What is the multiplicative inverse of 4 plus i?

Answer 1

To find the multiplicative inverse of a complex number z = (a + bi), divide its complex conjugate z* = (a - bi) by z* multiplied by z (and simplify):

z = 4 + i

z* = 4 - i

multiplicative inverse of z:

z* / (z*z)

= (4 - i) / ((4 - i)(4 + i)

= (4 - i) / (16 + 1)

= (4- i) / 17

= 1/17 (4 - i)

Answer 2

The multiplicative inverse of a complex number is the reciprocal of that number. To find the multiplicative inverse of 4 + i, we first need to find the conjugate of 4 + i, which is 4 - i. The product of a complex number and its conjugate is always a real number. Therefore, the multiplicative inverse of 4 + i is (4 - i) / ((4 + i)(4 - i)) = (4 - i) / (16 + 1) = (4 - i) / 17.