Reply to Re: I gave up to the Computers National Olimpiad
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The use of complex numbers in geometry:
You can express them in two ways:
The algebrical way: z=a+b*i.
The trygonometrical way: z=r*(cos(phi)+i*sin(phi))
Think this way: We have one point A(a,b). Sometimes, it's usefull to retain it's coordinates like this(for example, when we need to calculate distances). But, some other times, we need to retain polar coordinates: r-the distance from the origin. phi-the angle between OA and Ox, measured in trig way.
Here comes the multiplying and the division of complex numbers:
z1=r1*(cos(phi1)+i*sin(phi1))
z2=r2*(cos(phi2)+i*sin(phi2))
z1*z2=r1*r2*(cos(phi1+phi2)+i*sin(phi1+phi2))
z1/z2=(r1/r2)*(cos(phi1-phi2)+i*sin(phi1-phi2))
Now, let's find a good use for these: we have some nice formulas:
Let's consider three points: A(z1), B(z2), C(z3)
A, B and C are colinear if and only if arg((z1-z2)/(z2-z3))=0, where arg(z)=phi. Isn't this nice?
And there are a lot of formulas like this.
There are a lot of formulas even for the algebric form of complex numbers.
Let be eps a cube root of 1, that is not 1(in fact is a solution for the equation x^2+x+1=0).
A triangle ABC is echilateral if and only if z1+z2*eps+z3*eps^2=0.
In the end, I'll give you an example of a geometry problem that can be solved by complex numbers much more easy than using classical geometry:
Let be ABCDEF a hexagon. A1,A2,A3 are the middles of AB,CD and EF, while B1,B2,B3 are the middles of BC,DE and FA. Prove that A1A2A3 and B1B2B3 have the same center of mass(try it with and without complex numbers).
You can express them in two ways:
The algebrical way: z=a+b*i.
The trygonometrical way: z=r*(cos(phi)+i*sin(phi))
Think this way: We have one point A(a,b). Sometimes, it's usefull to retain it's coordinates like this(for example, when we need to calculate distances). But, some other times, we need to retain polar coordinates: r-the distance from the origin. phi-the angle between OA and Ox, measured in trig way.
Here comes the multiplying and the division of complex numbers:
z1=r1*(cos(phi1)+i*sin(phi1))
z2=r2*(cos(phi2)+i*sin(phi2))
z1*z2=r1*r2*(cos(phi1+phi2)+i*sin(phi1+phi2))
z1/z2=(r1/r2)*(cos(phi1-phi2)+i*sin(phi1-phi2))
Now, let's find a good use for these: we have some nice formulas:
Let's consider three points: A(z1), B(z2), C(z3)
A, B and C are colinear if and only if arg((z1-z2)/(z2-z3))=0, where arg(z)=phi. Isn't this nice?
And there are a lot of formulas like this.
There are a lot of formulas even for the algebric form of complex numbers.
Let be eps a cube root of 1, that is not 1(in fact is a solution for the equation x^2+x+1=0).
A triangle ABC is echilateral if and only if z1+z2*eps+z3*eps^2=0.
In the end, I'll give you an example of a geometry problem that can be solved by complex numbers much more easy than using classical geometry:
Let be ABCDEF a hexagon. A1,A2,A3 are the middles of AB,CD and EF, while B1,B2,B3 are the middles of BC,DE and FA. Prove that A1A2A3 and B1B2B3 have the same center of mass(try it with and without complex numbers).
