Oh, no... here she comes, Trevor thought. And she looks bored.

Brianna flounced into his room, throwing herself on Trevor's bed.

Trevor tried his best to ignore her, pretending to be distracted by the city he was constructing.

"Hey, Trevor... do you want to do some math puzzles?", she asked lazily,

not really even expecting an answer.

Trevor rolled his eyes. "Look out the window, sis. See the sunshine? It's S-U-M-M-E-R.

Summmm - mmmmer... I don't do math in summer."

'Yeah, but look at this book I found: 'Tips, tricks and games to tickle your imagination.' Don't you like imaginative games?”

"No", Trevor answered. "I play with my Legos. It's practical, manly stuff.

No imagination required. Just practical, manly ability."

"Yes, that's what you'll be", Bri began taunting. "Practically a man with practically

no imagination."

Trevor opened his mouth to retort, but just then, Djinn popped in. He was really, really agitated.

"I need your help! There's a war brewing!"

Now this sounded like manly stuff. "A war? Count me in!", he said, dropping his Legos.

"Where are we going?", both children chimed at once.

"We're taking a trip to the Imaginary Plane!", Djinn told them excitedly.

"A plane trip? Cool." Trevor paused. "Wait a minute." He looked skeptically at Djinn.

"This doesn't involve hieroglyphs, does it?"

"No, no, no. No hieroglyphs. Just imagination, which I know you have plenty of. And logic.

All we have to do to get there is to bridge The Gap."

"The Gap?", they both chimed at once.

"The Gap of Knowledge. To the World of Mathematics. But there's no time for questions. Let's go."

Travel was a lot easier now that Djinn's finger was fixed. Djinn snapped, and

<vlip>

they found themselves standing in front of a huge chasm.

Trevor gaped. “Uh, Djinn. Why didn't you just <vlip> us all the way to the other side?”

“Can't”, he replied. “On the other side, it's all Logic. Logic is the magic in the World of Mathematics. So my magic doesn't work there. We would get halfway across, and we'd then plunge haphazardly into the depths of the chasm. The only way across is to bridge the Gap.“

“The problem is”, Djinn went on, “that in order to get completely across, you first have to get halfway across. But before you get halfway across, you have to get one quarter the way across. And before you get one quarter of the way across, you have to get one eighth of the way across. And so on, and so on... so you see the problem, you can never get across, because it takes you forever to move the first infinitesimal distance.”

Now both Trevor and Bri were confused. They were sure they had crossed bridges before, and had never encountered such a dilemma. They began to discuss it, out loud. “We've crossed bridges haven't we? And we've never had to piece together the halves before doing it.”

“Aha!”. An impish little creature appeared beside them, pointing an elongated index finger accusingly. “You clearly are a Halve-not! “ He moved his wrinkly puke-colored face in between them, right up to each of their noses, and grimaced as he studied their faces.

“Well”, said Brianna tentatively, backing off a little. “We may have to halve. I'm not really sure. It sounds logical to me that we may have to halve.” She wished she had finished reading that book of math puzzles before embarking on this adventure.

“So it is just YOU”, the imp said, poking fiercely at Trevor.

This was exactly what Djinn wanted to avoid: the war between the Halves and the Halve-nots. They had better think of something quick!

“Wait a minute”, Trevor said. He would not be bullied by this wimpy little imp. “This is the Real World.” As long as he had one foot in the Real World, he had a leg to stand on. Actually, he had a Lego™ to stand on, for he was relying on his knowledge that he had built many bridges out of Legos. And he never had a problem getting all the way across.

He turned and strode confidently to Djinn. “Djinn, would you conjure me up a pile of giant Legos?”

Djinn bowed and complied, and in the blink of an eye, there were fifty six-foot-long Lego pieces piled high at the edge of the chasm. He watched Trevor to see where he was going with this.

Trevor affixed two thirds of the first Lego piece to the cliffs, with one third hanging over the chasm. He then confidently walked out, and attached another Lego on top of that, and so on, and so on, until he had constructed a Lego stairway-bridge that they could all walk across.

The imp watched them with a nasty scowl upon his face, for he had never before crossed the Gap.

On the other side, Trevor turned to look back at their starting point, and triumphantly folded his arms across his chest. “Just like I said. I deal with practical, manly stuff.”*

* Don't try this at home. It only works under conditions of “cartoon gravity”, the physics of which have been explored in great depth by Wile E. Coyote. See bibliography for details.

