Complex Numbers For High School Students

[Pages:29]Complex Numbers For High School Students

For the Love of Mathematics and Computing Saturday, October 14, 2017

Presented by: Rich Dlin

Presented by: Rich Dlin

Complex Numbers For High School Students

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About Rich

Graduated UWaterloo with BMath in 1993 Worked as a software developer for almost 10 years Began teaching in 2002 Taught in 4 different high schools over a span of 5 years Department head of mathematics for 10 years Earned MMT from UWaterloo in 2013 Visiting Lecturer at UWaterloo beginning September 2017 Part-time artist, musical theatre performer/director, bodybuilder, philosopher Fully believes that all of these are consistent with being a mathematician.

Presented by: Rich Dlin

Complex Numbers For High School Students

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Laying a Foundation For Complex Numbers

We'll start with some seemingly trivial exploration, because as it turns out, the exploration of the trivial can be decidedly non-trivial!

Most of us instinctively solve the following equation, either by inspection or more formally using the (cherished) algorithms we teach as part of an algebra curriculum:

x + 8 = 13

(I know ... shockingly difficult)

A basic understanding of the meaning of the statement allows us to see that the solution is x = 5. But as soon as we start teaching algorithms, we sneakily introduce significant yet abstract notions that are rarely if ever addressed.

So let's address them!

Presented by: Rich Dlin

Complex Numbers For High School Students

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Identities

Consider the following equations in x:

x+y =x

and xy = x, x = 0

Once again, in each case, we instinctively know the value of y that solves the equation, even though we do not know the value of x (nor can we).

Identities (Not the Triggy kind)

The number 0 is called the Identity Element under addition, because for any x, x + 0 = 0 + x = x.

Similarly, the number 1 is called the Identity Element under multiplication, because for any x, 1 ? x = x ? 1 = x.

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Complex Numbers For High School Students

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Inverses

Inverses

if a is the identity element for an operation , then the inverse of x under is x-1. where x x-1 = x-1 x = a.

Examples:

The

inverse

of

5

under

multiplication

is

1 5

.

We

say

that

5

and

1 5

are

multiplicative inverses of each other.

The inverse of 5 under addition is -5. We say that 5 and -5 are additive inverses of each other.

So then the operations of division and subtraction are really just

multiplication and addition, respectively, but using the appropriate

inverse.

For

example

5

-

2

=

5

+

(-2),

and

7

?

9

=

7

?

1 9

.

This is actually extremely significant.

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Complex Numbers For High School Students

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Solving Equations

Let's solve some equations. But let's really think while we do.

3a - 7 = 42

(1)

3a - 7 + 7 = 42 + 7 (2)

3a + 0 = 49

(3)

1

1

? 3a = ? 49 (5)

3

3

49

1?a =

(6)

3

3a = 49

(4)

49

a=

(7)

3

The above is really a logical argument

It begins with the assumption that "There exists a R, such that 7 less than triple the value of a results in 42.

So we have proven this implication (given the Real numbers as a

universe of discourse):

If

3a

-

7

=

42

then

a

=

49 3

.

(Interesting thought: What are we proving when we "check the answer"?)

We did it by using some fundamental numeric concepts, not the least of which is the power of identity elements and inverses.

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Complex Numbers For High School Students

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Solving Harder Equations

Here's one that, in Ontario, is usually seen first in grade 10.

Baby's First Quadratic Equation

3x2 - 7 = 41 3x2 = 48 x2 = 16

(Ok ... maybe not baby's first ...)

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Solving Harder Equations

So we have x2 = 16 But what now? The truth is at this stage most people think we just use square root to determine the solution. Students instinctively deduce x = 4. Why? Because they implicitly use a universe of discourse of non-negative Real numbers. We then remind them that we may consider negatives also. Many students (and teachers?) then say something like "16 has two square roots". But the number 16 does not have two square roots (entirely because square root is not defined that way)!

Presented by: Rich Dlin

Complex Numbers For High School Students

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