The Emperor watched the procession gloomily from his balcony. The mathematician worked his abacus, sitting in a tangle of scrolls at the Emperor’s feet. He scribbled and muttered while the beads of the abacus clicked and clacked.
“My figures must be wrong!” he shouted at last. “There couldn’t be that much rice in the world!” So he started his calculations again. He stammered and swore and spluttered and broke his brushes and tore up paper and grew more and more frustrated until finally the Emperor had to send him away.
Alone in his private chamber, the Emperor sat with his head in his hands. Soon his treasury would be empty.
He called his lords to his Court. And he summoned Pong Lo.
The young man arrived at the palace dressed richly in the clothes he had bought with his rice. Twenty servants of his own attended him. His step was light and the expression on his face was as good-natured as ever. He bowed to the Emperor and smiled at the Princess, who stood in her place at the Emperor’s side. Chang Wu’s eyes sparkled as she returned Pong Lo’s smile.
“Greetings, most honorable Pong Lo,” the Emperor began. “You are looking well. It has been some time since we have met.”
“Thank you, Your Majesty,” said Pong Lo. “To be precise, it has been forty days.”
“Only forty days,” lamented the Emperor. “Life at the summer palace is pleasant, I hope?”
“The view is wonderful,” returned Pong Lo. “And my days are filled with activity. Counting, storing, selling rice—”
“Yes!” the Emperor interrupted. “They must be. I imagine you are growing tired of rice?” he asked hopefully.
“Oh no, Your Majesty,” said Pong Lo. “It will make a fine barrier against the winter wind. And there are so many things to be made with it. Rice paper, rice wine, rice cakes, rice noodles, rice syrup—I could go on and on.”
The Emperor looked gloomy. “I do not doubt it,” he said. “Things have not gone as well for me, however. I have been having a problem with my daughter, Chang Wu. The mere mention of marriage sends her into a fit of melancholy.”
The Princess blushed.
The Emperor shifted uncomfortably on his throne. “Honorable Pong Lo,” he began again, “you have become a rich man.”
Pong Lo smiled modestly.
“Richer than any nobleman in China,” said the Emperor. “At last you can care for my daughter as a Princess should be cared for. I have therefore decided to make you a Prince and grant you her hand in marriage.” Here the Emperor leaned forward and lowered his voice. “But no more rice!” he said.
The clever Pong Lo bowed again. “I humbly accept your offer, Father,” he said.
A sigh went through the Court. “When is dinner?” someone whispered.
Pong Lo and Chang Wu were married. The wedding feast was wonderful, its preparation supervised by the new Prince himself. There was bean soup, and bean curd. Bean paste and sprouted beans. Pressed duck and steamed dumplings. Fish with millet and pheasant with millet. There were barley cakes and barley candies. Wheat noodles, potato noodles, corn noodles, fried noodles. But—out of respect for the Emperor’s feelings—there was not a single grain of rice.
The Emperor lived to be an old man. At his death, Prince Pong Lo and Princess Chang Wu inherited his kingdom. They lived happily, and ruled wisely, all their days.
THE MATH BEHIND THE STORY
A Grain of Rice is a beautifully told tale about love, ingenuity…and math! In this story, we learn how young Pong Lo uses a little mathematical thinking to win over Princess Chang Wu’s father, the Emperor. Pong Lo tries all kinds of things to impress the Emperor, but it’s not until he uses his smarts to make numbers—and the amount of rice he owns—grow quickly that he gains the Emperor’s respect and builds a fortune!
While I can’t guarantee that being clever with numbers is a surefire path to marrying a prince or princess, I can assure you that being comfortable with numbers, seeing patterns in them, and learning to understand equations will pay off someday. It might be in the quality of your work, or it might be in the fun and satisfaction of understanding something interesting. That something might involve money, but it might also involve science and other real-world phenomena. Math is everywhere.
