Express the following in a + bi form: i / (2 + i)
Answer
a. 2 + i
b. (-1/5) - (2/5)i
c. 2 - i
d. (1/5) + (2/5)i
To express a complex number in the form a + bi, where a and b are real numbers, we need to rationalize the denominator of the expression i / (2 + i).
Step 1: Multiply the numerator and denominator by the conjugate of the denominator.
The conjugate of 2 + i is 2 - i, so we have:
(i / (2 + i)) * ((2 - i) / (2 - i))
Step 2: Simplify the expression by multiplying the numerator and denominator.
Using the FOIL method:
(i * 2) + (i * (-i)) / ((2 * 2) + (2 * (-i)) + (i * 2) + (i * (-i)))
= (2i - i^2) / (4 + 2i - 2i - i^2)
= (2i - (-1)) / (4 + 1)
= (2i + 1) / 5
So the expression i / (2 + i) can be expressed as:
(2i + 1) / 5
Therefore, the correct answer is (d) (1/5) + (2/5)i.