My Favorite One-Liners: Part 86

In this series, I’m compiling some of the quips and one-liners that I’ll use with my students to hopefully make my lessons more memorable for them.

To get students comfortable with i = \sqrt{-1}, I’ll often work through a quick exercise on the powers of i:

i^1 = i

i^2 = -1

i^3 = -i

i^4 = 1

i^5 = i

Students quickly see that the powers of i are a cycle of length 4, so that i^5 = i \cdot i \cdot i \cdot i \cdot i is the same thing as just i. So I tell my students:

There’s a technical term for this phenomenon: aye-yai-yai-yai-yai.

See also http://mentalfloss.com/article/52790/where-did-phrase-aye-yai-yai-come for more on the etymology of this phrase.

My Favorite One-Liners: Part 82

In this series, I’m compiling some of the quips and one-liners that I’ll use with my students to hopefully make my lessons more memorable for them.

In differential equations, we teach our students that to solve a homogeneous differential equation with constant coefficients, such as

y'''+y''+3y'-5y = 0,

the first step is to construct the characteristic equation

r^3 + r^2 + 3r - 5 = 0

by essentially replacing y' with r, y'' with r^2, and so on. Standard techniques from Algebra II/Precalculus, like the rational root test and synthetic division, are then used to find the roots of this polynomial; in this case, the roots are r=1 and r = -1\pm 2i. Therefore, switching back to the realm of differential equations, the general solution of the differential equation is

y(t) = c_1 e^{t} + c_2 e^{-t} \cos 2t + c_3 e^{-t} \sin 2t.

As t \to \infty, this general solution blows up (unless, by some miracle, c_1 = 0). The last two terms decay to 0, but the first term dominates.

The moral of the story is: if any of the roots have a positive real part, then the solution will blow up to \infty or -\infty. On the other hand, if all of the roots have a negative real part, then the solution will decay to 0 as t \to \infty.

This sets up the following awful math pun, which I first saw in the book Absolute Zero Gravity:

An Aeroflot plan en route to Warsaw ran into heavy turbulence and was in danger of crashing. In desparation, the pilot got on the intercom and asked, “Would everyone with a Polish passport please move to the left side of the aircraft.” The passengers changed seats, and the turbulence ended. Why? The pilot achieved stability by putting all the Poles in the left half-plane.

My Favorite One-Liners: Part 67

In this series, I’m compiling some of the quips and one-liners that I’ll use with my students to hopefully make my lessons more memorable for them.

Here are a couple of similar problems that arise in Precalculus:

  1. Convert the point (5,-5) from Cartesian coordinates into polar coordinates.
  2. Convert the complex number 5 - 5i into trigonometric form.

For both problems, a point is identified that is 5 steps to the right of the origin and then 5 steps below the x-axis (or real axis). To make this more kinesthetic, I’ll actually walk 5 paces in front of the classroom, turn right face, and then walk 5 more paces to end up at the point.

I then ask my class, “Is there a faster way to get to this point?” Naturally, they answer: Just walk straight to the point. After some work with the trigonometry, we’ll establish that

  1. (5,-5) in Cartesian coordinates is equivalent to (5\sqrt{2}, -\pi/4) in polar coordinates, or
  2. $5-5i$ can be rewritten as 5\sqrt{2} [ \cos(-\pi/4) + i \sin (-\pi/4)] in trigonometric form.

Once this is obtained, I’ll walk it out: I’ll start at the origin, turn clockwise by 45 degrees, and then take 5\sqrt{2} \approx 7 steps to end up at the same point as before.

Continuing the lesson, I’ll ask if the numbers 5\sqrt{2} and -\pi/4, or if some other angle and/or distance could have been chosen. Someone will usually suggest a different angle, like 7\pi/4 or 15\pi/4. I’ll demonstrate these by turning 315 degrees counterclockwise and walking 7 steps and then turning 675 degrees and walking 7 steps (getting myself somewhat dizzy in the process).

Finally, I’ll suggest turning only 135 degrees clockwise and then taking 7 steps backwards. Naturally, when I do this, I’ll do a poor man’s version of the moonwalk:

For more information, please see my series on complex numbers.

