Rolfe Schmidt

We’re packed. We’ve disconnected our phone and canceled our cable. Tomorrow we get on a plane and start traveling.

We’re flying to London, then we’ll be getting on a boat and going around the Baltic sea with my parents. After that, we’ll work our way down to Paris to visit my brother. We’ll end the trip spending a week at a backpackers in Switzerland and hanging out with some friends. We’re excited about it.

Here are some of the things we plan to do with the kids:

London - Visit the science and natural history museums — they are free! Drop by Downing Street and Parliament (Gunnar wants to see all of the national capitols, so we’ll make a point of seeing government buildings when we can).

Copenhagen - Hang out at Tivoli for a while. Eat Danishes. Peruse the Tycho Brahe planetarium.

Stockholm - Visit Skansen , roam around Djurgården, and see the Vasa ship.

Paris - Go up the Eiffel tower. Eat crepes and macaroons. Visit the Pompidou center. Ride a carousel. Run around the Jardin de Tuileries. Maybe the Cité des Sciences or something like that. We’ll see.

Schwyz - Hike a lot. Figure out whether William Tell really existed. Walk the Weg der Schweiz. See the Rutli meadow where Switzerland started and learn a little bit about why. Let the kids play with their friends.

We’re also visiting Tallinn, St. Petersburg, and Helsinki, but I don’t have many plans there besides roaming around and enjoying the towns.

The boys are excited about the trip. Well, Niels doesn’t care, but Gunnar and Soren are excited. Soren has been drawing maps of Europe and practicing his French and German. I’ve started trying to speak French around the house some, so maybe they’ll pick some up. He’s practicing Italian and Spanish too, but I don’t think that will come in very handy.

Gunnar has been learning all about the Cantons of Switzerland and can give you a nice little report about the order they joined the Confederation. He must have drawn at least 20 maps of Switzerland and just as many cartoon histories, with Uri, Schwyz, and Unterwalden fighting off the Hapsburgs, then being joined by all of their friends. One of my favorite projects of his was this set of cantonal flags:

This poster is about 7 feet tall. It’s hard to make out the names, but you might notice that they are listed in the order they joined the confederation.

This should be fun.

Gunnar started getting interested in playing chess last week.  He played my dad a couple of times, and he asks to play me every day.  I’ve never been much of a player, and it isn’t always easy to drop things and sit down for an hour to play.  I like to see him get excited about it, but chess just doesn’t thrill me.  And no, it’s not just because he wins.

What did thrill me was that another kid brought his Lego chess set to our homeschool park day yesterday.  Gunnar loved it and within a couple of minutes he was in a game.  It was great to see him playing, joking around,  and having fun.  Other kids would gather around, ask questions or give advice, then disperse back into the park.   

Soren was napping, Hoa was chasing Niels around, so I had some time to sit and think. I wished I had brought a book. 

Gunnar, Soren, and I did a fun little experiment with water, some salt, and a D-cell battery. We saw how little electric current ran through distilled water, then slowly added salt and watched the current increase. We recorded our observations and started making guesses about what the value we’d see next. We also tried running current through our tap water to see how much “stuff” it had in it. Gunnar also figured out that Chlorine stinks. It’s nothing very novel, but it’s easy and interesting. Here’s what we did:

Materials

1. 1 cup of distilled water in a small glass bowl
2. 2 paper clips
3. 3 wires with alligator clip ends
4. a rubber band
5. a small measuring spoon
6. salt
7. A multimeter to measure current and voltage (I got mine at Radio Shack)

Setup

1. Bend the paper clips so that they can hang over the edge of the bowl and have one end in the water. Hang them on opposite ends of the bowl (just make sure they don’t touch).
2. Clip an alligator clip to each paper clip
3. Now your cup of water has two wires coming out of it with unused alligator clips. Clip one of these to a lead for the multimeter, then clip your 3rd (unused) alligator clip wire to the other multimeter lead
4. Now your cup of water has more stuff coming out of it, but you still have just two free alligator clip leads. Use the rubber band to secure one of these to the ‘+’ and ‘-’ terminals of your battery.

