BA0 ENGINEERING MATHEMATICS 0 CHAPTER COMPLEX NUMBER. INTRODUCTION TO COMPLEX NUMBERS.. Quadratic Equations Examples of quadratic equations:. x + 3x 5 = 0. x x 6 = 0 3. x = 4 i The roots of an equation are the x-values that make it "work" We can find the roots of a quadratic equation either by using the quadratic formula or by factoring. We can have 3 situations when solving quadratic equations. Case : Two roots Example: x + 3x 5 = 0 We proceed to solve this equation using the quadratic formula as we did earlier: Case : One root Example: 4x x + 9 = 0 Notice what happens when we use the quadratic formula this time. Under the square root we get 44 44 = 0.
BA0 ENGINEERING MATHEMATICS 0 So it means we only have one root. We can also say that this is a repeated root, since we are expecting roots. Case 3: No Real Roots Example: x 4x + 0 = 0 This example gives us a problem. Under the square root, we get (-64), and we have been told repeatedly by our teachers that we cannot have the square root of a negative number. Can we find such a root?. Imaginary Numbers To allow for these "hidden roots", around the year 800, the concept of (-) was proposed and is now accepted as an extension of the real number system. The symbol used is and is called an imaginary number...3 Powers Of Since stands for, let us consider some powers of.
BA0 ENGINEERING MATHEMATICS 0 Recall: And, for any value of a. Using these, we can derive the following: i i i even power odd power Example : Simplify each of the following equation. 9 9 9i 3i.. 50 50 3. 5 4. 5 5. 3 Example : Simplify. i 3. i i i i i 4 4 8 3. i i 5 7 7 4. 5. 6. i i i i i i 3
BA0 ENGINEERING MATHEMATICS 0 Example 3: Simplify the expressions below.. i 3i 7 4 i i 3i 8 i 3 i 3 8 i 3 3 i. 3. 4.. COMPLEX NUMBERS Complex numbers have a real part and an imaginary part. Example:. Real part: 5, Imaginary part:. Real part: -3, Imaginary part: - Some examples of complex numbers are 3 i, 5 i, i, 0 3 i, 5 0 i, 0 0i 3 NOTE: We can write the complex number as. There is no difference in meaning. 4
BA0 ENGINEERING MATHEMATICS 0.. Solving Equations with Complex Numbers We now return to our problem from above. We didn't know then what to do with. Now we can write the solution using complex numbers, as follows: 4 64 x 4 i 64 4 i8 4 i@ 4i Exercise:. Express in terms of : 4. Simplify: a. b. c. d. 3. 4. 5. 5x 6x5 0 x 4x3 0 x 6x0 0 5
BA0 ENGINEERING MATHEMATICS 0.3 ADDITION AND SUBTRACTION OF COMPLEX NUMBERS If and, Then, x u yi vi z w x yi u vi x u yi vi z w x yi u vi Example: Solve the problem below:...4 MULTIPLICATION OF COMPLEX NUMBERS Expand brackets as usual, but care with j! If, and So, a p a qi bi p bi qi z w a bi p qi ap aqi pbi bqi ap aqi pbi bq ap aqi pbi bq 6
BA0 ENGINEERING MATHEMATICS 0 Example: Solve each equation in the form... 3. 4..4. Multiplying By The Conjugate Example 6 is a special case. is the conjugate of. In general: is the conjugate of and is the conjugate of. Notice that when we multiply conjugates, our final answer is real only (it does not contain any imaginary terms). We use the idea of conjugate when dividing complex numbers. 7
BA0 ENGINEERING MATHEMATICS 0.5 DIVISION OF COMPLEX NUMBERS If, and. So, z a bi w p qi a bi p qi p qi p qi ap aqi pbi bqi p pqi pqi q i Multiply with conjugate to get a real number. If conjugate of is ap aqi pbi bqi p pqi pqi q ap aqi pbi bqi p q Example:. Express in the form. Simplify: 4i 9i 3. i i 4. 3 5i 5 3i 8
BA0 ENGINEERING MATHEMATICS 0 Exercise:. Express in the form : a) i 39 3i b) c) 3 i i i d) 9.4 6.i.5.5i e) f) g) 3 4i i 4 i 8 i 4i h) ( 3 i) (3 5i). If and, solve the following and write each answer in standard form. a) b) c) 3. Express in the form. 9
BA0 ENGINEERING MATHEMATICS 0.6 EQUIVALENT COMPLEX NUMBERS Two complex numbers and are equivalent if: The real parts are equal (x = a), and The imaginary parts are equal (y = b). Example:. Given that, then a = 3 and b =.. Find the value of and, ( x y) ( x - y) i 4.8 6.i ( xy) 4.8 () ( x- y) 6. () x4.8 y (3) 4.8 y y 6. 4.8 y 6. y 6. 4.8 y.4.4 y y 0.7 if, y 0.7 x 4.8 0.7 x 5.5 x5.5 y 0.7 Exercise: x yi 7i 4 i ans : x 0., y.6. 5i 3 7. x yi ans : x, y i 5 3i x yi 4 i ans : x 6, y 3. 4. x yi 3i 5 i ans : x 4, y x yi 7 i ans : x 45, y 8 5. 6. 7 8 3 a bi i 5 3i 7. a bi 3 3 i 0
BA0 ENGINEERING MATHEMATICS 0.7 GRAPHICAL REPRESENTATION OF COMPLEX NUMBERS We can represent complex numbers in the complex plane. We use the horizontal axis for the real part and the vertical axis for the imaginary part. Represent Complex Number Imaginary axis Argument of z Y P(X,Y) X Modulus of z R P (X,-Y) Real axis Example: The number is represented by: A is the representation of the complex number.