“Yes, mathematics is practical, manly stuff.”, said a rumbling voice behind them. “But like your bridge, it is also beautiful and connective. It spans both the Real and Imaginary worlds, which are also known as the worlds of Practice and Theory.” They turned and saw a stout, genial-looking man, who waved his hands in a welcoming gesture. “Allow me to introduce myself. I am Mayor Fibonacci, Ruler of Numbers, and you are welcome here in the World of Mathematics.”

Trevor looked around, becoming a bit overwhelmed. “Djinn, I thought you said we were taking a plane trip.”

Mayor Fibonacci laughed. “You are. You're on the plane. But like all planes in the World of Mathematics, this one doesn't start or stop. It's infinite, and it can take you anywhere you want to go.”

How would you like to start with a visit to the Moebius Amusement Park? You can look at your reflections in the Hall of Symmetry. You can slide down an Asymptote. There's even a pi-eating contest that's totally irrational.

Trevor cheered up. “Sounds like fun to me.”

“Lead the way”, Djinn said, and they all happily followed the Mayor down the road.

“Ah”, said Mayor Fibonacci, “just one stop I have to make before we get there... one of our factories is having trouble with their products, and I promised to stop in and see what I could do. It's just down the road a bit, at #1597 Cartesian Way.”

“#1597? Do we have to walk?”, Bri complained. “We are only at #8. It will take us forever to get there.”

“Not the way I count”, smiled Mayor Fibonacci. “I developed a numbering system that ascends much more rapidly than normal.”

Brianna was intrigued. “What do you mean?”

“It's simple”, said the mayor. “What comes next is the sum of the previous two numbers. We'll be there is no time at all.”

Brianna read the addresses as they walked past, numbered exactly as the mayor had said. ”8...13...21...34...55...89...144...233...377...610...987...1597.” Twelve addresses later, they were at their destination.

Just then, Brianna happened to look down at the road. It was made of brick, and painted a bright yellow. Looking back at where they started, she noticed they hadn't walked in a straight line. The buildings were laid out along a spiral. “Hey, Trev”, she said, “Did you notice? We just walked around a spiral! And we've been following a yellow brick road.”

Trevor smirked at her. “Yeah, and, Dorothy? Toto called. You left your mind in Kansas.”

Brianna was too busy hopscotching along to reply.

The group was welcomed into the factory by an excitable but friendly manager. “How nice to meet you all. Please come in and see our facility. We are responsible for ALL the Real products here on the Imaginary Plane. We're the biggest supplier of products in the entire world!”

“Excuse me”, said Brianna timidly, “but what do you make?”.

“Products”, the manager exclaimed proudly.

“What kind of products?”, asked Trevor, just as confused as his sister.

“Why, every imaginable product!”, exclaimed the manager.

“How can you make every imaginable product?”, Trevor asked skeptically.

Seeing the confused looks on everyone in the group, the manager said, “Let me explain. When you multiply two numbers together you get a product. For example, 3 x 4 gives you a product of 12. So as long as we have a good supply of whole numbers, we can make any product imaginable.”

“So what is the problem? Why did you need me to stop by?”, asked the Mayor.

“Well, the new recycling laws are the problem. We get a product back, but then how do we break it down? We've tried breaking it down into digits, and just look at the mess”, he said, holding up a 12 that was smashed into two pieces, sharp edges and chips gouged into it where it was separated.

“You could try factoring the product”, said Brianna.

“Factoring? That sounds exactly like what a factory should do!”, exclaimed the manager. Then he paused, and frowned. “But that sounds soooo complicated.”

“Not really”, she said. “If the number is even (and half of all numbers are!), it will have a factor of two. Then what's left is a number that's half as much, and you can check if that is even, etc. If not, you can check if it is divisible by 3 and 5 and 7, etc., until you have broken it down into parts that are smaller than 100. Then it is easy to see the final answer using the multiplication tables that you use to build your products in the first place!”

“Hmmm... sounds plausible. But give me an example, so I can see it in action.”

“Well”, Brianna said thoughtfully, “for example, start with 120. It ends in a 0, so you know right away it breaks down into 10 x 12. Both 10 and 12 are even (all numbers ending in 2, 4, 6, 8 or 0 will be even), so we can break it down further into 2 x 5 x 2 x 6. Now six is even, so the final factors will be 2 x 5 x 2 x 3 x 2. So that one product can be recycled into three 2s, one 5, and one 3.”

“Brilliant!”, the manager exclaimed. “And enormously reusable. One product gives us back 5 components that can be used in other products. With no waste at all! I must get right to it.”

“Well, I learned something”, Djinn said happily. “And time spent learning is never wasted.”