Input a complex number and compute its reciprocal instantly.
The reciprocal (or multiplicative inverse) of a complex number is a specific transformation. For a complex number \( z = a + bi \), its reciprocal is given by: \( \frac{1}{a + bi} = \frac{a - bi}{a^2 + b^2} \) Here, \( a \) is the real part, and \( b \) is the imaginary part of the complex number. The reciprocal is calculated by flipping the sign of the imaginary part and dividing by the square of the modulus.
Solution:
\( \frac{1}{3 + 4i} = \frac{3 - 4i}{3^2 + 4^2}\)
\( = \frac{3 - 4i}{9 + 16}\)
\( = \frac{3 - 4i}{25}\)
\( = \frac{3}{25} - \frac{4}{25}i\)
Result: The reciprocal of \( 3 + 4i \) is \( \frac{3}{25} - \frac{4}{25}i \).
Solution:
\( \frac{1}{-2 + 5i} = \frac{-2 - 5i}{(-2)^2 + 5^2}\)
\( = \frac{-2 - 5i}{4 + 25}\)
\( = \frac{-2 - 5i}{29}\)
\( = \frac{-2}{29} - \frac{5}{29}i\)
Result: The reciprocal of \( -2 + 5i \) is \( \frac{-2}{29} - \frac{5}{29}i \).