LC25-42SP Leaving Cert Maths Complex Numbers

Complex Numbers– 4 Sessions, 1 hour each

Note: Typed notes and instructional videos with worked solutions to exam questions will be

provided as part of the course material.

Session 1 March 19th @ 7p.m.

• Number Systems – Natural, Integer, Rational, Real

• Introduction to Imaginary number i

• Complex Numbers, Argand Diagram

• Conjugate of Complex Number

• Modulus of Complex Number

• Addition/Subtraction/Multiplication/Division of Complex Numbers

• Exam Questions

Session 2 March 26th @ 7p.m.

• Conjugate Roots Theorem and roots of polynomials

• Solving quadratics with complex roots

• Polar Form

• De Moivre’s Theorem

• Exam Questions

Session 3April 2nd @ 7p.m.

• Proof of De Moivre’s Theorem using Induction

• General Polar Form

• Exam Questions

Session 4 April 9th @ 7p.m.

• nth roots of unity, n ∈ N

• identities such as Cos 3θ = 4 Cos3 θ – 3 Cos θ

• Transformations using Complex Numbers

• Exam QuestionsLearning Outcomes tied to Leaving Cert Hons Maths Syllabus (Strand 3 & 4)

Strand 3

– recognise irrational numbers and appreciate that R ≠ Q

– work with irrational numbers

– revisit the operations of addition, multiplication, subtraction and division in the following

domains:

• N of natural numbers

• Z of integers

• Q of rational numbers

• R of real numbers and represent these numbers on a number line

– investigate the operations of addition, multiplication, subtraction and division with complex

numbers C in rectangular form a+ib

– illustrate complex numbers on an Argand diagram

– interpret the modulus as distance from the origin on an Argand diagram and calculate the

complex conjugate

– calculate conjugates of sums and products of complex numbers

Strand 4

– use the Conjugate Root Theorem to find the roots of polynomials

– work with complex numbers in rectangular and polar form to solve quadratic and other equations

including those in the form zn = a, where n ∈ Z and z = r (Cos θ + iSin θ )

– use De Moivre’s Theorem

– prove De Moivre’s Theorem by induction for n ∈ N

– use applications such as nth roots of unity, n ∈ N, and identities such as Cos 3θ = 4 Cos3 θ – 3 Cos θ