Complex Numbers For High School Students For the Love of Mathematics - PowerPoint PPT Presentation

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 1 / 29

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 2 / 29

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 3 / 29

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 . Presented by: Rich Dlin Complex Numbers For High School Students 4 / 29

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. Presented by: Rich Dlin Complex Numbers For High School Students 5 / 29

Solving Equations Let’s solve some equations. But let’s really think while we do. 1 1 3 a − 7 = 42 (1) 3 × 3 a = 3 × 49 (5) 3 a − 7 + 7 = 42 + 7 (2) 49 1 × a = (6) 3 a + 0 = 49 (3) 3 49 3 a = 49 (4) = (7) a 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 3 a − 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. Presented by: Rich Dlin Complex Numbers For High School Students 6 / 29

Solving Harder Equations Here’s one that, in Ontario, is usually seen first in grade 10. Baby’s First Quadratic Equation 3 x 2 − 7 = 41 3 x 2 = 48 x 2 = 16 (Ok ... maybe not baby’s first ...) Presented by: Rich Dlin Complex Numbers For High School Students 7 / 29

Solving Harder Equations So we have x 2 = 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 8 / 29

Solving Harder Equations In fact the solution is the logical statement √ √ � � � � x = 16 = 4 OR x = − 16 = − 4 . We use properties of multiplication and negative numbers to deduce the second root of the equation, namely − 4. This is not intuitive. At first. Understanding this requires a background in understanding negative numbers, and how they behave under multiplication. We take these “understandings” for granted, which is interesting - why should we? ◮ Are negative numbers intuitive? ◮ We call them R eal, but are they truly realistic? Presented by: Rich Dlin Complex Numbers For High School Students 9 / 29

Thinking More Deeply About Equations Now let’s turn to a somewhat mystifying equation: x 2 = − 1 Is there a solution? In the Real numbers, there is no number capable of squaring to a negative. Therefore, no R eal solution. But wait! We have already shown that we are capable of imagining unrealistic numbers ... ... so surely we can imagine a universe of discourse with a solution? Presented by: Rich Dlin Complex Numbers For High School Students 10 / 29

Imagining Numbers Consider this symbol: Is it the number three? If we were to poll a class of thirty students, it is extremely likely that unless they thought it was a trick question, they would all answer yes. However it is not so simple. The actual answer is NO. It is a symbol that we have all agreed would represent the number three, so that when we see it, or write it, it is tied to the notion of three in our minds. But what is three? Can you define this number? Presented by: Rich Dlin Complex Numbers For High School Students 11 / 29

Imagining Numbers An Attempt to Define “Three” A counting concept used to represent the total number of elements in a set which could be subdivided into non-empty sets, each containing fewer elements than the original, exactly one of which could be subdivided again in the same manner, none of which could be subdivided in the same manner again. This is cumbersome A clever eye will realize that this definition depended very much on the definitions of “one” and “zero” With these numbers already defined, the definition for three could be made more concise. Presented by: Rich Dlin Complex Numbers For High School Students 12 / 29

Imagining Numbers Definition of 3 Three is a counting concept used to represent a quantity of objects that is one more than one more than one, and not more than that. In other words ... ... but what is 1? Presented by: Rich Dlin Complex Numbers For High School Students 13 / 29

I magining Numbers Definition of i i is a number defined so that i 2 = − 1. Note: Despite numerous humorous t-shirts, mugs and beach blankets, it is not the case that √− 1 = i . In the Real number system, the square root function is defined only on non-negative numbers. Additionally, if it were the case that √− 1 = i , then we would have √ − 1 = i 1 ( − 1) = i 2 � 2 � 1 i 2 ( − 1) = 2 ( − 1) 2 � 1 � 2 i 2 = 1 i 2 [1] = 2 1 = − 1 which would be ... problematic. Presented by: Rich Dlin Complex Numbers For High School Students 14 / 29

I magining Numbers Some rules for i : i has magnitude 1. Like vectors, i can be scaled using Real numbers, via scalar multiplication. E.g., 3 i , where | 3 i | = 3. We define a new number set, the Imaginary numbers, as I = { ai : a ∈ R , i 2 = − 1 } . This set is unfortunately named, since it implies that to conceive of it requires proprietary implementation of imagination over previously defined number sets. Adding two Imaginary numbers works the way we would hope: For all a , b ∈ R , ai + bi = ( a + b ) i . I and R share exactly one number. I ∩ R = { 0 } . Negatives work the way we would hope: For all a ∈ R , ai + ( − a ) i = 0. i.e., the additive inverse of ai is ( − a ) i . To multiply two Imaginary numbers, multiply the scalars and square the i . For example (3 i )(2 i ) = 6 i 2 = − 6. Presented by: Rich Dlin Complex Numbers For High School Students 15 / 29

Back to Equations Once Imaginary numbers have been defined, and we can solve equations like x 2 = − 9, we quickly want to use this tool to solve harder quadratic equations that previously seemed to have no solution. Here’s an example. Solve: x 2 − 6 x + 10 = 0 (1) x 2 − 6 x = − 10 (2) x 2 − 6 x + 9 = − 1 (3) ( x − 3) 2 = − 1 (4) This yields two possibilities: x − 3 = i or x − 3 = − i . We would love to add 3 to both sides. But we don’t have any way to add non-zero Real numbers to non-zero Imaginary numbers. So we define a way! And in so doing, define a new universe of discourse. Presented by: Rich Dlin Complex Numbers For High School Students 16 / 29