1/765 09-01 13:43:26-09-01 13:43:35 Have you ever thought about,
2/765 hp 09-01 13:43:40-09-01 13:43:46 Why a normal person can clearly calculate,
3/765 09-01 13:42:54-09-01 13:42:59 Counting money,
4/765 hp 09-01 13:42:55-09-01 13:42:59 Look at the clock.
5/765 aaa 09-01 13:42:56-09-01 13:42:59 But if you delve a little deeper into mathematics,
6/765 xss 09-01 13:42:59-09-01 13:43:11 I immediately felt like an idiot.
7/765 de 09-01 13:42:58-09-01 13:43:11 This is not your problem.
8/765 09-01 13:43:03-09-01 13:43:11 At least, it's not a completely fictional story.
9/765 n 09-01 13:43:04-09-01 13:43:11 In the 1980s, there was a fast-food chain.
10/765 hp 09-01 13:53:08-09-01 13:53:14 Launched a quarter-pound burger.
11/765 09-01 13:43:05-09-01 13:43:11 Selling well,
12/765 n 09-01 13:46:45-09-01 13:46:54 The competitors see that this won't work.
13/765 aaa 09-01 13:43:09-09-01 13:43:11 Therefore, the cheaper one-third pound burger uses more ingr...
14/765 09-01 13:43:06-09-01 13:43:11 Cheaper.
15/765 09-01 13:43:07-09-01 13:43:11 According to common sense,
16/765 hp 09-01 13:43:11-09-01 13:43:23 Consumers should rush to buy.
17/765 bg 09-01 13:43:12-09-01 13:43:23 The results were disastrous.
18/765 09-01 13:43:14-09-01 13:43:23 The company sent someone to investigate,
19/765 09-01 13:43:15-09-01 13:43:23 Many customers sincerely believe that one-quarter is larger ...
20/765 n 09-01 13:43:16-09-01 13:43:23 Because 4 is greater than 3.
21/765 09-01 13:43:17-09-01 13:43:23 This is the real math ability of American consumers.
22/765 de 09-01 13:43:19-09-01 13:43:23 This company has lost a lot of money as a result.
23/765 xss 09-01 13:43:23-09-01 13:43:46 Finally, I had to rename "one-third pound" into a more under...
24/765 aaa 09-01 13:43:20-09-01 13:43:23 Look at the torture that mathematics inflicts on people.
25/765 09-01 13:43:22-09-01 13:43:35 Start with simple fractions.
26/765 09-01 13:43:29-09-01 13:43:35 Here's another example that is more relatable.
27/765 bg 09-01 13:43:25-09-01 13:43:35 You work one day a week,
28/765 hp 09-01 13:43:27-09-01 13:43:35 Six hours of rest,
29/765 n 09-01 13:43:30-09-01 13:43:35 Your friend works on a double-shift,
30/765 09-01 13:43:31-09-01 13:43:35 Half a day off,
31/765 de 09-01 13:43:35-09-01 13:43:46 Do you think that taking only one day off is just a little l...
32/765 aaa 09-01 13:43:33-09-01 13:43:35 Right?
33/765 xss 09-01 13:44:30-09-01 13:44:40 Calculate the amount of time off,
34/765 09-01 13:43:37-09-01 13:43:46 Your break time is one-sixth of your working day.
35/765 bg 09-01 13:43:36-09-01 13:43:46 On weekends,
36/765 n 09-01 13:43:38-09-01 13:43:46 Your rest time is two-fifths of the workday.
37/765 09-01 13:43:42-09-01 13:43:56 Your rest time is not just a little bit less,
38/765 09-01 13:43:43-09-01 13:43:46 Instead, it plummeted by 58 percent.
39/765 09-01 13:43:45-09-01 13:43:46 What is the extent of this difference?
40/765 aaa 09-01 13:43:46-09-01 13:43:55 People who work long hours will feel that life is for work.
41/765 09-01 13:43:49-09-01 13:43:56 And people who have double weekends will feel that outside o...
42/765 bg 09-01 13:43:47-09-01 13:43:56 There is still life.
43/765 09-01 13:44:02-09-01 13:44:07 Same city,
44/765 n 09-01 13:43:48-09-01 13:43:56 Same job title,
45/765 09-01 13:43:51-09-01 13:43:56 Just because of the different rest policies,
46/765 09-01 13:43:52-09-01 13:43:56 Their life experiences were completely different.
47/765 de 09-01 13:43:53-09-01 13:43:57 Mathematics plays the role of a silent referee here,
48/765 hp 09-01 13:43:55-09-01 13:43:57 He doesn't speak.
49/765 xss 09-01 13:43:56-09-01 13:44:18 But he tells you with cold numbers that...
50/765 aaa 09-01 13:43:57-09-01 13:44:07 You are being exploited to a much greater extent than you re...
51/765 bg 09-01 13:43:58-09-01 13:44:07 Looking at the income again,
52/765 09-01 13:44:00-09-01 13:44:07 Your monthly salary is six thousand,
53/765 09-01 13:44:01-09-01 13:44:07 Subtract 5,000 rent.
54/765 n 09-01 13:44:03-09-01 13:44:07 Electricity, food, and commuting.
55/765 de 09-01 13:44:07-09-01 13:44:18 I have 1000 left.
56/765 09-01 13:44:05-09-01 13:44:07 You're getting paid eight thousand,
57/765 bg 09-01 13:44:08-09-01 13:44:18 Subtract the same 5,000.
58/765 09-01 13:44:10-09-01 13:44:18 Fixed expenses,
59/765 aaa 09-01 13:44:11-09-01 13:44:18 Three thousand left in hand,
60/765 aaa 09-01 13:45:13-09-01 13:45:15 Your salary increased by 33 percent.
61/765 bg 09-01 13:54:26-09-01 13:54:36 But your actual disposable income has doubled.
62/765 n 09-01 13:44:12-09-01 13:44:18 This is why the rich are getting richer,
63/765 hp 09-01 13:45:04-09-01 13:45:13 Because mathematics is about income,
64/765 09-01 13:44:13-09-01 13:44:18 It has never been a linear process.
65/765 09-01 13:44:20-09-01 13:44:30 When a person's fixed expenses only account for a small part...
66/765 09-01 13:44:15-09-01 13:44:18 He earns one more dollar,
67/765 hp 09-01 13:44:16-09-01 13:44:18 Almost all are disposable income.
68/765 09-01 13:44:17-09-01 13:44:30 When you are a freelancer,
69/765 bg 09-01 13:44:19-09-01 13:44:30 Most of the palm is occupied by fixed costs.
70/765 09-01 13:44:22-09-01 13:44:30 You hardly feel any change in the quality of life.
71/765 n 09-01 13:44:23-09-01 13:44:30 Mathematics has subtly solidified class.
72/765 09-01 13:44:25-09-01 13:44:30 Not through conspiracy, but through exponential calculation.
73/765 hp 09-01 13:44:26-09-01 13:44:30 Stocks are the same,
74/765 bg 09-01 13:44:31-09-01 13:44:40 First, increase by 5%.
75/765 aaa 09-01 13:44:28-09-01 13:44:30 Down by another 5%.
76/765 de 09-01 13:44:29-09-01 13:44:40 After repeated attempts,
77/765 09-01 13:44:32-09-01 13:44:40 Your money is gone, you thought it was offset by the gains a...
78/765 n 09-01 13:44:35-09-01 13:44:40 Actually, each time it's multiplied by 0.95.
79/765 aaa 09-01 13:44:36-09-01 13:44:40 Can't do it a few times and then return to zero.
80/765 09-01 13:44:34-09-01 13:44:40 Similarly,
81/765 hp 09-01 13:44:38-09-01 13:44:40 First, it fell by 5 percent.
82/765 09-01 13:45:29-09-01 13:45:36 Another 5% increase,
83/765 09-01 13:44:39-09-01 13:44:40 You've really managed to break through the intuitive defense...
84/765 09-01 13:44:40-09-01 13:44:52 Based on your feelings, you operate the stock market.
85/765 bg 09-01 13:44:42-09-01 13:44:52 Math is silently mocking from behind.
86/765 n 09-01 13:44:43-09-01 13:44:52 Okay,
87/765 09-01 13:44:45-09-01 13:44:52 If you think these are just everyday tips,
88/765 09-01 13:44:44-09-01 13:44:52 Then let's go deeper.
89/765 aaa 09-01 13:44:46-09-01 13:44:52 Let's see how math starts from childhood.
90/765 09-01 13:44:48-09-01 13:44:52 I'm setting up the traps step by step.
91/765 de 09-01 13:44:49-09-01 13:44:52 When you were a child,
92/765 hp 09-01 13:44:51-09-01 13:44:52 Teacher told you that three multiplied by two is three twos.
93/765 xss 09-01 13:44:52-09-01 13:45:04 I believe you.
94/765 n 09-01 13:44:54-09-01 13:45:04 Then one day you learned about negative numbers,
95/765 bg 09-01 13:44:53-09-01 13:45:04 Three multiplied by negative one,
96/765 09-01 13:44:56-09-01 13:45:04 Honestly, it's a -3.
97/765 09-01 13:44:57-09-01 13:45:04 You start to be confused,
98/765 09-01 13:44:58-09-01 13:45:04 Three minus one equals negative three.
