# It’s all fun and games until someone loses an i

How’s your Thursday going? Keeping it real? And by real, of course, I mean somewhere in the $0$ or $\pi$ direction in the complex plane.

I’ve been teaching about complex numbers in my Algebra 2 class, so I thought I’d share this groaner with you (HT: Doug McDonald).

# Great NCTM problem

Yesterday I presented this problem from NCTM’s facebook page:

Solve for all real values of $x$:

$\frac{(x^2-13x+40)(x^2-13x+42)}{\sqrt{x^2-12x+35}}$

Don’t read below until you’ve tried it for yourself.

Okay, here’s the work. Factor everything.

$\frac{(x-8)(x-5)(x-7)(x-6)}{\sqrt{(x-5)(x-7)}}=0$

Multiply both sides by the denominator.

$(x-8)(x-5)(x-7)(x-6)=0$

Use the zero-product property to find $x=5,6,7,8$. Now check for extraneous solutions and find that $x=5$ and $x=7$ give you $\frac{0}{0}\neq 0$ and $x=6$ gives $x=\frac{0}{\sqrt{-1}}=\frac{0}{i}=0$. This last statement DOES actually hold for $x=6$ but we exclude it because it’s not in the domain of the original expression.The original expression has domain $(-\infty,5)\cup(7,\infty)$. We could have started by identifying this, and right away we would know not to give any solutions outside this domain. The only solution is $x=8$.

Does this seem problematic? How can we exclude $x=6$ as a solution when it (a) satisfies the equation and (b) is a real solution? This is why we had such a lively discussion.

But this equation could be replaced with a simpler equation. Here’s one that raises the same issue:

Solve for all real values of x:

$\frac{x+5}{\sqrt{x}}=0$

Same question: Is $x=-5$ a solution? Again, notice that it DOES satisfy the equation and it IS a real solution. So why would we exclude it?

Of course a line is drawn in the sand and many people fall on one side and many fall on the other. It’s my impression that high-school math curriculum/textbooks would exclude $x=-5$ as a solution.

Here’s the big question: What does it mean to “solve for all real values of x“? Let’s consider the above equation within some other contexts:

Solve over $\mathbb{Z}$:

$\frac{x+5}{\sqrt{x}}=0$

Is $x=-5$ a solution? No, I think we must reject it. If we try to check it, we must evaluate $\frac{0}{\sqrt{5}}$ but this expression is undefined because $\sqrt{5}\notin\mathbb{Z}$. Here’s another one:

Solve over $\mathbb{Z}_5$:

$\frac{x+5}{\sqrt{x}}=0$

Is $x=-5$ a solution? No. Now when we try to check the solution we get $\frac{0}{\sqrt{5}}=\frac{0}{\sqrt{0}}=\frac{0}{0}$ which is undefined.

The point is that, if we go back to the same question and ask about the solutions of $\frac{x+5}{\sqrt{x}}=0$ over the reals, and we check the solution $x=-5$, we must evaluate $\frac{0}{\sqrt{-5}}$ which is undefined in the reals.[1]

So in the original NCTM question, we must exclude $x=6$ for the same reason. When you test this value, you get $\frac{0}{i}$ on the left side which YOU may think is 0. But this is news to the real numbers. The reals have no idea what $\frac{0}{i}$ evaluates to. It may as well be $\frac{0}{\text{moose}}$.

There’s a lot more to say here, so perhaps I’ll return to this topic another time. Special thanks to all the other folks on facebook who contributed to the discussion, especially my dad who helped me sort some of this out. Feel free to comment below, even if it means bringing a contrary viewpoint to the table.

________________________

[1] This last bit of work, where we fix the equation and change the domain of interest touches on the mathematical concept of algebraic varieties, which I claim to know *nothing* about. If someone comes across this post who can help us out, I’d be grateful!

# Nine important equations

Certain Uncertainty: Schrödinger Equation

From Wired.com

9 Equations True Geeks Should (at Least Pretend to) Know

By Brandon Keim

Even for those of us who finished high school algebra on a wing and a prayer, there’s something compelling about equations. The world’s complexities and uncertainties are distilled and set in orderly figures, with a handful of characters sufficing to capture the universe itself.

For your enjoyment, the Wired Science team has gathered nine of our favorite equations. Some represent the universe; others, the nature of life. One represents the limit of equations.

We do advise, however, against getting any of these equations tattooed on your body, much less branded. An equation t-shirt would do just fine.

