Ben Jenkins loved spending his summers at his grandparents' farm. Ben, who would be thirteen in a few days, looked back through the kitchen window at his grandmother. Grandma, who never seemed to rest or stop doing things for him, had already begun baking a special pie for the big day. His mother, on the other hand, seemed quiet and somewhat sad. She was always this way when his father was away. Ben's dad was an astronomer and a teacher at the local college. He was in Chile at an observatory getting data about galaxies that are only visible from the southern hemisphere.

Ben's grandmother was a professor of physics at the same school where his parents taught. She and her associates were trying to find a way to generate and control energy using the same process as the sun, a process called nuclear fusion.

"I prefer baking atoms to pies, except when it's for my favorite grandson," he heard his grandmother say. Actually Ben was her only grandchild, but she called him her favorite anyway.

"Now don't spoil your grandson, mother. He's perfectly capable of helping with the cooking." Ben wondered if his mother would call him back to help in the kitchen. There was only silence. It was unusual for it to be so quiet when these two were together. They were often found playfully teasing each other about the strangest of topics: what is the true definition of "mathematics."

Cathy, his mother, was an assistant professor of mathematics. Ben's parents met in a math class. His father, Jim, liked to joke that "it was love at first sight…of a summation sign." Cathy believed mathematics was an orderly process for developing step-by-step methods for reaching solutions. "Not so!" grandma would assert. "It's a language with which to talk to the universe."

Outside, Ben prided himself on his knowledge of the night sky. After all, his dad had taught him. That is why what happened seemed so strange. He had just thrown his frisbee into the air for his dog Chip to catch. As Chip jumped, Ben's attention was drawn to a meteor blazing a path low on the horizon. It was difficult to see in the twilight, but Ben thought two things seemed unusual. The light tracing the meteor's path did not end at some obvious point in the sky. it continued to rise from below the horizon. He guessed that it had not burned up in the atmosphere as most meteors do, but had actually hit the ground (which would make it a meteorite). If he could find it, it might make a great science project.

"C'mon, Chip," he called. With Chip running by his side he took a shortcut through the corn field toward where he guessed the meteor had come down. He thought about the other unusual thing he had seen. The meteor had followed an erratic path across the sky. Most meteors move in an almost straight line until the friction with our atmosphere causes them to burn up and disappear. Maybe this one had an irregular shape that caused it to move so strangely. It would be an even better exhibit if that were true. From the steep angle of descent he calculated it might have landed just beyond the creek. He and Chip would have to go through the corn field and the woodland that bordered it, down the little gully, across the creek, and up the hill on the other side. If he had guessed right, it would have landed somewhere in the pasture just beyond the hill. That would be about a mile and a half from the farm house.

Jogging the whole distance to the creek took almost ten minutes. Chip had run ahead, however, and was now playing in the creek by biting the water. Ben motioned "come" with his hand. The dog ran to his side wagging its tail. It was precisely then that Ben noticed a faint beam of blue light coming from over the hill. It moved in a circular fashion much like a searchlight. The hum of a machine also came from beyond the top of the hill.

"Why would anyone use a searchlight near the farm?" Ben wondered. Whispering, he said, "Chip, down! Stay." Ben had often wondered why he talked to Chip as though the dog completely understood him, but the dog lay down at the foot of the hill while he slowly climbed up. Reaching the top he took a quick look at the pasture and threw himself on the ground. Grabbing Chip, who now had disobeyed by coming up the hill after him, he pushed the dog down at his side.

Ben crawled slowly to the edge of the hill and looked over. He could hear his heart pounding, his own rapid breathing, and the louder hum of the machine. That was no meteor he had seen. Right before him was a spaceship-what some people might call a flying saucer. The base was round like a saucer. On the top was a small dome that rotated, sending out the beam of light. Several days later he would describe it to his best friend Tim as one-half a tennis ball resting on a coffee cup saucer.

As Ben watched, a small passageway opened in the spaceship. Two aliens walked out. They appeared to be wearing some type of metallic clothing that might have been a space-suit. He could not get a good look at them just yet because it was too dark. Soon the rotating blue light would provide the extra light he needed. He waited anxiously. All at once he saw them clearly. They had faces much like those of pandas only they were light green with black spots. But that was where the resemblance ended. They walked erect and their hands had flexible fingers like his. Well, not exactly like his. It was their hands that Ben had the most difficulty believing. They were extremely thin, more like those of a raccoon than of a person. Ben waited again for the light to return. Yes, he had seen it. The pandas had seven fingers on each hand.

