Category Archives: Precalculus

Interesting Cube Problem

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If the cube has a volume of 64, what is the area of the green parallelogram? (Assume points I and J are midpoints.)

Go ahead, work it out. Then, go here for a more in depth discussion, including a video explanation. Also, see my very simple solution in the comments on that page. (My Precalculus students should especially take note!)

And, welcome, SAT Math Blog, to the internet! Thanks for pointing us to this great problem and creating the nice diagram above.

π is Transcendental

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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

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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:

y=x+1

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.

 

 

 

 

 

 

Why does the harmonic series diverge?

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My Precalculus students have been asking me this question. I don’t really have a great answer, except that it’s true. Granted it’s not very intuitive. But nothing about infinite series is intuitive. For those not in my class or not familiar with the harmonic series, the question is:

Does \sum_{n=1}^{\infty}\frac{1}{n}=1+\frac{1}{2}+\frac{1}{3}+\frac{1}{4}+\cdots converge?

And the answer is no.  This is surprising, because the terms of the series approach zero.

The proof that it diverges is due to Nicole Oresme and is fairly simple. It can be found here. There are at least 20 proofs of the fact, according to this article by Kifowit and Stamps.

Interestingly, the alternating harmonic series does converge:

\displaystyle \sum_{n=1}^{\infty}\frac{(-1)^n}{n}=1-\frac{1}{2}+\frac{1}{3}-\frac{1}{4}+\cdots =\ln{2}

And so does the p-harmonic series with p>1. For instance:

\displaystyle \sum_{n=1}^{\infty}\frac{1}{n^2}=1+\frac{1}{2^2}+\frac{1}{3^2}+\frac{1}{4^2}+\cdots =\frac{\pi^2}{6}

Besides looking at the sequence of partial sums, I’m not sure I can help you with the intuition on any of these facts. Can you?

The Saint Louis Arch and y=coshx

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NPR’s Science Friday had an episode highlighting the mathematics behind the Saint Louis Arch. You can watch a little video on the subject at their website, here.

The shape of the arch is the same shape of a hanging chain, called a catenary. “Catenary” is another name for the hyperbolic cosine function,

\cosh {x} = \frac {e^{x}+e^{-x}} {2}

It’s not the world’s most riveting video, but it does highlight this important function that doesn’t get much press in our high school math curriculum. If you watch the video, you’ll learn something about this function, and you’ll learn that the catenary is not only the shape of a hanging chain but also the shape of the most stable arch. For more on the mathematics of the Saint Louis Arch, visit the wikipedia article.

In particular, the Saint Louis Arch has the equation

y = 693.8597 - 68.7672\cosh {(0.0100333x)}

Now, of course you want to know about the hyperbolic sine function, too. I’ll let you look it up yourself (or maybe you can take a guess, first?). Then ask yourself some questions you might be dying to know: Which functions are odd/even? What trig identities are associated with these functions? For Calculus students: What is the derivative of each of these functions? What is the power series? And there are some exciting connections with these functions and complex numbers, too. Go play, and tell me what you learn!