Variable Compleja (inverso aditivo)
Usa i=(0,1) y y=(y,0) para verificar que -(iy)=(-i)y. Por lo tanto mostrar que el inverso aditivo de un numero complejo z=x+iy puede ser escrito como -z=-x-iy
Respuestas
Respuesta dada por:
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Recordemos los siguiente ![(a_1,b_1)\cdot (a_2,b_2)=(a_1a_2-b_1b_2,a_1b_2+a_2b_1) (a_1,b_1)\cdot (a_2,b_2)=(a_1a_2-b_1b_2,a_1b_2+a_2b_1)](https://tex.z-dn.net/?f=%28a_1%2Cb_1%29%5Ccdot+%28a_2%2Cb_2%29%3D%28a_1a_2-b_1b_2%2Ca_1b_2%2Ba_2b_1%29)
Luego se tiene
![-(iy)=-[(0,1)\cdot(y,0)]=-(0,y) = (0,-y)\\
(-i)y=(0,-1)\cdot(y,0)=(0,-y)\\ \\
\Longrightarrow -(iy)=(-i)y -(iy)=-[(0,1)\cdot(y,0)]=-(0,y) = (0,-y)\\
(-i)y=(0,-1)\cdot(y,0)=(0,-y)\\ \\
\Longrightarrow -(iy)=(-i)y](https://tex.z-dn.net/?f=-%28iy%29%3D-%5B%280%2C1%29%5Ccdot%28y%2C0%29%5D%3D-%280%2Cy%29+%3D+%280%2C-y%29%5C%5C%0A%28-i%29y%3D%280%2C-1%29%5Ccdot%28y%2C0%29%3D%280%2C-y%29%5C%5C+%5C%5C%0A%5CLongrightarrow+-%28iy%29%3D%28-i%29y)
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Luego
. También ![(-x)+x=0 (-x)+x=0](https://tex.z-dn.net/?f=%28-x%29%2Bx%3D0)
![[(-i)y+(iy)]+[(-x)+x]=0+0i\\ \\
\left[(-i)y+(iy)\right]+[x+(-x)]=\mathbf0\\ \\
(-i)y+[(iy)+x]+(-x)=\mathbf0\\ \\
(-i)y+(-x)+[(iy)+x]=\mathbf0\\\\
\left[(-i)y+(-x)\right]+[(iy)+x]=\mathbf0\\ \\
\left[(-x)+(-i)y\right]+[x+(iy)]=\mathbf0\\ \\
\left[(-x)-(iy)\right]+[x+(iy)]=\mathbf0\\ \\
\text{Por ende el inverso aditivo de }z\text{ es }(-x)-(iy)=-x-iy=-z [(-i)y+(iy)]+[(-x)+x]=0+0i\\ \\
\left[(-i)y+(iy)\right]+[x+(-x)]=\mathbf0\\ \\
(-i)y+[(iy)+x]+(-x)=\mathbf0\\ \\
(-i)y+(-x)+[(iy)+x]=\mathbf0\\\\
\left[(-i)y+(-x)\right]+[(iy)+x]=\mathbf0\\ \\
\left[(-x)+(-i)y\right]+[x+(iy)]=\mathbf0\\ \\
\left[(-x)-(iy)\right]+[x+(iy)]=\mathbf0\\ \\
\text{Por ende el inverso aditivo de }z\text{ es }(-x)-(iy)=-x-iy=-z](https://tex.z-dn.net/?f=%5B%28-i%29y%2B%28iy%29%5D%2B%5B%28-x%29%2Bx%5D%3D0%2B0i%5C%5C+%5C%5C+%0A%5Cleft%5B%28-i%29y%2B%28iy%29%5Cright%5D%2B%5Bx%2B%28-x%29%5D%3D%5Cmathbf0%5C%5C++%5C%5C%0A%28-i%29y%2B%5B%28iy%29%2Bx%5D%2B%28-x%29%3D%5Cmathbf0%5C%5C+%5C%5C%0A%28-i%29y%2B%28-x%29%2B%5B%28iy%29%2Bx%5D%3D%5Cmathbf0%5C%5C%5C%5C%0A%5Cleft%5B%28-i%29y%2B%28-x%29%5Cright%5D%2B%5B%28iy%29%2Bx%5D%3D%5Cmathbf0%5C%5C+%5C%5C%0A%5Cleft%5B%28-x%29%2B%28-i%29y%5Cright%5D%2B%5Bx%2B%28iy%29%5D%3D%5Cmathbf0%5C%5C+%5C%5C%0A%5Cleft%5B%28-x%29-%28iy%29%5Cright%5D%2B%5Bx%2B%28iy%29%5D%3D%5Cmathbf0%5C%5C+%5C%5C%0A%5Ctext%7BPor+ende+el+inverso+aditivo+de+%7Dz%5Ctext%7B+es+%7D%28-x%29-%28iy%29%3D-x-iy%3D-z)
Luego se tiene
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Luego
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