This fact shouldn’t have shocked you.

Today we’ll see that we can raise an irrational number to an irrational power and get a rational number.

The result is surprising but the proof is, in my opinion, downright delightful.

I first read this proof in an article by Paul Halmos.

Yesterday we showed that the square root of two is irrational. What about the square root of two raised to the square root of two power?

If that was a rational number then we’d be done.

If not, take this number (the square root of two raised to the square root of two power) and raise it to the square root of two power. This is a rational number – it’s the number two. (a^b)^c = a ^(bc). Therefore we’re done.

I love this proof.

What I love the most is that we don’t know whether the square root of two raised to the square root of two is rational or not but in either case the theorem is proved.

A rational number can be expressed as a fraction p/q where p and q are integers, q is not zero, and p and q have no factors in common.

In other words, p/q is a fraction that can’t be reduced.

We like rational numbers. Rational numbers, as we also learned in school, can be expressed as decimals that terminate or repeat.

What if we want to live in a world of rational numbers?

Add two rational numbers together and we get a rational number. The same is true of subtractions, multiplication, and division.

The Greeks noticed that if you have a square that has length one on each side, the diagonal was not rational.

The Pythagorean theorem tells us that the length of the diagonal squared is two. So the length of the diagonal is the square root of two.

Another way to write the square root of two is two raised to the power of (1/2). It turns out that the square root of two is an irrational number.

So this would mean that a rational number raised to a rational power is irrational.

Tomorrow we’ll look at one of my favorite proofs – the proof that you can raise an irrational number to an irrational power and get a rational number – let’s finish today with a quick look at the classic proof of how we know the square root of two is irrational.

If the square root of two is rational then it can be written as p/q where p and q are integers with no factors in common.

This means that 2 = p^2/q^2 or, multiplying both sides by q^2, we see that 2 q ^2 = p^2.

Now, this means that p^2 is an even number. The only way for p^2 to be even is if p is even. So p is an even number and therefore it can be written as two times another number. In other words there is an integer r so that p = 2 r so p^2 = (2 r)^2 = 4 r^2.

2 q^2 = 4 r^2. Dividing both sides by 2 we see that q^2 = 2 r^2. This tells us that q^2 is an even number which tells us that q is an even number.

So this tells us that p/q wasn’t in lowest terms since both p and q were divisible by 2. Therefore the square root of two is not a rational number.

You have to love a proof like that.

Twice each day I’d walk across the square from my apartment to enjoy a cup of coffee with a diverse group of friends.

One of them was a priest. He was one of those “one of the guys” priests. You never felt awkward around him. You never felt there were topics you couldn’t bring up when he was there.

Every once in a while the priest would talk about good and evil. He believed deeply in the strength of the force of good. I remember being surprised at how strongly he believed in evil. A priest. He believed that evil was all around us and yet he believed in the power of good over evil.

In mathematics there is a lot more evil than good. We just spend most of our time focusing on the good.

For example, consider the graph of the function f(x) = 3x.

The graph is a straight line with slope three that passes through the origin. If you know what these words mean, the graph is continuous and smooth everywhere.

Let’s define a new function g that has the same value as f everywhere but x = 0. When x = 0, choose any value for g(0) except 0.

Now g is not continuous or smooth at 0. This is true for any value you can choose for g(0) except 0. In other words, of the infinite values available for you to choose for g(0), only one of them makes g continuous. This is true about every other x value.

Much more evil than good — we just spend our time on the good.

As another example, there are a lot of rational numbers. You could, however, count them.

Actually, you couldn’t. But you could organize them in a way where there is a first, a second, a third, … you could build a list of all of the rational numbers so that all of them appear on the list at least once. We say that the rational numbers are countable.

Real numbers can’t be counted. There is a famous proof by Cantor that shows that any time you think you’ve listed all of the real numbers, I can find one that is not on your list. So there are more Real numbers than rational numbers.

We’ll look at proofs on rational and irrational numbers in our next couple of posts.