Now, back to Pong Lo. Pong Lo starts by asking the Emperor for a single grain of rice, but then—after a little encouragement—further asks that the Emperor “double the amount every day” for him, for one hundred days. It doesn’t sound like a large request to the Emperor, so he agrees.
Would you have agreed? Quick: imagine holding a grain of rice in your hand. It might not look like much, but in just forty days the Emperor is forced to deliver more than five hundred billion grains of rice on the backs of one hundred elephants! (I did some checking, and that’s about twenty thousand tons of rice! For comparison, your car weighs around two tons, so this is the weight of roughly ten thousand cars.) Faced with the idea of doubling the amount of rice again the next day, the Emperor realizes he can’t keep his promise to Pong Lo. He simply cannot deliver that much rice. As quickly as the rice accumulates, the poor Emperor’s good mood disintegrates.
Is it hard to picture that much rice? Let’s make the point another way. Suppose that this doubling rule were instead a rule about your height and how you grow. Say that in the first year of your life, you are just an inch tall, but then each year you grow twice as many inches as the year before. You might say, “No way! I don’t want to be tiny my whole life!” But what does doubling do? In the second year of your life, you grow two inches, and in the third year, four more. In the tenth year of your life, you will grow—get ready for this—512 inches. That is more than 42 feet! It’s 42 feet and 8 inches, to be exact, which is taller than a school bus is long! You’ll definitely make your school basketball team if you grow like that. Whether with grains of rice or inches per year, you can see that the numbers may start small, but they get big very quickly once you start doubling them.
Another way to look at this progression is with a table. Here is a table that shows how Pong Lo accumulates grains of rice over the first ten days:
A list of numbers is fine, but a picture’s worth a thousand words, as they say, so below, we show the growth using a graph.
In this picture, going from left to right, we mark off the days from 1 to 10. Going from bottom to top, we mark off the number of grains of rice that Pong Lo receives from 0 to 600. Since the numbers going from bottom to top have a wide range, we don’t have space to make a mark for each number. If you look closely, you will see that we’ve divided each section of 100 grains into 5 chunks, so that each mark represents another 20 grains of rice. Above the marker for each day, we draw a bar in the graph that is level with the number of grains of rice received on that day. For example, above 10, the top of the bar is positioned between the marks representing 500 and 520. If you had a microscope and a ruler, you would be able to confirm that, in fact, the top of the bar is 512 units above the line on the bottom, equal to the number of grains of rice received on day 10.
The takeaway from this picture is how quickly the curve connecting the tops of the bars swoops up as the numbers increase. And it gets even steeper as you go out farther and the number of days increases. Between days 9 and 10, the number of grains received increases by 256. Between days 10 and 11, by 512, and so on. From 1 grain to over 500 billion over the course of just 40 days—that is extraordinarily fast growth.
What do we call this kind of growth? We could call it “really fast growth,” or perhaps something more precise like “growth by doubling.” But this fast growth has a special name: “exponential growth.” The term exponential growth has to do with the mathematical operation we use to calculate the number of grains on any given day: the process of “exponentiation.” (Don’t worry. I’ll explain!)
Exponentiation is another way of saying “using powers.” Have you learned abou
t powers yet? Well, you might have, but maybe under different names. Have you been introduced to the idea of squaring a number? That’s the operation of multiplying a number by itself. 1 squared is 1 x 1, which equals 1. 2 squared is 2 x 2, which equals 4. 3 squared is 3 x 3, which equals 9, and so on. (Quick: What is 4 squared?) The reason we call it “squaring” is because the area of a square is equal to the number you get when you multiply the length of one side of the square by itself. For example, if you have a square with sides that are a length of 2 (and it could be 2 of anything: feet, meters, yards, etc.), then the area of that square would be 2 x 2 = 4 (square feet, square meters, square yards, etc.).