The moon was starting to rise. It would be much brighter soon because there would be a full moon tonight. Ben was afraid he might be seen. The "gurgols" were walking away from him. (He would later come to call them gurgols because all he could understand of what they seemed to be saying to each other sounded like gurgle, gurgle, gurgle.) Just then Chip pulled away from him and charged, barking at the two aliens.

Ben instinctively jumped to his feet and yelled, "Chip, stop!" The dog turned and looked at his master. Not only was his dog looking at him, but Ben immediately realized that the two other creatures were looking right at him.

"Chip, home. Now!" The command was given with such force that the dog immediately headed for the farm house. "Home" was the one command Chip always obeyed, as it usually meant time for food. Just why he had chosen to protect the dog and leave himself alone he would wonder about for years. One of the gurgols headed for its spaceship, running from Ben. The other just stared at him. It and Ben remained motionless.

At least ten minutes had passed since Ben had revealed himself. For those ten minutes he and this strange creature had just gazed at each other. The moon was now up. He could see much better. Something told him as he looked into this creature's eyes that it was as scared as he was. Somehow he just knew it. Both remained quiet. Finally the panda-faced alien began to talk. Gurgle, gurgle, and then a high-pitched whistle. It reminded Ben of the time he had heard on television the sound of killer whales talking to each other. It was definitely not English. It came as no shock then that when he finally had enough nerve to say, "I'm friendly. Can you understand me?", the only answer was a gurgle from a gurgol. Five more minutes of silence passed before Ben would utter what he later considered the silliest remark of his life. He took a deep breath and said, "Take me to your leader." Now the last thing Ben wanted was to go to this creature's leader. Go home, yes. Go to his mother and grandparents, definitely. To this thing's leader, never. But there it was, Take me to your leader. Fortunately, all he heard from the gurgol was a high-pitched whine and a gurgle.

The other gurgol walked back slowly. Ben had forgotten all about it. As he looked, it suddenly occurred to him that if they intended to harm him, they had already had about fifteen minutes in which to do it. But with all the power of their spaceship at their command, neither one had made the slightest move to hurt him. Maybe, just maybe, they were friendly. Ben was starting to feel excited. Just possibly he had made the most exciting discovery of the century: friendly creatures from another planet. More and more he felt the need to try to communicate with them. But how?

Very slowly he knelt down on one knee. "What would Dad do?" he asked himself. His father had often told him that math was the common language among all scientists. With his finger Ben wrote the following in the dirt:

"What is your name? I'm Ben," he said, pointing at himself. The little panda-faced creature pointed to itself and made a sound that sounded to Ben like static from a radio. "Well, you're Pan to me." Again he wrote the same symbols in the dirt. Pan looked without seeming to understand. Ben wondered what was wrong as he stared at his figures. A minute later he shouted, "Of course! They don't understand the equal sign." This time he wrote:

Pan looked and then picked up a small stick. He added the following below Ben's writing:

Contact! Ben realized Pan was trying to communicate the zero to him. Wanting to work in his own number system, Ben wrote:

He then crossed out the X's and replaced them with zeros. Pan watched as Ben went back to his first drawing:

Pan finished Ben's writing. With the stick he wrote:

Guessing that Ben was writing the numbers in sequence, Pan showed that it understood the "=" sign and what "+" means. Ben could feel his heart pounding again. His breathing came rapidly, but this time it was from the absolute joy of realizing he was actually "talking" to a being from another world. That's right, talking, because that is just what he decided this writing was. If it was talk, then it must also be in a language. He stopped to consider what this new language needed.

"Now let's see. I've named numbers. The stick signs were the original numbers. The equal sign made two marks equal the name '2.' That means I've established a way of assigning names to nouns." Ben smiled. He had wondered in school why he had to learn what nouns and verbs were. Now he realized the little he knew about the structure of the English language was going to help him with the most important scientific discovery of his lifetime-a new language in which to talk to aliens.

If a noun is the name of a thing, a verb shows the condition of a thing. A verb might be active, as in the word "running." If he could make verbs, Ben thought, he could put two words together into a sentence like "Pan runs," which consists of a noun, "Pan," and a verb, "runs," that shows what the noun is doing. With sentences he could communicate complete thoughts. With complete thoughts the entire knowledge of two different civilizations could be shared.

"How do I make a verb?" he said to himself out loud. All at once he realized something important. He had already done it. The equal sign doesn't have to apply just to math. He could think of it as the words "is made equal to." It represents a verb phrase all by itself, an entire group of words that describes the condition of a noun more completely. His excitement grew!