OK, I’m sorry, that is a mouthful. You just want to make a circuit that runs current through the water and multimeter in sequence. There are loads of ways to do it, just play with it. Here you can see what it looked like once we had it running (yes we were still wearing pajamas):

The Experiment

Once we got this set up, we set the multimeter to measure the current. For distilled water, we saw almost no current at all — just about 0.01 mA. Then we started adding 1/4 teaspoon of salt at a time. For each addition, we’d take the wires out of the water, add the salt, stir, replace the wires, and wait 2 minutes to measure the current.

We noticed that the current always started high and dropped steadily. At 2 minutes it was still dropping, but seemed fairly stable. We probably should have waited longer.

The first spoonful of salt gave us a big jump in current up to 0.68 mA. Each spoon after that gave us progressively smaller increases in current. After 4 additions, we had these results:

We plotted our results on graph paper and drew a nice curve through it. We extended the curve to 1 1/4 teaspoons and guessed that we’d see a current of 1.24 mA. The actual result came in at 1.27 mA. Not too bad for sloppy techniques and curve-fitting by hand.

Here is a graph of what we recorded:

Notice that the best-fit curve I plotted with gnuplot slightly overestimated the current for 1.25 tsp of salt, while our hand-drawn curve underestimated it.

After that, we dumped the salt water and tried out tap water. We found that it did cunduct electricity and we got a current of 0.27 mA. Going back to our graph, we figured that meant there was about 0.07 teaspoons of “salt” in each cup of our water. Of course we don’t know that it is salt, but there is something salt-like in there.

We also mixed up some really salty water and let the current run for a long time. We watched the bubbles forming on one of the paper clips and saw the water turn yellow and develop an awful smell. Gunnar concluded that it must be the Chlorine separated out from the salt and that Chlorine must stink.

Kids interpretations

I did most of the data recording and graphing. Gunnar read off observations from the multimeter and kept time. Soren help mix and stir. They both took notes while I took my notes. Here are some of Gunnar’s notes:

And here are Soren’s:

They seemed to get the idea that the Na and Cl atoms split apart in the solution at least.

Improvements for next time

If I were going to do this again, I would be a little more careful. Most importantly, I would

• Weigh the salt instead of using a measuring spoon. Then we could really talk about the concentration of salt in the solution.
• Wait longer for the current to stabilize before recording results

I also want to see if I can dump the data into our XO computer so a capture the way the current drops over time. I have no idea why that happens and it seems worth investigating.

I’m still here. Barely.

I’m spending all of my time clearing out the house and sorting through our belongings. Since we are downsizing from a 2800 square foot house to a minivan I have to be pretty vicious about it. When I look at something I have to decide “Would I take this in the car?” and “Would I pay to store it?”. Usually the answer should be no.

Still, I’m not doing a good enough job. We’ve packed 22 boxes for storage so far, and seeing them all stacked up is a little overwhelming. I thought we were going to start traveling light!

What is so important that we’ll pay to store it? 14 boxes are full of of books. Most of these are technical books that I really don’t want to part with, and all the books we kept are really good. I still have about 30 books on my shelf that I just couldn’t seal away yet: Fulton and Harris, Davenport, Weyl, Chalmers, Plato, etc.

I have a box of winter clothes that I plan to open in the fall, and I have a box of ski clothes and accessories for all of us that I hope to use again one day. I have a box of summer clothes that I didn’t like enough to pack for our travel. I should probably just take this box directly to the Salvation Army. We have some linens and comforters, shoes, old records, rollerblades, and more.  So much stuff, and we don’t really need much of it at all.  We just want it.

One of the most interesting things I’ve done in this process was to sort through my old grad school notes. I dumped most of them, but only after I scanned through each page. When I found something interesting I tore it out, labeled it, and filed it. It amazed me how much of my time was spent on completely mundane problems like “how can I get this paper published?” or “how can I convince my advisor to let me graduate?” Less than five percent of my notes were interesting, but they were really fun to read. I was trying to model the way our brains recognize and reproduce hand movements. I was playing with self-assembling DNA tiles (look here to see how it’s really done). I was exploring how zeta-functions popped out of physical systems as thermodynamic quantities.

