Transcript Document

OCF.01.6 - Introducing
Complex Numbers
MCR3U - Santowski
(A) Review - Number Systems:
(i) The set of natural numbers are counting numbers N =
(ii) The set of whole numbers includes all counting numbers as well
as 0 W = {0,1,2,3,4,5,6,....}
(iii) The set of integers includes all whole numbers as well as
negative “natural” numbers I = {...,-3,-2,-1,0,1,2,3,....}
(iv) The set of rational numbers includes any number that can be
written in fraction form Q = {a/b|a,bεI, b≠0}
(v) The set of irrational numbers which includes any number that
cannot be expressed as a decimal ex. π, 2
The set of real numbers (R) is comprised of the combination of
rational and irrational numbers
(B) Review - Discriminant
You can use part of the Quadratic Formula, the discriminant
(b2 - 4ac) to predict the number of roots a quadratic equation has.
If b2 - 4ac > 0, then the quadratic equation has two zeroes
– ex: y = 2x2 + 3x - 6
If b2 - 4ac = 0, then the quadratic equation has one zeros
– ex: y = 4x2 + 16x + 16
If b2 - 4ac < 0, then the quadratic equation has no zeroes
– ex: y = -3x2 + 5x - 3
If the value of the discriminant is less than zero, then the parabola
has no x-intercepts.
(C) Negative Discriminants
Sketch the quadratic y = x2 + 2x + 2 by:
(i) completing the square to find the vertex
y = (x + 1)2 + 1 so V(-1,1)
(ii) partial factoring to find two additional points on the parabola
y = x(x + 2) + 2 so (0,2) and (-2,2)
(iii) Try to find the zeroes by using the quadratic formula and the
completing the square method. What problem do you encounter?
x = ½[ -2 + (22 – 4(1)(2))] = ½[-2 + (-4)]
(D) Internet Links
Read the story of John and Betty's Journey Through Complex
Complex Numbers Lesson - I from Purple Math
Complex Numbers
(E) Quadratics and Complex Numbers
To work around the limitation of being unable to solve for a negative
square root, we “invent” another number system.
We define the symbol i so that i2 = -1,
Thus, we can always factor out a -1 from any radical.
The “number” i is called an imaginary number.
Re-consider the example f(x) = x2 + 2x + 2,
when using the quadratic formula, we get ½[ -2 + (-4)]
which simplifies to ½[ -2 + (4i2)]
which we can simplify further as ½[ -2 + 2i] = -1 + i
(F) Properties of Complex Numbers
From the example on the previous slide, we see that the resultant
number (-1 + i) has two parts: a real number (-1) and an imaginary
This two parted expression or number is referred to as a complex
The number 4 can also be considered a complex number if it is
rewritten as 4 + 0i
Likewise, the number -2i can also be considered a complex number
if it is written as 0 - 2i
When working out complex roots for quadratic equations, you will
notice that the roots always come in “matching” pairs i.e. -1 + i which
is -1 + i and -1 - i. The numbers are the same, only a sign differs.
As such, these two “pairs” are called conjugate pairs.
(G) Examples
ex 1. Find the square root of -16 and –7
ex 2. Solve 2x2 + 50
ex 3. Find the complex roots of the equation 5d2 + 10d = -70
ex 4. Find the roots of f(x) = 4x2 - 2x + 3
ex 5. If a quadratic has one root of 2 + 5i, find the other root. Write
the equation in factored form
(H) Homework
Nelson text, p331-2, Q2ce, 3cdf, 4d, 5be, 7,9,10,11,12 do eol, 19,21