The use of complex numbers in number theory and geometry - Cao Zexian

Jane Medium ： Teacher Cao has a lot of materials for the course , I hope to have the opportunity to listen to teacher Cao's lecture on site .

key word ： The plural , aggregate , number theory

 

§01 complex The use of numbers

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   today Tian saw Cao Zexian's course in the watermelon video The National University of science does not divide its subjects —— Cao Zexian ： What's the use of plural numbers Explain the use of plural .  It is interesting to , Compare last semester when sorting out complex number related knowledge in signal and system course , Also thinking about the use of plural expression .  The example given here can just be borrowed .

  

One 、 Application of number theory

   The first example given by teacher Cao ,  It is an application in number theory .  That is, the result of the sum of the squares of any two integers ,  It must be the sum of the squares of two integers .  Frankly speaking , At the beginning of this example ,  I didn't even hear the meaning of the question .  Later, I looked carefully PPT  I found that there are four integers involved .  about a,b,c,d Four integers ,  Add the last two squares ,  The result of multiplication ,  Equal to the sum of the squares of two integers .

  

   Proof of this problem ,  Teacher Cao's PPT Given in .  about a,b,c,d Four certificates ,  The result of their sum of squares is equal to the sum of the four product squares .  And for the sum of the squares of these two integers , After simplification, it is also equal to the square sum of the same four grades .  So the original question is . But what does this proof have to do with the plural ？  Here is the multiplication formula of two complex numbers .  If both sides take the square of the mold at the same time ,  The module of the product of the two complex numbers on the left is equal to the result of their modules ,  Then according to the definition of complex modulus , Then we can get the following formula .  therefore , According to the complex number product formula and the definition of module ,  It can be skillfully proved that this is the application of sun drying varieties .
  

   Teacher Cao's course sounds interesting and humorous , Although I didn't pay attention to the above in class Why the use of plural numbers can prove this problem Detailed description , But after class, students can figure it out as long as they think about it .
  

Two 、 Geometric proof

   The second example given by teacher Cao , About the proof of triangle center of gravity .

  

   For triangles ,  Prove that its three centers are collinear , Form the center of gravity of the triangle .  Put the triangle on the complex plane ,  Its three vertices correspond to three complex vectors .  The center point of the three sides is the average value of the complex vector of the two adjacent fixed points .  Now is the most interesting time .  You can prove , The average value of the three complex vectors falls on the three central lines ,  Just see a third of each center ,  Are the same numbers ,  That is, the average of the complex number of three vertices .  The collinear problem of three centerlines of a triangle has been proved silently .

  

 

※ total    junction ※

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   Cao There are many materials in the teacher's course , I hope to have the opportunity to listen to teacher Cao's lecture on site .

  

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■ Links to related literature :

• The National University of science does not divide its subjects —— Cao Zexian ： What's the use of plural numbers