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原文：joanwestenberg.com/smart-people-d…

Number theory is the foundation of encryption. Without encryption, bank transactions would be highly insecure. Algebra and logic are integral parts of information science. Thanks to graph theory, the Global Positioning System can find the shortest path between two points.

The "beauty of math" refers to the spiritual pleasure it brings to people. If a formula can connect two far - apart fields, we can call it "beautiful". If a proof is concise and wonderful, and takes an unexpected path, it is very beautiful. Math Academy So MA is beautiful.

Those who are truly awake are always in the minority, but there are also some who pretend to be asleep. Today, you not only need to filter out what is correct, but also instinctively say no to what is wrong. If this is difficult to achieve, then learn mathematics.

The 6C standards, refer to the following principles: Concise, Clear, Complete, Consistent, Correct, Colorful I think this also applies to mathematics.

I missed the restriction of the y-axis, I got 4M/3, but it wasn't among the options. So I tried another method, still got 4M/3. Surprisingly, I gained new insights. Then I noticed the y-axis, quickly worked out the correct answer 7M/6—and got multiple benefits from this process!

Terence Tao says today's AIs pass the eye test -- but fail miserably on the smell test. They generate proofs that look flawless. But the mistakes are subtle, and strangely inhuman. “There's a metaphorical mathematical smell.. it's not clear how to get AI to duplicate that.”

Justin Skycak I believe mental math in advanced math has the same value. Not simple number - crunching, but complex problems needing 20 + steps on paper. My students & I try to solve them purely mentally. Tough, but super rewarding when done!

Justin Skycak Long-term memory is key to Automaticity: 1. It strengthens neural pathways, enabling automated execution of complex tasks like mental math. 2. Reduces cognitive load by eliminating the need to rederive knowledge. Turning deliberate actions into effortless, automated performance.

Beautiful mathematics is that which can "resonate with the mind". A beautiful proof must be unexpectedly simple, or simple relative to its achievements. It is reflected in the high "compression factor" achieved through cognitive modules, namely the so-called "chunking" process.

陶哲轩：人类并没有专门数学脑区， 不同的人处理数学问题有他们自己的方法。 * 有些人调用视觉中心，所以他们以视觉化的方式思考数学。 * 有些人调用语言中心，所以他们以符号化、逻辑化的方式思考。 * 有些人调用解谜/游戏中心，所以他们擅长竞赛和策略性的思考。

I have just completed 100% of the Multivariable Calculus course at Math Academy. When the progress was 85%-95%, I was a bit slack, which resulted in the time span for completing this course being as long as 36 days. The whole process was very challenging and extremely valuable.

Multivariable Calculus, 100％! When the progress reached 85%-95%, I slacked off a bit, which led to the course taking me as long as 36 days to finish.Math Academy Overall, I gained a lot from it, and I will teach the essence of the course to my high school students.

Niels Hoven 🐮 In fact, learning is an extremely private matter. It's hard for outsiders to understand whether someone has truly learned. Assessment results are for reference only, so don't take them too seriously. Perhaps few people care about the truth—too many are pretending to be asleep.