The Beautiful Equation: Euler’s Identity

Also called Euler’s relation, or the Euler equation of complex analysis, this bit of mathematics enjoys accolades across geeky disciplines.

Swiss mathematician Leonhard Euler first wrote the equality, which links together geometry, algebra, and five of the most essential symbols in math — 0, 1, i, pi and e — that are essential tools in scientific work.

Theoretical physicist Richard Feynman was a huge fan and called it a “jewel” and a “remarkable” formula. Fans today refer to it as “the most beautiful equation.”

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I was glad to see that the first “must have” equation was Euler’s Identity (note that “Euler’s Identity” is the accepted name for this, not to be confused with Euler’s Formula or Euler’s Polyhedron Formula or any of the other amazing facts named for Euler). I think there’s large consensus in the math community that this is, indeed, a breathtaking equation. It may not be the most fundamentally important, but it definitely showcases why mathematicians delight in math.

I’m ashamed to say it, but I hardly knew any of the other equations. I knew Boltzman’s equation; Maxwell’s equations and Schrödinger’s equation have come up in some of my graduate coursework, but the others I hadn’t ever seen. One might argue that the other equations are not so important. (If you like arguing about such things, join those commenting on the article).  You should still look through the list yourself; how many of these equations do you know?

Granted, this was a general article that encompased all “true geeks” not just math geeks. But still, don’t we all want to be a true geek?

(Oh, and happy birthday to Johan (III) Bernoulli, who had no notable equations named for him :-))

# Geogebra has new skills

A new version of Geogebra has been released, in beta. It’s called Geogebra 5.0, and you can see the news about it here. Or, here’s a direct link to launch it right away. Thanks to The Cheap Researcher for the lead on this. As readers of this blog may already know, I love Geogebra!

One of the main highlights is that Geogebra now supports 3D manipulations. Awesome! However, don’t get too excited–it doesn’t let you graph anything except planes. No surfaces. It will do geometric constructions, like spheres and prisms. Using parametric equations and the locus feature, you can coax it into rendering spirals or other space curves. [edit: I figured this was possible, but it actually wasn't. Not sure why.]

Another highlight, which I find even more exciting, is that Geogebra now has a built in CAS. Here’s a screen shot of me playing around with a few of its features. It also has a ways to go, especially for those who are used to more robust systems like Mathematica/Maple/Derive/TI-89. But this is a great step in the right direction, and 10 points for the open-source camp!

Notice that it can work with polynomials in ways you would expect, it can symbolically integrate and derive (simple things), perform partial fraction decomposition, evaluate limits, and find roots. Here are a few more things it can do. Strangely, it had problems finding the complex roots of a quadratic (easy), but not a cubic (hard). Just take a look at my screen shot. Seeing that it did okay finding the complex roots, I wondered if it could also plot them for me. I started by entering (copying and pasting) the complex zeros as points in Geogebra, which worked. But then I discovered the new ComplexRoot[] function which approximates the roots and plots them on the coordinate plane all at once. Cool! Here’s the screenshot:

The seven complex roots of f(z)=z^7+5z^4-z^2+z-15

As you can see, I asked for the roots of a 7th degree polynomial. Since the polynomial had real coefficients, notice that every zero’s conjugate is also a zero, as we’d expect. And we also expect that at least one solution of an odd-degreed polynomial will be real (notice this one has only one real root, approximately 1.22).

That’s all I’ve discovered so far. I’ll let you know if I come across anything else exciting. Keep in mind that this is beta, so the final release will likely have all the bugs worked out and more features.

# Very Nice Java Applets

I’m always on the look out for nice java applets (or flash, or javascript, or whatever) that help visualize tough math concepts. The best applets are not merely gimmick, but truly make the mathematics more accessible and invite you to explore, predict, and play with the concepts.

Most recently, my dad pointed me to this website.  At this site, you’ll find well over one hundred very nice applets that show things I haven’t seen around the web before. For instance, play around with your favorite polyhedra in 3d, then let the applet truncate or stellate the polyhedron, all with nice animation. And of course, you’ll want to mess with all the settings on the drop down menus, too! Additionally, you can see a mobius strip, klein bottle, cross cap, or a torus. Also, I was just recently blogging about complex functions. Some of these applets allow you to visualize with animation and color various complex functions. Having just completed a grad class in differential equations, I was also interested in these ODE applets. You may also have fun with fractals, like this one.

