"Modulo arithmetic" can be a handy tool.  For example, if you want to know what day of the week it will be, say, 10 days from now, you can tick them off on your fingers.  For 20, you need to take off your shoes, and 30 means finding a friend.

Another way is to divide 30 by 7 and take the remainder, which is always less than seven, for your ticking off.  Works for any number.  If today is Tuesday, and you want to know what day it will be 101 days from now, divide 101 by 7 and register the remainder (3) on your fingers.  Then just say "Wednesday, Thursday, Friday."  Better than counting through 14 weeks worth of fingers.  In mathematical parlance, that's "101 modulo 7."

The population of the U.S. according to the 1990 Census was 248709873.  To make large numbers easier to read, we like to put commas separating every third digit: 248,709,873.  But it's common to write the number starting on the left.  How many digits do you write before the first comma?  For small numbers the answer is easy.  The population of Idaho is a million and seven thousand, which is 7 digits long.  Since "7 modulo 3" equals 1, you can write 1,007,000 straight away.

What if you wanted to punctuate a googol?

y father made it a point to be busy.  Almost too busy.  In retrospect I think it was part of his approach to teaching.  On a Saturday, my dad worked on projects in the garage, which was a treasure house filled with the wonderments of technology.  My friends and I would hang around after playing hit-the-bat or work-ups (before Little League existed) and ask questions.  My dad always took time to answer.  "Code-practice oscillator," he might say then resume his soldering and snipping.  Another question, another answer: "Morse code."  Ten-year-olds are veritable question machines.

My dad's replies were deliberately limited to the scope of each query.

"Telegraph."
"A hundred years ago."
"Samuel Morse."

"Here it is, Jimmie!"  There it was: all the dots and dashes that make up the letters in Morse Code.

"How do you make 'SOS'?" Billie asked.  Dirty fingers pressed the page, freckled faces gaped.  "Dot-dot-dot, dash-dash-dash, dot-dot-dot."

My dad put down his pliers.  "Use 'dit' and 'dah'," he said.  "And say it fast."

We remembered V-for-Victory, "Dit-dit-dit-dah," which  marked station breaks as a civilian morale builder during the war.  Soon my friends and I were spelling things to each other with dits and dahs.  Later we took turns pressing the key my dad had fashioned out of a hacksaw blade and listened to our own names beeping in bakelite headphones.  By then, my dad was tinkering with something else.  "Gyroscope," he answered.

ne Saturday stands above all others in my memory despite the passage of more than half a century.  My friends and I played marbles and mumblety-peg until peanutbutter time.  After lunch, we abandoned our circle in the dirt and invaded the garage.  My dad was repairing an electric motor.  Without looking up he greeted us with a question: "Ever hear of a googol?"

"Sounds like baby talk," scoffed one of us.

"It's a number," my dad said.  "One with a hundred zeros."

Someone made woo-woo sounds.  We snickered and exchanged glances.  My dad would have let it go at that.  So would my friends and I.  All but one, a kid we called Mort.  "I wonder what a googol looks like," he said.

"Yeah," said Billie.  "I know where there's some paper." He rode off on his bike and returned with a piece of butcher paper fluttering in the wind.

Soon Mort lay prone on the garage floor, tongue protruding.  He pencilled zeroes while Jimmie counted.  When he was finished, the four of us stood over the paper in silent amazement.

"But what is a googol good for?" I asked.

My dad shrugged.  "How many grains of sand are there?"

"Nobody knows that!" exclaimed Billie.

"Which beach?" asked Mort, which brought a round of hoots.  Absurdity was still new to us (see literally).

"Why not all of them?" my dad asked, solemnly.

"That's not possible!" I exclaimed.

My dad nodded toward the paper.  "Is a googol bigger?"

"Not more than all the sand in the world," Billie said.

"You're sure?" asked my dad.  "Why not figure it out?"

Billie punched me on the shoulder.  "Have your father drive us to the beach, and let's start counting."

"Figuring," said my dad, "doesn't mean counting."

That was the closest my dad ever came to a lecture.

"Find a handful of sand," he said.  "I'll help you."

"My sister's sandbox!" Mort cried.  We mounted our bikes and returned with sand in our pockets.  My dad had cleared a space on his bench for our butcher paper.  He handed Mort a ruler and told him to draw a square one inch on a side and cover it with sand.

"How deep?" I asked, while scraping away the excess.

"Thin as possible."

