Thanks, Math! Four Times Numbers Changed the World

About 4,000 years ago, the ancient city of Ur was one of the great capitals of Sumeria in southern Mesopotamia. The agricultural revolution had enabled settlements of unprecedented size, but it was increasingly difficult for priests and kings to keep track of grain harvests, storage and disbursements to feed both the gods and men.

What they needed, of course, was math. Nothing fancy at first, just some basic arithmetic (addition, subtraction, multiplication, division). And one of the earliest champions of arithmetic, according to ancient clay tablets recovered from Ur, was a guy named King Shulgi.

"There are records of hymns that were sung about his ability to add stuff up," says Brooks. "He made his subjects worship him for his mathematical abilities."

Shulgi didn't just show off his godlike math skills; he built what scholars recognize as "the first mathematical state," says Brooks. Math was mainly put to use in bookkeeping, which allowed Shulgi and his scribes to maintain tight control of Ur's finances and prevent people from defrauding the state.

You could argue that Shulgi and his scribes were nothing but glorified auditors, but auditing, Brooks writes in "The Art of More," is "the true cradle of civilization."

"Shulgi recognized that once you've got control of the numbers, it starts to be very financially lucrative," says Brooks. "This math thing works."

By putting math to work, Shulgi and Ur grew tremendously wealthy and used that wealth to develop one of the world's earliest and greatest civilizations. Shulgi is credited with constructing the Great Ziggurat of Ur, building an extensive road network and expanding his trade empire to include Arab and Indus communities.

The 18th-century French monarch Louis XVI helped bankroll the American Revolution, but it left France mired in debt. The king needed a good accountant to help balance the books, so he appointed a Genevan banker named Jacques Necker as his minister of finance.

But Necker was a little "too good" at his job. He made the budget public — unusual for an absolute monarchy — and used double-entry bookkeeping to carefully track expenses. Brooks says Necker believed that balanced books were not only good accounting, but also the basis of a moral, prosperous, happy and powerful government.

"The profligate French royal court didn't like it, because it meant they couldn't spend money on anything they wanted to," says Brooks. "So, Necker was sacked."

The king's cronies may have despised Necker, but the revolutionaries loved him. It was Necker's dismissal, in fact, that was the spark that lit the powder keg of the French Revolution.

"They carried a bust of Jacques Necker on their shoulders as they were storming the Bastille," says Brooks. "He's kind of a cool accountant."

The German astronomer Johannes Kepler is famous for his laws of planetary motion, which proved that the planets in our solar system orbited around the sun in elliptical paths, but he also wrote an entire book about the proper shape of wine barrels.

The story goes that Kepler ordered a barrel of wine for his second wedding in the town of Linz, Austria, but when it came time to pay for the wine an argument broke out. Kepler didn't like the method that the wine merchant used to price the barrel.

According to custom, the wine merchant laid the barrel on its side and poked a long rod through a hole in the center of the barrel until it struck the opposite corner. The rod was removed, and the cost of the wine was determined by how much of the rod was wet.

Kepler quickly saw where the method fell short: The price for the same amount of wine would change with the dimensions of the barrel. A long and thin barrel would cost less than a short and fat one. Kepler grumpily paid the bill, but he couldn't drop the question of how to construct a barrel that results in the most wine for your money.

Kepler's method was to calculate the volume of a curved wine barrel by imagining it as a stack of flat cylinders. To get the most accurate result, though, you need to use lots of cylinders. In fact, they need to become infinitesimally small to fill up every inch of space inside the barrel. "And when we slice time, distance or anything else into infinitesimals," Brooks writes in his book, "we are in the realm of calculus."

In 1615, Kepler published "Nova Stereometria Dolorium Vinariorum " or "New Solid Geometry of Wine Barrels," now considered the foundational text of integral calculus.

In showing how to maximize the dimensions of a wine barrel to minimize its price (the Austrian barrels were spot on, by the way), Kepler pointed the way to using calculus to maximize the efficiency of all manner of things. Brooks uses the modern examples of calculating the right dose of a cancer drug to deliver the most effective response, or how much fuel a 747 should carry to travel the farthest without being weighed down.

Nothing is as triggering to math-phobes as imaginary numbers. Math is hard enough when we're using actual numbers! Now you want us to mess around with imaginary numbers?

Calm down, Brooks says. It turns out that imaginary numbers are very real; they just have a really stupid name.

The problem started when mathematicians tried to solve quadratic equations that required the square root of a negative number. Since it's impossible for any number times itself to equal a negative (even a negative times a negative equals a positive), mathematicians started calling numbers like that "imaginary numbers."

Imaginary numbers might have remained a curious mathematical oddity if not for a 4-foot, 9-inch (1.45-meter) phenom born Karl August Rudolf Steinmetz, but better known as Charles Proteus Steinmetz.

Steinmetz discovered how to use imaginary numbers to solve one of the most challenging engineering problems of the 1890s: how to harness the exciting new power of electricity and deliver it to homes and businesses. While bigwigs like Thomas Edison and Nikola Tesla argued over the benefits of AC vs. DC current, engineers wrestled with the incredibly complex math required to build functioning electrical circuits.

"Charles Steinmetz came up with a formula for taking all of these really difficult calculations and making them into really easy ones that used imaginary numbers," says Brooks. "That's basically how we electrified America."

Steinmetz's formulas powered the electrical age and great leaps forward in industrialization and scientific discovery. Half a century later, Bill Hewlett and David Packard used imaginary numbers to design their first product, an audio oscillator, in their garage in Palo Alto, California, known as the "birthplace of Silicon Valley."

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