## All posts tagged **complex numbers**

# Day After Thanksgiving

*Posted by John Quintanilla on November 24, 2017*

https://meangreenmath.com/2017/11/24/day-after-thanksgiving/

# Not Real

*Posted by John Quintanilla on June 18, 2017*

https://meangreenmath.com/2017/06/18/not-real/

# 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’ll often work through a quick exercise on the powers of :

Students quickly see that the powers of are a cycle of length 4, so that is the same thing as just . 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.

*Posted by John Quintanilla on April 27, 2017*

https://meangreenmath.com/2017/04/27/my-favorite-one-liners-part-86/

# 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

,

the first step is to construct the characteristic equation

by essentially replacing with , with , 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 and . Therefore, switching back to the realm of differential equations, the general solution of the differential equation is

.

As , this general solution blows up (unless, by some miracle, ). 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 or . On the other hand, if all of the roots have a negative real part, then the solution will decay to 0 as .

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.

*Posted by John Quintanilla on April 23, 2017*

https://meangreenmath.com/2017/04/23/my-favorite-one-liners-part-82/

# 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:

- Convert the point from Cartesian coordinates into polar coordinates.
- Convert the complex number 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 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

- in Cartesian coordinates is equivalent to in polar coordinates, or
- $5-5i$ can be rewritten as 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 steps to end up at the same point as before.

Continuing the lesson, I’ll ask if the numbers and , or if some other angle and/or distance could have been chosen. Someone will usually suggest a different angle, like or . 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.

*Posted by John Quintanilla on April 8, 2017*

https://meangreenmath.com/2017/04/08/my-favorite-one-liners-part-67/

# My Favorite One-Liners: Part 59

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.

*Posted by John Quintanilla on March 31, 2017*

https://meangreenmath.com/2017/03/31/my-favorite-one-liners-part-59/

# My Favorite One-Liners: Part 54

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 “axis” becomes the real axis and the “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:

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:

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

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

*Posted by John Quintanilla on March 26, 2017*

https://meangreenmath.com/2017/03/26/my-favorite-one-liners-part-54/

# 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 , the proof of the triangle inequality is no joke, as it requires the following as lemmas:

With all that as prelude, we have

In other words,

.

Since and are both positive, we can conclude that

.

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!!!

*Posted by John Quintanilla on March 18, 2017*

https://meangreenmath.com/2017/03/18/my-favorite-one-liners-part-46/

# My Favorite One-Liners: Part 34

Suppose that my students need to prove a theorem like “Let be an integer. Then is odd if and only if 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 is odd, and show that is odd.
- Assume that is odd, and show that 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 with real coefficients has a complex root , then is also a root. It’s a blue-light special: two for the price of one.

*Posted by John Quintanilla on March 6, 2017*

https://meangreenmath.com/2017/03/06/my-favorite-one-liners-part-34/

# My Favorite One-Liners: Part 17

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 , then .

**Proof**. Let , where , and let , where . Then

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 , then .

**Proof**. Let , where , and let , where . Then

.

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”:

.

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 , where , and let , where . Then

.

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

*Posted by John Quintanilla on February 17, 2017*

https://meangreenmath.com/2017/02/17/my-favorite-one-liners-part-17/