Soon we had a square inch of sand, one grain deep.  Mort shook his head.  "It will take all afternoon to count them."

"Let's take a quarter of an inch," Billie said.

"Still too many," Mort complained.

"Try guessing," my dad suggested.

Adults don't tell kids to guess.  After some debate, we estimated 100 grains along a quarter of an inch, 400 grains per inch.  One of us remembered how to calculate an area by multiplying.  That gave us 160,000 grains of sand to cover a square inch.  My dad told us to multiply by 400 again.  He showed us how to keep track of all the zeroes.  "Cubic inch," said he.

"Wow!" cried Jimmie.  "Sixty-four million grains of sand in just one cubic inch."

"Not a googol yet," my dad mused.

"No," said Billie.  "But this isn't the beach either."

"Anybody have a map?" my dad asked.

Back on the bikes again.  We returned from Billie's house with a world atlas.  Four 10-year-old boys studied a map and concluded that beaches were wrapped about 6 times around the earth.  My dad nodded.  "How big is the earth?"

The dictionary gave us the diameter of the earth as 7,918 miles.  My dad suggested that we round that off to 8,000 and multiply by 3 to calculate the circumference of the earth, 24,000 miles.  "Approximation, it's called."

Jimmie remembered to multiply that by 6 for the beaches.  Each new number made headlines.  "The beaches are 144,000 miles long," we exulted.  "Let's get the inches."  Back to the dictionary for 5,280 feet in a mile.  Multiplying by that and by 12, Jimmie announced the result: "Nine billion, one-hundred twenty-three million, eight hundred and forty thousand inches!"

There was no mistaking the power of numbers.  The butcher paper was half covered with them.  And there was no stopping us.

"How wide is the beach?" I asked.

"A hundred feet is my guess," said Billie.

"And ten feet deep, maybe more," Mort commented.

For the next hour, we worked as a team building bigger and bigger numbers by multiplications and counting zeroes.  A hundred times twelve times ten times twelve again and that times our other number gave us 1,313,830,000,000,000.  Billie checked the calculation.  "Where are we?" I asked.

Jimmie pointed to that number and said, "That's how many cubic inches of sand are on all the beaches of the world -- "

"And each one has 64 million grains of sand!" Mort exclaimed.

"Here we go," said Billie.  One more multiplication and we would know how many grains of sand are on all the beaches of the world!  Which is 84,085,120,000,000,000,000,000.

"Not a googol, is it," said my dad.

n one afternoon, my friends and I acquired some of the most powerful tools of problem solving, including breaking a big problem into smaller problems, approximating and estimating instead of brute-force determinism, and the most powerful of all: how to make assumptions (see Discovering Assumptions).  My dad's instinctive methods of stimulating discovery enabled us to experience a sense of achievement which was -- which is -- beyond counting.

Next time you want to write a googol, which has 101 digits (the one plus the one hundred zeros), and put the commas in as you go, just calculate "101 modulo 3," which is 2.  A googol gets its first two digits, 10, before the first comma and looks like this:

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.

That quotient (33), which you set aside, represents the number of triple-digit "clusters."  Each cluster is preceded by its comma and has a name: thousands, millions, billions, trillions, quadrillions, quintillions,...  Most of us do not have need of the rest of them, although I have seen "quads" in measurements of worldwide energy consumption.

The numerical values embedded in the Latin roots seem to be one less than the number of clusters to the left of the decimal point (for example four clusters minus one gives us "trillion").  A googol has 33 clusters, so a person needs to embed a Latin expression for 32, which is "tridecabillion."  I don't need to do that, though.

My dad gave me "googol."  And a whole lot more.
 

Epilog: Orders of Magnitude

A reader recently sent me the link to a wondrous website supported by Florida State University.  I was delighted by the experience, which begins with a telescopic view of the Milky Way from 10 million light years away (10⁺²³ meters).  Then my screen moved through space towards the Solar System and Earth in successive orders of magnitude until I zoomed in on a tall oak tree just outside the buildings of the National High Magnetic Field Laboratory in Tallahassee, Florida.  My eyes and I were automatically drawn close enough to view a leaf in actual size, then I entered a microscopic world that reveals leaf cell walls, the cell nucleus, chromatin, DNA and finally, I was plunged into the subatomic universe of electrons and protons -- hey, and quarks (10^-15 meters).  The whole trip spans a ratio of dimensions (largest to smallest) of only 10³⁸  -- nowhere near a googol (10¹⁰⁰).  My dad would have liked that.