Here’s another high school flashback. We punish students by making them factor quadratics for several weeks during Algebra and Algebra II. We give them ax^2 + bx + c and ask them to factor it as (Ax + B) (Cx + D).

The sad thing is that there are relatively few choices of a, b, and c that we can factor by hand. So we teach students the few ones we know how to do handle and then we only ask them those. For most choices of a, b, and c — even where a, b, and c are positive integers — we’re stuck using the quadratic equation to find the roots of the quadratic y = ax^2 + bx + c.

More evil than good.

We spend most of the student’s time on good because, when we go to higher ordered equations, we don’t have formulas like the quadratic equation. By the way, the reason we don’t have these formulas is not that no one has figured them out yet. We don’t have these formulas because someone has proved that they can’t exist.

I first heard this explained by Joan Countryman twenty-some years ago when she held up a calculator and said “this calculator could get a B+ in my Algebra II class. Shouldn’t we change what we’re teaching?”

Imagine we live in a world where only rational numbers exist. Tomorrow we’ll see that we quickly are led to a world where they aren’t enough.

Mathematics, as Paul Lockhart writes in A Mathematician’s Lament, is “wondering, playing, amusing yourself with your imagination.”

If you haven’t read Lockhart’s piece, please do so. It’s way more interesting than anything I am about to say. It begins with first a musician and then a painter waking up in a cold sweat after their respective nightmare’s on how their field has come to be taught.

Lockhart compares Math education to these nightmare scenes of music and art because he sees math as being the same sort of endeavor. Math is a playful, creative art.

Only later after we have discovered something new and worth sharing do we go back and formalize it. We make sure that what we stumbled on is really true. Then we present it to others in a way that they can see its truth.

Unfortunately, this means that we often present new findings to others in a way that they don’t see its beauty.

That’s a shame.

There have been a bunch of blogs recently that have talked about what should be taught in math classes. Most of them focus on answering the question “when will I ever use this.”

My daughter has been taking band since fifth grade. At her school they have music every day. In the high school they begin the year with marching band and then audition and split into five smaller bands. They play an array of interesting music and, in my opinion, play well beyond their years. Their top band was the only band in regionals to receive the top marking from the judges.

Of the three hundred kids taking band this year, a handful will continue in music or music education beyond college. Most of the rest will never pick up their instrument again. Most will say “I used to play trumpet (or flute, or …).”

The marching band plays at half-time of the football games. The football program has well over a hundred kids on the teams if you count varsity, junior varsity, and the freshman team. The team is not very good. Two years ago we only saw one victory. Last year they won a couple of games and almost won a couple more.

Some of the students will play in college. It would be very unusual for any of them to play beyond college.

The band kids and the football kids never ask their teachers and coaches, “when will I ever use this”.

Never.

Both sets of kids have to show up in August two weeks before the school year begins to prepare for the season. Families cut their vacations short so that their kids can spend all day out in the heat of the summer working hard at something they’ll never use in life.

Never.

And yet they come into my math class and raise their hand half-way through my demonstration of the mean value theorem to ask me when they will ever use this. They want me to justify that I’m not wasting their time.

I never mind answering this question.

Never.

It allows me to give them a glimpse into the beauty that I see when I play with mathematics. To me math is a game. We have a set of rules and we take turns playing this game. When the rules change — say we relax that fifth postulate so that a line can have no parallel lines through a given point — we are playing a different game.

The reason for learning mathematics should never be restricted to how it will help us do something else.

Never.

The hard part is that math is so darned useful. There is math everywhere. It’s easy for us to think about learning the math we need to do science or economics.

A recent blog post questioned why so much of the math we learn is continuous as opposed to discrete. We should learn more discrete math, but we shouldn’t ignore the continuous. The continuous describes a lot of the world we live in.

