2 BIG Numbers
264 - 1

	On this page I will discuss two of my favorite big numbers that
have interested and intrigued me for some time.  I hope you find them
interesting as well.

GOOGOL The first one is a number that was created, as it were, by a 9-year-old boy. His uncle, a famous mathematician, was talking one day with the lad and asked him what was the largest number the boy could think of. The youngster thought a moment and said, "A one with a hundred zeros after it." The man said, "My, that certainly is big, but what shall we call it?" The boy replied, "A googol!" And so it has been since that day. (This is a true story.) Let's see that number: 10,000,000,000,000,000,000,000,000,000,000,000,- 000,000,000,000,000,000,000,000,000,000,000,- 000,000,000,000,000,000,000,000,000,000,000 Now that's what I call BIG! Of course, a more compact and efficient (if less impressive) way to write this number is to use exponents:


Though it doesn't look as big as the first way, it does represent the same quantity. Related to the googol is its bigger brother, the googolplex. Now here is one really terrific -- or, googol-ific -- number, because it is so big that it can't be written out in the standard, decimal notation. You see, it it a "1" followed by a googol of zeros!!! Most of my students have a hard time grasping this concept. In fact, many take out some paper and say they will do it over the weekend. Little do they know what they are in for. We can, however, express it in exponential form, thus:


Perhaps a good way to illustrate just how large this number is would be to compute how long it would take you, or you and a team of friends taking turns, to write it. Assume a writing speed something on the order of one or two zeros per second. Amazing what you will discover. But as for me, a googol is big enough for most of my needs. I often say to my wife, "I love you a googol!" (Who says mathematicians can't be romantic?)
Wheat on a Chessboard Another big number comes to us in a legend about the invention of chess. The story goes:

	A long time ago Sissa ben Dahir, the Grand
Vizer to the Indian king, Shirham, presented his
latest creation to his ruler.  It was a game called
chess.  The king was so pleased, that he told Sissa
that he could name his own reward.  Sissa replied,
"Majesty, give me the sum of 10,000 rupees; or give
me some wheat in the following manner: I need 1
grain to place on the first square of the chessboard,
2 grains to place on the second square, 4 grains
for the third square, and 8 grains for the 4th
square; and to continue in like manner, oh Mighty
and Generous One, let me cover each of the 64
squares of the board."

   Now, Shirham was not too good at arithmetic;
that's why he had advisors, you see.  And his
realm was famous for its wheat production; the
storage bins were always full.  And as many
kings are, he did not wish to part with so much
money either.  So he decided on the choice of
the wheat and exclaimed, "Is that all you wish,
Sissa, you fool?  I shall grant your wish of
the wheat."

   So the king ordered a bag of wheat to be
brought to the throne room, and the counting began.
1, 2, 4, 8, 16, 32, 64, 128 and so on.  Before
long the bag was empty.  And then one bag, two
bags, four bags, and so on were soon necessary,
sometimes not even being sufficient for one square.
This process did begin to take a long time; in fact,
they soon quit counting individual grains, instead
they were counting by bags.  Later, believe it or
not, even an entire granery bin wasn't sufficient.

   When the king began to realize just how much
wheat was involved, his heart sank.  He knew it
was he who was the fool.  Sissa then admitted,
"Oh Sire, I have calculated that more wheat is
needed than you have in your kingdom, nay, more
wheat than there is in the entire world, verily,
enough to cover the whole surface of the earth
to the depth of the twentieth part of a cubit."

   At this point in our story, we are sad to say,
nothing much is known of the whereabouts of that
clever inventor of chess or whatever happened to
him.  There is no mention of his name in the
official court records.  Kings are not known for
treating their subjects in a kindly and loving
manner when the latter have made a fool of the
Royal Sovereign.  Some are of the opinion that
Sissa was banished from the kingdom forever.
Others feel he is languishing in some dark and
remote prison.  Yet many people who know more
about the inner workings of the court are sure
that Sissa was beheaded.  (Ouch!)

But, be that as it may, we still have a problem to solve: just how many grains of wheat were they talking about? It appears at first glance to be a rather straightforward addition:
1 + 2 + 4 + 8 + 16 + 32 + 64 + ...
until a mere 64 addends have been used, right? Ah, but you see, that's what King Shirham thought, too. And you saw where that led him. What makes this problem unique and important for teaching problem solving in the math classroom is that if it is approached intelligently, it can be solved without any adding at all! Yes. Observe this table: Square | Grains | Total so far ---------|------------|-------------- 1 | 1 | 1 2 | 2 | 3 3 | 4 | 7 4 | 8 | 15 5 | 16 | 31 6 | 32 | 63 7 | . | . 8 | . | . 9 | . | . etc. | etc. | etc. Can you begin to discern some definite patterns in the table? Especially in the values of the second and third columns? Once you achieve that you are well on your way to the ultimate solution -- which is, by the way, found at the top of this article in the upper right hand corner. But, as we said before, exponential notation tends to minimize the true "size" of a big number; so we will give the "real" number here for you to contemplate further:

Finally, as a personal aside: I have always thought this value to be an unusually ODD number. Oh no, not for the fact that it ends with the odd digit 5, nothing that mundane even crossed my mind. But, recall how it was constructed -- from a series of powers of 2. Yet in the final answer of 20 digits nary a 2 appears. Every other digit does, but not the one on which its foundation rests. (Hmmm...) Also, how about a connection between the googol and the wheat count number. Try solving this: If one had a googol of grains of wheat [remember: I just said IF.], how many times could the chessboard problem be solved? [Whew! Big numbers can really bring on some big work, can't they?]
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