As I looked through all of this, I realized how much time I had wasted. I should have been doing stuff that was interesting instead of making stuff I had already done look interesting. I was putting more value on the degree than I was on the study. No wonder I got bored and dropped out. Ah well, all those old drafts and notes are now being recycled.

But now I should get back to work. I have a few carloads of stuff to give away and I lot more packing to do.

We have too much stuff. Our stuff ties us down. Whenever we go somewhere, we worry about our stuff. Will our stuff at home be OK? How will we carry all our stuff that we’re bringing? We want to own our stuff, but it feels like our stuff owns us.

We need to fix that.

We’ve been trying to fix it by getting rid of all of our excess toys and clothes, but that is just dodging the real problem. So we’ve decided to face the real problem head on. We’re selling our house.

I guess I should say that we’re putting our house on the market, and hoping to sell it. It doesn’t seem like there’s much certainty these days. But we’re starting the process.

Where will we go? We’re not too sure. The kids want to see all 50 states, so we may do an extended road trip. We want to go to South America for a few months. We plan to get really good at cooking camp food and cleaning out our tent. We have friends who live in nice places, and we’d like to find short term rentals and hang out with them for a while. We’re planning to be vagabonds.

We’re a little concerned about what we’ll do if we get tired of it. But we have to sell the house before we worry about that too much.

Whenever my kids ask a good question, I like to do my best to give them a good answer. So I was pretty frustrated when Gunnar asked me

If I take one half, plus one third, plus one fifth, plus for all of the prime numbers, will that make one?

The answer is no, the sum just keeps growing, getting larger than any number you can think of. But I have no idea how to explain it to him.

Here I’ll give the simplest proof I can think of, relying only on a little bit of Algebra II. If anyone can do better, please let me know! I’m not aiming for rigor here, although I’m happy to provide rigor on request. I’m just thinking about how I can get my boys interested in this stuff when the time comes. Or at least explain to them why I love it.

Adding All the Numbers

I want to start by playing some games with multiplication and addition.

First, notice that

For the moment, don’t try to add the numbers. Just think of addition as a way to write a list of numbers. This example shows how we can use multiplication to take two short lists and make a longer list.
What if we want a 4 to show up in the list? We can just write

Now we are missing 5, so we fix it like this

We can keep going like this: multiply by (1 + 7) to put 7 in the list, multiply by (1 + 2 + 4 + 8 ) instead of (1 + 2 + 4) to get 8 in the list, multiply by (1 + 3 + 9) instead of (1 + 3) to put 9 in the list. Play around with it, and you’ll see that

We can keep going like this, and will hit a hole in our list every time we hit either (1) a new prime number, or (2) a power of a prime number that we’ve already seen. We know how to fill in those holes

1. When we hit a new prime number, , multiply by
2. When we hit a power of a prime number that we’ve already seen, say , we will already have a term in our product that looks like , so just add to this term, giving you

If we keep going this way, we’ll put every number in our long list and no number will show up twice (this is another way to say that numbers can be uniquely factored into primes). We’ll end up with this interesting equation

Of course this is meaningless. Interesting, but meaningless. But interesting is good enough for me, so let’s charge forward.

In my first post in this series I introduced the Sigma notation for sums, so we didn’t have to write out all of those +-signs and ‘…’ marks. We can use that to make our formula a little more concise and precise here:

But we still have an infinite number of multiplications cluttering up the equation. We can use the Greek capital Pi, , just like we used Sigma before to take care of this. So, for example

is just a shorthand way to write

Using the Pi notation, we can write

Still meaningless, but it looks impressive!

Back to the zeta function

Now let’s make it meaningful. We weren’t interested in adding up whole numbers anyway. In our last post we found that when we can define the Riemann zeta function

Well we can play the same sort of list-making games with the numbers that we did with the whole numbers. For example

And so on. Try it.

