In his last book Billions & Billions: Thoughts on Life and Death at the Brink of the Millennium, the late great astrophysicist Carl Sagan retells an old tale about the invention of the chessboard. The story goes like this: many years ago, in a kingdom, somewhere far away, the King’s chief advisor, the Counsellor, presented him with a new game. This game was played by moving pieces back and forth on a board consisting of 64 squares and the most important piece was, of course, the one modeled after the King (the Counsellor was the next most important). The game was won by capturing the opponent’s King piece. The King was overjoyed by this new invention, and asked his Counsellor how he would like to be rewarded for his ingenuity. Sagan continues:
"The [Counsellor] had his answer ready: He was a modest man, he told the [King]. He wished only for a modest reward. Gesturing to the eight columns and eight rows of squares on the board he had invented, he asked that he be given a single grain of wheat on the first square, twice that on the second square, twice that on the third, and so on, until each square had its complement of wheat. No, the King remonstrated, this is too modest a reward for so important an invention. He offered jewels, dancing girls, palaces. But the Counsellor, his eyes becomingly lowered, refused them all. It was little piles of wheat that he craved. So, secretly marveling at the humility and restraint of his counselor, the King consented."
Is the Counsellor just a big dummy or does he know something the King doesn’t?
We’ll revisit our tale later on, but let’s do a little exercise first. Go grab a calculator and turn it on. Punch in the number “10”. Multiply it by “10” and press the equal sign. It should read “100” (which I’m sure was not a discrepant event for you). Multiply by 10 again and again, and see how quickly the number grows. Eventually, your calculators won't be able to display a number so large.
If we wanted to write such large numbers out, it would be easy, but time consuming (and potentially hand-cramping). We can save time using exponents, which you may have heard of or seen before. Exponents tell us how many times a number has been multiplied by itself. For instance, the beginning of our calculator equation, 10 x 10 = 100, would look like this in exponential notation: 102 = 100.
That little tiny number is the exponent. Aren’t exponents cute? All tucked away up there. In the case of multiples of 10, the exponent also tells us how many zeroes are after the 1:
Now we have a much easier way to think and talk about numbers so large, but why is this important?
Sagan Speaks Again
Considering the enormous scale of the universe, astrophysicists like Carl Sagan frequently work with gigantic numbers. Exponents help us think about numbers outside of everyday life. Sagan writes:
"Once you’ve mastered exponential notation, you can deal effortlessly with immense numbers, such as the rough number of microbes in a teaspoon of soil (108); of grains of sand on all the beaches of the Earth (maybe 1020); of living things on Earth (1029); of atoms in all the life on Earth (1041); of atomic nuclei in the Sun (1057); or of the number of elementary particles (electrons, protons, neutrons) in the entire Cosmos (1080). This doesn’t mean you can picture a billion or a quintillion objects in your head—nobody can. But, with exponential notation, we can think about and calculate with such numbers. Pretty good for self-taught beings who started out with no possessions and who could number their fellows on their fingers and toes."
And that’s just talking about really big stuff. Exponents help us understand little tiny things too. When the exponent is a negative number, it tells us how many times to divide (instead of multiply) 1 by the base number. For instance, taking our previous equation:
10-2 means 1 ÷ 10 ÷ 10 = 0.01
This lets us think on a scale that’s as big as it is small. This website helps you explore the full sizes range of the entire universe, which you are now even more mathematically equipped to do!
Back to the Chessboard
Remember the Counsellor and the King? Armed with exponential thought, we can see now that the Counsellor may have put one over his King. Sagan concludes:
"When, however, the Master of the Royal Granary began to count out the grain, the King faced an unpleasant surprise. The number of grains starts out small enough: 1, 2, 4, 8, 16, 32, 64, 128, 256, 512, 1024 . . . but by the time the 64th square is approached, the number of grains becomes colossal, staggering. In fact, the number is near 18.5 quintillion. Perhaps the Counsellor was on a high-fiber diet."
That much grain would weigh around 75 billion metrics tons, which would take today’s wheat producers about 150 years to accumulate, far more than could be held in the King’s kingdom. Score one for the Counsellor and exponents! I wonder what the King thought of that.