My Favorite One-Liners: Part 59

In this series, I’m compiling some of the quips and one-liners that I’ll use with my students to hopefully make my lessons more memorable for them.

Often I’ll cover a topic in class that students really should have learned in a previous class but just didn’t. For example, in my experience, a significant fraction of my senior math majors have significant gaps in their backgrounds from Precalculus:

  • About a third have no memory of ever learning the Rational Root Test.
  • About a third have no memory of ever learning synthetic division.
  • About half have no memory of ever learning Descartes’ Rule of Signs.
  • Almost none have learned the Conjugate Root Theorem.

Often, these students will feel somewhat crestfallen about these gaps in their background knowledge… they’re about to graduate from college with a degree in mathematics and are now discovering that they’re missing some pretty basic things that they really should have learned in high school. And I don’t want them to feel crestfallen. Certainly, these gaps need to be addressed, but I don’t want them to feel discouraged.

Hence one of my favorite motivational one-liners:

It’s not your fault if you don’t know what you’ve never been taught.

I think this strikes the appropriate balance between acknowledging that there’s a gap that needs to be addressed and assuring the students that I don’t think they’re stupid for having this gap.

 

My Favorite One-Liners: Part 54

In this series, I’m compiling some of the quips and one-liners that I’ll use with my students to hopefully make my lessons more memorable for them.

The complex plane is typically used to visually represent complex numbers. (There’s also the Riemann sphere, but I won’t go into that here.) The complex plane looks just like an ordinary Cartesian plane, except the “x-axis” becomes the real axis and the “y-axis” becomes the imaginary axis. It makes sense that this visualization has two dimensions since there are two independent components of complex numbers. For real numbers, only a one-dimensional visualization is needed: the number line that (hopefully) has been hammered into my students’ brains ever since elementary school.

While I’m on the topic, it’s unfortunate that “complex numbers” are called complex, as this often has the connotation of difficult. However, that’s not why our ancestors chose the word complex was chosen. Even today, there is a second meaning of the word: a group of associated buildings in close proximity to each other is often called an “apartment complex” or an “office complex.” This is the real meaning of “complex numbers,” since the real and imaginary parts are joined to make a new number.

When I teach my students about complex number, I tell the following true story of when my daughter was just a baby, and I was extremely sleep-deprived and extremely desperate for ways to get her to sleep at night.

I tried counting monotonously, moving my finger to the right on a number line with each number:

1, 2, 3, 4, ...

That didn’t work, so I tried counting monotonously again, but this time moving my finger to the left on a number line with each number:

-1, -2, -3, -4, ...

That didn’t work either, so I tried counting monotonously once more, this time moving my finger up the imaginary axis:

i, 2i, 3i, 4i...

For the record, that didn’t work either. But it gave a great story to tell my students.

 

My Favorite One-Liners: Part 46

In this series, I’m compiling some of the quips and one-liners that I’ll use with my students to hopefully make my lessons more memorable for them. Today’s one-liner is something I’ll use after completing some monumental calculation. For example, if z, w \in \mathbb{C}, the proof of the triangle inequality is no joke, as it requires the following as lemmas:

  • \overline{z + w} = \overline{z} + \overline{w}
  • \overline{zw} = \overline{z} \cdot \overline{w}
  • z + \overline{z} = 2 \hbox{Re}(z)
  • |\hbox{Re}(z)| \le |z|
  • |z|^2 = z \cdot \overline{z}
  • \overline{~\overline{z}~} = z
  • |\overline{z}| = |z|
  • |z \cdot w| = |z| \cdot |w|

With all that as prelude, we have

|z+w|^2 = (z + w) \cdot \overline{z+w}

= (z+w) (\overline{z} + \overline{w})

= z \cdot \overline{z} + z \cdot \overline{w} + \overline{z} \cdot w + w \cdot \overline{w}

= |z|^2 + z \cdot \overline{w} + \overline{z} \cdot w + |w|^2

= |z|^2  + z \cdot \overline{w} + \overline{z} \cdot \overline{~\overline{w}~} + |w|^2

= |z|^2 + z \cdot \overline{w} + \overline{z \cdot \overline{w}} + |w|^2

= |z|^2 + 2 \hbox{Re}(z \cdot \overline{w}) + |w|^2

\le |z|^2 + 2 |z \cdot \overline{w}| + |w|^2

= |z|^2 + 2 |z| \cdot |\overline{w}| + |w|^2

= |z|^2 + 2 |z| \cdot |w| + |w|^2

= (|z| + |w|)^2

In other words,

|z+w|^2 \le (|z| + |w|)^2.