This happens because the function is a homomorphism: . [Excercise: Verify this.] If we repeat our argument from above, we’ll be able to write down a very fancy looking formula that actually means something

But wait a minute. Look at all of those sums on the left hand side — the ones that look like . These are just geometric series, and we know that this just equals ! So we can simplify the expression and write

This is called Euler’s product formula for the zeta function.

This is starting to look good. We’ve found a way to write the zeta function that only involves the prime numbers. And it is a neat expression, an analytic way to say that every number can be uniquely factored into primes. Any number that can’t be factored wouldn’t show up in our list-building exercises, and any number that can be factored two ways would have shown up in our list twice. I never proved that this can’t happen, so you might want to think about it a little more.

Turning products into sums

The problem is that we have an infinite product, and I had set out to study an infinite sum of primes. Luckily there is a function called the logarithm that turns multiplication into addition: . I’m going to use the natural logarithm because, well, it’s natural. (Unless you’re doing computer science, when it’s natural to let e = 2.) Applying logs gives us the equation

But it’s easy to check that (just look at ), so we can now write

Approximate, simplify, and control the error

Now as p gets big, the value is going to get really small. We know that , so we would expect to be really close to zero. If we knew how fast the logarithm function was changing near zero, we could get a pretty good estimate for these terms in the sum. Well here is a graph of the function and the function near .

It turns out that is a pretty good approximation. If we could just swap this function for the log function it would be great! We’d have

Which is just the sort of sum we wanted to study. This is why we used the natural logarithm instead of , say a base 10 or base 2 logarithm: the natural logarithm is the only one that is well approximated by a line with slope 1. If we used a different logarithm, we’d get some messy looking constants in our approximation.

Now we should make sure that our approximation is good enough!

To do that, let’s plot the error we make when we swap for and compare it to the function .

Notice that if x is close to 1, our error is much bigger than , but as long as , the error we make is certainlyless than . For prime numbers p, is always going to be less than a half, so we can write

But now we know that because the second sum ranges over all of the numbers, not just the primes. We also know, from our last post, that

So we can write

And we have a pretty good idea how big our sum over the primes must be.

Now here is the fun part: , as we saw in our last post. This means that

This can only happen if is infinite.

Earlier I said this means that there are “a lot” of primes — enough to make this sum go to infinity. In a later post I’ll go into some more detail about exactly what that means, but this is enough for today.

Now how do I explain that to a 5-year-old?

As I was sketching out my argument that goes to infinity, I realized that there are a few other infinite sums that I need to discuss first. This is turning into a series of longish posts, so I want to make my goal here clear. I believe that any interested student with background in what americans call “Algebra II” can understand many problems and techniques that are usually reserved for university Math majors and grad students. Most of what I’m saying in these posts can be dispensed with in a few lines of Calculus, but I don’t want to use that crutch. I am writing to that interested student starting out, and trying to convince them that the big ideas are accessible to them now. You don’t need years of training to get to the good stuff, you should spend your years of training doing the good stuff.

Hopefully by writing posts like this and refining them, I’ll do a better job helping my own kids out when they are ready.

On that note, back to work. I’ll start by looking at another sum that goes to infinity, the harmonic series.

The harmonic series

If we want to get a good idea about how might behave, we can look at the simpler and larger sum

This is called the harmonic series. This is just like , except now we are summing over all whole numbers, not just primes. If “a lot of numbers are prime”, then the difference shouldn’t be too great. If “very few numbers are prime”, then the difference should be large. So measuring the difference between these two sums is really the same as understanding the distribution of the prime numbers. More on that later, now let’s get to work.

I claim that the harmonic series diverges. It will keep growing and growing. To see why, let’s break it into chunks. The first chunk will be pretty small: it is just the first term, 1. The second chunk has two terms: . The next chunk has 4 terms: . The next chunk has 8 terms, then 16, and so on. Each chunk has twice as many terms as the one before it.

Now look at the size of the terms in each chunk. The first chunk has one term, and it has a value of 1. The second chunk has 2 terms — and and both of them are certainly bigger than . So the sum of terms in this chunk has to be bigger than .

The next chunk has 4 terms and they’re all bigger than , so the sum of terms in the next chunk must be bigger than .

