Questions and Answers #5 Complex Numbers

Questions and Answers Sheet 5 Complex Numbers Question #1 How do you simplify (−10𝑖)(9𝑖) + 8 ? Answer: Explanation: (−10𝑖)(9𝑖) + 8 First, do multiplication: = −90𝑖 2 + 8 We know that 𝑖 2 equals to -1: = −90(−1) + 8 Simplify: =90+8 =98 Question #2 How do you simplify 𝑖 6 (2𝑖 − 𝑖 2 − 3𝑖 3 )? Answer: Solution: Expand expression. 𝑖 6 (2𝑖 − 𝑖 2 − 3𝑖 3 ) = 2𝑖 7 − 𝑖 8 − 3𝑖 9 But we know that: 𝑖1 = 𝑖 𝑖 2 = −1 (by definition) 𝑖 3 = 𝑖 2 ⋅ 𝑖 = −𝑖 𝑖 4 = 𝑖 2 ⋅ 𝑖 2 = (−1) ⋅ (−1) = 1 𝑖 5 = 𝑖 4 ⋅ 𝑖 = 1 ⋅ 𝑖 = 𝑖 and starts again 𝑖 6 = 𝑖 5 ⋅ 𝑖 = 𝑖 ⋅ 𝑖 = 𝑖 2 = −1 𝑖 7 = 𝑖 6 ⋅ 𝑖 = −𝑖 for this cliclic result, we can write: 𝑖 6 (2𝑖 − 𝑖 2 − 3𝑖 3 ) = 2𝑖 7 − 𝑖 8 − 3𝑖 9 = −2𝑖 − 1 − 3𝑖 = −1 − 5𝑖 Question #3 How do you simplify (−𝑖)3 ? Answer: (−𝑖)3 = (−𝑖) × (−𝑖) × (−𝑖) But (−𝑖) × (−𝑖) = +𝑖 2 = −1 Answer: (−𝑖)3 = (−𝑖) × (−𝑖) × (−𝑖) = −𝑖 × −1 = 𝑖 Question #4 How do you find the absolute value of 5+12i ? Answer: Quick method The first couple of Pythagorean triples are: 3,4,5 5,12,13 So a right angled triangle with legs 5 and 12 will have hypotenuse 13 Standard formula using Complex conjugate |𝑧| = √𝑧𝑧 |5 + 12𝑖| = √(5 + 12𝑖)(5 − 12𝑖) = √25 + 144 = √169 = 13 Two dimensional distance formula The absolute value of a Complex number is its distance from 0, which is given by the distance formula: |𝑥 + 𝑖𝑦| = √𝑥 2 + 𝑦 2 In our example: |6 + 12𝑖| = √52 + 122 = √25 + 144 = √169 = 13 Question #5 How do you simplify √−2156 ? Answer: √−2156 = √−1 ⋅ √2156 𝑖 2 = −1 𝑜𝑟 𝑖 = √−1 𝑖 ⋅ √2156 ≈ 𝑖 ⋅ 46.43 ≈ 46.43𝑖 Question #6 How do you simplify 𝑖 100 ? Answer: 𝑖 100 = (𝑖 2 )50 From the fact that 𝑖 2 = −1 , we get (−1)50 = 1 as -1 raised to any even power is 1. Alternatively, we can rewrite in trigonometric form and then in the form 𝑟𝑒 𝑖𝜃 : 𝜋 𝜋 2 2 𝑖 = cos ( ) + 𝑖 sin ( ) 𝑖 𝜋 2 =𝑒 Raise the exponential to the power of 100: 𝜋 100 (𝑒 𝑖 2 ) = 𝑒 50𝜋 = cos(50𝜋) + 𝑖 sin(50𝜋) = cos 2 𝜋 + 𝑖 sin 2 cos 2 𝜋 = 1, sin 2 𝜋 = 0 Answer: =1 Question #7 How do you simplify √−4 − √−25? Answer: Explanation: We know for the complex numbers: 𝑖 2 = −1 Let, 𝑧 = √−4 − √−25 ⇒ 𝑧 = √4(−1) − √25(−1) ⇒ 𝑧 = √4𝑖 2 − √25𝑖 2 ⇒ 𝑧 = √(2𝑖)2 − √(5𝑖 2 ) ⇒ 𝑧 = (±2𝑖) − (±5𝑖) ⇒ 𝑧 = (2𝑖) − (5𝑖) or 𝑧 = (−2𝑖) − (−5𝑖) ⇒ 𝑧 = −3𝑖 or z=3i Question #8 How do I find the complex conjugate of 12 5𝑖 ? Answer: To find a conjugate of a complex number we first have to convert it to a form a+bi . To do this here we can multiply both numerator and denominator by i. 