Yes, That's a Thing

"Just one more problem. The last one."

Shen Qi was confident about his answers to the first five problems, but he had no idea how the other contestants were doing.

The surest way to win a gold medal was to answer every question correctly.

After carefully examining the last problem, Shen Qi thought whoever had set it was a complete bastard.

The last problem read:

"Travel back in time to 500 BC. You are Hippasus's junior fellow student. Prove that there is no ratio of two integers whose square is 2."

"Be careful. Your senior fellow student, Hippasus, was just drowned by your teacher, Pythagoras. Do not try to complete the proof using geometric constructions, or you will be drowned too."

"If you get drowned, you won't get even a single point."

Yes, this was the final problem of the National Mathematics League finals. It was that devious.

Put into mathematical language, the problem was actually very simple: prove that the square root of 2 is irrational.

An irrational number is a nonterminating, nonrepeating decimal, like 1.41421356... It has no pattern, makes no sense, and goes on endlessly without ever repeating.

Even middle school students knew that the square root of 2 was irrational, and could write down at least one proof.

Shen Qi could write at least eight different proofs that the square root of 2 was irrational.

This problem was so easy. Even eighth graders could solve it.

Really?

Was that actually true?

No, it wasn't.

This was the final problem of the National Finals. It wasn't as low-level as you might think.

The problem setters' premise was that Shen Qi had traveled back to ancient Greece and become Pythagoras's student and Hippasus's junior fellow student.

Anyone who studied mathematics had to know about the Pythagorean school and its founder, Pythagoras.

Pythagoras was an ancient deity in the history of mathematics. He established a mysterious organization on the island of Samos, combining science, religion, and philosophy. In modern terms, it was probably the legendary "Church of Science."

The core tenet of the Pythagorean school was that mathematical research dealt with abstract concepts.

Even in the twenty-first century, mathematicians acknowledged the view Pythagoras had put forward 2,500 years ago: mathematics studied abstract concepts.

Pythagoras had two great passions in life: studying mathematics and killing his students. The smarter they were and the better their grades, the more he wanted to kill them.

Hippasus was Pythagoras's favorite student. Using geometric constructions, he proved that no ratio of two integers had a square equal to 2. This method was recorded in eighth-grade textbooks, in the introductory chapter on irrational numbers.

Then Pythagoras had Hippasus bound and thrown into the sea to feed the fish. "You think you can show off? Show-offs have to die."

After Pythagoras's death, the geometric proof method created by Hippasus eventually became known to the world. The ingenious idea he had paid for with his life was what middle school textbooks today called the "infinite repeated-division algorithm for finding the greatest common divisor of a square's side length."

In the unusual setting of the National Finals' last problem, the problem setter had made Shen Qi Hippasus's junior fellow student. So he couldn't use geometry to prove that the square root of 2 was irrational. Otherwise, the problem setter would "drown" him, and he wouldn't get a single point.

Of the at least eight proof methods Shen Qi knew, there were, of course, other options. But he was Hippasus's junior fellow student, living 2,500 years in the past. In that era, the prime-number method did not yet exist, and even square roots had not been conceived of, so all the other methods were automatically ruled out.

The problem said, "Prove that there is no ratio of two integers whose square is 2," not, "Prove that the square root of 2 is irrational."

That was why the problem was so twisted.

It also bore out an old saying in the Mathematics Community: simple is hard.

The simpler it was, the harder it became.

"This is driving me crazy. With all these ridiculous restrictions, how am I supposed to solve this?"

Shen Qi looked a little anxious. Crack! He snapped his pencil in two with too much force, his palm slick with sweat.

While working through the National Preliminary and the first five problems of the National Finals, Shen Qi hadn't been free of difficulties.

But whenever he ran into trouble, he could at least grasp a small thread of an idea, follow it, and eventually arrive at the right answer.

The final problem of the National Finals, "Hippasus's Curse," left Shen Qi helpless. Pythagoras's death stare seemed to pierce through time, making Shen Qi feel as if thorns were pricking his back.

"What should I do? What can I do? This problem is absurdly tricky. It's way beyond what a high schooler or even a college student knows about mathematics. Hell, maybe only a graduate student or PhD student in the Mathematics Department could solve it."

