Mathematics is a tool for solving problems

Master class — Tao zhe Xuan Terence Tao

Definition and purpose

definition and purpose of Mathematics?
Mathematics is a method for solving problems （ Tools ）; meanwhile , Like language , It is also a way of thinking

Mathematics is a systematic method ： Accept complex problems - > Break it down into simple parts -> So that each independent part can work independently -> after , Then combine them

The purpose of Mathematics ： Communicate ideas and concepts in a very precise way

Connection

Mathematics can be divided into many separate parts , Like algebra 、 The geometric 、 Differential and integral calculus
But to some extent , They are interconnected , It's like different parts of the same body , There are many connections between disciplines , As your research deepens , You will see more and more connections
The most wonderful thing in Mathematics ： At first , You may not realize , Until after years of training

Through mathematics , You can get Another way to understand the world , Find a connection between things

Intuition

Some people think they are not good at maths , I am not the material to learn mathematics , Actually , Everyone in their own heart , Have an intuition about mathematics

intuition ： When you see something , You can think of something immediately

The main reason is ： Mathematics usually uses a In a very standard way professor , This is very unnatural , We must learn all the rules 、 Do a lot of practice , But it doesn't seem to have much to do with our intuition
People can pass through many Different ways To learn mathematics , Some people like intuitive , Some people are very logical , Some people are very systematic , These are all ways to learn mathematics

Proof by Contradiction

If an opposite is true , Then your assumption is wrong

such as , If one says the maximum number is XXX, then , You just have to refute ：“ that XXX + 1 Well ？”

Abstraction

Real world problems -> Mathematical problems The first step is abstraction
Abstraction is a process , take Problematic elements in the real world Pulled out , Turn it into the most basic mathematical form , after , You can use algebra 、 Geometry or any other mathematical form to deal with it .
You must distinguish between the problems What characteristics be necessary , Those who are not , Then try to translate all the data you get into mathematical language

Less scary

If you have a mathematical mind , Something you don't know , You can also gain some confidence from it , Such as mobile phone 、 Computer 、 Internet 、 How the economy works

It gives you a tool , So that you can feel , If you really need to , You can In essence To understand how these things work

Curtain

Almost anything in real life can be turned into a small problem , Find problems in your life that are less risky , It doesn't matter if you screw it up , You can start over
Just holding such a state of mind ： Solving problems quickly and effectively is not necessarily the most important ,just enjoy yourself, And learn from it , Failure , It's no big deal ~

Freedom

in real life , You may need a question like this , Resources are limited , You only have 100 element To solve this problem

But in Mathematics , You can change the parameters , such as , Suppose I have 1 Billion element , Can I solve this problem ？

Mathematics gives you a lot of flexibility , Turn the problem into Suppose you can solve the problem , Then go back and solve the practical problems

This is a kind of freedom that you don't have in real life

abstract , Can provide a lot of creative freedom

Create bite-size problems

How to determine the orbit of earth and Mars around the sun ？

It takes the earth to go round the sun 365 God , Mars needs 687 God , Time is set

Before , People put forward , The orbit of the earth around the sun is round , But this is not consistent with the observed situation

Kepler knew about triangulation , Through two coordinates , You can measure the angle between the current position and the other two coordinates , You can draw a triangle , Then mark your position on the map

Take the sun and Mars as coordinates , Can measure the trajectory of the earth , But Mars is also mobile , So we have only one coordinate of the sun

So Kepler hypothesized , If Mars is fixed , Then it can be used as a coordinate
Kepler's idea is , On the sun and Mars 687 Sampling at intervals of days , For certain days , Mars is in a fixed position , therefore , Mars is effectively fixed

Using this, we can take Mars and the sun as coordinates , Calculate the orbit of the earth , Knowing the orbit of the earth , Through triangulation technology , You can launch the orbit of Mars , Even the orbits of other planets
Kepler through this mage , We finally discovered what we call Kepler's law of planetary motion

Group Testing

Breaking down a complex problem into a simple one is called group testing , The technology originated from 1950 year , The U. S. Army is testing soldiers for a rare disease , So a blood test , But the number of blood tests is limited , So there are not enough tests to test each individual soldier

The solution they came up with was to take samples from different soldiers , Then mix the blood together , So if there is a positive result , It means that one of those people got the disease

To turn it into a mathematical problem is ： What is the minimum number of inspections required ？
adopt 2 Division , At most $lo{g}_{2}n$ The number of times is enough

Good Stories Come From Dumb Questions

Describe things in a narrative way , It activates a part of your brain

Look for analogies , Find connections , The most basic but important way is Ask questions , in addition , The more stupid the question, the better
The process of speaking out the problem , Can make you really realize what the problem is

$-3\times -4=12$ Negative is positive , It is not very intuitive for many people

You can use an economic analogy , Suppose the hourly rate is 3 dollar , Work 4 Hours , You will get $3\times 4=12$ dollar ; Now suppose that using something costs... Per hour 3 dollar , Work 4 Hours later, , You get a total of $-3\times 4=-12$ dollar , How to compare $-3\times -4$ Well ？

Suppose your water is always on , It costs... An hour 3 dollar , You try to turn off the water 4 Hours , You will save 4 The cost of an hour

By studying these analogies , Ask some questions ：“ What does this mean ？” You can get the true meaning of negative to positive , Your intuition is better than before

Good Guys and Bad Guys

Personification is a useful technique , They're not just $x,y$

If you want to solve an unknown $x$, You think of it as a detective game , There are some mysterious rivals , But you don't know where your opponent is , He is located in $x$, So you need to make sure $x$ Where is the

It puts you into a mode of thinking , Start thinking about clues , Use exclusion ,“$x$ It can't be like this , Because if $x$ This is it , Then this will happen , But the truth is not ”

Good Narratives

poll , Such as voting for a political party ,2 Drawn from 100 million people 500 People to sample , It is only a small part of the population , The accuracy of the poll is unbelievably accurate , This seems very non intuitive . The fact is that , The size of the population is not very important , What matters is the sample size ,1000 Human investigation is more than 500 Human surveys are more accurate .