The next chunk has 8 terms bigger than . And so on. Here is a little picture of the first few terms. The red bars show the values and the green boxes show the lower bounds.

The n-th chunk has terms, and they are all bigger than . This means that when we add up the terms in each chunk we get some number bigger than one half. But there are an infinite number of chunks! If we add an infinite number of times, of course the sum will keep growing and growing. It diverges.

And the harmonic series is even bigger:

So we can safely say that . The sum diverges.

This process of breaking a sum into manageable chunks, each one twice as big as the last, is called a dyadic decomposition.

An almost-harmonic series

The numbers get smaller and smaller, but they don’t get small fast enough to make the harmonic series converge. what if we made the terms smaller? Consider the series

We can play a similar game with this series, breaking the sum into chunks. Only this time we are going to show that the sum of terms in each chunk is smaller than some small number.

The first chunk is easy, it has one term and the sum is just one. In the next chunk, notice that each term is smaller than . There are two terms, so the sum for this chunk is less than .

For the third chunk, there are 4 terms, and each of them is smaller than so the sum of terms in this chunk is less than . For the fourth chunk, we find the sum is smaller than , the sum of terms in the fith chunk is smaller than and so on.

Putting these together we find that

This is just the geometric series we saw in the last post, and int converges to . So we know that

Not only does the sum converge, it is pretty small. The exact value of the sum was found by Leonhard Euler (it was called the Basel problem), and is .

We can also look at series that lie between this one and the harmonic series. Consider any number . We can look at the series

A slight variation of the argument I just gave shows that this sum converges as long as . (Excercise: [1] Prove this for . [2] Prove this for all .) So we can talk sensibly about this sum , and study the way it changes as changes.

is known as the Riemann zeta function. It is the function that lies at the heart of the Riemann Hypothesis — perhaps the most famous unsolved Math problem today. As we’ll start to see in the next post, if we can understand zeta, then we understand the primes.

When we were walking back from the playground on Saturday, Gunnar asked me

“If I take one half, plus one third, plus one fifth, plus for all of the prime numbers, will that make one?”

What a question! The answer is no, when you add it up you’ll find out that the sum just keeps growing and eventually is bigger than any number you can think of. It is infinite. I told him that. He said

“Do you mean it gets bigger than a googolplex? a googolplex googolplexes?”

Yes and yes. And bigger still. I just couldn’t think of an easy way to explain why I know this. In my next post, I’m going to give the simplest argument I know. As a warm-up, I’m going to talk a little bit about a simpler problem.

Some sums are finite: Geometric Series

I didn’t try to explain to Gunnar why is infinite, but I did want to make sure he knew it was a really good question. Sometimes you can add an infinite number of terms and get a finite result. To convince him, I had him stop at the next street corner.

There wasn’t any traffic around, so I said “Let’s just cross halfway”.

Once we were in the middle of the street, I said “OK, that’s half. Now how much is half of what is left?”

“A fourth.” said Gunnar.

“OK, let’s cross a fourth more. There, that’s a half plus a fourth. Now how much is half of what is left?”

“An eighth!” said Gunnar, starting to get it.

“OK, let’s cross an eighth more. There, that’s half plus a fourth plus an eighth.”

And so we continued until we only had a sixty-fourth of the street left. To Zeno’s dismay, we just stepped up onto the sidewalk after that. Gunnar got the point: the process would never stop, but once we added all the terms we would have crossed exactly one street. .

This is an example of a geometric series. To get a general geometric series, pick any number . Now we can look at the sum

If then the series is , which obviously just keeps getting bigger. If , the problem just gets worse. (Excercise: whate happens when ?)

But if then this sum will get closer and closer to a number that we will call the “sum of the series”. We can figure out exactly what this number is. If we call it then:

So , or

All these long sums get tedious to write, so we will start using some shorthand. We will use a Greek capital Sigma, , to denote a sum, like this:

(Remember, ! ) For an infinite series, we’ll need to write

It looks fancy, but it’s just a way to save some writing and make things more precise. Whenever I put those “…” marks in, I was trusting that you’d know what was supposed to come next. That will cause trouble down the road, and our new notation avoids it. Have a look here if you want a bit more detail.