99/765 aaa 09-01 13:45:00-09-01 13:45:04 I understand,
100/765 de 09-01 13:45:01-09-01 13:45:14 But -1 multiplied by -1,
101/765 bg 09-01 13:45:05-09-01 13:45:14 Why is it equal to one?
102/765 09-01 13:45:07-09-01 13:45:14 The teacher might say that this is a rule.
103/765 n 09-01 13:45:08-09-01 13:45:14 Why is this the case?
104/765 09-01 13:45:10-09-01 13:45:14 Honestly, just remember it.
105/765 09-01 13:45:11-09-01 13:45:14 Actually,
106/765 hp 09-01 13:46:10-09-01 13:46:11 The underlying meaning of this sentence,
107/765 xss 09-01 13:45:14-09-01 13:45:26 Is your intuition,
108/765 hp 09-01 13:45:15-09-01 13:45:26 I can no longer keep up with the logic of mathematics.
109/765 09-01 13:45:18-09-01 13:45:26 The concept of negative numbers, from its initial proposal t...
110/765 bg 09-01 13:45:17-09-01 13:45:26 It took over a thousand years for Dicaar,
111/765 09-01 13:45:20-09-01 13:45:26 Those high-level experts call the root of a complex number "...
112/765 bg 09-01 13:46:35-09-01 13:46:45 Think of something that even Dick Carle would find awkward.
113/765 09-01 13:45:21-09-01 13:45:26 Don't you feel tortured?
114/765 n 09-01 13:45:22-09-01 13:45:26 The commutative property of addition also applies.
115/765 09-01 13:45:24-09-01 13:45:36 Six multiplied by seven equals seven multiplied by six.
116/765 de 09-01 13:45:25-09-01 13:45:35 Adults consider this to be normal and natural.
117/765 aaa 09-01 13:45:26-09-01 13:45:35 But have you ever thought about it?
118/765 09-01 13:45:31-09-01 13:45:36 Six sevens and seven sixes are completely different in their...
119/765 hp 09-01 13:45:32-09-01 13:45:36 Six sevens make six groups.
120/765 bg 09-01 13:45:28-09-01 13:45:36 Seven in each group,
121/765 09-01 13:45:34-09-01 13:45:39 Seven six is seven groups,
122/765 n 09-01 13:45:35-09-01 13:45:39 Six in a group.
123/765 09-01 13:55:51-09-01 13:56:04 Why would these two completely different operations produce ...
124/765 xss 09-01 13:45:36-09-01 13:45:50 On the surface,
125/765 aaa 09-01 13:45:38-09-01 13:45:46 This is the commutative property of multiplication,
126/765 09-01 13:46:36-09-01 13:46:45 However, at a deeper level,
127/765 de 09-01 13:45:39-09-01 13:45:49 This is because linear transformations in one-dimensional sp...
128/765 bg 09-01 13:45:41-09-01 13:45:49 Arrived in the two-dimensional space,
129/765 09-01 13:45:42-09-01 13:45:49 Multiplication does not satisfy the commutative property.
130/765 09-01 13:45:43-09-01 13:45:49 Two rotation matrices multiplied in different orders,
131/765 09-01 13:45:45-09-01 13:45:50 Different results.
132/765 hp 09-01 13:45:46-09-01 13:45:50 That is to say,
133/765 09-01 13:45:47-09-01 13:45:59 Your math education from primary school to university,
134/765 aaa 09-01 13:45:50-09-01 13:46:00 Basically, you're being told that the intuitive understandin...
135/765 n 09-01 13:45:49-09-01 13:45:54 This constant cognitive disruption,
136/765 09-01 13:45:54-09-01 13:46:01 One of the core mechanisms that makes mathematics so frustra...
137/765 bg 09-01 13:45:52-09-01 13:46:01 Zero is also a torturous concept.
138/765 09-01 13:45:53-09-01 13:46:00 Ling said,
139/765 de 09-01 13:45:56-09-01 13:46:01 I don't understand this well.
140/765 hp 09-01 13:45:57-09-01 13:46:01 But zero cannot be a divisor.
141/765 09-01 13:45:59-09-01 13:46:11 This is confusing.
142/765 xss 09-01 13:46:00-09-01 13:46:11 Why?
143/765 aaa 09-01 13:46:05-09-01 13:46:11 Because the definition of division is the inverse operation ...
144/765 09-01 13:46:03-09-01 13:46:11 If zero can be a divisor,
145/765 09-01 13:46:04-09-01 13:46:11 What is the result of dividing by zero?
146/765 bg 09-01 13:46:07-09-01 13:46:22 equal to a number x such that zero times x equals one.
147/765 de 09-01 13:46:11-09-01 13:46:22 But zero multiplied by any number is zero.
148/765 09-01 13:46:08-09-01 13:46:11 It cannot be equal to one.
149/765 09-01 13:46:12-09-01 13:46:22 Therefore, 1 divided by zero does not exist.
150/765 09-01 13:46:17-09-01 13:46:33 What about zero divided by zero? Zero multiplied by any numb...
151/765 09-01 13:46:14-09-01 13:46:22 Therefore, zero divided by zero can be any number.
152/765 09-01 13:46:15-09-01 13:46:22 Math refuses to be ambiguous,
153/765 aaa 09-01 13:46:18-09-01 13:46:22 Therefore, he simply stipulated that zero cannot be used as ...
154/765 hp 09-01 13:46:53-09-01 13:46:56 This regulation seems arbitrary,
155/765 n 09-01 13:46:19-09-01 13:46:22 The purpose is to maintain logical consistency.
156/765 hp 09-01 13:46:21-09-01 13:46:22 But ordinary people don't need to understand the underlying ...
157/765 xss 09-01 13:46:22-09-01 13:46:45 They only remembered one rule.
158/765 bg 09-01 13:46:24-09-01 13:46:33 Then, I was confused for a lifetime.
159/765 09-01 13:46:25-09-01 13:46:33 But what truly elevated mathematics to the throne of torment...
160/765 hp 09-01 13:46:30-09-01 13:46:33 It was a scholarly tragedy that occurred two and a half mill...
161/765 09-01 13:46:27-09-01 13:46:33 The ancient Greek school of Pitagoras,
162/765 09-01 13:46:28-09-01 13:46:33 These people believe that,
163/765 aaa 09-01 13:46:31-09-01 13:46:33 A shocking truth,
164/765 n 09-01 13:46:32-09-01 13:46:33 Everything in the universe can be represented by integers or...
165/765 09-01 13:46:38-09-01 13:46:45 This is the cornerstone of their entire school's world.
166/765 de 09-01 13:46:33-09-01 13:46:45 Pythagoras was a legendary figure.
167/765 09-01 13:46:39-09-01 13:46:45 He traveled to Egypt and Babylon.
168/765 aaa 09-01 13:46:43-09-01 13:46:45 Studied a large amount of Eastern mathematics knowledge,
169/765 hp 09-01 13:46:41-09-01 13:46:45 Upon returning to Greece,
170/765 09-01 13:46:42-09-01 13:46:56 Founded this school.
171/765 09-01 13:46:47-09-01 13:46:56 The school not only studies mathematics, but also is a relig...
172/765 hp 09-01 13:48:03-09-01 13:48:06 I believe that trees are the origin of the universe.
173/765 bg 09-01 13:46:46-09-01 13:46:56 For them,
174/765 de 09-01 13:48:05-09-01 13:48:16 The ratio of integers is a symbol of the sacred order.
175/765 09-01 13:46:49-09-01 13:46:56 One day, someone in the academy discovered a problem.
176/765 09-01 13:46:50-09-01 13:46:56 A square with a side length of one.
177/765 aaa 09-01 13:46:52-09-01 13:46:56 What is its diagonal length?
178/765 n 09-01 13:52:05-09-01 13:52:09 According to the Pythagorean theorem, the square of a number...
179/765 de 09-01 13:46:54-09-01 13:47:07 However, this number cannot be expressed as the ratio of any...
180/765 xss 09-01 13:46:56-09-01 13:47:16 No matter how you try,
181/765 n 09-01 13:46:57-09-01 13:47:07 You can't find two integers,
182/765 bg 09-01 13:46:59-09-01 13:47:07 p and q such that p divided by q squared equals two.
183/765 09-01 13:47:00-09-01 13:47:07 Because if such scores exist,
184/765 aaa 09-01 13:48:00-09-01 13:48:06 You can simplify it to its lowest terms.
185/765 09-01 13:47:01-09-01 13:47:07 Then derive the numerator and denominator.
186/765 09-01 13:47:03-09-01 13:47:16 The definition of "all even numbers" is contradictory.
187/765 aaa 09-01 13:47:04-09-01 13:47:07 This proof is concise and effective.
188/765 09-01 13:47:06-09-01 13:47:07 Called reductio ad absurdum.
189/765 bg 09-01 13:47:08-09-01 13:47:16 It is said that the person who discovered this secret was th...
190/765 09-01 13:47:12-09-01 13:47:16 This was the first major crisis in mathematical history.
191/765 hp 09-01 13:47:07-09-01 13:47:16 From this moment on,
192/765 09-01 13:47:10-09-01 13:47:16 Humans are forced to admit that geometric intuition is unrel...
193/765 09-01 13:47:11-09-01 13:47:16 You can draw a diagonal of a square.
194/765 de 09-01 13:57:21-09-01 13:57:25 But you can never write its exact length.
