Ivan Shishkin, Rye (1878)

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Additivité de la conjugaison complexe

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Solution

Solution by darktoaster · FR

On note x1x_{1} (resp. y1y_{1}) la partie réelle (resp. imaginaire) de z1z_{1} : z1=x1+iy1z_{1} = x_{1} + iy_{1}
On note x2x_{2} (resp. y2y_{2}) la partie réelle (resp. imaginaire) de z2z_{2} : z2=x2+iy2z_{2} = x_{2} + iy_{2}
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D’une part, on additionne z1z_{1} et z2z_{2} : z1+z2=(x1+x2)+i(y1+y2)z_{1} + z_{2} = (x_{1}+x_{2} ) + i(y_{1}+ y_{2})
puis on conjugue la somme : z1+z2‾=(x1+x2)−i(y1+y2)\overline{z_{1}+z_{2}} = (x_1 + x_2) - i(y_1 + y_2)
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D’une autre part, on conjugue z1{z_{1}} et z2z_{2} : z1‾=x1−iy1\overline{z_{1}} = x_1 - i y_1 et z2‾=x2−iy2\overline{z_{2}} = x_{2} - i y_{2}
puis on somme les conjugés : z1‾+z2‾=x1−iy1+x2−iy2=(x1+x2)−i(y1+y2)\overline{z_{1}} + \overline{z_{2}} = x_{1} - i y_{1} + x_{2 } - i y_{2 } = (x_{1} +x_{2}) - i (y_{1} + y_{2})

On conclut : z1+z2‾=z1‾+z2‾\overline{z_{1}+z_{2}} = \overline{z_{1} } + \overline{z_{2}}

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