1 OCF Introducing Complex Numbers MCR3U - Santowski

2 (A) Review - Number Systems: (i) The set of natural numbers are counting numbers N = {1,2,3,4,5,6,..…} (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

3 (B) Review - Discriminant You can use part of the Quadratic Formula, the discriminant (b 2 ‑ 4ac) to predict the number of roots a quadratic equation has. If b 2 ‑ 4ac > 0, then the quadratic equation has two zeroes –ex: y = 2x 2 + 3x ‑ 6 If b 2 ‑ 4ac = 0, then the quadratic equation has one zeros –ex: y = 4x x + 16 If b 2 ‑ 4ac < 0, then the quadratic equation has no zeroes –ex: y = ‑ 3x 2 + 5x ‑ 3 If the value of the discriminant is less than zero, then the parabola has no x-intercepts.

4 (C) Negative Discriminants Sketch the quadratic y = x 2 + 2x + 2 by: (i) completing the square to find the vertex y = (x + 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 +  (2 2 – 4(1)(2))] = ½[-2 +  (-4)]

5 (D) Internet Links Read the story of John and Betty's Journey Through Complex Numbers John and Betty's Journey Through Complex NumbersJohn and Betty's Journey Through Complex Numbers Complex Numbers Lesson - I from Purple Math Complex Numbers Lesson - I from Purple Math Complex Numbers Complex Numbers

6 (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 i 2 = -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) = x 2 + 2x + 2, when using the quadratic formula, we get ½[ -2 +  (-4)] which simplifies to ½[ -2 +  (4i 2 )] which we can simplify further as ½[ i] = -1 + i

7 (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 number. This two parted expression or number is referred to as a complex number. 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 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.

8 (G) Examples ex 1. Find the square root of -16 and –7 ex 2. Solve 2x ex 3. Find the complex roots of the equation 5d d = -70 ex 4. Find the roots of f(x) = 4x 2 - 2x + 3 ex 5. If a quadratic has one root of 2 + 5i, find the other root. Write the equation in factored form ex 5. If a quadratic has one root of 2 + 5i, find the other root. Write the equation in factored form

9 (H) Homework Nelson text, p331-2, Q2ce, 3cdf, 4d, 5be, 7,9,10,11,12 do eol, 19,21