Question

Express each of the following complex numbers in the form of x+iy 1+i/3+i

Answer 1

I assume you mean the number

(1 + i) / (3 + i)

Multiply the numerator and denominator by the complex conjugate of 3 + i, which is 3 - i :

(1 + i) / (3 + i) • (3 - i) / (3 - i)

Recall the difference of squares identity,

a² - b² = (a + b) (a - b)

which makes the denominator of the product simplify to

(3 + i) (3 - i) = 3² - i² = 9 + 1 = 10

and so

(1 + i) / (3 + i) = (1 + i) (3 - i) / 10

Expand the numerator:

(1 + i) (3 - i) = 3 + 2i - i² = 4 + 2i

Then

(1 + i) / (3 + i) = (4 + 2i) / 10 = 2/5 + i 1/5