195/765 aaa 09-01 13:47:14-09-01 13:47:16 This feeling is like discovering that your girlfriend, who y...
196/765 de 09-01 13:47:16-09-01 13:47:29 It's just like the feeling of being happy and content.
197/765 n 09-01 13:47:15-09-01 13:47:16 How to say it?
198/765 09-01 13:47:21-09-01 13:47:29 Both disgusting and exciting,
199/765 09-01 13:47:22-09-01 13:47:29 The existence of the square root of 2 is a milestone event.
200/765 09-01 13:47:19-09-01 13:47:29 Before this,
201/765 hp 09-01 13:47:23-09-01 13:47:29 Mathematics is intuitive geometry.
202/765 aaa 09-01 13:47:25-09-01 13:47:29 Carpenter makes furniture,
203/765 09-01 13:47:26-09-01 13:47:29 Architects,
204/765 n 09-01 13:47:28-09-01 13:47:29 Temple of the Gods,
205/765 xss 09-01 13:47:29-09-01 13:47:53 Mathematicians study shapes.
206/765 bg 09-01 13:47:30-09-01 13:47:42 Everything is tangible and can be seen or touched.
207/765 09-01 13:47:32-09-01 13:47:42 The appearance of the square root,
208/765 09-01 13:47:33-09-01 13:47:42 It's your pleasure.
209/765 aaa 09-01 13:47:35-09-01 13:47:42 Round suddenly opened his mouth and spoke.
210/765 hp 09-01 13:47:37-09-01 13:47:42 Hello brother,
211/765 n 09-01 13:47:36-09-01 13:47:42 Oh,
212/765 09-01 13:47:39-09-01 13:47:42 Adopted the voice of a male hormone,
213/765 09-01 13:47:40-09-01 13:47:53 Instantly shrink to the size of a peanut root,
214/765 09-01 13:47:43-09-01 13:47:53 Proves that there are things in the world that cannot be ful...
215/765 de 09-01 13:47:42-09-01 13:47:53 You can keep approaching,
216/765 09-01 13:47:47-09-01 13:47:53 He is approximating him with increasingly precise scores.
217/765 bg 09-01 13:47:45-09-01 13:47:53 But you can never grasp his essence.
218/765 09-01 13:47:48-09-01 13:47:53 This discovery has led to the creation of a completely new n...
219/765 hp 09-01 13:47:51-09-01 13:47:53 Rational numbers and irrational numbers together are called ...
220/765 aaa 09-01 13:47:50-09-01 13:47:53 This translation is also interesting.
221/765 n 09-01 13:47:53-09-01 13:48:05 Real numbers.
222/765 09-01 13:47:55-09-01 13:48:06 But the problem is,
223/765 bg 09-01 13:47:54-09-01 13:48:06 Irrational numbers are the most unreal things.
224/765 09-01 13:47:57-09-01 13:48:06 Because it is infinite and non-repeating,
225/765 09-01 13:47:58-09-01 13:48:06 You can never finish writing,
226/765 xss 09-01 13:48:06-09-01 13:48:36 What's even more torturous is that pi, which represents the ...
227/765 09-01 13:48:01-09-01 13:48:16 Circumference to diameter ratio,
228/765 n 09-01 13:48:08-09-01 13:48:16 This is an important constant that all civilizations indepen...
229/765 09-01 13:48:11-09-01 13:48:26 The ancient Egyptians calculated pi to approximately 3.16, w...
230/765 bg 09-01 13:48:17-09-01 13:48:26 Zu Chongzhi, a Chinese mathematician from ancient China, cal...
231/765 bg 09-01 13:48:09-09-01 13:48:16 However, all these approximations share a common problem:
232/765 09-01 13:48:40-09-01 13:48:49 Not a fraction,
233/765 bg 09-01 13:49:14-09-01 13:49:26 He is irrational like the square root of two.
234/765 de 09-01 13:48:47-09-01 13:48:49 Infinite loop.
235/765 aaa 09-01 13:48:12-09-01 13:48:16 In 1593,
236/765 hp 09-01 13:48:15-09-01 13:48:16 The French mathematician Fermat published an infinite produc...
237/765 09-01 13:48:16-09-01 13:48:26 This is the first time humans have used infinite algebraic o...
238/765 09-01 13:48:13-09-01 13:48:16 Vida's work marks a significant turning point.
239/765 n 09-01 13:48:19-09-01 13:48:26 No longer just splinters of wood,
240/765 aaa 09-01 13:48:20-09-01 13:48:26 It has become an algebraic structure.
241/765 09-01 13:48:22-09-01 13:48:26 Before this,
242/765 09-01 13:48:23-09-01 13:48:36 The original image was originally a logic.
243/765 de 09-01 13:48:26-09-01 13:48:36 This shift means that human understanding of mathematics has...
244/765 09-01 13:48:27-09-01 13:48:36 But the premise for thinking is that you must abandon visibl...
245/765 hp 09-01 13:48:25-09-01 13:48:26 For ordinary people,
246/765 bg 09-01 13:48:28-09-01 13:48:36 It's like asking you to give up your happiness.
247/765 xss 09-01 13:58:30-09-01 13:58:40 A distant obsession with Chinese characters.
248/765 09-01 13:48:33-09-01 13:48:36 Go and spend the night with him,
249/765 09-01 13:48:30-09-01 13:48:36 Together forever.
250/765 n 09-01 13:48:31-09-01 13:48:36 However,
251/765 aaa 09-01 13:48:35-09-01 13:48:36 The cruelty of mathematics lies in,
252/765 hp 09-01 13:48:36-09-01 13:48:49 He will never stop.
253/765 bg 09-01 13:48:37-09-01 13:48:49 After dealing with the square root of 2,
254/765 09-01 13:48:39-09-01 13:48:49 The medieval Europeans encountered another new demon negativ...
255/765 n 09-01 13:48:41-09-01 13:48:49 You said that debt can be represented by a negative number.
256/765 09-01 13:48:43-09-01 13:48:49 Borrowing can be represented by negative numbers.
257/765 aaa 09-01 13:48:44-09-01 13:48:49 This seems natural today,
258/765 09-01 13:48:46-09-01 13:48:49 In the 16th century in Europe,
259/765 xss 09-01 13:48:49-09-01 13:49:01 The position of the rich is similar to that of the magician.
260/765 n 09-01 13:48:53-09-01 13:49:01 The focus of the mathematicians' debate is a tree smaller th...
261/765 bg 09-01 13:48:50-09-01 13:49:01 Does zero exist or not?
262/765 hp 09-01 13:49:17-09-01 13:49:26 It's already so strange,
263/765 hp 09-01 13:48:51-09-01 13:49:01 Nothing,
264/765 09-01 13:48:56-09-01 13:49:01 How can something be less than nothing?
265/765 09-01 13:50:37-09-01 13:50:38 The prevailing mathematical view at the time was that negati...
266/765 09-01 13:48:54-09-01 13:49:01 There is no meaning in mathematics.
267/765 aaa 09-01 13:48:57-09-01 13:49:01 He only appears in the accounting records.
268/765 09-01 13:48:59-09-01 13:49:01 As a bookkeeping symbol,
269/765 09-01 13:49:00-09-01 13:49:13 The mathematician Cardano encountered negative roots when so...
270/765 de 09-01 13:49:01-09-01 13:49:13 He called it a fictional solution.
271/765 09-01 13:49:04-09-01 13:49:13 And clearly stated that this solution has no practical signi...
272/765 bg 09-01 13:49:03-09-01 13:49:13 Until Newton appeared,
273/765 hp 09-01 13:49:06-09-01 13:49:13 Hardinge, a mathematician, put negative numbers into the mat...
274/765 n 09-01 13:49:07-09-01 13:49:13 Why did Newton have this boldness?
275/765 09-01 13:49:09-09-01 13:49:13 Because he studies physics problems, he often encounters qua...
276/765 aaa 09-01 13:49:10-09-01 13:49:13 For example, upward is positive.
277/765 09-01 13:49:11-09-01 13:49:13 Down is negative,
278/765 xss 09-01 13:49:13-09-01 13:49:26 There are no negative numbers.
279/765 09-01 13:49:16-09-01 13:49:26 The physical equations cannot be written.
280/765 de 09-01 13:49:19-09-01 13:49:26 Newton's practical application of physics forced mathematici...
281/765 n 09-01 13:49:20-09-01 13:49:26 This indicates that even pure mathematicians are reluctant t...
282/765 09-01 13:49:21-09-01 13:49:26 It is the physicist's hammer,
283/765 aaa 09-01 13:49:23-09-01 13:49:26 A concept that even the most brilliant mathematicians have r...
284/765 09-01 13:50:50-09-01 13:50:59 An ordinary person would think that being tortured is perfec...
285/765 09-01 13:49:26-09-01 13:49:39 After all, you can choose not to care about happiness.
286/765 09-01 13:49:24-09-01 13:49:26 Round is both male and female,
287/765 n 09-01 13:51:43-09-01 13:51:47 But I advise parents to accept the dowry of gold and silver ...
288/765 bg 09-01 13:49:27-09-01 13:49:39 It will be difficult to find a wife.
289/765 09-01 13:49:29-09-01 13:49:39 What's even more extreme than negative numbers is,
290/765 aaa 09-01 13:49:32-09-01 13:49:39 The translation "imaginary imaginary" actually indicates the...