We pour a cup of coffee and it is 180 degrees F. When we remember to drink it is is cold. Blech. It dropped 120 degrees while we weren’t looking. It didn’t drop suddenly. It followed Newton’s Law of Cooling. It’s kind of cool that the temperature of the coffee at any given time is roughly governed by a law that only describes the rate at which the liquid cools.

Here’s the email I received that set me off on this post. Several people thought that I would be interest in this mathematical oddity and sent me this:

It made me sad.

The email described this amazing phenomenon of having five Fridays, Saturdays, and Sundays in the month of July. The text of the email said that this only happens every 823 years.

The people who sent it to me were impressed that they were about to live through an event that wouldn’t happen again for more than 800 years.

Look again at the calendar.

You can see that this happens any time the month of July begins on a Friday. You can reason that this happens about as often as July beginning on any other day. It would be shocking to think that this won’t happen for another 823 years. In fact, if July 1, 2012 is Friday this year then it will be Saturday next year. The day on which July 1st falls advances by one day each non-leap year and by two each leap year. On average every seven years July 1 falls on a Friday. In reality, it will sometimes take more than seven years to return to Friday and sometimes take more – but these numbers are close to seven and not close to 823.

In order to confirm that July 1 advances by one day in a non-leap year and two days in a leap year I had to look at the actual calendar. There I discovered that the premise of the email was wrong. July 1st doesn’t fall on a Friday this year.

In other words, the claim that this is the only time in your life that this will occur is wrong on two counts. First, it’s not going to occur this year. Second, if you’re reading this blog post it is likely to have already occurred in your life and it will happen again in 2016.

This, to me, is mathematics. Playing, confirming, questioning — even in my real life. It gave me a much sadder answer for people when they ask me “when am I ever going to use this”.

Never.

And that’s sad.

Pull your trumpet out of the drawer and play it now and then. Toss a ball with a neighbor.

Look at the world with the eyes of a mathematician “wondering, playing, amusing yourself with your imagination.”

Actually, Apple’s explanation is more clear than Godin’s. Apple explains in their terse explanation that they don’t allow linking to any website that sells electronic publications.

I know this because I got the same error message on one of my books last Friday.

I had just re-uploaded my book so that I could make my Kiwi book available in more markets. Within ten minutes I got a note from Apple telling me that my book was rejected because of links to Amazon.

Like Godin, I was initially a bit put off. The books I was linking to were (1) not available as pubs and (2) not available from iBookstore. In other words, I hadn’t seen any conflict.

On the other hand, I’ve learned that I’m not going to convince Apple so I removed the links and uploaded the fixed files within five minutes of receiving the note from Apple.

It’s been five days and Apple still hasn’t rerun my files to verify the problem isn’t there. My books are still pulled from the iBookstore. I hit the “Contact” button as instructed and was told that if it takes more than ten business days I should contact them again.

Two weeks. They found the issue in ten minutes, I replied five minutes later and it’s going to take them up to two weeks to run that same ten minute check to put me back on the store.

I actually have more issues with the length it times to respond than to their rule about links to Amazon.

They could fix the Amazon rule. They could decide that they will allow links but Apple gets listed as the affiliate so that Apple at least benefits from the link. I don’t have a problem with that. They could allow links to books not available on iBookstore. Probably difficult to enforce but I don’t mind that. I don’t even mind them pulling my book if the linked book becomes available on iBookstore. They could just decide they don’t care where links go – if the deals are better for the reader then it’s better for iBookstore. This is the lesson from Santa in Miracle on 4th street.

I don’t know how they can fix the long wait time to get my book back on the store. I don’t remember Santa having an answer for that one.

By the way, see where it says “Pending” in the status field in the screenshot? That is my reminder of how long it’s been since I did all that I can.

I want to take my iOS or Mac app and easily generate a file that I can send to my local translators. I then want to take their translations and easily incorporate them in my project file.

Of course, easy means different things to different people. Some folks would say that I can do this now. I can. I can used something called NSLocalizedStrings in my code and I can open up a terminal window and run genstrings against the files that contain these NSLocalizedStrings. Then I can take the file created by NSLocalizedStrings and add them to my project. I then select this newly added file and add a localization. I select it and add another localization. I send these files to my translators and then add them back in when they have finished with them.

