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To evaluate this, we need a common denominator. The easy way to get a common denominator is to multiply the two denominators together like so:
(1+i)*(1-i) = 1+i-i-i^2
Combining like terms gives us a denominator of 1-i^2 (the positive and negative i's cancel each other out).
Multiply each numerator by the denominator of the other fraction as we just multiplied the denominators of the other fractions by each other and we have:
Combine like terms in the numerator and we have 2/(1-i^2)
Recall that i^2 = -1 and the denominator becomes (1-(-1)) or (1+1) or 2
Our fraction is now 2/2 = 1
Supposing that the expression which has to be calculated looks like:
E = 1/(1+i) + 1/(1-i)
We'll consider the followings:
- we have to have real numbers at the denominators and not complex numbers;
- we'll transform the complex numbers from the denominators into real numbers, by multiplying them with their adjoints.
If a complex number is:
z = a + b*i
It's adjoint is:
z' = a - b*i
We'll multiply the first ratio with the conjugate number (1-i) and the second ratio with (1+i).
[1/(1+i)]*(1-i) + [1/(1-i)]*(1+i)=(1-i)/2+(1+i)/2=1
To calculate 1/(1+i) +1/(1-i).
Rationalising the denominators, we get:
1(1-i)/(1-i^2) + (1+i)/(1-i^2)
= (1-i)/2 +(1+i)/2 , as i^2 = -1.
= 2/2 =1.
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