Since |z+w| and |z| + |w| are both positive, we can conclude that

|z+w| \le |z| + |w|.

QED

In my experience, that’s a lot for students to absorb all at once when seeing it for the first time. So I try to celebrate this accomplishment:

Anybody ever watch “Home Improvement”? This is a Binford 6100 “more power” mathematical proof. Grunt with me: RUH-RUH-RUH-RUH!!!

My Favorite One-Liners: Part 34

In this series, I’m compiling some of the quips and one-liners that I’ll use with my students to hopefully make my lessons more memorable for them.

Suppose that my students need to prove a theorem like “Let n be an integer. Then n is odd if and only if n^2 is odd.” I’ll ask my students, “What is the structure of this proof?”

The key is the phrase “if and only if”. So this theorem requires two proofs:

  • Assume that n is odd, and show that n^2 is odd.
  • Assume that n^2 is odd, and show that n is odd.

I call this a blue-light special: Two for the price of one. Then we get down to the business of proving both directions of the theorem.

I’ll also use the phrase “blue-light special” to refer to the conclusion of the conjugate root theorem: if a polynomial f with real coefficients has a complex root z, then \overline{z} is also a root. It’s a blue-light special: two for the price of one.

 

My Favorite One-Liners: Part 17

In this series, I’m compiling some of the quips and one-liners that I’ll use with my students to hopefully make my lessons more memorable for them.

Sometimes it’s pretty easy for students to push through a proof from beginning to end. For example, in my experience, math majors have little trouble with each step of the proof of the following theorem.

Theorem. If z, w \in \mathbb{C}, then \overline{z+w} = \overline{z} + \overline{w}.

Proof. Let z = a + bi, where a, b \in \mathbb{R}, and let w = c + di, where c, d \in \mathbb{R}. Then

\overline{z + w} = \overline{(a + bi) + (c + di)}

= \overline{(a+c) + (b+d) i}

= (a+c) - (b+d) i

= (a - bi) + (c - di)

= \overline{z} + \overline{w}

\square

For other theorems, it’s not so easy for students to start with the left-hand side and end with the right-hand side. For example:

Theorem. If z, w \in \mathbb{C}, then \overline{z \cdot w} = \overline{z} \cdot \overline{w}.

Proof. Let z = a + bi, where a, b \in \mathbb{R}, and let w = c + di, where c, d \in \mathbb{R}. Then

\overline{z \cdot w} = \overline{(a + bi) (c + di)}

= \overline{ac + adi + bci + bdi^2}

= \overline{ac - bd + (ad + bc)i}

= ac - bd - (ad + bc)i

= ac - bd - adi - bci.

A sharp math major can then provide the next few steps of the proof from here; however, it’s not uncommon for a student new to proofs to get stuck at this point. Inevitably, somebody asks if we can do the same thing to the right-hand side to get the same thing. I’ll say, “Sure, let’s try it”:

\overline{z} \cdot \overline{w} = \overline{(a + bi)} \cdot \overline{(c + di)}

= (a-bi)(c-di)

= ac -adi - bci + bdi^2

= ac - bd - adi - bci.

\square

I call working with both the left and right sides to end up at the same spot the Diamond Rio approach to proofs: “I’ll start walking your way; you start walking mine; we meet in the middle ‘neath that old Georgia pine.” Not surprisingly, labeling this with a catchy country song helps the idea stick in my students’ heads.

Though not the most elegant presentation, this is logically correct because the steps for the right-hand side can be reversed and appended to the steps for the left-hand side:

Proof (more elegant). Let z = a + bi, where a, b \in \mathbb{R}, and let w = c + di, where c, d \in \mathbb{R}. Then

\overline{z \cdot w} = \overline{(a + bi) (c + di)}

= \overline{ac + adi + bci + bdi^2}

= \overline{ac - bd + (ad + bc)i}

= ac - bd - (ad + bc)i

= ac - bd - adi - bci

= ac -adi - bci + bdi^2

= (a-bi)(c-di)

= \overline{(a + bi)} \cdot \overline{(c + di)}

\overline{z} \cdot \overline{w}.

