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Ontario Tech acknowledges the lands and people of the Mississaugas of Scugog Island First Nation.

We are thankful to be welcome on these lands in friendship. The lands we are situated on are covered by the Williams Treaties and are the traditional territory of the Mississaugas, a branch of the greater Anishinaabeg Nation, including Algonquin, Ojibway, Odawa and Pottawatomi. These lands remain home to many Indigenous nations and peoples.

We acknowledge this land out of respect for the Indigenous nations who have cared for Turtle Island, also called North America, from before the arrival of settler peoples until this day. Most importantly, we acknowledge that the history of these lands has been tainted by poor treatment and a lack of friendship with the First Nations who call them home.

This history is something we are all affected by because we are all treaty people in Canada. We all have a shared history to reflect on, and each of us is affected by this history in different ways. Our past defines our present, but if we move forward as friends and allies, then it does not have to define our future.

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Complex Numbers

INTRODUCTION

A complex number is a number that can be represented by an expression of the form:  a + bi

where a and b are real numbers, and i is a symbol with the property 

i2  = −1

We call a the real part of the complex number and we call b the imaginary part.

ADDING AND SUBTRACTING COMPLEX NUMBERS

To add or subtract complex numbers, simply add or subtract their real parts and their imaginary parts separately.

(a + bi) ± (c + di) = (a ± c) + (b ± d )i

Example:  If x = 3 + 2i and y = −2 − 5i , find x + y .

Solution:

x + y = (3 + 2i) + (−2 − 5i)

         = (3 − 2) + (2 − 5)i

         = (1) + (−3)i

         = 1 − 3i

MULTIPLYING COMPLEX NUMBERS                         

Multiplication is defined so that the usual laws hold: i.e., do FOIL as usual, but simplify at the end using the fact that i2  = −1 .

(a + bi) × (c + di) = ac + adi + bci + bdi2

                             = (ac − bd ) + (ad+ bc)i

Example:  If x = 3 + 2i and y = −2 − 5i , find xy.

Solution:
xy = (3 + 2i)(−2 − 5i)
     = (3)(−2) + (3)(−5i) + (2i)(−2) + (2i)(−5i)
     = (−6) + (−15i) + (−4i) + (−10i2 )
     = −6 − 15i − 4i − 10(−1)
     = −6 − 15i − 4i + 10
     = 4 − 19i