Find:
Re[1/(2+j)]
Im[2+j/(3+4j)]
Rationalize the denominator by multiplying top and bottom by 2-j
1/(2+j)
=(2-j)/((2+j)(2-j))
=(2-j)/(4-j²)
=(2-j)/5
=(2/5)-(j/5)
Can you take it from here?
The next one is similar.
Yes, thank you. I was doing it that way, but didn't know if I was correct.
Good, keep up the good work!
To find the real (Re) and imaginary (Im) parts of complex numbers, we need to manipulate the given expressions and use the properties of complex numbers.
Let's find Re[1/(2+j)] first:
Step 1: Simplify the expression.
To simplify 1/(2+j), we multiply the numerator and denominator by the conjugate of the denominator, which is (2-j):
1/(2+j) * (2-j)/(2-j) = (2-j)/(4+2j-2j-j^2) = (2-j)/(4+1) = (2-j)/5
Step 2: Separate into real and imaginary parts.
The real part, Re[2-j)/5], is obtained by taking the real part of the complex number (2-j) and dividing it by 5:
Re[2-j)/5] = 2/5 = 0.4
The imaginary part, Im[2-j)/5], is obtained by taking the imaginary part of the complex number (2-j) and dividing it by 5:
Im[2-j)/5] = -1/5 = -0.2
Therefore, the real part of 1/(2+j) is 0.4, and the imaginary part is -0.2.
Now let's find Im[2+j/(3+4j)]:
Step 1: Simplify the expression.
To simplify 2+j/(3+4j), we need to rationalize the denominator by multiplying the numerator and denominator by the conjugate of the denominator, which is (3-4j):
(2+j) * (3-4j) / (3+4j) * (3-4j) = (2+j)(3-4j) / (9+12j-12j-16j^2)
= (2+j)(3-4j) / (9+16) = (2+j)(3-4j) / 25
= (6-8j+3j-4j^2) / 25 = (6-5j-4(-1)) / 25
= (6-5j+4) / 25 = (10-5j) / 25
Step 2: Separate into real and imaginary parts.
The real part, Re[(10-5j)/25], is obtained by taking the real part of the complex number (10-5j) and dividing it by 25:
Re[(10-5j)/25] = 10/25 = 0.4
The imaginary part, Im[(10-5j)/25], is obtained by taking the imaginary part of the complex number (10-5j) and dividing it by 25:
Im[(10-5j)/25] = -5/25 = -0.2
Therefore, the real part of 2+j/(3+4j) is 0.4, and the imaginary part is -0.2.