BA0 ENGINEERING MATHEMATICS 0.7. ADDING COMPLEX NUMBERS GRAPHICALLY We can add complex numbers graphically. Example:. Add and graphically. We add the complex numbers by setting up a parallelogram. The solution is (4 + i ).. Subtract from graphically.
BA0 ENGINEERING MATHEMATICS 0 Exercise:. Perform graphically: i. 3 i i ii. i 3 4i iii. 5i 4 i. Given state each of the following function in the Argand Diagram. a) z, z, z3, z 4 b) z z c) z3 z4 d) z z e) z4 z3 3
BA0 ENGINEERING MATHEMATICS 0.8 MODULUS AND ARGUMENT Modulus, R z x y Argument, y x - tan Example: Find Modulus and Argument for each Complex Number. a. z i b. z 3 4i c. z3 4i d. z i 4
BA0 ENGINEERING MATHEMATICS 0.9 FORM OF COMPLEX NUMBERS We can write complex numbers in 4 different ways: Example: Cartesian form x yi 5 6i r cos isin 8 cos4 isin 4 Trigonometric form Polar form z R z 84 Exponential form z Re i.5 ( z 6e i.9. Polar Form of a Complex Number Our aim in this section is to write complex numbers in terms of a distance from the origin and a direction (or angle) from the positive horizontal axis. We find the real (horizontal) and imaginary (vertical) components in terms of r (the length of the vector) and θ (the angle made with the real axis): From Pythagoras, we have: r = x + y and basic trigonometry gives us: x = r cos θ y = r sin θ 5
BA0 ENGINEERING MATHEMATICS 0 Multiplying the last expression throughout by gives us: So we can write the polar form of a complex number as: r is the absolute value (or modulus) of the complex number θ is the argument of the complex number. There are two other ways of writing the polar form of a complex number: r cis θ [means r (cos θ + j sin θ)] r θ [means once again, r (cos θ + j sin θ)] NOTE: When writing a complex number in polar form, the angle θ can be in DEGREES or RADIANS. Example : Find the polar form and represent graphically the complex number. Solution: We need to find r and θ. To find θ, we first find the acute angle α : 6
BA0 ENGINEERING MATHEMATICS 0 Now, is in the fourth quadrant, so θ = 360-35.54 = 34.46 So, expressing in polar form, we have: 7-5j = 8.6 (cos 34.5 +j sin 34.5 ) We could also write this answer as 7-5j = 8.6 cis 34.5. Also we could write: 7-5j = 8.6 34.5 The graph is as follows: Example : Express 3(cos 3 + j sin 3 ) in cartesian form Represent graphically and give the cartesian form of 7.3-70 Represent i 3 graphically and write it in polar form 7
BA0 ENGINEERING MATHEMATICS 0 Exercises:. Represent i graphically and write it in polar form. Represent graphically and give the cartesian form of 6(cos80 + i sin80 ) And the good news is... Now that you know what it all means, you can use your calculator directly to convert from cartesian to polar forms and in the other direction, too. HOW TO CONVERT USING CALCULATOR. Polar Form Cartesian Form Example: Convert 56 7 to rectangular form Shift Rec 56, 7 ) = RCL tan Answer: 49.9+5.4i 8
BA0 ENGINEERING MATHEMATICS 0. Cartesian Form Polar Form Example: Convert to polar form r Shift Pol, -4 ) = RCL tan Answer: 43.68-74.05.9. Exponential Form of a Complex Number IMPORTANT: In this section, θ MUST be expressed in radians. The exponential form of a complex number is: re i (r is the absolute value of the complex number, the same as we had before and θ is in radians.) Example : Express in exponential form. Solution: We have r = 5 from the question. We must express θ = 35 in radians. Recall: 9
BA0 ENGINEERING MATHEMATICS 0 So (or.36 radians) So we can write Example : Express in exponential form. 0
