Day: June 10, 2017

Sharing Kelsey Houston-Edward’s complex number video with kids

I didn’t have anything planned for our math project today, but both kids asked if there was a new video from Kelsey Houston-Edwards! Why didn’t I think of that 🙂

The latest video is about the pantograph and complex numbers:

Here’s what the boys thought about the video:

They boys were interested in the pantograph and also complex numbers. We started off by talking about how the pantograph works. With a bit more time to prepare (and probably a bit more engineering skill than I have), building a simple pantograph would make a really fun introductory geometry project.

Next we talked about complex numbers. We’ve talked about complex number several times before, so the idea wasn’t a new one for the boys. I started from the beginning, though, and tried to echo some of the introductory ideas that Kelsey Houston-Edwards brought up in her video.

To finish up today’s project we looked at some basic geometry of complex numbers. The specific property that we looked at today was multiplying by i. At the end of this short talk I think that the boys had a pretty good understanding of the idea that multiplying by i was the same as rotating by 90 degrees.

Complex numbers are a topic that I think kids will find absolutely fascinating. I don’t know where (if at all) they come into a traditional middle school / high school curriculum, but once you understand the distributive property you can certainly begin to look at complex numbers. It is such a fun topic with many interesting applications and important ideas – many of which are accessible to kids. Just playing around with complex numbers seems like a great way to expose kids to some amazing math.

A night with Cut the Knot, Nassim Taleb, and some Supernova

Please note the correction at the bottom of the post

A further correction – there is still an error. Ugh. This approach may not work, unfortunately . . .

Saw a neat problem from Alexander Bogomolny earlier today:

I actually missed the problem when it was initial posted, but saw it via Nassim Taleb’s solution:

The problem sort of gnawed at me all day and I figured it was in the maybe 1 in 10 problems that Bogomolny posts that I might be able to solve.

So, tonight I poured a glass of Supernova and gave it a go

One thing on my mind all day with this problem was Jensen’s inequality. What I would *love* to be able to do is say that by Jensen’s inequality:

(1/3) \sqrt{x^2 + 3} + (1/3) \sqrt{y^2 + 3} + (1/3) \sqrt{xy + 3}

\geq \sqrt{ (1/3)( x^2 + y^2 + xy) + 3}

Which is easily seen to be \geq 2 because of the constraint x + y = 2. That work would show the original inequality was \geq 6.

The approach has a tiny bit of merit since \sqrt{x^2 + 3} is concave up for x between 0 and 2 -> here’s a little Mathematica plot showing that the second derivative is indeed positive on 0 to 2:

Plot1

But . . . the problem is that folding in the 3rd term in the sum is stretching the rules of Jensen’s inequality a bit, I think, since it is not of the form \sqrt{a^2 + 3}.

With the first two terms, though, applying Jensen’s inequality seems ok, but I now need (1/2)’s instead of (1/3)’s since there are only two terms. So, I’ll use Jensen’s on the first two terms only and try to show that

(1/2) \sqrt{x^2 + 3} + (1/2) \sqrt{y^2 + 3} + (1/2) \sqrt{xy + 3} \geq 3

By Jensen’s inequality this new sum is

\geq \sqrt{ \frac{x^2 + y^2}{2} + 3} + (1/2) \sqrt{xy + 3}

A bit of algebra and the fact that x + y = 2 allows us to simplify this expression to:

\sqrt{5 - xy} + (1/2) \sqrt{xy + 3}

and then further to:

\sqrt{ (x - 1)^2 + 4} + (1/2) \sqrt{4 - (x - 1)^2}

Now we are just down to a fairly straightforward calculus problem, and I’ll let Mathematica do the heavy lifting since the algebra isn’t that interesting:

Plot2

We can see visually that the minimum occurs at x = 1 from the plot, and the plot of the derivative further confirms that there is only one critical point. The value of the last expression at x = 1 is indeed 3 as we were hoping.

So, Jensen’s inequality, a bit of calculus, and a nice glass of scotch shows that the original inequality is indeed true.

Thanks to Alexander Bogomolny for the problem, and to Nassim Taleb for his solution that got me thinking about the problem.


Correction

I received a note from Alexander Bogomoly over night. He spotted an error in the calculation:

[/embed]https://twitter.com/CutTheKnotMath/status/873413279749722113[/embed]

and I thought my kids having trouble sleeping and waking me up at 5:00 am today was a bad start to the day!

But it seems that I’ve gotten very lucky – both learning from my carelessness in applying Jensens inequality and that the path forward from Bogomolny’s correction is easier than the path I actually took.

Starting here – we wish to show that:

(1/2) \sqrt{x^2 + 3} + (1/2) \sqrt{y^2 + 3} + (1/2) \sqrt{xy + 3} \geq 3

The correction shows that the expression on the left hand side is \geq than

\sqrt{ (\frac{x + y}{2})^2 + 3} + (1/2) \sqrt{xy + 3}

but since x + y = 2, the first piece of this expression is equal to 2 and the 2nd expression simplifies as before. So we are left with

2 + (1/2) \sqrt{4 - (x - 1)^2}

and this expression has a maximum of 3 at x = 1.

That means that the expression we were trying to show to be greater than 3 is indeed greater than 3, and the expression in the original tweet is greater than 6.

I’m grateful to Alexander Bogomolny for spotting the error in my original argument.