So let’s get back to our street crossing experiment. At each step, we were going half as far as we had in the previous step. So . As we crossed the street, we crossed half, then a fourth, then and eighth, and so on. So we can write the number of streets we crossed like this:

This almost looks like the geometric series in our formula above, but it starts with one-half, not one. We’ll fix that by adding one to both sides:

So

Whew! I’d have been a little worried otherwise.

When I write about prime numbers in my next post, we’ll use these geometric series a lot. It’s amazing how many places they pop up.

Keeping her vision for the country at the forefront, Hillary has endorsed John McCain. Classy.

It’s funny because most of the Texans I’ve talked to thought that Hillary’s “3AM ads” were for John McCain anyway. Oh well, there’s no need to worry much about all of it, because the Onion already broke the big story:

Diebold Accidentally Leaks Results Of 2008 Election Early

“I want to go to college”

That’s what Gunnar told us out of the blue a few days ago.

“Fine,” I said “But you’re going to have to convince a college that you need their school to learn what you want to learn. And there are going to be things they expect you to know before you go.”

“Like what?” Gunnar asked.

“Well, Algebra and Geometry are a start.”

“OK, I want to learn Algebra. Can we do an Algebra class?”

I wondered what I had just done. But I figured hey, basic Algebra is simpler than most Arithmetic so it just might work.

I like talking about things for a while before we put anything on paper, so our first Algebra class was just a discussion. I started with a question that went something like this:

I’m thinking of a number, and 4 times this number plus 3 is 103. What is it?

After a little bit of thought, he said “It’s 25!” That went pretty well, so I asked a few more — always set up so the arithmetic was easy — and he seemed to be able to answer that sort of question consistently. I told him that in the next class we’d need to start learning how to write these problems on paper because we wouldn’t be able to do them in our heads when they got tricky, then wee called it a day.

For our next class, I asked him to think of an Algebra problem for me. He said “I’m thinking of a number and 5 times it plus 5 is 105.”

I went to the easel and wrote .

I told him that I would call the missing number , and that what I just wrote means that 5 ’s plus 5 equals 105. Then I started writing out the steps: subtract 5 from each side, simplify, etc.

Gunnar just said “Well I already know it is 20.”

I finished writing it out anyway, but he made it clear that he doesn’t like showing unneeded work any more than I do.
So I wrote a more complicated problem on the easel — — only this time I didn’t say anything.

He stared at it for a while and said “Well… I don’t know, but that means that 99 is made of 3’s”

So I asked him “How many 3’s make 9?”, “How many 3’s make 90?”, and “Can you put those together to figure out how many 3’s make 99?”

He got a big smile and said “I know, is 33!

I went ahead and showed him how to write out the steps to solve the problem, and this time he didn’t interrupt me. I told him how we were really solving a Geometry problem — finding out where two lines intersect — and plotted the lines and on a big sheet of paper. He used it to solve a lot of other problems, like , , etc.

Then I screwed up. I wrote a few “easy” problems on the easel, like and asked him to write the answers. He started writing random stuff that looked like my work and was clearly getting frustrated. When I offered to help he stomped off and said he didn’t want to do it anymore. He hated the idea that he knew the answers but was getting the problem wrong.

I hated that when I was in school, and I just did it to my son.

We started doing taekwondo practice and he cheered up again. After that he told me that he didn’t like Algebra that much because he could only solve problems with little numbers.

“I bet you can solve 3,000,000 times a number plus 1,000,000 equals 4,000,000.” I said.

“Those are just like little numbers.” he humphed.

“Well, I can teach you how to use a calculator to solve the problems.”

“But then I’ll have to take a calculator with me EVERYWHERE” he lamented.

“OK, then do you want me to start teaching you how to do Math with bigger numbers?”

“Yes, let’s do that next.”

So we’ll be learning Arithmetic after all. But it was a fun diversion, and it seemed to stick with him. This morning I saw him plotting lines and marking intersections, solving his own Algebra problems.