Yes, There's This Kind of Operation

"The last question, only one question left."

Although Shen Qi was confident in his answers to the first five questions, he didn't know how the other contestants were doing.

To win the gold medal, the safest bet was to answer every question correctly.

After Shen Qi carefully reviewed the last question, he felt the person who created it was a complete jerk.

The last question was written as follows:

"Time travel to 500 BCE. You are Hippasus' junior fellow student, and you must prove that there is no ratio of integers whose square is 2."

"Be careful, your senior fellow student Hippasus was just drowned by your teacher Pythagoras. Do not attempt to use geometric construction to complete the proof, or you will also be drowned."

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

Yes, this was the final question of the National Mathematics League finals, and it was just that infuriating.

The problem, translated into mathematical language, was actually very simple: Prove that the square root of 2 is an irrational number.

Irrational numbers are infinite, non-repeating decimals, like 1.41421356... They have no pattern, no logic, just extending infinitely without ever repeating.

Even a middle schooler knows that the square root of 2 is irrational and can provide at least one method to prove it.

Shen Qi could write at least eight methods to prove that the square root of 2 is irrational.

This question is so easy, even a second-year middle schooler can do it.

Really?

Is that the truth?

No, it's not.

This is the final question of the National Finals; it's not as simple as you might think.

Because in the problem setter's scenario, Shen Qi has time-traveled to ancient Greece and become a student of Pythagoras, Hippasus' junior fellow student.

Anyone who studies mathematics knows about the Pythagorean school and its founder, Pythagoras.

Pythagoras is an ancient giant in the history of mathematics. He established a mysterious organization on the island of Samos, combining science, religion, and philosophy. In today's terms, this organization was likely the legendary "Cult of Science."

The core tenet of the Pythagorean school was: mathematics studies abstract concepts.

Even today, mathematicians acknowledge the views put forth by Pythagoras 2,500 years ago: mathematics studies abstract concepts.

Pythagoras had two great hobbies in his life: studying mathematics and killing his students. The smarter and better-performing students were the ones he was most eager to kill.

Hippasus was Pythagoras' prize student. He used geometric construction to prove that there was no ratio of integers whose square was 2. This method is recorded in second-year middle school textbooks and is the introductory chapter for middle schoolers learning about irrational numbers.

Then Hippasus was tied up by Pythagoras and thrown into the sea to feed the fish. "You think you're so smart? Those who show off must die."

After Pythagoras' death, the geometric proof method created by Hippasus was eventually passed down. The ingenious idea he exchanged for his life is what is known today in middle school textbooks as the "Euclidean algorithm for finding the greatest common divisor."

In the special scenario of the National Finals' final question, Shen Qi is set by the problem setter as Hippasus' junior fellow student, so he cannot use the geometric method to prove that the square root of 2 is irrational. Otherwise, he would be "drowned" by the problem setter and wouldn't get a single point.

Among the at least eight proof methods Shen Qi knows, there are, of course, other ways. But he is Hippasus' junior fellow student, living 2,500 years ago. At that time, the prime number method didn't exist, and even the square root symbol hadn't appeared yet, so other proof methods were automatically invalidated.

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

So, this question is incredibly twisted.

This also confirms an old saying in the mathematics community: simple-is-hard.

The simpler it is, the more difficult it becomes.

"Stuck, I'm stuck. Under so many twisted restrictions, how can this problem possibly be solved?"

Shen Qi appeared anxious. Crack. He applied too much force and accidentally broke his pencil, his palm covered in sweat.

During the National Preliminary and the first five questions of the National Finals, Shen Qi had encountered difficulties.

Even when he faced trouble, Shen Qi could always grasp a slight clue and follow it to the correct answer.

But for the final question of the National Finals, "Hippasus' Curse" left Shen Qi helpless. Pythagoras' death glare seemed to pierce through time and space, making Shen Qi feel like a target.

"What should I do? What can I do? This question is too devious. It far exceeds the mathematical understanding of a high school or even a university student. This might only be solvable by a graduate student or even a doctoral student in mathematics!"

This was the biggest predicament Shen Qi had faced in months. It reminded him of his days as a slacker, where he understood all the words in the problems but had no idea how to solve them.