𝑧= 12 5𝑖 = 12𝑖 5𝑖 2 = 12𝑖 −5 =− 12𝑖 5 Now to calculate the conjugate we just have to change sign of the imaginary part: 𝑧= 12𝑖 5 Question #9 Write the complex number −5−3𝑖 4𝑖 in standard form? Answer: −5−3𝑖 4𝑖 3 5 4 4 =− + 𝑖 Explanation: We want the complex number in the form a+bi . This is a bit tricky because we have an imaginary part in the denominator, and we can't divide a real number by an imaginary number. We can however solve this using a little trick. If we multiply both top and bottom by i, we can get a real number in the bottom: −5−3𝑖 4𝑖 = 𝑖(−5−3𝑖) 𝑖⋅4𝑖 = −5𝑖+3 −4 3 5 4 4 =− + 𝑖 Question #10 How do you simplify 1 1+𝑖 ? Answer: 1 1+𝑖 You multiply by the complex conjugate the denominator with the sign changed: 1−𝑖 1−𝑖 1 1−𝑖 ⋅ 1+𝑖 1−𝑖 1−𝑖 1−𝑖+𝑖+𝑖 2 1−𝑖 1−𝑖 2 1−𝑖 1−√−12 1−𝑖 1−(−1) 1−𝑖 2 Question #11 How do you simplify 7𝑖 8+𝑖 ? Answer: 7𝑖 8+𝑖 7 = 65 + 56 𝑖 65 Explanation: 7𝑖 To simplify , we should multiply numerator and denominator by the complex conjugate of the 8+𝑖 denominator. As complex conjugate of the denominator (8+i) is (8-i) , we have 7𝑖 7𝑖×(8−𝑖) 8+𝑖 = = = = = (8+𝑖)(8−𝑖) 56𝑖−7𝑖 2 64+8𝑖−8𝑖−𝑖 2 56𝑖−7×(−1) 64+1 7+56𝑖 65 7 56 65 + 65 𝑖 Question #12 How do you divide 3𝑖 1+𝑖 + 2 2+3𝑖 ? Answer: 3𝑖 1+𝑖 2 + 2+3𝑖 3𝑖(1−𝑖) 2(2−3𝑖) = (1+𝑖)(1−𝑖) + (2+3𝑖)(2−3𝑖) = = = 3𝑖−3𝑖 2 + 4−6𝑖 1−𝑖 2 4−9𝑖 2 3𝑖−3(−1) 4−6𝑖 + 1−(−1) 4−9(1) 3𝑖+3 4−6𝑖 [as, 𝑖 2 = −1 ] + 4+9 3𝑖 + 3 4 − 6𝑖 = + 2 13 3 3 4 6 = 𝑖+ + − 𝑖 2 2 13 13 27 47 = 𝑖+ 26 26 1+1 Question #13 −2−5𝑖 How do you divide 3𝑖 ? Answer: To solve −2−5𝑖 3𝑖 multiply numerator and denominator by i (note 𝑖 2 = −1 ) and we get or −2𝑖−5(−1) 3⋅−1 𝑖⋅(−2−5𝑖) 𝑖⋅(3𝑖) 5−2𝑖 i.e. 5 −3 2 3 3 or Question #14 Answer: 3𝑖 2 , which is equivalent to Answer: − + ( ) 𝑖 How do you divide −2𝑖−5𝑖 2 3 5𝑖 ? Keep in mind that 𝑖 = √−1 which means that 𝑖 2 = −1 and therefore 𝑖 4 = 1. So we can rewrite the question as: 3𝑖 4 5𝑖 3 = 3𝑖 {3] 5 𝑖 = −√−1 = −𝑖 and so: 3𝑖 4 5𝑖 = 3𝑖 3 3 =− 𝑖 5 5 Question #15 How do you write the following quotient in standard form 2+𝑖 2−𝑖 ? Answer: My preferred way to divide complex numbers is to multiply numerator and denominator by the complex conjugate of the denominator and then simplify. Given: 2+𝑖 2−𝑖 Multiply numerator and denominator by the complex conjugate