Non-Euclidean Geometry
The Italy Fibonacci was a master of mathematics, and his translation skills were at least master level.
Italy, located in the heart of the Mediterranean Sea, connects Europe, Asia, and Africa, where goods and knowledge from all over the world converge.
Fibonacci, a multilingual scholar and the son of a Venice merchant, translated mathematical texts from Egypt, Arabia, India, and China, integrating them to create a European Renaissance- era Mathematical Encyclopedia.
European mathematics was thus revived, and in the following centuries, a great number of mathematical geniuses emerged in Europe, ushering in the golden age of mathematics, the profound influence of which continues to this day.
"Next, I will get to know Luca Philbinotti." Shen Qi focused on a white boy with dark curly hair, thick eyebrows, and big eyes. According to the information, Luca Philbinotti was an Italy exchange student, and this boy was the only one in the classroom with Southern Europe features.
A boy with dark curly hair, thick eyebrows, and big eyes raised his hand to indicate that he was Luca Philbinotti.
Shen Qi asked, "Luca, please tell me when the First Mathematical Crisis occurred and why."
"Around 400 BC, the square root of 2 was discovered." Luca, in a dramatic, Greek epic style, recited. "All things are number, the universe is measurable, Hippasus, lord of evil."
Wow, you're quite the actor! You must have studied at the Milan Academy of Dramatic Arts, Luca?
Shen Qi's acting skills were ignited. It was his first time seriously tutoring students, and he hoped to become one of the young men.
"My dear friend." Shen Qi walked up to Luca, slowly turning the pages of a book on non- Non-Euclidean Geometry. He looked up at the ceiling and spoke in a weighty, Socrates tone. "Your words are biased, my heart is desolate, the square root of 2 exists rationally."
Luca and Shen Qi exchanged a knowing smile, their understanding unspoken.
Two drama queens, they're really putting on an act!
The other students were surprised by Shen Qi's teaching style—humorous and flexible, composed and generous—much more interesting than those old men.
The dialogue between Shen Qi and Luca, switching between Greek epic styles, illustrates the First Mathematical Crisis through a question-and-answer format.
The Pythagorean school believed that "all things are numbers," and that everything in the universe can be measured in terms of commensurability, that is, expressed as a ratio of two integers.
The appearance of the square root of 2 terrified Pythagoras, for this wicked number could not be expressed as the ratio of any two integers!
No, the square root of 2 or any evil number like it absolutely does not exist.
However, Pythagoras's student Hippasus successfully proved the existence of the square root of 2, thus giving birth to the concept of irrational numbers.
"Irrational" refers to irrationality relative to rational numbers; irrational numbers themselves are rational.
This was the mathematical crisis that occurred more than two thousand years ago, caused by the square root of 2 and irrational numbers, and is known as the "First Mathematical Crisis".
"I am very satisfied with Luca's answer." Shen Qi walked back to the blackboard and said, "In exchange for an equivalent exchange, I will reveal my age. I am 21 years old this year, and will turn 22 in a few months."
"My God, I'm also 21, and I'm still in second grade!" Stephen shouted exaggeratedly. There's no harm in comparison.
"Stephen, Colorado guy, when you're 23 and in your fourth year of high school, Qi, who's also 23, could very well be your thesis advisor." Luca boldly predicted this scenario.
Luca is 19 years old this year. Youth represents hope, and he has more potential and an age advantage than those older guys who are over 20.
"What a talkative Italy!" Stephen comes from a small, fifth-tier Xiao Cheng in Colorado and is one of the few native-American students in the class.
While 90% of Princeton's undergraduate students are from the United States, the Mathematics Department is an exception, with more than half of its undergraduate students being international students.
"The exchange on the History of Mathematics is considered an extension activity and will not be included in the evaluation report for the Recitation Class." Shen Qi smoothly transitioned to the main topic, stating that the History of Mathematics was just an appetizer, while Non-Euclidean Geometry was the main course.
The atmosphere of the project returned to seriousness. After a brief moment of relaxation, the 21-year-old teaching assistant decided to get serious in the end.
The professors of large lectures are responsible for explaining the basic theories, while the teaching assistants of Recitation Class are responsible for tutoring students in their review, assigning homework, and checking homework.
After going through a series of tests, including large lectures, Recitation Class, group discussions, completing assignments, and exams, these second-year students can earn credits for a course.
Princeton has a large number of faculty and a small number of students, giving it one of the best undergraduate educations in the world.
Princeton University face immense pressure and endure a tough life, but those who have received such a high-level, high-quality education possess a unique charm and demeanor that sets them apart after leaving Princeton.
Princeton boasts an employment rate of over 94% for its undergraduate graduates, a remarkable achievement in the United States, where unemployment is so high that "Princeton University graduates are highly sought after."
Shen Qi got to the point and began his tutoring session, starting with a professional question, "John M. Gross, please answer me in one sentence: what is the difference between Non-Euclidean Geometry and Euclidean geometry? I think Professor Muller has already taught you this; let's review this most basic concept."
John M. Gross from England immediately replied, "In Euclidean geometry, if a line intersects two other lines such that the sum of the interior angles on one side is less than two right angles, then the two lines, when extended along that side, will necessarily intersect. However, in Non-Euclidean Geometry..."
Shen Qi interrupted John M. Gross's answer, shaking his head in dissatisfaction. "Hey, John, I don't want to hear a verbatim recitation of Euclid's Fifth Postulate; that's the passing grade for a high school student. And you, John M. Gross, are a second-year Princeton Mathematics Department. I need a more creative answer."
"Jack -Ma, please continue answering this question." Shen Qi focused on a student with an Asian face. He remembered a student named "Jack-Ma" on the student roster, whose nationality was American.
The name Jack-Ma clearly indicates Asian descent, and there's a high probability he's of Chinese descent.
Sure enough, the young man with the Asian face was Jack Ma. He was not tall, had dark skin, and sported a fashionable perm.
Jack Ma gave his answer, his fluent American English suggesting he had spent a long time in the US and likely grew up there. "A sphere in Euclidean geometry is a Non-Euclidean Geometry plane to an ant crawling on its surface; the ant wouldn't perceive the curvature of the sphere while crawling."
Shen Qi snapped his fingers, very satisfied with Jack Ma's answer, "Hey, hey, Jack, Jack, perfect answer, A+! That's the Princeton style."
"Hehehe, yes, thank you for the compliment." Jack Ma showed no humility or politeness after receiving the praise, showing that he has been deeply influenced by American culture.
After entering the classroom, Shen Qi quickly asked several questions, including two questions about the History of Mathematics and several questions about Non-Euclidean Geometry. Each of the twelve students answered Shen Qi's questions.
Shen Qi memorized the face of every student in the tutoring class and matched it with their English name on the roster.
So far, the best performing students are Luca, an Italy exchange student, and Jack Ma, an Asian American.
Excellent students always receive more attention from teachers. Shen Qi unconsciously changed his role, and his first Recitation Class was quite successful.
So far, at least twelve second-year Mathematics Department students have given positive feedback to their Recitation Class instructor Shen Qi.
Shen Qi earned the $120 tuition fee easily and comfortably. He enjoyed the classroom atmosphere and hoped to share some of his mathematical experience and insights with his juniors, hoping to help them.
Lively discussions and the sharing of diverse opinions are the main themes of the Recitation Class. From time to time, there will be a brainstorming session where students will conduct divergent logical deductions on a particular topic.
The topics in the Mathematics Department are often just a math problem. Shen Qi, who started out by doing a lot of math problems, no longer does that; he's now in charge of creating problems and assigning homework.