\square

 For further reading, here’s my series on complex numbers.

Engaging students: Adding, subtracting, multiplying, and dividing complex numbers

In my capstone class for future secondary math teachers, I ask my students to come up with ideas for engaging their students with different topics in the secondary mathematics curriculum. In other words, the point of the assignment was not to devise a full-blown lesson plan on this topic. Instead, I asked my students to think about three different ways of getting their students interested in the topic in the first place.

I plan to share some of the best of these ideas on this blog (after asking my students’ permission, of course).

This student submission again comes from my former student Daniel Adkins. His topic, from Algebra: adding, subtracting, multiplying, and dividing complex numbers.

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How has this topic appeared in pop culture?

Robot chicken aired a television episode in which students were being taught about the imaginary number. Upon the instructor’s completion of his definition of the imaginary number, one student, who understands the definition, immediately has his head explode. One student turns to him and says, “I don’t get it. No wait now I-“, and then his head also explodes.

This video can be used as a humorous introduction that only takes a few seconds. It conveys that these concepts can be difficult in a more light-hearted sense. At the same time it satirically exaggerates the difficulty, and therefore might challenge the students. All the while the video provides the definition as well.

 

 

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How did people’s conception of this topic change over time?

The first point of contact with imaginary numbers is attributed to Heron of Alexandria around the year 50 A.D. He was attempting to solve the section of a pyramid. The equation he eventually deemed impossible was the sqrt(81-114). Attempts to find a solution for a negative square root wouldn’t reignite till the discovery of negative numbers, and even this would lead to the belief that it was impossible. In the early fifteenth century speculations would rise again as higher degree polynomial equations were being worked out, but for the most part negative square roots would just be avoided. In 1545 Girolamo Cardono writes a book titled Ars Magna. He solves an equation with an imaginary number, but he says, “[imaginary numbers] are as subtle as they would be useless.” About them, and most others agreed with him until 1637. Rene Descartes set a standard form for complex numbers, but he still wasn’t too fond of them. He assumed, “that if they were involved, you couldn’t solve the problem.” And individuals like Isaac Newton agreed with him.

Rafael Bombelli strongly supported the concept of complex numbers, but since he wasn’t able to supply them with a purpose, he went mostly unheard. That is until he came up with the concept of using complex numbers to find real solutions. Over the years, individuals eventually began to hear him out.

One of the major ways that helped aid with people eventually come to terms with imaginary numbers was the concept of placing them on a Cartesian graph as the Y-axis. This concept was first introduced in 1685 by John Wallis, but he was largely ignored. A century later, Caspar Wessel published a paper over the concept, but was also ignored. Euler himself labeled the square root of negative 1 as i, which did help in modernizing the concept. Throughout the 19th century, countless mathematicians aided to the growing concept of complex numbers, until Augustin Louis Cauchy and Niels Henrik Able make a general theory of complex numbers.

This is relevant to students because it shows that mathematicians once found these things impossible, then they found them unbelievable, then they found them trivial, until finally, they found a purpose. It encourages students to work hard even if there doesn’t seem to be a reason behind it just yet, and even if it seems like your head is about to blow.

 

 

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How has this topic appeared in high culture?

The Mandelbrot set is a beautiful fractal set with highly complex math hidden behind it. However it is extremely complicated, and as Otto von Bismarck put it, “laws are like sausages. Better not to see them being made.”

Like most fractals, the Mandelbrot set begins with a seed to start an iteration. In this case we begin with x2 + c, where c is some real number. This creates an eccentric pattern that grows and grows.

For students, this can show how mathematics can create beautiful patterns that would interest their more artistic senses. Not only would this generate interest in complex numbers, but it might convince students to investigate recurring patterns.

