Algebra Notes
Arithmetic and Geometric series
Introduction to arithmetic series. Deriving the formula for the sum of an arithmetic series. Questions on arithmetic series and answers.
pdf of notes solution to questionsIntroduction to geometric series. Eating cake and geometric series. Sigma notation. Derivation of the sum formula and sum of an infinite geometric series. Fractional form of recurring decimals.
pdf of notesComplex numbers
The complex conjugate solutions of quadratic functions. The Argand diagram. The product of a complex number and it’s conjugate is real. The modulus of a complex number and its relation to the complex conjugate. Review of addition, subtraction, multiplication, and division of complex numbers. Polar form of a complex number. The unit circle and complex numbers, rotation by multiply complex numbers on the unit circle.
pdf of notesThe Euler Formula
Maclaurin series proof of the Euler Formula. Proof by differentiation. Convert from Cartezian form to polar form and then exponential form. Multiplying complex numbers in exponential form, much more simpler than using the Cartezian form. The complex numbers proof for the double and trigonometrical identities.
pdf of notesMore Euler Formula Applications
Formulas for triple angle trigonometrical identities and higher powers. Proof for compound angle formulas. The unit circle quadrants I and IV, using the symmetric and antisymmetric properties of cos(x) and sin(x) respectively. Application to the powers of cos(x) and sin(x), to make such functions easier to integrate.
pdf of notesThe roots of Unity
Find the roots of unity by factorisation compared to using the exponential form for complex numbers. Finding square roots of complex numbers using the exponential form.
pdf of notes