For a complex number z = x + jy:

Here's how to determine the modulus and argument for each complex number: For a complex number z = x + jy: • The modulus is |z| = sqrt(x^2 + y^2). • The argument is (z) = , where = (y)/(x). The quadrant of (x, y) determines the correct angle. We typically use the principal argument in the range (-, ] radians or (-180^, 180^]. a) For z = 2+j4: Here, x=2 and y=4. This complex number is in the first quadrant. Step 1: Calculate the modulus. |z| = sqrt(2^2 + 4^2) = sqrt(4 + 16) = sqrt(20) = sqrt(4 × 5) = 2sqrt(5) Step 2: Calculate the argument. = (4)/(2) = 2 = (2) ≈ 1.107 radians (or 63.43^) Since x>0 and y>0, is in the first quadrant, so this is the principal argument. Modulus = 2sqrt(5), Argument ≈ 1.107 rad b) For z = -5-j2: Here, x=-5 and y=-2. This complex number is in the third quadrant. Step 1: Calculate the modulus. |z| = sqrt((-5)^2 + (-2)^2) = sqrt(25 + 4) = sqrt(29) Step 2: Calculate the argument. First, find the reference angle = (|(y)/(x)|). = (|(-2)/(-5)|) = ((2)/(5)) ≈ 0.3805 radians (or 21.80^) Since x<0 and y<0, the argument is in the third quadrant. For the principal argument, we subtract from the reference angle (or 180^). = - ≈ 0.3805 - ≈ -2.761 radians (or 21.80^ - 180^ = -158.20^) Modulus = sqrt(29), Argument ≈ -2.761 rad c) For z = j(2-j): First, simplify the complex number to the form x+jy. z = j(2-j) = 2j - j^2 Since j^2 = -1: z = 2j - (-1) = 1 + 2j Here, x=1 and y=2. This complex number is in the first quadrant. Step 1: Calculate the modulus. |z| = sqrt(1^2 + 2^2) = sqrt(1 + 4) = sqrt(5) Step 2: Calculate the argument. = (2)/(1) = 2 = (2) ≈ 1.107 radians (or 63.43^) Since x>0 and y>0, is in the first quadrant, so this is the principal argument. Modulus = sqrt(5), Argument ≈ 1.107 rad 3 done, 2 left today. You're making progress.