply a logarithm to a complex number, the argument (input) of the log becomes directly related to the argument (angle) of the complex number.

Viète: the relationship between coefficients and roots — and in the exam, don't quote "Vieta's formula" by name; say "the relationship between coefficients and roots of a polynomial." The sum rules for cubes of a quadratic's roots versus a cubic's are very different — the latter is completely new. The sum always starts with something negative.

Volumes of revolution: revolution means circular or elliptical movement about an axis — horizontal, vertical, oblique. One rule: integrate with respect to the axis of rotation. Think of infinitesimal slicing into thin apples: if the thickness dx runs along the x-axis, revolve about x, so ∫πy²dx; vice versa for y. Heuristic: d£ where £ is the axis revolved about, the perpendicular variable gets squared, π for the circle's area. A cylinder's volume is three times its cone.

Matrices: the elements language of sets permeates here too — each number in a matrix is an element; an array is an arrangement of terms. Transposes and inverses. Det(A) is like the discriminant of a quadratic — it determines existence. Consistency of linear equations: at least one solution set means consistent; none means inconsistent — and discern between parallel non-identical planes versus identical ones. Linear transformations sound fancy but it's just re-expressing what we know in matrix notation; start 2D, extend 3D — rotation and reflection are in the formula booklet, only the y-axis rotation is the weirdo, the top-right sign flips. Order of multiplication matters. Eigenvalues and eigenvectors: find eigenvalues from det(A − λI) = 0; substitute λ into Ax = λx, equate elements, find the Cartesian relationship; choose values keeping the elements as integers; and state that any non-zero multiple of an eigenvector is also an eigenvector. A normalised vector is the eigenvector divided by its magnitude. For any symmetric matrix the normalised eigenvectors are mutually perpendicular, so the matrix P formed from them satisfies P⁻¹ = Pᵀ — an orthogonal matrix. Don't need the why; know the properties. And DON'T FORGET the determinant multiplied into the transpose of the matrix of cofactors: find the determinant, find the matrix of minors.

Proof by induction: base case (the basis), the assumption — let the proposition be true for k — the inductive step built from that assumption, then the verbal conclusion. Everything follows: basis, assumption, inductive logic, conclusion.