If z=1+i, then the multiplicative inverse of z esqaire is:
What does z esqaire mean?
According to Wikipedia,
"In mathematics, a multiplicative inverse or reciprocal for a number x, denoted by 1/x or x^−1, is a number which when multiplied by x yields the multiplicative identity, 1."
Accordingly, the multiplicative inverse of 1 + i is
1/(1+i) = (1-i)/[1-i)(1+i)]
= (1-i)/2
If you want the multiplicative inverse of z^2, where z = 1+i,
that would be 1/z^2 = 1/(1+i)^2
To find the multiplicative inverse of a complex number, you need to find its reciprocal.
Let's start by squaring z: z^2 = (1+i)^2
To simplify this, we can use the FOIL method:
z^2 = (1+i)(1+i)
= 1 + i + i + i^2
= 1 + 2i + i^2
Since i^2 is defined as -1, the equation becomes:
z^2 = 1 + 2i - 1
= 2i
Now we want to find the multiplicative inverse, or reciprocal, of z^2:
1/z^2 = 1/(2i)
To rationalize the denominator, we multiply the numerator and denominator by the conjugate of 2i, which is -2i:
1/z^2 = (1/(2i)) * (-2i)/(-2i)
= -2i/(-4i^2)
Since i^2 is -1, we simplify further:
1/z^2 = -2i/(-4(-1))
= -2i/4
= -i/2
Therefore, the multiplicative inverse, or reciprocal, of z^2 is -i/2.