One analogy is the amount of salt in the ocean , All you need is a drop of sea water , Have a taste of , I know the sea is salty

The reason is that salt is well distributed in seawater , This analogy cleverly explains why population size is not so important , meanwhile , And tell you about the limitations of opinion polls , If the ocean has fresh water and salt water , If you take the wrong sample , Misunderstanding of the overall salt content

therefore , Random sampling is important , Try to draw people from different countries , Different regions , Different demographics , Different social levels , Half men and half women

Analogy is a , Can make good use of , The rational and emotional parts of the brain , Method

Thinking over and over is one of the most interesting parts of mathematics , Because you can connect things , such as , The numbers in the equation relate to your intuition

Playing with Transformations

seek 15 The rules of the game are ： from 1~9 Select a number in turn to take out , The sum of all the numbers collected first is 15 Your player wins

This problem can be translated into # Word games ：$\left[\begin{array}{ccc}2& 7& 6\\ 9& 5& 1\\ 4& 3& 8\end{array}\right]$

Horizontal row $\left[\begin{array}{ccc}2& 7& 6\end{array}\right]$, Vertical row $\left[\begin{array}{ccc}2& 9& 4\end{array}\right]$, Oblique row $\left[\begin{array}{ccc}2& 5& 8\end{array}\right]$ The sum of all is 15

You can see , Transformation makes seek 15 This abstract game becomes # word This specific game

The Counterfeit Coin Puzzle

12 A coin , among 1 One is counterfeit money , The weight is different from others , You have a scale , If you can only say 3 Time , If you find counterfeit money ？

 Divide coins into 3 Share , label 1~12
 First ：1+2+3+4 and 5+6+7+8 Compare （ The first comparison ）
--  Balance the balance  ->  Counterfeit money is in 9~12 in 
----  take 1+2 and 9+10 Compare （ The second comparison ） Real money is compared with the unknown 
------  Balance the balance  ->  Counterfeit money is in 11~12 in 
--------  take 1 and 11 Compare （ The third comparison ） Real money is compared with the unknown 
----------  Balance the balance  ->  Counterfeit currency is 12
----------  The balance is unbalanced  ->  Counterfeit currency is 11
------  The balance is unbalanced  ->  Counterfeit money is in 9~10 in 
--------  take 1 and 9 Compare （ The third comparison ） Real money is compared with the unknown 
----------  Balance the balance  ->  Counterfeit currency is 9
----------  The balance is unbalanced  ->  Counterfeit currency is 10
--  The balance is unbalanced  ->  Counterfeit money is in 1~8 in  ->  Rebranding , Heavy is 1~4, Light ones are 5~8
----  take 1+2+5 and 3+4+10 Compare （ The second comparison ）
------  The balance is unbalanced , Heavy or heavy  ->  Counterfeit money is in 1~2 in （ Counterfeit money is heavy money ）
--------  take 1 and 10 Compare （ The third comparison ） Unknown and real money comparison 
----------  Balance the balance  ->  Counterfeit currency is 2
----------  The balance is unbalanced  ->  Counterfeit currency is 1
------  The balance is unbalanced , Heavy becomes light  ->  Counterfeit money is in 3、4、5 in （3 or 4 It is a heavy currency 、 or 5 It is light money ）
--------  take 3 and 4 Compare （ The third comparison ） Unknown and real money comparison 
----------  Balance the balance  ->  Counterfeit currency is 5
----------  The balance is unbalanced  ->  The counterfeit money is the heavy one （3 or 4）
------  Balance the balance  ->  Counterfeit money is in 6、7、8 in （ Counterfeit money is light money ）
--------  take 6 and 7 Compare （ The third comparison ） Unknown and real money comparison 
----------  Balance the balance  ->  Counterfeit currency is 8
----------  The balance is unbalanced  ->  The counterfeit money is light （6 or 7）

Scan your organs with an MRI scan , Find out whether there are tumors or other abnormal objects , But the machine is very slow , For children , You may not be able to stay still for a long time , that , Can we measure the presence of tumors with fewer scans ？ This problem is very similar to weighing coins

Now? , The latest model uses a technology called compressed sensing , Increase the speed 1 even to the extent that 10 times , It used to take 3~5 minute , Now it's just a matter of 20 second

Cheat

In mathematics , Often make the problem simple , It's usually much simpler , Just to get started

It is similar to playing computer games by cheating , Give your character unlimited life , Or instant displacement or some absurd thing , after , Go back and turn off the cheater

Failure is Cheap

The great thing about mathematics is that failure doesn't have to pay a huge price

Don't care about success , Just enjoy it , If you think in this way , that , It will be part of your mind

Some accidents that didn't go as planned , This is not a failure , This is often some exciting new clue