"The plus sign doesn't just have to mean add two numbers," he said to Pan. Pan just turned its head to the side, which indicated that he did not understand. "It can also mean 'bring together,' another verb phrase. And if the plus sign means 'bring together,' the minus sign can mean 'take apart,' another verb." Ben had begun to talk to Pan the same way he talked to his dog Chip. Without the slightest chance that the other one would understand, he just babbled away as if they could.

Pan broke the stick it was writing with into two parts, handing one-half to Ben. Pan took the initiative now in shaping this new language. He began to write in a new type of numbers.

Ben looked and looked. One can not equal four or even two. Ben began to wonder if Pan had understood at all. Next Pan wrote:

"Binary," shouted Ben. "But you have it backwards. We write it like this." He traced "4 = 100" in the dirt. Pan indicated he understood by changing "01 = 2" into "10 = 2." Ben came back with "B=2." Pan replied by pointing to himself and writing "B =." Pan held up twelve fingers, keeping his two thumbs down. They had a word now for number base. Gurgols counted in the number base 12. "Maybe 'B' for base could be used to represent the foundation or basic building block of anything," thought Ben. (His dad would later tell him that the gurgols were wise in choosing base 12. Since more numbers divide evenly into 12 than 10, they would have fewer fractions to deal with.)

Ben and the gurgol quickly expressed three more concepts. For the first one, Ben wrote:

Ben wrote a note in the dirt.

After drawing an arrow that pointed at Pan, he walked toward Pan who quickly jumped back. Ben stopped. Pan now drew an arrow pointing in the other direction. Ben followed it back to where he started. They now had the words "to" and "from" in the form of arrows. Those words formed Ben's next message:

Ben was trying to say that he would go home and return, but alone each time. Pan did not understand at first. Later, when Ben pointed toward the farmhouse, he seemed to understand.

Ben sought a symbol for the all important word "yes." He wrote:

Pan did not understand. Ben next wrote:

It meant "OK equals not the word 'no.'" The combination of the two explanations of "OK" seemed to get the idea across to Pan, who replied by writing "OK." His hand slowly, gently touched Ben's arm. Seven alien fingers gently closed around his arm, as if to say that they were friends. Pan let go and Ben started toward home.

When he returned on his bike about an hour later, Ben was carrying many items. Some were in a knapsack on his back. Some were in the basket of his bike tied down with rope. He carefully unloaded what he had brought. There were his computer and monitor, a box of disks, a circuit board that allowed six things to be plugged into it, and the portable electric generator his dad sometimes used when they went camping. Ben set everything up. On went the computer with a Logo disk in the drive. He was going to teach Pan "Logo", a computer programming language. Perhaps this knowledge would expand their ability to talk to each other.

The other gurgol joined them at the keyboard. Ben showed both of them a few commands, which Pan tried. All fourteen fingers began to type out words like "PRINT," "IF," "FD," "RT," "LT," and "REPEAT." As he typed, Pan began to giggle. The other gurgol made the same laugh. The only way to describe the sound, Ben later told Tim, was as if they had been breathing helium at the fair and began to giggle. He loved listening to these gurgols giggle.

Pan made a great effort to get Ben to show him a special command. Pan drew a star map on the computer screen and sent the cursor (called a turtle) to the home position. Next he drew a line in the x direction. Then he turned the turtle ninety degrees to the left and moved in the y direction, ending at his star system. By pointing, Pan indicated that Earth was in the home position. He put a finger on the screen at his star and another at home. With his other hand he typed "= ?" It meant "Help."

Ben understood the math concept called the Pythagorean Theorem. Pythagoras discovered 2,500 years ago that if a triangle has a 90° angle, the length of the longest side of the triangle times itself is equal to the sum of the lengths of the two other sides each times itself. Ben knew how to use an x- and y-coordinate system, in which the x and y directions always make a 90° angle with each other, to find the distance between any two points.

Ben guessed what it was Pan wanted to learn. With the turtle at the gurgol's star, he typed "PRINT XCOR." The number 50 appeared on the screen. Next Ben typed "PRINT YCOR." The number 80 appeared. These values represented the lengths of the two short sides of a 90° triangle. Ben now typed "PRINT SQRT (50 * 50) + (80 * 80)." The Logo program printed 94.3. An excited alien now typed "SQRT = 2RT."

The second root or square root is written this way.

Jim Jenkins looked pleased as he watched his son Ben at work on the new computer. "I want you to know how proud of you your mother and I are. We just wish you hadn't risked the danger of going back alone."

"I was OK, Dad," Ben replied. "They were really friendly. I'm only a little sad that I'll never see them again."