291/765 09-01 13:49:30-09-01 13:49:39 What is "empty"?
292/765 hp 09-01 13:49:33-09-01 13:49:39 There's no way,
293/765 de 09-01 13:49:34-09-01 13:49:39 i is equal to the square root of -1.
294/765 09-01 13:49:37-09-01 13:49:39 However, the square of any real number is positive.
295/765 xss 09-01 13:49:39-09-01 13:49:51 Therefore, there is no real number whose square is equal to ...
296/765 n 09-01 13:49:36-09-01 13:49:39 What is that?
297/765 hp 09-01 13:59:33-09-01 13:59:35 It is a symbol created by a mathematician.
298/765 n 09-01 13:49:42-09-01 13:49:51 When the Italian mathematician Cardano solved cubic equation...
299/765 bg 09-01 13:49:40-09-01 13:49:51 First time I encountered this thing.
300/765 aaa 09-01 13:49:44-09-01 13:49:51 He mentioned this tree in his book "The Great Tree," publish...
301/765 09-01 13:49:43-09-01 13:49:51 But then he said,
302/765 09-01 13:49:46-09-01 13:49:51 This is a useless and bizarre thing.
303/765 09-01 13:49:47-09-01 13:49:51 Like a subtle calculation.
304/765 de 09-01 13:49:48-09-01 13:49:51 The problem lies in the fact that...
305/765 hp 09-01 13:49:51-09-01 13:50:04 There is no such strange thing.
306/765 09-01 13:49:50-09-01 13:49:51 Three equations cannot be solved.
307/765 bg 09-01 13:49:53-09-01 13:50:04 In the cubic equation formula,
308/765 n 09-01 13:49:54-09-01 13:50:04 Even if all three roots are real numbers,
309/765 09-01 13:49:55-09-01 13:50:04 This is unavoidable during the calculation process.
310/765 aaa 09-01 13:49:57-09-01 13:50:04 After the imaginary number,
311/765 09-01 13:49:58-09-01 13:50:04 That is, imaginary numbers are an inevitable path in mathema...
312/765 09-01 13:50:03-09-01 13:50:16 Even if you have a hundred objections in your heart,
313/765 09-01 13:50:00-09-01 13:50:04 I have to walk through this road.
314/765 de 09-01 13:50:01-09-01 13:50:04 Mathematicians reluctantly accepted the falsehood.
315/765 xss 09-01 13:50:04-09-01 13:50:16 But he gave him a name that was extremely insulting.
316/765 bg 09-01 13:50:07-09-01 13:50:16 imaginary number The term "imaginary number" has been used i...
317/765 n 09-01 13:50:09-09-01 13:50:16 It's already considered polite to make vague statements.
318/765 hp 09-01 13:50:05-09-01 13:50:16 Imagine this scene,
319/765 de 09-01 13:50:16-09-01 13:50:28 A group of scholars who claim to represent the highest level...
320/765 09-01 13:50:11-09-01 13:50:16 Admit,
321/765 09-01 13:50:12-09-01 13:50:16 They have to cram a fictional thing into their subject.
322/765 09-01 13:50:14-09-01 13:50:16 Otherwise, the equation cannot be solved.
323/765 aaa 09-01 13:50:15-09-01 13:50:16 This kind of torment,
324/765 bg 09-01 13:50:17-09-01 13:50:28 How could an ordinary person endure it?
325/765 hp 09-01 13:50:19-09-01 13:50:28 Later,
326/765 09-01 13:50:20-09-01 13:50:28 People discovered that,
327/765 09-01 13:50:21-09-01 13:50:28 Abstraction is indispensable in electrical quantum mechanics...
328/765 n 09-01 13:50:23-09-01 13:50:28 No imaginary numbers.
329/765 09-01 13:50:24-09-01 13:50:28 The foundation of modern physics will collapse directly.
330/765 09-01 13:50:26-09-01 13:50:38 A mathematician believes that imaginary numbers are an essen...
331/765 aaa 09-01 13:50:25-09-01 13:50:28 This cognitive shift,
332/765 xss 09-01 13:50:28-09-01 13:50:47 Just like you always thought the round source was a figment ...
333/765 n 09-01 13:50:33-09-01 13:50:38 The result showed that he was currently sleeping next to you...
334/765 bg 09-01 13:50:29-09-01 13:50:38 In the 17th century,
335/765 hp 09-01 13:50:30-09-01 13:50:38 Mathematics has launched another large-scale attack on human...
336/765 09-01 13:50:32-09-01 13:50:38 Infinite.
337/765 09-01 13:50:34-09-01 13:50:47 Have you heard of the Zeno paradox?
338/765 aaa 09-01 13:50:36-09-01 13:50:38 The ancient Greek philosopher, Aristotle, proposed a series ...
339/765 de 09-01 13:50:38-09-01 13:50:47 The most famous is Achilles and the tortoise.
340/765 bg 09-01 13:50:39-09-01 13:50:47 Achilles is the Greek hero who runs fastest,
341/765 hp 09-01 13:50:43-09-01 13:50:47 He made the turtle run a certain distance first.
342/765 09-01 13:50:41-09-01 13:50:47 Then he started chasing.
343/765 aaa 09-01 13:50:45-09-01 13:50:47 When he chased the turtle to its starting point,
344/765 09-01 13:50:42-09-01 13:50:47 The turtle crawled forward again.
345/765 n 09-01 13:50:46-09-01 13:50:47 When he chased the turtle to its nest,
346/765 09-01 13:50:47-09-01 13:50:59 The turtle crawled a little further forward.
347/765 09-01 13:51:28-09-01 13:51:34 In the same way,
348/765 bg 09-01 13:50:49-09-01 13:50:59 He seems to always be chasing the turtle.
349/765 09-01 13:50:54-09-01 13:50:59 This paradox has troubled humanity for two thousand years,
350/765 09-01 13:50:51-09-01 13:50:59 Because he exploited a weakness in human intuition.
351/765 de 09-01 13:50:52-09-01 13:50:59 We don't understand infinity.
352/765 aaa 09-01 13:50:55-09-01 13:50:59 Until the 17th century,
353/765 hp 09-01 13:50:56-09-01 13:50:59 Newton and Leibniz independently invented calculus.
354/765 n 09-01 13:50:58-09-01 13:50:59 Humans have barely found a way to deal with infinity.
355/765 xss 09-01 13:50:59-09-01 13:51:11 The core concept of calculus is the limit and infinitesimal.
356/765 aaa 09-01 13:51:54-09-01 13:51:57 Newton calculated planetary orbits using calculus.
357/765 aaa 09-01 13:51:04-09-01 13:51:11 Leibniz solved the problem of finding the shortest path usin...
358/765 bg 09-01 13:51:00-09-01 13:51:11 The power of calculus is immediately apparent.
359/765 09-01 13:51:01-09-01 13:51:11 But he has a flaw in his theory,
360/765 09-01 13:51:03-09-01 13:51:11 What is an infinitesimal?
361/765 09-01 13:51:05-09-01 13:51:11 Newton sometimes treated it as zero.
362/765 de 09-01 13:51:07-09-01 13:51:11 Sometimes he has to be the divisor.
363/765 09-01 14:00:08-09-01 14:00:20 Leibniz said,
364/765 hp 09-01 13:51:09-09-01 13:51:11 It is a number smaller than any positive number.
365/765 n 09-01 13:51:08-09-01 13:51:11 But not zero.
366/765 09-01 13:51:11-09-01 13:51:23 Both of them couldn't explain what they were doing.
367/765 bg 09-01 13:51:12-09-01 13:51:23 But they calculated,
368/765 09-01 13:51:13-09-01 13:51:23 The result is surprisingly correct.
369/765 aaa 09-01 13:51:17-09-01 13:51:23 In 1734, the Archbishop of Canterbury wrote a pamphlet.
370/765 09-01 13:51:14-09-01 13:51:23 Title: Analyst
371/765 09-01 13:51:16-09-01 13:51:23 To a non-believing mathematician,
372/765 hp 09-01 13:51:20-09-01 13:51:23 Some mock infinitesimals as ghosts of dead quantities.
373/765 n 09-01 13:51:19-09-01 13:51:23 He said,
374/765 de 09-01 13:51:21-09-01 13:51:34 Calculus is less reliable than religious beliefs.
375/765 xss 09-01 13:51:23-09-01 13:51:34 This sentence, although containing a religious bias,
376/765 bg 09-01 13:51:24-09-01 13:51:34 But he hit on a key point.
377/765 09-01 13:51:25-09-01 13:51:34 At the very beginning of calculus,
378/765 09-01 13:51:26-09-01 13:51:34 Indeed, it is based on a vague foundation.
379/765 09-01 13:51:29-09-01 13:51:34 This feeling,
380/765 aaa 09-01 13:51:31-09-01 13:51:34 What is it like?
381/765 n 09-01 13:51:33-09-01 13:51:34 You brought the joy back home,
382/765 hp 09-01 13:51:34-09-01 13:51:47 Do you know he is a man?
383/765 bg 09-01 13:51:35-09-01 13:51:47 He also knew that he was a man.
384/765 09-01 13:51:37-09-01 13:51:47 Your parents also know that he is a man.
385/765 09-01 13:51:38-09-01 13:51:47 As a result, he had a child.
386/765 bg 09-01 13:52:32-09-01 13:52:40 The little Fen is this child.
