簡約 (1+i)/(1-\frac{1+i){1-(1+i)/(1-i)}}

解

簡約

\frac{1 + i}{1 - \frac{1 + i}{1 - \frac{1 + i}{1 - i}}}

i

解答ステップ

\frac{1 + i}{1 - \frac{1 + i}{1 - \frac{1 + i}{1 - i}}}

\frac{1 + i}{1 - \frac{1 + i}{1 - i}} = i

= \frac{1 + i}{1 - i}

複素数の算術規則を適用する:

\frac{a + bi}{c + d i} = \frac{( c - d i ) ( a + bi )}{( c - d i ) ( c + d i )} = \frac{( a c + b d ) + ( b c - a d ) i}{c ^{2} + d ^{2}}

a = 1, b = 1, c = 1, d = - 1

= \frac{( 1 \cdot 1 + 1 \cdot ( - 1 ) ) + ( 1 \cdot 1 - 1 \cdot ( - 1 ) ) i}{1 ^{2} + ( - 1 ) ^{2}}

改良

= \frac{2 i}{2}

簡素化

\frac{2 i}{2} : i

= i