Time ticked by, and there was only half an hour left until the papers were collected.

Shen Qi had spent 2 hours on the final question without writing a single word, while he had spent 2 hours on the first two questions combined.

"Teacher Zhang, Teacher Cao, Teacher Tian, please teach me how to solve this. What approach should I take? I have absolutely no clue!" Students naturally think of their teachers when they encounter difficult problems. But Shen Qi realized that none of his math teachers, from elementary school to high school, had ever taught a method to prove that the square root of 2 is irrational without using Hippasus' infinite geometric method or later algebraic methods.

We all know that when a person is born, they come with one head and two arms. The difficult part is proving this universally accepted fact. Why not three heads and six arms? What is the real reason? Is it due to the quality of one's reincarnation? If reincarnation quality is the true cause, please prove it.

Simple-is-hard.

The predicament Shen Qi was in was roughly the same: he knew the conclusion but couldn't prove it.

"Teacher Zhang, Teacher Cao, Teacher Tian, I might have to disappoint you. I know that if you show off too much, you'll eventually be thrown into the sea to feed the fish. Teacher Zhang, Teacher Cao, Teacher Tian... Damn it, Teacher Tian!" Shen Qi jolted, a fleeting spark of inspiration zipping through his brain like an electric shock.

"Yes, that's right! Teacher Tian, the Ancient Babylonian numeral system, the sexagesimal system!"

A thrill of near-death escape surged through Shen Qi. Before coming to the Capital, during the provincial team training, Teacher Tian had taught him about the sexagesimal system of the Ancient Babylonian numeral system.

The Babylonians had used the ancient sexagesimal system to calculate an approximate value for the square root of 2. This was a method from 5,000 years ago, a personal tidbit from Teacher Tian.

The sexagesimal system is older than Pythagoras, so I'm not breaking any rules by using it! Shen Qi picked up his pen and began to write:

▲

▲▲

▲▲▲

◆

▼

▲▲▲-▏◆-▼

What Shen Qi was writing was cuneiform script. He was using cuneiform script to provide a proof, a pure Ancient Babylonian sexagesimal numeral system proof, the oldest branch in mathematics' five-thousand-year history.

In the Ancient Babylonian numeral system, ▲ represented 1, ▲▲ represented 2, ▲▲▲ represented 3... The same cuneiform mathematical symbols could be stacked up to 9, representing 1-9.

◆ represented 10, and ▼ represented 60.

▏◆ represented the multiplication sign, which was read as "aire" in the Babylonian language.

▲▲▲▏◆▼ represented 3 multiplied by 60. Shen Qi needed to perform a sexagesimal "aire," which would allow him to smoothly enter the special reciprocal table of the Ancient Babylonian numeral system.

The Babylonians converted reciprocals into sexagesimal "decimals." In reality, they didn't realize they were decimals at the time, hence the quotation marks.

After entering the decimal domain of the Babylonian reciprocal table, Shen Qi became increasingly excited. His intuition told him that he was using an awesome method to prove an incredibly absurd problem, and he was about to succeed!

"Hahaha, this is simply a divine operation, a heavenly show!"

Shen Qi's entire proof process was written in cuneiform script. Finally, he wrote down the answer: ▲▲◆▼▲▲▲▲▲▲▲▲...

At this moment, the urgent bell rang. The 4.5 hours of competition time were up.

Shen Qi submitted his paper in a hurry, with no time to check it.

This was the only competition he had participated in among so many math competitions where he didn't have time to check his work – the National Final.

Regardless of the outcome, this year's National Final was over. All Shen Qi could do was wait for the results.

At 3 PM, the Chinese Mathematical Society unsealed all the National Final exam papers, and the grading work began.

At 7 PM, in the grading room, one of the graders stared blankly. He was Cadre Liu from the Chinese Mathematical Society.

Cadre Liu was grading Shen Qi's National Final paper. When he saw that Shen Qi had answered the last question entirely in cuneiform script, he felt completely unwell. "Xiao Wen, quickly... quickly bring me my quick-acting savior pills... they're in my briefcase..."