387/765 de 09-01 13:51:40-09-01 13:51:47 No one can explain how this man gave birth to the child.
388/765 09-01 13:51:45-09-01 13:52:02 But he really is the one who, along with engineers and physi...
389/765 09-01 13:51:41-09-01 13:51:47 But the mathematicians saw him looking nervous.
390/765 aaa 09-01 13:51:44-09-01 13:51:47 This state of knowing something without understanding why ha...
391/765 bg 09-01 13:51:48-09-01 13:51:57 It was not until the 19th century that it was solved.
392/765 xss 09-01 13:51:46-09-01 13:51:57 Nineteenth-century mathematicians were finally fed up.
393/765 09-01 13:51:49-09-01 13:51:57 When Will Strauss, Dettmann, and Cantor come out to speak?
394/765 aaa 09-01 13:52:59-09-01 13:53:02 We can't keep playing around like this anymore.
395/765 hp 09-01 13:51:50-09-01 13:51:57 Must completely and rigorously formalize the concept of infi...
396/765 09-01 13:51:52-09-01 13:51:57 WeiErSiTeLaSiFaMingLe epsilon delta YuYan，
397/765 09-01 13:51:53-09-01 13:51:57 Describe dynamic limit processes using static inequalities.
398/765 de 09-01 13:51:57-09-01 13:52:08 This definition is extremely precise.
399/765 n 09-01 13:51:56-09-01 13:51:57 Strictly speaking,
400/765 09-01 13:51:59-09-01 13:52:09 The first time someone sees it, they feel like they're readi...
401/765 bg 09-01 13:51:58-09-01 13:52:08 Let's explore a sequence.
402/765 09-01 13:52:01-09-01 13:52:09 The limit of a as n approaches infinity is defined as follow...
403/765 bg 09-01 13:53:39-09-01 13:53:50 There exists a positive integer n such that when n is greate...
404/765 09-01 13:52:02-09-01 13:52:09 a minus the absolute value of a is less than epsilon.
405/765 n 09-01 14:08:43-09-01 14:08:50 FanYiChengDaBaiHua，
406/765 aaa 09-01 13:52:03-09-01 13:52:09 No matter how high the accuracy you want,
407/765 bg 09-01 13:52:10-09-01 13:52:19 I can always find a position where all subsequent terms are ...
408/765 hp 09-01 13:52:06-09-01 13:52:09 Describe infinite processes using pure finite language.
409/765 de 09-01 13:52:11-09-01 13:52:27 The revolutionary aspect of this definition lies in its use ...
410/765 xss 09-01 13:52:08-09-01 13:52:19 From a mysterious thing to an operational process.
411/765 09-01 13:52:09-09-01 13:52:19 Infinity is not just a vague, large, large thing in your min...
412/765 de 09-01 13:55:42-09-01 13:55:54 It is a set of precise logical rules.
413/765 09-01 13:52:12-09-01 13:52:29 But the price is that ordinary people can no longer understa...
414/765 09-01 13:52:14-09-01 13:52:19 When x approaches a, as described in your middle school text...
415/765 n 09-01 14:00:11-09-01 14:00:20 The definition behind it is like a set of incomprehensible g...
416/765 aaa 09-01 13:52:15-09-01 13:52:19 So when you take your college entrance exam,
417/765 09-01 13:52:16-09-01 13:52:19 Tortured to death by difficult problems,
418/765 n 09-01 13:52:18-09-01 13:52:19 You are actually just cleaning up after mathematicians from ...
419/765 hp 09-01 13:52:19-09-01 13:52:29 They are repairing something you don't even know is broken.
420/765 09-01 13:52:22-09-01 13:52:29 Professor Qian Xuesen once said that if a person is 14 years...
421/765 bg 09-01 13:52:20-09-01 13:52:29 Haha, you still have a lot to learn, Mr. Qian.
422/765 aaa 09-01 13:52:23-09-01 13:52:29 Another thing that makes me crazy is...
423/765 09-01 13:52:25-09-01 13:52:29 0.99 multiplied by 99 equals 1,
424/765 09-01 13:52:26-09-01 13:52:40 Many people can't accept this fact in their lifetime.
425/765 n 09-01 13:52:27-09-01 13:52:30 0.999 followed by an infinite number of 9s,
426/765 xss 09-01 13:52:29-09-01 13:52:51 Ming watched the pair of birds,
427/765 de 09-01 13:52:30-09-01 13:52:40 How does it equal one?
428/765 09-01 13:52:33-09-01 13:52:40 All the proof methods are right there.
429/765 hp 09-01 13:52:35-09-01 13:52:40 One-third equals 0.3 repeating,
430/765 09-01 13:52:36-09-01 13:52:40 Both sides multiplied by three, one equals 0.9 repeating.
431/765 09-01 13:52:37-09-01 13:52:40 Or set x to 0.9, a repeating decimal.
432/765 n 09-01 13:52:40-09-01 13:52:51 Ten times equals 9.9,
433/765 aaa 09-01 13:52:39-09-01 13:52:40 Subtracting nine from x equals nine.
434/765 09-01 13:53:54-09-01 13:54:01 Therefore, x is a perfect proof.
435/765 bg 09-01 13:52:41-09-01 13:52:51 But intuitively, I felt that something was wrong.
436/765 09-01 13:52:43-09-01 13:52:51 Why?
437/765 09-01 13:52:47-09-01 13:52:51 Because "approaching" and "equal" are two concepts that are ...
438/765 09-01 13:52:44-09-01 13:52:51 We believe that the value of 0.9 is approaching 1,
439/765 de 09-01 13:52:50-09-01 13:53:02 But it's always just a little bit short.
440/765 hp 09-01 13:52:46-09-01 13:52:51 Math tells you not to,
441/765 aaa 09-01 13:52:48-09-01 13:52:51 In the infinite world, infinitely close.
442/765 bg 09-01 13:52:52-09-01 13:53:02 That is, your brain hasn't evolved the ability to process in...
443/765 09-01 13:52:51-09-01 13:53:02 So you think this is torture.
444/765 09-01 13:52:53-09-01 13:53:02 However, all of these are just appetizers.
445/765 n 09-01 13:52:55-09-01 13:53:02 A truly great meal.
446/765 09-01 13:53:05-09-01 13:53:14 It was only brought up in the late 19th and early 20th centu...
447/765 09-01 13:52:56-09-01 13:53:02 In 1872,
448/765 09-01 13:52:58-09-01 13:53:02 German mathematician Wierstrass presented the world with a m...
449/765 hp 09-01 13:53:01-09-01 13:53:02 A function that is continuous everywhere but nowhere differe...
450/765 xss 09-01 13:53:02-09-01 13:53:14 What does it mean?
451/765 bg 09-01 13:53:04-09-01 13:53:14 The image of this function has no breakpoints.
452/765 09-01 13:53:06-09-01 13:53:14 You can finish drawing without even lifting your pen.
453/765 09-01 13:53:09-09-01 13:53:14 But there was no place that was smooth.
454/765 09-01 13:53:42-09-01 13:53:50 Every corner is sharp,
455/765 aaa 09-01 13:53:10-09-01 13:53:14 It's really sharp,
456/765 n 09-01 13:53:12-09-01 13:53:14 No pattern.
457/765 09-01 13:53:13-09-01 13:53:25 Since the Newtonian era, humans have...
458/765 de 09-01 13:53:14-09-01 13:53:25 Always assume that things are continuous.
459/765 bg 09-01 13:53:16-09-01 13:53:25 At least in some places, it is smooth.
460/765 09-01 13:53:21-09-01 13:53:25 After all, there must be a flat place somewhere when you tou...
461/765 09-01 13:53:19-09-01 13:53:25 The objects around us,
462/765 hp 09-01 13:53:17-09-01 13:53:25 Table,
463/765 09-01 13:53:20-09-01 13:53:25 Ball
464/765 n 09-01 13:53:23-09-01 13:53:25 Are all the slopes not continuous and smooth?
465/765 aaa 09-01 13:53:24-09-01 13:53:25 Wells said,
466/765 xss 09-01 13:53:25-09-01 13:53:38 No,
467/765 09-01 13:53:28-09-01 13:53:38 Math doesn't care about your handwriting.
468/765 bg 09-01 13:53:27-09-01 13:53:38 This function is perfectly valid in mathematics.
469/765 09-01 13:53:29-09-01 13:53:38 But you can never draw his true image.
470/765 hp 09-01 13:53:31-09-01 13:53:38 Because it is jagged in any scale,
471/765 aaa 09-01 13:53:32-09-01 13:53:38 If you magnify it a hundred times,
472/765 09-01 13:53:33-09-01 13:53:38 It is still jagged.
473/765 de 09-01 13:53:35-09-01 13:53:38 Magnify 10,000 times
474/765 n 09-01 13:53:36-09-01 13:53:38 Still jagged.
475/765 09-01 13:53:38-09-01 13:53:50 Forever so,
476/765 09-01 13:53:40-09-01 13:53:50 No end.
477/765 09-01 13:53:43-09-01 13:53:50 Why is this function so frustrating?
478/765 aaa 09-01 13:53:45-09-01 13:53:50 Because he completely destroyed the concept of continuity.
479/765 hp 09-01 13:53:46-09-01 13:53:50 In the beautiful image in human hearts.
