Summary Notes & Proofs for Test 5 (MAM1000W)
Summarised Notes & Proofs made according to the scope given to us for Test 5 in 2019.
• Factorials - Resource book 2
• 1-1 functions and permutations - Resource book 2
• Falling factorials - Resource book 2
• Combinations - Resource book 2
• Binomial coefficients and Pascal’s Triangle - Resource book 2
• The Binomial Theorem - Resource book 2
• Maclaurin and Taylor polynomials - Stewart 11.10 (polynomials, and not series).
• Elementary properties of complex numbers (all complex numbers sections are taken from
the resource book).
• Modulus, the complex plane, and Argand Diagrams
• The complex conjugate and division of complex numbers
• Multiplication and De Moivre’s Theorem
• Argument and modulus-argument form
• The complex exponential function
• The complex trigonometric functions
• n^th roots
• Roots of polynomials
And the proofs that you are expected to know (potentially among others):
• Pascal's identity
• Be able to prove the properties of various aspects of complex numbers (modulus,
complex conjugate, etc.).
• De Moivre’s Theorem (Note that you can’t use the complex exponential function in your
• The non-real roots of a polynomial with real coefficients occur in complex conjugate pairs
• Every polynomial with real coefficients can be written as a product of real linear and
irreducible quadratic factors