Jim handed Ben the blue object and an envelope. "Or they may return. That is what I want to talk to you about. We've made as complete an analysis of the composition of this object as we could. The decision has been made to return it to you. It was a decision made at the highest levels of government."

"What is the envelope for, Dad?" Ben looked at the picture on the object of a seven fingered hand holding the arm of a five finger hand. The edge near the picture had been chipped off.

"A small piece of that thing is back at the university, son. We'll continue tests. One thing we know is we can't reproduce it with our current technology. It's strength is incredible. But that's not why you are getting it back. X-ray analysis suggests that this thing seems to be a beacon. It could relay a message, possibly in Logo, that only you may understand. This envelope is to be kept locked in a safe. It contains instructions for you in case they set the beacon off. The beacon, however, should always stay with you."

Ben looked puzzled. "How do you know it's a beacon? Why do you think they may come back?"

Ben's father took the object from Ben. "We weren't learning too much with X-rays until we got lucky. Something we did produced just the right effect. Several times, for about five seconds each, the beacon went on. An arrow making a circle pointing to its beginning appeared. In the center of the circle was the number 360. Next to the arrow in a circle was an equal sign. I know enough Logo to figure that one out. It's the Total Trip Theorem of Logo. Return to the starting point in 360 degrees. Something after the equal sign will tell you when they are returning. I'm sure of it. I'm so sure that I convinced everyone not to try to cut open your blue beacon. We might have destroyed the mechanism inside."

Ben took the beacon from his father and put it upstairs on the dresser near the window in his bedroom. Next he went with his Dad to the bank to put the envelope in a safe deposit box.

The next day

When Jim Jenkins returned home, he was anxious to talk to his son. "Tell me one last time, Ben, everything that happened. There may be something important that I've overlooked." Ben told his father the entire story one more time. When he got to the part about his first problem, the cube root of 18, he told Jim that the answer on his first attempt never seemed to come.

Ben's problem was to find the cube root of 18.

"Testing the answer I started to get panicky. I realized that the computer must be in an endless loop, calculating the same thing over and over again. I couldn't find a number that could be multiplied by itself twice-as in 3 * 3 * 3-and equal 18. I had to use the concept of a limit to find the answer. Why did it go into an endless loop?"

Jim Jenkins stared at the computer while the room became quiet for a few seconds. "I suppose it is because the computer keeps track of digits in an unusual way. It must use a small amount of memory for each number, if it is to avoid running out of memory. It must have kept too few spaces in its memory banks for the decimal fraction part of the number to make a calculation with the needed amount of precision. With too little memory, it never could find the exact answer."

"Dad, how does the computer keep track of digits?"

"I can show you best on the computer," said Jim.

Ben's father loaded the Logo programming language. He went to a new page and wrote a procedure called RAISE.

"This procedure will help us. Ben, type any number to two different powers. How about 3 to the third power and 3 to the fifth power?"

Ben followed his fathers instructions. He typed out:

"Now Ben, tell me what you think 3³ times 3⁵ becomes in terms of a power of 3."

Ben stared at the screen and thought carefully about the problem. "That's easy, Dad. I'll bet it's 3 to the fifteenth power."

"Very logical, Ben. Unfortunately you're way off. Watch."

Taking over the computer Jim Jenkins typed out PRINT RAISE 3 15. The answer shocked Ben. It was 14348907.

"I guess the answer has nothing to do with the powers of 3."

"Wrong again, son." Jim then typed PRINT 27 * 243. The answer of 6561 appeared. Next Jim typed:

"Wow! How did you know to use 3 to the eighth power?"

Jim smiled and said, "That's the point of all this. The computer changed a complicated multiplication into a simpler one. It added the powers 3 and 5 to get 8. Look at it this way. We could have written the problem as this:

"Remove the parentheses, which we don't really need, and you see that it's just a string of eight 3s multiplied together. The only restriction before adding the powers is that the numbers, which are called the base of the powers, must be the same. Now guess what 4 to the fourth power divided by 4 to the second power is."

"It's 4 to the second, Dad. Division is related to subtraction as multiplication is to addition."

"Now you've got it." Jim showed Ben two ways of looking at this type of division. First he typed the following:

Next, on paper, Jim wrote:

"Two of the 4s in the divisor cancel two of them above in the numerator," said Ben.