Sources:

https://www.youtube.com/watch?v=oENQ2jlHpfo

History of imaginary numbers:

http://rossroessler.tripod.com/

 

Mendelbrot sets:

https://plus.maths.org/content/unveiling-mandelbrot-set

 

 

Exponents and the decathlon

During the Olympics, I stumbled across an application of exponents that I had not known before: scoring points in the decathlon or the heptathlon. From FiveThirtyEight.com:

Decathlon, which at the Olympics is a men’s event, is composed of 10 events: the 100 meters, long jump, shot put, high jump, 400 meters, 110-meter hurdles, discus throw, pole vault, javelin throw and 1,500 meters. Heptathlon, a women’s event at the Olympics, has seven events: the 100-meter hurdles, high jump, shot put, 200 meters, long jump, javelin throw and 800 meters…

As it stands, each event’s equation has three unique constants — $latex A$, $latex B$ and $latex C$— to go along with individual performance, $latex P$. For running events, in which competitors are aiming for lower times, this equation is: $latex A(BP)^C$, where $latex P$ is measured in seconds…

B is effectively a baseline threshold at which an athlete begins scoring positive points. For performances worse than that threshold, an athlete receives zero points.

Specifically from the official rules and regulations (see pages 24 and 25), for the decathlon (where P is measured in seconds):

  • 100-meter run: 25.4347(18-P)^{1.81}.
  • 400-meter run: 1.53775(82-P)^{1.81}.
  • 1,500-meter run: 0.03768(480-P)^{1.85}.
  • 110-meter hurdles: 5.74352(28.5-P)^{1.92}.

For the heptathlon:

 

  • 200-meter run: 4.99087(42.5-P)^{1.81}.
  • 400-meter run: 1.53775(82-P)^{1.88}.
  • 1,500-meter run: 0.03768(480-P)^{1.835}.

Continuing from FiveThirtyEight:

 

For field events, in which competitors are aiming for greater distances or heights, the formula is flipped in the middle: $latex A(PB)^C$, where $latex P$ is measured in meters for throwing events and centimeters for jumping and pole vault.

Specifically, for the decathlon jumping events (P is measured in centimeters):

  • High jump: 0.8465(P-75)^{1.42}
  • Pole vault: 0.2797(P-100)^{1.35}
  • Long jump: 0.14354(P-220)^{1.4}

For the decathlon throwing events (P is measured in meters):

  • Shot put: 51.39(P-1.5)^{1.05}.
  • Discus: 12.91(P-4)^{1.1}.
  • Javelin: 10.14(P-7)^{1.08}.

Specifically, for the heptathlon jumping events (P is measured in centimeters):

  • High jump: 1.84523(P-75)^{1.348}
  • Long jump: 0.188807(P-210)^{1.41}

For the heptathlon throwing events (P is measured in meters):

  • Shot put: 56.0211(P-1.5)^{1.05}.
  • Javelin: 15.9803(P-3.8)^{1.04}.

I’m sure there are good historical reasons for why these particular constants were chosen, but suffice it to say that there’s nothing “magical” about any of these numbers except for human convention. From FiveThirtyEight:

The [decathlon/heptathlon] tables [devised in 1984] used the principle that the world record performances of each event at the time should have roughly equal scores but haven’t been updated since. Because world records for different events progress at different rates, today these targets for WR performances significantly differ between events. For example, Jürgen Schult’s 1986 discus throw of 74.08 meters would today score the most decathlon points, at 1,384, while Usain Bolt’s 100-meter world record of 9.58 seconds would notch “just” 1,203 points. For women, Natalya Lisovskaya’s 22.63 shot put world record in 1987 would tally the most heptathlon points, at 1,379, while Jarmila Kratochvílová’s 1983 WR in the 800 meters still anchors the lowest WR points, at 1,224.

FiveThirtyEight concludes that, since the exponents in the running events are higher than those for the throwing/jumping events, it behooves the elite decathlete/heptathlete to focus more on the running events because the rewards for exceeding the baseline are greater in these events.

Finally, since all of the exponents are not integers, a negative base (when the athlete’s performance wasn’t very good) would actually yield a complex-valued number with a nontrivial imaginary component. Sadly, the rules of track and field don’t permit an athlete’s score to be a non-real number. However, if they did, scores might look something like this…