I don’t expect the first step or the last step to change, but I would like the steps in the middle to become easier. Apple clearly cares about localization. It’s been one of their distinguishing characteristics for as long as I can remember. It’s time for them to bring the tools up to date so that we can embrace languages other than our own as well.

There is a bit of an irony in this. (Uggh – I’ve used irony twice in a week. That makes me uncomfortable.)

I released my first book in my new Editor’s Cut series on Wednesday and so far the only complaints I’ve received is “why isn’t this available in my local store.” Some have asked why I don’t care about their country and others have asked don’t I want to make money from their country.

I do care about these countries. I just didn’t understand the interface when I was uploading my book and so I didn’t select countries where English wasn’t the native language. I didn’t know that I could offer my book in English in a country where English is a second or third language.

Also this isn’t just about English. It turns out I left off countries that do have English as their primary language.

I’m sorry about that. Really.

I’m working to correct that right now and I really do apologize. I do care about your country.

It also amazes me how easy Apple has made it for me to offer this book for other countries. It’s just a checkbox. I hope to be checking yours today.

I’m enjoying this end of the process and learning a lot every day. Thank you for your patience as I make a bunch of mistakes and thank you for pointing out the issues I need to address.

As I mentioned yesterday, I’m launching a new series of short, focused books.

This first one scratches my own itch.

I came from Java development and was a strong believer in TDD. Test Driven Development, for me, wasn’t about testing my code so much as it was about driving a better design of my code by using it first.

I’ve tried to embrace OCUnit for Unit testing Cocoa and iOS apps but I never like STAsserts and think that Apple should have given us a test runner by now.

A couple of years ago I saw David Chelimsky speak at a small conference in Chicago about RSpec. His book, The RSpec Book was in beta so I picked up a copy and it made a lot of sense to me.

Allen Ding and Luke Redpath have created a version of RSpec for iOS development named Kiwi that sits on top of OCUnit but uses Behavior Driven Development instead of TDD. It feels right to me.

My book Test Driving iOS Development with Kiwi is designed to get you up and running with Kiwi quickly. You won’t peer into every nook and cranny – in fact, you don’t peer into any nooks or crannies. This is the book that lets you spend half a day exploring Kiwi and deciding if it’s for you.

I hope you enjoy.

The guys I worked for felt that separate imprints were too ambitious and that we should aim for new series instead. They also felt that two was too much and that I should first get one up and going.

Both made sense, and after almost two years of talking about it we launched the series. During those two years we each made a number of compromises. In the post mortem for the series after they killed it two years ago, one of them observed that it was like boiling a frog. We had made many decisions that individually weren’t bad but collectively left the series as not worth continuing.

They were right. The series as it then stood was not consistent with my vision and it was not worth them maintaining. It no longer made sense for either of us.

Unfortunately, the series had a very personal subplot for me. I wanted to fill it with all of the books I would have wanted to share with my youngest daughter had she lived.

Today is the sixth anniversary of Elena’s death and although the pain is still real, I have had a little more space to think less emotionally about a book series. First, and hardest to realize, is the fact that she died before she was seven. I can’t possibly know what books she would have grown up to be interested in. Second, the series must be steered by what I’m interested in and not by my trying to guess what someone else might be interested in.

I’ve been working on a new series of books that I’m writing and publishing through Apple’s iBookstore using the iBooks Author tool.

Coincidentally, Apple approved the first book in the series late last night.

Perhaps it’s even ironic – although I try to be careful in using that word. I have made the decision not to base the series on what Elena might have been interested in and I end up launching it on the anniversary of her death.

So it goes.

I’ll talk about the book and the imprint more in the coming week.

Now I have memories to share and pancakes to eat.

