This mathematics problem involves applying core mathematical principles and formulas. Below you will find a complete step-by-step solution with detailed explanations for each step, helping you understand not just the answer but the method behind it.
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Here are the solutions to your complex numbers problems. 1. Complete the following table We will fill in the missing values for each row. Row a: Given z = -2 + 2sqrt(3)i Step 1: Calculate the modulus r. r = sqrt((-2)^2 + (23))^2 = sqrt(4 + 12) = sqrt(16) = 4 Step 2: Calculate the argument . The number is in the second quadrant. = (|2sqrt(3)-2|) = (sqrt(3)) = ()/(3) = - = - ()/(3) = (2)/(3) Step 3: Write the polar and exponential forms. Polar form: 4(((2)/(3)) + i((2)/(3))) Exponential form: 4e^i(2)/(3) Row b: Given modulus r = 3 and argument = -(5)/(4) Step 1: Calculate the rectangular form x+yi. x = r = 3(-(5)/(4)) = 3(-sqrt(2)2) = -3sqrt(2)2 y = r = 3(-(5)/(4)) = 3(sqrt(2)2) = 3sqrt(2)2 Rectangular form: -3sqrt(2)2 + 3sqrt(2)2i Step 2: Write the polar and exponential forms. Polar form: 3((-(5)/(4)) + i(-(5)/(4))) Exponential form: 3e^-i(5)/(4) Row c: Given polar form 6((()/(6)) + i(()/(6))) Step 1: Identify the modulus and argument. r = 6 = ()/(6) Step 2: Calculate the rectangular form x+yi. x = r = 6(()/(6)) = 6(sqrt(3)2) = 3sqrt(3) y = r = 6(()/(6)) = 6((1)/(2)) = 3 Rectangular form: 3sqrt(3) + 3i Step 3: Write the exponential form. Exponential form: 6e^i()/(6) Row d: Given exponential form e^i()/(2) Step 1: Identify the modulus and argument. r = = ()/(2) Step 2: Calculate the rectangular form x+yi. x = r = (()/(2)) = (0) = 0 y = r = (()/(2)) = (1) = Rectangular form: i Step 3: Write the polar form. Polar form: ((()/(2)) + i(()/(2))) Here is the completed table: |l|c|c|c|c| Complex Number x+yi & Modulus r & Argument & Polar form r( + i) & Exponential form re^i \\ a) -2+2sqrt(3)i & 4 & (2)/(3) & 4(((2)/(3)) + i((2)/(3))) & 4e^i(2)/(3) \\ b) -3sqrt(2)2 + 3sqrt(2)2i & 3 & -(5)/(4) & 3((-(5)/(4)) + i(-(5)/(4))) & 3e^-i(5)/(4) \\ c) 3sqrt(3) + 3i & 6 & ()/(6) & 6((()/(6)) + i(()/(6))) & 6e^i()/(6) \\ d) i & & ()/(2) & ((()/(2)) + i(()/(2))) & e^i()/(2) \\ 2. Express the following in the form re^i with - < a) z = -i Step 1: Identify x=0, y=-1. Calculate the modulus. r = sqrt(0^2 + (-1)^2) = 1 Step 2: Calculate the argument. Since z is on the negative imaginary axis, = -()/(2). Step 3: Write in exponential form. z = e^-i()/(2) b) z = 1+i Step 1: Identify x=1, y=1. Calculate the modulus. r = sqrt(1^2 + 1^2) = sqrt(2) Step 2: Calculate the argument. The number is in the first quadrant. = ((1)/(1)) = ()/(4) Step 3: Write in exponential form. z = sqrt(2)e^i()/(4) c) z = -5 - 12i Step 1: Identify x=-5, y=-12. Calculate the modulus. r = sqrt((-5)^2 + (-12)^2) = sqrt(25 + 144) = sqrt(169) = 13 Step 2: Calculate the argument. The number is in the third quadrant. = (|(-12)/(-5)|) = ((12)/(5)) = - + = - + ((12)/(5)) Step 3: Write in exponential form. z = 13e^i(- + ((12)/(5))) d) z = 3 - 4i Step 1: Identify x=3, y=-4. Calculate the modulus. r = sqrt(3^2 + (-4)^2) = sqrt(9 + 16) = sqrt(25) = 5 Step 2: Calculate the argument. The number is in the fourth quadrant. = (|(-4)/(3)|) = ((4)/(3)) = - = -((4)/(3)) Step 3: Write in exponential form. z = 5e^i(-((4)/(3))) e) z = -2(sqrt(3) - 1) Step 1: Simplify the number. z = -2sqrt(3) + 2. This is a real number. Since sqrt(3) ≈ 1.732, z ≈ -2(1.732) + 2 = -3.464 + 2 = -1.464. This is a negative real number. Step 2: Calculate the modulus. r = |-2(sqrt(3) - 1)| = 2(sqrt(3) - 1) Step 3: Calculate the argument. For any negative real number, = . Step 4: Write in exponential form. z = 2(sqrt(3) - 1)e^i Got more? Send 'em! 📸