There are lots of other nice applet websites out there. As you know, I’m a huge fan of GeoGebra (free, open source, able to be run online without installation, easy to use, powerful). Someday maybe I’ll post more extensively about its merits. In the meantime, you can check it out here.  Many of these applets use GeoGebra. This site has a lot of nice applets too.  And here. This one’s rather elementary, but fun for educators. These are great, too. These are all sites I’ve used in class. Like I said, I think they’re more than just ‘cool’–they really do help elucidate hard-to-understand topics.

Feel free to share other applet sites you like.

# π is Transcendental

Passing on a post from my dad…I think some of the math is accessible for my readers. In fact, for my precalculus students, it ties together some of the nice stuff we’ve studied this semester (infinite series, complex numbers).

Teaching History of Math this past semester gave me an excuse to read carefully two Dover Publications books that I have owned since high school, but only skimmed then. Imagine my delight to discover that if you are given a theorem that is hard to prove beforehand, you can prove that $\pi$ is transcendental in just a couple of lines. The hard theorem gives many other corollaries too, corollaries that I’ve known in my gut but never had a handle on how to prove.

Here are the details, from p. 76 of Felix Klein’s book Famous Problems of Elementary Geometry. You can read it on-line at Google Books.

(more)

Go check it out!

# Functions of a Complex Variable

My precalculus classes just finished a unit on polar coordinates and complex numbers. When I teach about complex numbers,  I mention functions of a complex variable in passing, but we don’t really give it much thought. We do complex arithmetic and that’s all; that is, problems like these:

Evaluate.

$\frac{i^5(2-i)}{1+3i}$

$\left(2+2i\sqrt{3}\right)^{10}$

$\sqrt[4]{40.5+40.5i\sqrt{3}}$

In our precalculus class, we also understand how to plot complex numbers. Complex numbers must be plotted on a two-dimensional plane because complex numbers are…well…two dimensional! The real number line has no place for them. For instance, we represent the complex number $w=2-3i$ as the point $(2,-3)$

But we don’t ask questions about complex functions. This is sad! Because functions of a complex variable are fairly accessible.  That is, we want to consider functions like

$f(z)=z+1$

$f(z)=z+4i$

$f(z)=3z$

$f(z)=iz$

$f(z)=z^2$

The first thing you’ll notice is that I’ve written these functions in terms of $z$, to indicate that they take complex arguments and (possibly) return complex values. Here’s where the problem comes. Take for instance, $f(x)=x+1$. We’re used to visualizing it this way:

Notice we’re wired (because of schooling, perhaps) to understand the $x$ coordinate as being the “input” to the function and the $y$ coordinate as being the “output” from the function. Now, think about $f(z)=z+1$ where $z$ is complex. Do you see the problem? Remember, complex numbers are two-dimensional.  A function $f(z)$ that operates on complex numbers has two-dimensional input and two-dimensional output. For example, take the function $f(z)=z+1$. If we try putting a few complex numbers into the function for $z$, what happens? If $z=-2+4i$, then $f(z)=-1+4i$. Geometrically, what is happening to a complex number on the complex plane when we apply $f(z)$? If you thought “the function translates complex numbers to the right by one unit in the complex plane,” you’re right!

I’ve built another Geogebra applet to help you visualize this kind of function. Make sure you use it with purpose, rather than just dragging things around randomly. Try making predictions about what will happen before revealing the result. Read the directions.

Have fun, and I hope you learn something about complex functions! I’m sure to post more on them someday. There’s a lot more to say.

# “Real” Cool Solution

This week I start complex numbers with my Precalculus class. This is a fun problem I found here (you can also go there for the solution when you’re ready).

Given $x^2+x+1=0$, find $x^3$.

# Be Rational, Get Real

One present I got for my dad this Christmas was a t-shirt with this funny image on it. Perhaps you’ve seen this on a poster or t-shirt before. It’s very amusing. But my dad pointed out a bit of the irony I hadn’t noticed before.

For $i$ to tell $\pi$ to “be rational” is a bit like the pot calling the kettle black. Can you think why? If you remember what a rational number is, you’ll remember it’s a subset of the real numbers.  No imaginary number is “rational” in the typical sense.  So $\pi$ could just as easily say “be rational” to $i$.

You might have an intuitive sense for what might make a complex number “rational” though. You might say, if the real and imaginary part are both rational, then the complex number is “kind of” rational. Mathematicians call these “Gaussian Rationals.”

That’s your interesting thought for the day. Happy new year!
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