BA0 ENGINEERING MATHEMATICS 0 Exercises:. Express in exponential form:. Express in exponential form: 3. Express in polar and rectangular forms:.0 Products and Quotients of Complex Numbers When performing addition and subtraction of complex numbers, use rectangular form. (This is because we just add real parts then add imaginary parts; or subtract real parts, subtract imaginary parts.) When performing multiplication or finding powers and roots of complex numbers, use polar and exponential forms. (This is because it is a lot easier than using rectangular form.) We start with an example using exponential form, and then generalise it for polar and rectangular forms..0. Multiplying Complex Numbers in Polar Form We can generalise the example we just did, as follows: i i re re rr e i From this, we can develop a formula for multiplying using polar form: cos sin cos sin r r cos isin r i r i or with equivalent meaning:
BA0 ENGINEERING MATHEMATICS 0 r r rr In words, all this confusing-looking algebra simply means... To multiply complex numbers in polar form, Multiply the r parts Add the angle parts Example : Find 3(cos 0 isin 0 ) 5(cos 45 i sin 45 )
BA0 ENGINEERING MATHEMATICS 0.0. Division As we did before, we do an example in exponential form first, then generalise it for polar form. Example in Exponential Form: 8e e 4e 3.6i.i 3.6i.i 4e.4i [We divided the number parts, and subtracted the indices, just using normal algebra.] From this, we can conclude the following: r cos isin r cos r cos isin r or r r r r In words, this simply means... isin To divide complex numbers in polar form, Divide the r parts and Subtract the angle parts Example: i. Evaluate the following by first converting numerator and denominator into polar form. ii. Then check your answer by multiplying numerator and denominator by the conjugate of the denominator. 5i i 3
BA0 ENGINEERING MATHEMATICS 0 Solution: 4
BA0 ENGINEERING MATHEMATICS 0 Exercise: Express 5cos 59 isin59 in Cartesian form, polar form and exponential form. Find 8i Find using polar 7 i form Evaluate: (0.5 40 )(6 0 ) Evaluate: 30 50 Express 65e i5.45 in Cartesian form, polar form and trigonometric form. 5
BA0 ENGINEERING MATHEMATICS 0 POLITEKNIK KOTA BHARU JABATAN MATEMATIK, SAINS DAN KOMPUTER BA 0 ENGINEERING MATHEMATICS PAST YEAR FINAL EXAMINATION QUESTIONS A. Algebraic Operations Of Complex Number 3 i 7 5i. Simplify i. Given u 3 i, v 3i, and w 5i i. u v ii. iii. w u vw u 3. State each of the complex number in form a ib i. 3 ii. 4 4. State the value of Z Z and Z Z in each case i. Z 3i ; Z i ii. 5 Z i ; Z 6 3 i 4 5. Given z 3 7i and w 5i, find: i. z w ii. iii. z w z w 6. Simplify each of the complex numbers: i. 5 7i 3 4i 6 3i ii. iii. i i 7. Given that x y x yi 80 5i. Find the value of x and y. 6
BA0 ENGINEERING MATHEMATICS 0 8. If 3i z, find z in form a ib. i 9. Simplify each of the complex numbers in form a jb. i. ii. 6 j 5 7 j 4 5 j 3 j iii. 3 7 j4 5 j B. Graphical Representation and other Form of Complex Numbers 0. Given Z 3 i and Z i. Find the modulus and argument of: i. Z Z ii. Z Z. Given Z 4 i, Z 0, and Z rad (Given 80 ).. Given z i, z 3 i 0.35i 3 4e. Find i. Calculate modulus and argument of z, z and z z. ii. Sketch Argand diagram for z z. 3. Find the modulus and argument for each complex number: i. 4 3i ii. 3i 4. Given z 8cos 75 sin 75 dan z 6cos 54 sin 54 i i. z z ii. z z Z Z Z i 3 in the polar expression.. Find: iii. Find the sum of z 3 4i and z 5 3i by using an Argand diagram. 7
BA0 ENGINEERING MATHEMATICS 0 5. Given x yj j, where x and y are real numbers. 3 j i. Find the value of x and y ii. Refer to the answer in (i). Find the modulus and the argument. 6. Given Z 5cos 37 i sin 37 i. Cartesian/ Rectangular Form. ii. 7. Simplify i. Exponential Form. 5,express Z in: cos 60 i sin 60 4cos 30 i sin 30 cos 50 i sin 50 8. State each of the complex number in form a ib i. 6 j7 8 j5 4 j0 ii. 4 j44 j3 iii. 0 36 0 j5 iv. 3 j4 j5 j j3 v. 545 4 j 9. Given z 4 3i and z 4 3i. Find the modulus and argument for each complex number and show it in Argand diagram: i. z z ii. z z 3 0. Express each complex numbers to the polar form. i. z 5 8 j ii. z 3 4 j 8