"Now you know why any number to the zero power is the number 1. Look closely at that problem. If it were 4 to the fourth power divided by 4 to the fourth power, we could write it as 1. It is something divided by itself. But that is the same as a power of 4 minus another identical power of 4. Look, it's like this," he said as he wrote:

"In other words when we subtract the powers we get 0. Computers use this method of handling numbers because addition is faster than multiplication for them and the storage needed inside the computer is smaller this way. Here is an example. Fractional numbers are written as a decimal fraction times the base to a power. Like this."

Jim wrote 1.25 x 10⁵. "It looks funny this way but it takes very little space inside the computer." At the computer he typed PRINT 1.25 * RAISE 10 5. The answer 125000 appeared. Next he typed PRINT 2 * RAISE 10 3. The number 2000 was displayed.

"When the computer wants to multiply 125,000 times 2,000, it represents the problem as (1.25 * 2) * RAISE 10 (5 + 3). It's easy and quick to multiply 1.25 times 2. It's even easier to add 5 and 3 to get 10 to the eighth power."

Ben tried the problem both ways at the computer to prove to himself that the answer was the same both ways. First he typed

The answer was 250000000.

ROOT: Ben's procedure

Ben took a piece of paper and a pencil and told his father, "This is how my procedure worked. I needed to find a number which would equal 18 when multiplied by itself twice (number * number * number). I remembered that you told me one way to solve a problem on a computer is to find a way to get closer to the answer. Then let the computer do the same operation over and over again. Here is what I did."

Ben drew a diagram on the paper. "I knew that 2 * 2 * 2 = 8, too low. And I knew that 3 * 3 * 3 = 27, too high. I placed the 2 in the box in the computer for the variable TOO.LOW. The procedure put the 3 in a box called TOO.HIGH. Here is the part about the limit. I knew I could get closer by going exactly in between TOO.HIGH and TOO.LOW. By adding 2 + 3 and dividing the answer in half, I got 2.5. Multiplying 2.5 by itself twice-2.5 * 2.5 * 2.5-I got 15.6. This was still below 18, so I made the procedure put this amount in the TOO.LOW box. Now the right number had to be between 2.5 and 3. Adding them and dividing by 2, I got 2.75 as the new number to try. Multiplying 2.75 * 2.75 * 2.75 gave me a new number, 20.8, to put in the TOO.HIGH box. The answer had to be between 2.5 and 2.75. I just kept letting the computer get closer and closer.

"When I realized that the procedure wasn't going to find the exact answer, only get extremely close, I had the procedure stop looking for a number that would exactly equal 18 when multiplied by itself twice. Instead I set up a box in the computer called TRY. This was to hold the number the program produced on each attempt to reach 18. I had the procedure print out the difference between each TRY and the previous TRY. It got smaller each time. Eventually the procedure produced the same TRY each time. Since a limit is found when a process stops changing, I knew the procedure had reached the limit and an answer very close to 18 was found. Thank goodness it worked!"

Fifteen years later

Ben Jenkins was combing his hair in the mirror attached to his dresser. He looked out the window and back. Then he saw it. Just as his father had described, there was an arrow in a circle with 360 in its center, an equal sign, and the number 1111110 which kept flashing. He yelled, "Judy! Judy! It happened!"

Judy Jenkins was downstairs reading. Ben was so excited he jumped down the last four steps. In his hand was a blue object. Judy had a big smile as she looked at it and said, "I guess we're going to meet some aliens." Ben called his boss and asked if he could work over the weekend instead of that day. He said it was very important and the reason was personal. Would he please understand? Ben had been working as a computer programmer for this firm for some years. It was the first time he had asked for a favor. It was granted. A quick trip to the bank and he returned with the envelope. Opening it he found the following instructions: "1. You are sworn to secrecy. 2. As soon as you understand the meaning of the message, contact the number below and identify yourself as indicated."

That night the number changed to 1111101. The next morning it was 1111100. It appeared to be counting down every sundown and sunrise. Ben was very nervous as he wondered who would answer the telephone.

"This is emergency priority security clearance BJEN 7865."

He heard someone typing something into a computer. There was a short wait. Then, "The President of the United States will be with you in one moment."

"Young Mr. Jenkins, I presume. I'm the third president to hope to receive this call. When do you think they will arrive and where?"

"In sixty-two days, Mr. President, at the old landing site."

"Hmmm. Not much time. It will have to do. I would like to talk to you in person. How about coming to the White House one week from tomorrow at noon?

"Thank you, sir, for the invitation."

"One more thing, Mr. Jenkins. We will want to give the gurgols a gift, some things to symbolize our culture and technology. You may pick up to ten items. They can be books, video tapes, movies-anything you believe they might value." Ben wondered what to bring. What would you bring?