We no longer watch enough movies that the names mean the same things to us that they used to, but often there’s something or other that we see in the credits that’s interesting. Usually it’s something from the soundtrack that sounds familiar but we can’t put our finger on who that was.

I’ve long hated the practice on US television to shrink the credits for television shows and movies so they can run a promo for an upcoming show they want to promote. Even worse, the credits are sped up so that even if my eyes were good enough to read the shrunken and distorted type on the screen, the moment passes so quickly that I can’t read fast enough.

In An Attribution Failure Theory, Duncan wonders if this sort of model leads people to not give credit where credit is due.

He writes about the problem that there are a huge number of creatives who use others’ works as part of their own and aren’t giving credit where credit is due. Worse, he says, “many of them when confronted with the problem, just shrug and don’t see the issue.”

He’s right. They don’t see the issue.

Yesterday I followed a link I won’t share to a woman’s blog to read what she had to say about following a bread recipe I happened to like. I expected her link to talk about her experiences with the recipe, and it did. But it also reprinted the recipe. All of it. It reprinted all of the ingredients and the directions for how they were supposed to be mixed and treated right up to the time they came out of the oven.

She gave credit to the original.

To me, that wasn’t enough. She went on to reprint his work without permission. She even created a printer friendly version of his recipe.

Now no one needs to buy his work – they can just read hers.

The next step is inevitable. I’ve seen it over and over. Someone quotes her work. That person may or may not provide attribution to her contribution but they inevitably leave out the original author of the recipe. It’s too messy to quote the author that another author cites. Now the original author’s work is being reprinted without attribution or concern for how it might effect his income.

More importantly, just as I want to know who performed that piece from the movie’s soundtrack, I want to know whose recipe I may or may not be following.

One of the things that’s striking when you watch credits at the end of a television show or movie is how many people were involved. This wasn’t just a period piece with a few actors. A group of people dressed the actors while another group dressed the set. There were people running the cameras who pointed it at people and sets that other people lit and moved and there were other people deciding which cameras to use.

I agree with Duncan that credit is easy and its the least we should do. I think, though, credit is not enough.

Maybe credit is the wrong word for me. Maybe what I want is an appreciation of those whose work we enjoy and build on.

In comparing the EULA with the agreement I’ve signed with other publishers, I decided that Apple’s EULA wasn’t so bad.

I don’t believe the iBooks Author EULA has anything to do with Apple wanting to make money off of your creation. I don’t think it’s even about keeping you from creating for the Kindle bookstore with their software either.

Apple is wary of allowing authors to “Write Once, Publish Anywhere”.

Back in the early days of programming with Java, “Write Once, Run Anywhere” or WORA was the central Java value proposition. Write your app in Java and you’ll be able to target all of the platforms with a single code base.

Do you remember those client apps that ran on the Mac in the old days? There weren’t many of them and they either looked like Windows apps on the Mac or they were just plain hideously generic. There was nothing particularly Mac about them.

So Apple has released iBooks2 and created this new freely available tool for producing high quality books that include video, audio, slideshows, photo galleries, 3D images, annotated images, and more.

Apple wants to see high quality interactive books that include these widgets. If you use iBooks Author to target many different platforms you’re going to have to only include the least common denominator of features. If any one of the platforms you’re targeting don’t support a particular feature then you can’t use it.

Suddenly, the books produced for their store using their tool aren’t so special anymore.

You can see that in the initial offerings from the text book publishers. The digital books are essentially paper books that you can load on your iPad.

That’s not very exciting.

I’ve been asking the question from the opposite perspective. If my book will never be printed on paper, then what can I do differently? How can I tell stories that better explain the topics I’m teaching?

I may want to target multiple platforms but I completely understand why Apple wants me to deliver the best book possible for their platform.

It’s the same difference of opinions they had with Adobe. Adobe wanted the Flash experience to be the same on all platforms. Apple wanted the experience on the iPhone to be consistent with other iPhone apps.

For now, this approach is meeting my needs. I’m excited about what I can author with Author.