of the denominator: 2+𝑖 2+𝑖 2−𝑖 2+𝑖 The denominator becomes the difference of two squares: (2+𝑖)(2+𝑖) 4−𝑖 2 Use the F.O.I.L method to multiply the the numerator: 4+2𝑖+2𝑖+𝑖 2 4−𝑖 2 Replace 𝑖 2 with -1 : 4+2𝑖+2𝑖−1 4−(−1) Combine like terms: 3+4𝑖 5 Convert do a+bi form: 3 4 + 𝑖 5 5 Question #16 How do you write the complex conjugate of the complex number √−15 ? Answer: √−15 = −√15𝑖 Given a complex number a+bi , the complex conjugate of that number, denoted 𝑎 + 𝑏𝑖, is given by 𝑎 + 𝑏𝑖 = 𝑎 − 𝑏𝑖 In our case, then, we have √−15 = √15𝑖 = 0 + √15𝑖 = 0 − √15𝑖 = −√15𝑖 Question #17 How do you divide Answer: 3+9𝑖 −6−6𝑖 ? Let 𝑧 = 𝑧= 3+9𝑖 −6−6𝑖 3(1+3𝑖) 1 1+3𝑖 −6(1+𝑖) 1 1+3𝑖 2 1+𝑖 1−𝑖 =− ( 𝑧=− ( )( ) 2 1+𝑖 1−𝑖 1 1−𝑖+3𝑖−3𝑖 {2] 𝑧=− ( 12 −𝑖 2 2 𝑧=− ) ) 1 (1+2𝑖−3(−1)) 2 1−(−1) 1 4+2𝑖 𝑧=− ( 2 ) 2 𝑖 𝑧 = −1 − . 2 Question #18 How do you simplify −9 − 9𝑖 ÷ −1 − 8𝑖 ? Answer: −9 − 9𝑖 ÷ −1 − 8𝑖 = 81 65 − 63 65 𝑖 Explanation: −9 − 9𝑖 ÷ −1 − 8𝑖 = = −9−9𝑖 ⋅ −9−9𝑖 −1−8𝑖 −1+8𝑖 −1−8𝑖 −1+8𝑖 9+9𝑖−72𝑖+72 = 1+64 81−63𝑖 65 Question #19 What is the sum of this imaginary number series problem: 𝑖 + 2𝑖 2 + 3𝑖 3 + 4𝑖 4 + 5𝑖 5 + ⋯ + 2002𝑖 2002 ? Answer: The Complex multipliers can be simplified into a cyclic pattern of period 4: 𝑖=𝑖 𝑖 2 = −1 𝑖 3 = −𝑖 𝑖4 = 1 Since 𝑖 4 = 1 , we have 𝑖 = 𝑖 5 = 𝑖 9 =... and 𝑖 2 = 𝑖 6 = 𝑖 10 =... etc. and we find: 𝑛 ∑2002 𝑛=1 𝑛𝑖 500 500 499 499 = (∑(4𝑘 + 1)) 𝑖 − (∑(4𝑘 + 2)) − (∑(4𝑘 + 3)) 𝑖 + (∑(4𝑘 + 4)) 𝑘=0 𝑘=0 𝑘=0 𝑘=0 Each of these four sums is an arithmetic sequence, so is equal to the number of terms multiplied by the average term. The average term is equal to the average of the first and last terms, hence: = (501 ⋅ 1+2001 2 ) 𝑖 − (501 ⋅ 2+2002 2 ) − (500 ⋅ 3+1999 2 ) 𝑖 + (500 ⋅ 4+2000 = (501 ⋅ 1001)𝑖 − (501 ⋅ 1002) − (500 ⋅ 1001)𝑖 + (500 ⋅ 1002) = ((501 − 500) ⋅ 1001)𝑖 − ((501 − 500) ⋅ 1002) 1001i-1002 Question #20 1 What is the complex function of 𝑓(𝑧) = ? 𝑧 2 ) Answer: 1 It is a function that takes a complex value 𝑧 ≠ 0 and returns the value . 𝑧 Because it is a complex function, it has alternate representations. First, as we can write any complex number z as z=a+ib , where a and b are real numbers, we have that 1 1 𝑎 − 𝑖𝑏 𝑎 − 𝑖𝑏 𝑎 𝑏 𝑓(𝑧) = = ⋅ = = −𝑖 2 𝑧 𝑎 + 𝑖𝑏 𝑎 − 𝑖𝑏 𝑎2 + 𝑏 2 𝑎2 + 𝑏 2 𝑎 + 𝑏2 The other representations occurs when we take 𝑧 = 𝑒 𝑖𝜃 ≔ cos 𝜃 + 𝑖 sin 𝜃. In this case, we have that 𝑓(𝑧) = 𝑒 −𝑖𝜃 .