By Meir Niv
“So, what would you like to receive as your reward?” the king of India asked the honoree of the evening, a man who had recently invented the game of chess. “A large tract of land or a sack of gold coins?”
“Well,” he replied, “for the first of the 64 squares I’ll take one kernel of wheat.”
“And for the second square?”
“And for the third square?” The king obviously took pleasure in the clever man’s little game.
“Twice as much, meaning four kernels. And I’ll continue that way, doubling my reward with each successive square.”
“So be it,” agreed the king, shrugging his shoulders. “I had in mind to pay you a fortune in real estate or gold, but if this is what you want, you can have it.”
An hour later, the king asked whether the inventor had been paid yet. He was told that they were still calculating the amount. An hour later he inquired again, and received the same answer.
“What is this?” he fumed. “How long does it take to figure out a paltry sum like that?”
At the end of the day, the royal treasurer asked to see the king. He was admitted, and the king expected to hear that everything was taken care of. The treasurer, however, did not mince words.
“Your Majesty, we have been trying to calculate the reward, but I must inform you that even if we would dry up all the world’s oceans and then sow the entire planet’s surface, we would not grow enough wheat to pay the man,” he stated.
“Why? How much wheat are we speaking of?”
Clearing his throat and looking down at his notes, the treasurer read, “The sum total for the final square alone is 9,223,372,036,854,775,808 kernels. Well over nine quintillion! And then we have to add the amounts for each of the previous squares, for a grand total of almost 20 quintillion.”
This quaint legend brings to life a discussion in the Midrash. In the beginning of Parashas Devarim, Moshe Rabbeinu blesses Bnei Yisrael, saying that Hashem should increase their numbers a thousand times. Over 400 years later, when Dovid Hamelech instructed Yoav ben Tzeruyah to count Bnei Yisrael, he protested, saying (Shmuel II 24:3), “Hashem should increase the nation’s numbers a hundred times as much as they are and again as much as they are, while my master the king looks on. But, my master the king, why do you want this [counting] done?”
Rashi, citing the Midrash, notes that Yoav’s brachah, wishing the population to increase a hundred times, seems to be much less generous than that of Moshe Rabbeinu, who blessed them to increase a thousand times. When we examine the language each one used a bit more closely, however, we discover that Yoav’s blessing was literally exponentially greater. While Moshe Rabbeinu blessed the nation to increase its numbers a thousandfold, Yoav blessed them to double their numbers one hundred times, or by 2 to the 100th power, much like the intentions of the tricky inventor of the chess game.
In real numbers, Moshe Rabbeinu wished for the 600,000 people to be increased by 1,000 times their number, for a total of 600,000 x 1,001 (600,600,000). If Yoav would have blessed them to double their numbers only ten times, he would already have surpassed Moshe’s blessing. That is because 2 to the 10th power is already 1,024. If the population would increase by 2 to the 100th power, it would be multiplied by the impossible number of 1,267,650,228,229,401,496,703,205,376 times. A single line of that many people would stretch many times beyond the length of the entire universe!
The King’s Wives
The Mishnah (Sanhedrin 21a) states that a king is allowed to have up to 18 wives. A dissenting opinion in a Braisa permits up to 24 wives. How do we attain these numbers? The pasuk states that Dovid Hamelech had six wives, and Hashem promised to add to that number “kaheinah v’chaheinah — as many and again as many.” According to the Mishnah, these words mean “another six and another six,” for a total of 18. According to the Braisa, however, the words “as many” mean another six, while the words “and again as many” mean doubling the six-plus-six another time, for a total of 24 wives.
Yoav’s expression of “as much as they are and again as much as they are” is parallel to the expression of kaheinah v’chaheinah. If so, then the interpretation we gave to his blessing is correct only according to the Braisa in Sanhedrin. According to the Mishnah, his blessing was only that the population increase to one hundred times as much as it was then, which is substantially less generous than the blessing of Moshe Rabbeinu.
Luckily, Rashi cites another resolution to this problem, which fits in with the Mishnah’s opinion as well. Although Moshe Rabbeinu blessed the people to increase a thousandfold, he did not say how long it should take for this blessing to be fulfilled. It could have taken centuries. Yoav, on the other hand, gave a blessing for the population to increase a hundredfold within the lifetime of Dovid Hamelech.
(We cannot overlook the interpretation of the Binah L’itim [darush 51] which is that Moshe Rabbeinu’s blessing was that the population should double itself a thousand times, meaning by two to the thousandth power. If this is so, then we must rely on the second resolution in the Midrash. The Binah L’itim points out, by the way, that this is an impossibly large number, and when Harav Alexander Zusia Friedman cited it in his Maayanah shel Torah, he brought the story of the chess inventor to illustrate it.)
A Word About Words
In Hebrew, the largest word representing a number is revavah, or ribo. The classic translation of the word is myriad, which means ten thousand. The word million comes from the Latin mil, and it means one thousand times a thousand.
What is the correct term for one thousand millions? Well, in European languages, as well as British English up to recent times, it is milliard. The word billion, in British English, was used to represent one million times a million, or 10 to the 12th power.
At some point, the Americans changed the system. They decided to use the word billion for a thousand millions, and the word trillion for a million millions. They discarded the term milliard altogether. This, as you can imagine, was the cause of much confusion. More than 40 years ago, the British gave up and officially adopted the American system.
Why am I telling you all this? There is an obscure passage in Pesikta Rabbasi (parashah 21) that, according to the unedited text, features a parenthetical line containing the words mi l’yardam. Several classic commentators struggled to interpret these words. Harav Dovid Luria suggested that the text be corrected to read milliyardim, and that it means millions of millions. Harav Ephraim Margalios, however, suggested that the correct text reads millyonim.
Now, I am hardly qualified to give my opinion about this, but I have a problem with Rav Dovid Luria’s interpretation: As far as we know, the word milliard came into use only 800 years ago, and the Pesikta was composed some 1,500 years ago. The word million is an even more recent invention. How could anyone have suggested that these are the words used in that ancient text? What’s more, the term milliard was never used to mean millions of millions, but thousands of millions.
Translator’s note: This passage, according to Rav Dovid Luria’s correction, reads alfei millyardim, which means thousands of milliards. That really is millions of millions, exactly as he interpreted it. Furthermore, this parenthetical statement appears to be an addition to the Pesikta, inserted by an editor many centuries later. In fact, it concludes with the disclaimer that this is lashon laaz, a language other than Hebrew or Aramaic. The editor, who was familiar with these words, used them to clarify the Pesikta’s obscure text.