How do you calculate 1+i1-i?
Calculate the given expression:
Given expression: 1+i1-i.
Rationalize the denominator by multiplying both numerator and denominator by 1+i.
1+i1-i=1+i1+i1-i1+i⇒1+i1-i=1+i212-i2⇒1+i1-i=12+21i+i21-i2⇒1+i1-i=1+2i+i21-i2
It is known that, i=-1, so i2=-1.
Thus,
1+i1-i=1+2i-11--1
⇒1+i1-i=2i1+1⇒1+i1-i=2i2⇒1+i1-i=i
Therefore, 1+i1-i=i.