480/765 n 09-01 13:53:47-09-01 13:53:50 Before Wells Street,
481/765 de 09-01 13:53:49-09-01 13:54:01 Mathematicians have a warm and affectionate image of continu...
482/765 09-01 13:54:32-09-01 13:54:36 They might encounter some difficulties.
483/765 xss 09-01 13:53:50-09-01 13:54:12 Overall, it should be mild.
484/765 hp 09-01 13:54:19-09-01 13:54:25 Predictable.
485/765 bg 09-01 13:53:51-09-01 13:54:01 Just like your wife, Yuan Yuan, is a man.
486/765 09-01 13:53:53-09-01 13:54:01 But it's really well-written.
487/765 n 09-01 13:53:58-09-01 13:54:01 But the Weirstrass function is not like that.
488/765 hp 09-01 13:53:56-09-01 13:54:01 He's more like your wife, Yuan Yuan.
489/765 aaa 09-01 13:53:57-09-01 13:54:01 The previous second was round,
490/765 09-01 13:54:00-09-01 13:54:01 In the next second, I became Zhang Fei.
491/765 09-01 13:54:01-09-01 13:54:12 The next second, I was round again.
492/765 bg 09-01 13:54:03-09-01 13:54:12 In the next second, he turned into Li Kui.
493/765 09-01 13:54:04-09-01 13:54:12 This pathological existence,
494/765 xss 09-01 13:55:22-09-01 13:55:42 It made mathematicians question their own definitions.
495/765 09-01 13:54:05-09-01 13:54:12 But check and re-check the definitions,
496/765 aaa 09-01 13:54:11-09-01 13:54:12 It's not that we are wrong, it is our intuition.
497/765 hp 09-01 13:54:07-09-01 13:54:12 We thought that continuity meant smoothness.
498/765 09-01 13:54:08-09-01 13:54:12 But the continuity in mathematical definitions,
499/765 de 09-01 13:54:10-09-01 13:54:12 Only require the function value to be continuous.
500/765 aaa 09-01 14:00:16-09-01 14:00:20 I don't require handsome appearance from Uncle Han.
501/765 n 09-01 13:54:12-09-01 13:54:25 The huge gap between this definition and intuition,
502/765 bg 09-01 13:54:14-09-01 13:54:25 Another source of torture in mathematics.
503/765 09-01 13:54:15-09-01 13:54:25 Subsequently,
504/765 aaa 09-01 13:54:17-09-01 13:54:25 The founder of the "What to Say" platform, Cantor, dropped a...
505/765 09-01 13:54:23-09-01 13:54:25 He proved that there are as many natural numbers as even num...
506/765 09-01 13:54:18-09-01 13:54:25 What's the same thing?
507/765 de 09-01 13:54:21-09-01 13:54:25 Natural numbers include both even and odd numbers,
508/765 09-01 13:54:22-09-01 13:54:36 A portion of the numbers are even.
509/765 xss 09-01 13:54:25-09-01 13:54:45 How could it still be the same?
510/765 de 09-01 13:54:58-09-01 13:55:09 Cantor said,
511/765 09-01 13:54:28-09-01 13:54:36 Determine infinity,
512/765 n 09-01 13:54:29-09-01 13:54:36 The size of the set cannot be determined by counting.
513/765 hp 09-01 13:54:34-09-01 13:54:36 It needs to be one-to-one.
514/765 de 09-01 13:54:35-09-01 13:54:46 Each natural number n corresponds to an even number 2n, whic...
515/765 09-01 13:54:31-09-01 13:54:36 So, the same amount.
516/765 aaa 09-01 13:54:36-09-01 13:54:46 This conclusion shocked the mathematical community at that t...
517/765 bg 09-01 13:54:38-09-01 13:54:46 Kronek directly called Cantor a scientist.
518/765 09-01 13:54:39-09-01 13:54:46 Scammers say,
519/765 09-01 13:54:42-09-01 13:54:46 His theory is a plague in mathematics.
520/765 n 09-01 13:54:41-09-01 13:54:46 But Cantor didn't stop,
521/765 hp 09-01 13:54:44-09-01 13:54:46 He also proved that real numbers are more than natural numbe...
522/765 aaa 09-01 13:55:17-09-01 13:55:22 And there are many more.
523/765 09-01 13:54:45-09-01 13:54:58 Natural numbers are countable infinity.
524/765 09-01 13:54:46-09-01 13:54:58 Real numbers are uncountable infinity.
525/765 xss 09-01 13:54:48-09-01 13:55:09 I'm so tired,
526/765 bg 09-01 13:54:49-09-01 13:54:58 There are indeed different sizes.
527/765 aaa 09-01 13:54:51-09-01 13:54:58 This completely subverts the understanding of infinity that ...
528/765 09-01 13:54:52-09-01 13:54:58 Before this,
529/765 hp 09-01 13:54:53-09-01 13:54:58 People think that infinity is a very large tree.
530/765 n 09-01 13:54:55-09-01 13:54:58 Once you've arrived, there is no turning back.
531/765 09-01 13:54:56-09-01 13:54:58 Cantor told me,
532/765 bg 09-01 13:54:59-09-01 13:55:09 There are countless layers,
533/765 09-01 13:55:00-09-01 13:55:09 And in a progressive manner,
534/765 09-01 13:55:02-09-01 13:55:09 No end.
535/765 hp 09-01 13:55:03-09-01 13:55:09 This discovery directly led to Cantor's own mental breakdown...
536/765 aaa 09-01 13:55:05-09-01 13:55:09 Multiple times in my later years, I was admitted to a mental...
537/765 09-01 13:55:36-09-01 13:55:42 A study of infinity,
538/765 09-01 13:55:06-09-01 13:55:09 Driven to the brink of madness.
539/765 n 09-01 13:55:07-09-01 13:55:09 This torture even the most brilliant mathematicians couldn't...
540/765 n 09-01 13:55:19-09-01 13:55:22 Moreover, Cantor's tragedy in the ordinary world symbolizes ...
541/765 bg 09-01 13:55:10-09-01 13:55:22 You are exploring a realm beyond human comprehension,
542/765 09-01 13:55:09-09-01 13:55:22 You will eventually pay the price.
543/765 09-01 13:55:12-09-01 13:55:22 There are bigger storms to come.
544/765 09-01 13:55:13-09-01 13:55:22 Around 1900,
545/765 hp 09-01 13:55:14-09-01 13:55:22 Mathematicians discovered that set theory has paradoxes.
546/765 09-01 13:55:16-09-01 13:55:22 The most famous is Russell's paradox.
547/765 hp 09-01 13:55:26-09-01 13:55:32 Does the set r of all sets that do not contain themselves, a...
548/765 de 09-01 13:55:20-09-01 13:55:32 If r contains itself,
549/765 bg 09-01 13:55:23-09-01 13:55:32 Then according to the definition, it should not include itse...
550/765 09-01 13:55:24-09-01 13:55:32 If r is not equal to itself,
551/765 hp 09-01 13:56:25-09-01 13:56:34 Therefore, according to the definition, r should include its...
552/765 aaa 09-01 13:55:29-09-01 13:55:32 No matter what the situation, it leads to conflict.
553/765 09-01 13:55:27-09-01 13:55:32 This paradox is so simple that even elementary school studen...
554/765 09-01 13:55:28-09-01 13:55:32 However, it is enough to destroy the entire foundation of se...
555/765 n 09-01 13:55:31-09-01 13:55:32 Because it means the carefully constructed logic that mathem...
556/765 09-01 13:56:48-09-01 13:56:55 There is a fundamental crack in the building.
557/765 09-01 13:55:32-09-01 13:55:42 You don't need to understand advanced mathematics,
558/765 bg 09-01 13:55:33-09-01 13:55:42 You only need to understand a little basic logic,
559/765 n 09-01 13:55:34-09-01 13:55:42 You can see mathematicians lying.
560/765 hp 09-01 13:55:39-09-01 13:55:42 Or, in other words, mathematics is self-contradictory.
561/765 09-01 13:55:37-09-01 13:55:42 Mathematicians are in a panic.
562/765 aaa 09-01 13:55:38-09-01 13:55:42 Start repairing the foundation.
563/765 09-01 13:55:41-09-01 13:55:42 Melro and Frankel developed a formal set theory,
564/765 bg 09-01 13:55:43-09-01 13:55:54 Avoid paradoxes with a few carefully designed axioms.
565/765 09-01 13:55:44-09-01 13:55:54 They prohibited all sets of sets.
566/765 09-01 13:56:20-09-01 13:56:25 This dangerous operation,
567/765 09-01 13:55:46-09-01 13:55:54 Only construct sets within the safe range.
568/765 09-01 14:01:10-09-01 14:01:14 This is like discovering that the foundation of your house h...
569/765 aaa 09-01 13:55:47-09-01 13:55:54 Then you laid a rug on top of the crack.
570/765 n 09-01 13:55:49-09-01 13:55:54 The carpet is beautiful.
571/765 09-01 13:59:54-09-01 13:59:57 But you know in your heart that the crack is still there.
572/765 09-01 13:55:50-09-01 13:55:54 Knowing that there is a problem,
573/765 09-01 13:55:56-09-01 13:56:04 However, the assumption of no problems has been a constant t...
574/765 hp 09-01 13:55:52-09-01 13:55:54 After the Poison Paradox,
575/765 xss 09-01 13:55:54-09-01 13:56:04 A bigger problem has surfaced.