It was the biggest predicament Shen Qi had faced in months. It reminded him of his days as a poor student: he recognized every character in the problem, but had no idea how to solve it.

Time ticked by, with half an hour left until the papers had to be handed in.

Shen Qi had spent two hours on the final problem without writing a single word. The first two problems had taken him two hours altogether.

"Teacher Zhang, Teacher Cao, Teacher Tian, please teach me how to solve this. What approach should I take? I have absolutely no idea!" When students ran into hard problems, they naturally thought of their teachers. But Shen Qi realized that none of his math teachers, from elementary through high school, had ever taught him a way to prove that the square root of 2 was irrational without using Hippasus's infinite geometric method or later algebraic methods.

We all knew that a person was born with one head and two arms. What was hard was proving this accepted fact. Why not three heads and six arms? What was the real reason? Was it because of the way we were reincarnated? If reincarnation was the true cause, prove it.

Simple is hard.

That was more or less the predicament Shen Qi was facing now: he knew the conclusion, but couldn't prove it.

"Teacher Zhang, Teacher Cao, Teacher Tian, I might disappoint you. I know that if you show off too much, sooner or later you'll end up thrown into the sea to feed the fish. Teacher Zhang, Teacher Cao, Teacher Tian... Whoa, Teacher Tian!" Shen Qi jolted. A fleeting flash of inspiration surged through his brain like an electric shock.

"That's right! Teacher Tian, the Ancient Babylonian numeral system, base sixty!"

A surge of relief and exhilaration rippled through Shen Qi. Before coming to the capital, during the provincial team's training camp, Teacher Tian had taught him about the Ancient Babylonian numeral system and its base-sixty notation.

The Babylonians had once used the ancient base-sixty system to calculate an approximation of the square root of 2. It was a method from 5,000 years ago, a little trick Teacher Tian had kept up his sleeve.

The base-sixty system was older than Pythagoras, so using it didn't break the rules! Shen Qi picked up his pencil and began to write:

▲

▲▲

▲▲▲

◆

▼

▲▲▲-▏◆-▼

Shen Qi was writing in cuneiform, using it to construct a proof: a pure proof in the ancient Babylonian base-sixty numeral system, from the oldest branch in the 5,000-year history of mathematics.

In the ancient Babylonian base-sixty system, ▲ stood for 1, ▲▲ for 2, ▲▲▲ for 3, and so on. The same wedge-shaped mathematical symbols could be stacked up to nine times, representing 1–9.

◆ stood for 10, and ▼ stood for 60.

▏◆ was the multiplication sign, pronounced "ari" in the Babylonian language.

▲▲▲▏◆▼ meant 3 times 60. Shen Qi needed to perform a multiplication in base sixty so he could proceed to the special reciprocal table in the Babylonian numeral system.

The Babylonians converted reciprocals into base-sixty "decimals." They hadn't realized these were decimals at the time, which was why the word was in quotation marks.

Once he reached the realm of decimals in the Babylonian reciprocal table, Shen Qi grew more and more excited. His intuition told him that he was using an incredible method to prove an utterly absurd problem—and that he was about to succeed!

"Hahaha! What a brilliant move! Absolutely dazzling!"

Shen Qi wrote his entire proof in cuneiform. At last, he wrote down his answer: ▲▲◆▼▲▲▲▲▲▲▲▲...

Then the urgent bell rang. The four-and-a-half-hour contest was over.

Shen Qi handed in his paper in a rush, with no time to check it.

In all the math competitions he had taken part in, this was the only one he hadn't had time to check: the National Finals.

Whatever happened, this year's National Finals were over. All Shen Qi could do now was wait for the results.

At three in the afternoon, the Chinese Mathematical Society opened all the National Finals exam papers, and grading began.

At seven that evening, one of the graders in the marking room stared in shock. He was Cadre Liu of the Chinese Mathematical Society.

Cadre Liu was grading Shen Qi's National Finals paper. When he saw that Shen Qi had answered the last question entirely in cuneiform, he nearly lost it. "Xiao Wen, quick... quick, get my emergency pills... They're in my briefcase..."