You could also multiply a number by itself three times! This is called “cubing.” 1 cubed is 1 x 1 x 1, which equals 1. 2 cubed is 2 x 2 x 2, which equals 8. 3 cubed is 3 x 3 x 3, which equals 27. And so on. Can you guess why we call it cubing? That’s right! It has to do with the volume of a cube. The volume of a cube is equal to the number you get when you multiply the length of one side by itself three times.
Why stop there? We could also multiply a number by itself four times or five times (or any number of times), but we don’t have good names for those specific operations. Instead, we say that we are “raising a number to the fourth power” if we multiply it by itself four times, or “raising it to the fifth power” if we multiply it by itself five times.
In other words, a “power” refers to how many times you multiply a number by itself. Squaring a number is called “raising it to the second power,” cubing is “raising a number to the third power,” and so on. By that definition, “2 to the second power” is 2 x 2, or 4, “2 to the third power” is 2 x 2 x 2, or 8, and “2 to the fourth power” is 2 x 2 x 2 x 2, or 16. Also, for that matter, “2 to the first power” is 2. (It’s even possible to make sense of “2 raised to the zero power!” But we won’t go there now….)
Now we’re finally reaching the end of our explanation, because when we raise 2 (or any number) to a power, that power is called the exponent, and we write it like this:
Because 2 is always the “base” in these equations, and only the powers change, these equations all represent different “powers of 2.” This is just a term that means 2 has been raised to a certain power. “Powers of 3” would look the same, except you would have 3 as your base number instead of 2:
To connect this all back to Pong Lo’s story, the amount of rice he receives each day can always be expressed as a power of 2, because it is always multiplied by 2 (or doubled) from the day before. On day 1, he gets one grain of rice. On day 2, he gets two grains of rice, which is 21 = 2. On day 3, he gets 22 = 2 x 2 = 4 grains of rice. On day 4, he gets 23 = 2 x 2 x 2 = 8, and so forth. Because the exponent increases by one each day as the rice doubles, we call the pattern of growth exponential growth.
On day 40, when the Emperor gives up, the number of grains of rice he has to come up with is 239 grains! This might not look significant, but there is a lot of power (get it? “power!”) packed into this notation:
I can’t even fit it on one line! And now we know that 239 is over 500 billion!
You still with me? I hope so! Give yourself a pat on the back for making it this far. In fact, if you want to stop reading now, that’s completely fine. You’ve already learned some pretty interesting stuff (and read at least one or two bad math puns). But if you want to learn something even more advanced, keep reading!
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Like so many things in math, the closer you look, the more there is to discover. What we’ve seen is just the beginning of the fascinating properties of exponents. I’m hoping you might be interested enough to join me on one extra observation, now that you are a master of powers of two. It’s about another pattern we can see if we take a moment to look at how much total rice Pong Lo has received by the end of each day.
Let’s assume that Pong Lo never parts with any of his rice as he receives it. How much rice does he have in total at the end of every day? That’s easy to figure out using the numbers we’ve already calculated for each day’s rice delivery. We just add them up! Here are the totals at the end of each of the first ten days:
Hmm…notice anything? Each day, the total number of grains of rice that Pong Lo has (assuming he didn’t give any away or sell any) is exactly one less than the number he will receive on the next day! If we write this as a collection of equations, we see the following:
Now let’s rewrite these using powers of two:
This is the beginning of a wonderful pattern possessed by powers of 2! If you use it to find the total on day 5, you will see that it still holds. In fact, it holds for any day that you have the patience to investigate.
The powers of 2 have all sorts of amazing properties and patterns, and with this little introduction we’ve seen what is just the first grain of rice in the vast and beautiful world of mathematics, full of mysteries, surprises, and most of all…fun.
—Daniel Rockmore, Professor of Mathematics and Computer Science, Dartmouth College
ABOUT THE AUTHOR-ILLUSTRATOR
HELENA CLARE PITTMAN is the author of numerous books for children, including the acclaimed A Grain of Rice, The Snowman’s Path, The Angel Tree, and Once When I Was Scared.
Visit Helena Clare Pittman at helenaclarepittman.com.
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