576/765 bg 09-01 13:55:55-09-01 13:56:04 1931,
577/765 hp 09-01 13:56:04-09-01 13:56:16 At the age of 25, Austrian logician Gödel published his inco...
578/765 de 09-01 13:55:58-09-01 13:56:04 In simple terms, this theorem states that...
579/765 09-01 13:56:00-09-01 13:56:04 There exists a statement that is true in any sufficiently po...
580/765 aaa 09-01 13:55:59-09-01 13:56:04 They are real.
581/765 n 09-01 13:56:01-09-01 13:56:04 But you can never prove what they mean within this system?
582/765 bg 09-01 13:56:05-09-01 13:56:16 This means that mathematics is not the perfect and flawless ...
583/765 09-01 13:56:03-09-01 13:56:04 Math has dead ends,
584/765 n 09-01 13:56:07-09-01 13:56:16 There are blind spots that can never be reached.
585/765 09-01 13:56:08-09-01 13:56:16 Hilbert once boasted that,
586/765 09-01 13:56:09-09-01 13:56:16 We must know that,
587/765 de 09-01 13:56:14-09-01 13:56:25 We will know that Gödel shattered this dream with his two th...
588/765 09-01 13:56:11-09-01 13:56:16 This isn't a technical issue.
589/765 aaa 09-01 13:56:12-09-01 13:56:16 This should be called a philosophical collapse.
590/765 09-01 13:56:13-09-01 13:56:25 Before this,
591/765 xss 09-01 13:56:16-09-01 13:56:34 Mathematicians felt that they were moving towards an ultimat...
592/765 bg 09-01 13:56:17-09-01 13:56:25 GeDeErZhiHou，
593/765 09-01 13:56:18-09-01 13:56:25 Everyone realized,
594/765 n 09-01 13:56:21-09-01 13:56:25 A formal system is not used to describe the only truth,
595/765 09-01 13:56:23-09-01 13:56:34 Just defining the different possibilities.
596/765 aaa 09-01 13:56:22-09-01 13:56:25 The same problem,
597/765 bg 09-01 13:56:26-09-01 13:56:34 This is correct in this axiom system,
598/765 09-01 13:56:27-09-01 13:56:34 What is right in one system may be wrong in another.
599/765 09-01 13:57:10-09-01 13:57:25 Mathematical truths have become multiple-choice questions.
600/765 09-01 13:56:30-09-01 13:56:44 GeDeErDingLiDeShaShangLiZaiYuTaGongJiDeBuShiShuXueDeMouGeFen...
601/765 de 09-01 13:56:29-09-01 13:56:34 Instead, it's about overall confidence in mathematics.
602/765 09-01 13:56:33-09-01 13:56:34 You have worked hard for ten years to learn mathematics,
603/765 09-01 13:57:12-09-01 13:57:15 Established the belief that mathematics equals truth.
604/765 aaa 09-01 13:56:31-09-01 13:56:34 Then someone told you,
605/765 n 09-01 13:56:34-09-01 13:56:44 Excuse me,
606/765 09-01 13:56:35-09-01 13:56:44 Some true statements cannot be proven.
607/765 bg 09-01 13:56:36-09-01 13:56:44 And this is not because you are not smart.
608/765 hp 09-01 13:56:38-09-01 13:56:44 Because logic itself has this flaw,
609/765 09-01 13:56:39-09-01 13:56:44 This feeling.
610/765 09-01 13:56:50-09-01 13:56:55 Just like you were taught from a young age to pursue truth a...
611/765 aaa 09-01 13:56:41-09-01 13:56:44 The result is reached,
612/765 09-01 13:56:42-09-01 13:56:44 There are multiple answers,
613/765 de 09-01 13:56:43-09-01 13:56:55 And some answers are impossible to reach.
614/765 xss 09-01 13:56:44-09-01 13:56:55 For an ordinary person,
615/765 n 09-01 13:56:46-09-01 13:56:55 This is a devastating blow.
616/765 bg 09-01 13:56:47-09-01 13:56:55 We learn mathematics,
617/765 aaa 09-01 13:56:53-09-01 13:56:55 Deep down, I always thought there was a right answer.
618/765 hp 09-01 13:56:54-09-01 13:56:55 Waiting for us at the finish line.
619/765 09-01 13:56:51-09-01 13:57:05 But mathematics tells you,
620/765 09-01 13:56:55-09-01 13:57:05 Some questions simply don't have a standard answer.
621/765 09-01 13:57:52-09-01 13:57:57 Because it's impossible to solve.
622/765 bg 09-01 13:56:56-09-01 13:57:05 More than the incompleteness theorem.
623/765 n 09-01 13:56:57-09-01 13:57:05 I instinctively knew about Banach-Tarski paradox.
624/765 09-01 13:56:59-09-01 13:57:05 In 1924,
625/765 de 09-01 13:57:00-09-01 13:57:14 Two Polish mathematicians have proven a seemingly absurd con...
626/765 09-01 13:57:01-09-01 13:57:14 A solid sphere can be divided into a finite number of pieces...
627/765 aaa 09-01 13:57:03-09-01 13:57:05 Reassemble two and the same as the original through rotation...
628/765 hp 09-01 13:57:04-09-01 13:57:05 The large ball doubles in volume out of nowhere.
629/765 xss 09-01 13:57:05-09-01 13:57:15 You didn't hear wrong,
630/765 bg 09-01 13:57:06-09-01 13:57:14 Mathematically, this has been rigorously proven.
631/765 09-01 13:57:08-09-01 13:57:14 Of course, these are extremely strange collections.
632/765 n 09-01 13:57:09-09-01 13:57:14 No concept of volume.
633/765 aaa 09-01 13:57:13-09-01 13:57:15 It is impossible to achieve in the physical world,
634/765 hp 09-01 13:57:14-09-01 13:57:25 But math doesn't matter.
635/765 09-01 13:57:16-09-01 13:57:25 Physical world,
636/765 bg 09-01 13:57:17-09-01 13:57:25 Mathematics only cares about whether it is logically consist...
637/765 09-01 13:57:18-09-01 13:57:25 This paradox completely devastated those who still believed ...
638/765 bg 09-01 13:58:31-09-01 13:58:40 Math starts with the square root of two,
639/765 09-01 13:58:10-09-01 13:58:18 Disconnection from sensory experience.
640/765 n 09-01 13:57:20-09-01 13:57:25 After two thousand years of development,
641/765 09-01 13:57:22-09-01 13:57:25 In the 20th century,
642/765 aaa 09-01 13:57:24-09-01 13:57:25 Completely immersed in the world of pure logic,
643/765 xss 09-01 13:57:25-09-01 13:57:37 No more contact with people on the ground.
644/765 bg 09-01 13:57:26-09-01 13:57:37 You can draw a geometric construction of the square root on ...
645/765 09-01 13:57:27-09-01 13:57:37 Can you understand the position of imaginary numbers in the ...
646/765 09-01 13:57:30-09-01 13:57:47 You can even use a computer to simulate an approximate image...
647/765 09-01 13:57:31-09-01 13:57:37 But the paradox of Banach-Tarski, which contains those fasts...
648/765 hp 09-01 13:57:29-09-01 13:57:37 You can't even imagine it.
649/765 09-01 14:02:52-09-01 14:02:58 They are the products of set theory axioms.
650/765 n 09-01 13:57:33-09-01 13:57:37 Not a mapping of the sensory world,
651/765 09-01 13:57:34-09-01 13:57:37 This marks a complete break between mathematics and intuitio...
652/765 de 09-01 13:57:37-09-01 13:57:47 In the 20th century, there was a significant discovery about...
653/765 aaa 09-01 13:57:35-09-01 13:57:37 Cantor proposed a question,
654/765 bg 09-01 13:57:38-09-01 13:57:47 Between the infinite sum of natural numbers and the infinity...
655/765 09-01 13:57:39-09-01 13:57:47 Does there exist a middle-sized infinity?
656/765 hp 09-01 13:57:43-09-01 13:57:47 Gödell proved in 1940 that using existing axioms,
657/765 n 09-01 13:57:40-09-01 13:57:47 It is impossible to prove that this intermediate infinity do...
658/765 09-01 13:57:41-09-01 13:57:47 1963
659/765 aaa 09-01 13:57:44-09-01 13:57:47 Koen has proven that using existing axioms,
660/765 09-01 13:57:45-09-01 13:57:57 Also, it cannot be proven.
661/765 n 09-01 13:57:49-09-01 13:57:57 Therefore, this statement cannot be verified.
662/765 xss 09-01 13:57:47-09-01 13:57:57 Cannot be disproven.
663/765 bg 09-01 13:57:48-09-01 13:57:57 Is it real or fake?
664/765 09-01 13:57:50-09-01 13:57:57 It depends on which set of kilometers you choose.
665/765 de 09-01 13:57:56-09-01 13:58:07 This is the first time in human history that a mathematical ...
666/765 aaa 09-01 13:57:53-09-01 13:57:57 However, the answer is not a definite thing.
667/765 09-01 13:57:54-09-01 13:57:57 And it is a choice.
668/765 hp 09-01 13:57:57-09-01 13:58:07 You can choose,
669/765 bg 09-01 13:57:58-09-01 13:58:07 You can choose,
670/765 bg 09-01 13:58:41-09-01 13:58:52 There are no two choices.
671/765 n 09-01 13:58:01-09-01 13:58:07 They can all establish self-consistent mathematical systems.
672/765 09-01 13:58:00-09-01 13:58:07 At this point,
673/765 09-01 13:58:02-09-01 13:58:07 Mathematics has become a language of definition, rather than...
674/765 09-01 13:58:03-09-01 13:58:07 In a possible world, ordinary people learn mathematics in sc...
675/765 aaa 09-01 13:58:04-09-01 13:58:07 To learn knowledge,
676/765 09-01 13:58:06-09-01 13:58:18 The mindset that comes from learning the truth.
677/765 xss 09-01 13:58:07-09-01 13:58:18 The results show that what I learned is not the truth.
678/765 bg 09-01 13:58:09-09-01 13:58:18 Just the rules.
679/765 09-01 13:58:11-09-01 13:58:18 Moreover, this rule is man-made.
680/765 n 09-01 13:58:16-09-01 13:58:18 You don't even have the right to choose your own rules.
681/765 hp 09-01 13:58:12-09-01 13:58:18 Because the textbook has already chosen for you,
682/765 aaa 09-01 13:58:14-09-01 13:58:18 This feeling of emptiness is no less than the feeling of los...
683/765 09-01 13:58:15-09-01 13:58:18 For ten years, he lived as a man.
684/765 de 09-01 13:58:18-09-01 13:58:30 Standing naked in front of you,
685/765 bg 09-01 13:58:53-09-01 13:59:02 And it's also bigger than yours.
686/765 bg 09-01 13:58:19-09-01 13:58:30 At this point,
687/765 09-01 13:58:21-09-01 13:58:30 I think you should have understood.
688/765 n 09-01 13:58:22-09-01 13:58:30 The reason why mathematics is so torturous for ordinary peop...
689/765 09-01 13:58:23-09-01 13:58:30 The root lies in a harsh reality.
690/765 09-01 13:58:28-09-01 13:58:40 The human brain is an organ that has evolved for survival.
691/765 09-01 13:58:24-09-01 13:58:30 Not designed for mathematics.
692/765 hp 09-01 13:58:25-09-01 13:58:30 Our brains are good at processing objects of medium size.
693/765 aaa 09-01 13:58:27-09-01 13:58:30 Limited quantity,
694/765 09-01 13:58:32-09-01 13:58:40 Continuous smooth movement,
695/765 09-01 13:58:33-09-01 13:58:40 The three-dimensional Euclidean space extends infinitely bey...
696/765 hp 09-01 13:58:34-09-01 13:58:40 High-dimensional discontinuity,
697/765 de 09-01 13:58:36-09-01 13:58:40 Random bending,
698/765 n 09-01 13:58:37-09-01 13:58:40 I've never seen time perception,
699/765 aaa 09-01 13:58:38-09-01 13:58:40 There is no reason to evolve corresponding perception abilit...
700/765 de 09-01 13:58:49-09-01 13:58:52 Math is different.
701/765 09-01 13:58:43-09-01 13:58:52 Mathematics is essentially about stepping out of the human c...
702/765 09-01 13:58:40-09-01 13:58:52 The things that are not visible to the naked eye,
703/765 09-01 13:58:50-09-01 13:59:02 The square root of 2 is a number that can be expressed as th...
704/765 09-01 13:58:44-09-01 13:58:52 Wu Qiong betrayed physical perception,
705/765 aaa 09-01 13:58:46-09-01 13:58:52 Gödell betrayed the truth,
706/765 hp 09-01 13:58:45-09-01 13:58:52 Intuition,
707/765 n 09-01 13:58:47-09-01 13:58:52 Probability has undermined causality.
708/765 bg 09-01 13:59:36-09-01 13:59:46 Every major breakthrough in mathematics,
709/765 xss 09-01 13:58:52-09-01 13:59:02 A violent breakthrough in the boundaries of human cognition,
710/765 n 09-01 13:58:54-09-01 13:59:02 The inevitable consequence is pain and discomfort.
711/765 09-01 13:58:57-09-01 13:59:02 But there is one fact that is often overlooked,
712/765 hp 09-01 13:58:55-09-01 13:59:02 Math is not monotonous.
713/765 aaa 09-01 13:59:32-09-01 13:59:35 More torturous,
714/765 09-01 13:58:58-09-01 13:59:02 He is also constantly being tamed by negative numbers.
715/765 09-01 13:59:00-09-01 13:59:12 In the thirteenth century, it was a tormenting thing.
716/765 09-01 14:00:03-09-01 14:00:08 Now elementary school students know about negative numbers.
717/765 aaa 09-01 13:58:59-09-01 13:59:02 In the 16th century, it was considered fiction.
718/765 de 09-01 13:59:02-09-01 13:59:12 Nowadays, engineers are using Microfit every day.
719/765 de 09-01 14:04:24-09-01 14:04:34 In the 17th century, it was mysticism.
720/765 bg 09-01 13:59:03-09-01 13:59:12 Now it's a basic science and engineering course.
721/765 09-01 13:59:05-09-01 13:59:12 Probability theory was a cutting-edge subject one hundred ye...
722/765 n 09-01 13:59:06-09-01 13:59:12 Now, the college entrance examination must include mathemati...
723/765 hp 09-01 13:59:07-09-01 13:59:12 The frontier is always a torment,
724/765 09-01 13:59:08-09-01 13:59:12 Because the frontier is still in the realm of intuition,
725/765 aaa 09-01 13:59:10-09-01 13:59:12 But the front line will constantly retreat.
726/765 09-01 13:59:11-09-01 13:59:23 Yesterday's torment has become today's commonplace.
727/765 xss 09-01 13:59:12-09-01 13:59:23 Today's suffering becomes tomorrow's commonplace.
728/765 hp 09-01 13:59:19-09-01 13:59:23 Human civilization is like this, it has emerged step by step...
729/765 bg 09-01 13:59:14-09-01 13:59:23 This process is slow and painful.
730/765 n 09-01 13:59:15-09-01 13:59:23 Irreversible.
731/765 09-01 13:59:16-09-01 13:59:23 So, when you feel like math is torturing you,
732/765 09-01 13:59:17-09-01 13:59:23 Don't be discouraged.
733/765 09-01 13:59:20-09-01 13:59:23 This is a sign that you are progressing.
734/765 bg 09-01 13:59:24-09-01 13:59:35 You are leaving that small, comfortable but narrow cognitive...
735/765 aaa 09-01 13:59:21-09-01 13:59:23 Math is not intentionally trying to make you difficult.
736/765 de 09-01 13:59:23-09-01 13:59:35 He just told you honestly,
737/765 09-01 13:59:27-09-01 13:59:35 The real world is much bigger than you think.
738/765 09-01 13:59:29-09-01 13:59:35 What you think is obviously only valid under a very small an...
739/765 n 09-01 13:59:26-09-01 13:59:35 Math helps you open that door,
740/765 09-01 13:59:28-09-01 13:59:35 It's just a bit rough.
741/765 09-01 13:59:30-09-01 13:59:35 Ask one more question,
742/765 n 09-01 13:59:38-09-01 13:59:46 When did mathematics start to be counterintuitive?
743/765 xss 09-01 13:59:34-09-01 13:59:46 Answer:
744/765 09-01 13:59:37-09-01 13:59:46 Starting from a vast knowledge base,
745/765 09-01 13:59:39-09-01 13:59:46 As long as there is no culture,
746/765 09-01 14:00:07-09-01 14:00:08 A pound of cotton and a pound of iron weigh the same.
747/765 aaa 09-01 13:59:41-09-01 13:59:46 All are counterintuitive.
748/765 hp 09-01 13:59:42-09-01 13:59:46 Actually, mathematics has never changed.
749/765 bg 09-01 14:00:10-09-01 14:00:20 The change is that every time you feel like math is torturin...
750/765 09-01 13:59:43-09-01 13:59:46 Actually, it's your cognition expanding.
751/765 de 09-01 13:59:45-09-01 13:59:57 This expansion is painful.
752/765 09-01 13:59:49-09-01 13:59:57 But this is the inevitable path for human civilization to mo...
753/765 09-01 13:59:46-09-01 13:59:57 We all feel confused when facing mathematics.
754/765 bg 09-01 13:59:48-09-01 13:59:57 Frustration and confusion,
755/765 n 09-01 13:59:50-09-01 13:59:57 These are all small steps in the grand process.
756/765 hp 09-01 13:59:55-09-01 13:59:57 So, the next time you're struggling with a math problem and ...
757/765 09-01 13:59:51-09-01 13:59:57 Please remember,
758/765 aaa 09-01 13:59:52-09-01 13:59:57 You're not just competing with math,
759/765 xss 09-01 13:59:57-09-01 14:00:08 You're just arguing with your own ignorance,
760/765 bg 09-01 13:59:58-09-01 14:00:08 And to compete with one's own ignorance,
761/765 09-01 14:00:00-09-01 14:00:08 That's the most rational thing a person can do.
762/765 n 09-01 13:59:59-09-01 14:00:08 Right?
763/765 de 09-01 14:00:01-09-01 14:00:08 Therefore,
764/765 aaa 09-01 14:00:04-09-01 14:00:08 Dare to love boldly, even from afar.
765/765 hp 09-01 14:00:05-09-01 14:00:08 Don't care if they are male or female.