MATHS (O) NOTES The Institute of Education 07 SUBJECT: Maths LEVEL: Ordinary Level TEACHER: Jean Kelly Topics Covered:
COMPLEX NUMBERS Strand 3(Unit ) Syllabus - Understanding the origin and need for complex numbers and how they are used to model D systems: as in computer games, alternating current and voltage. - How to interpret multiplication by i as a rotation of 90 anticlockwise. - How to express complex numbers in the rectangular form ( a bi ) and to illustrate complex numbers on an Argand diagram. - How to investigate the operations of addition and subtraction of complex numbers using the Argand diagram. - How to investigate the operations of addition, subtraction, multiplication and division with complex numbers C in the form a bi (rectangular form) and calculate the complex conjugate as a reflection in the real axis. - How to interpret the Modulus as distance from the origin on an Argand diagram. - How to interpret multiplication by a complex number as a multiplication of the modulus by a real number combined with a rotation. - How to solve Quadratic Equations having complex roots and how to interpret the solutions. Imaginary numbers There exists no real numbers that, when squared, result in a negative number: x x 0 x R To overcome this difficulty an imaginary number i was introduced, where i This allows for the square root of negative numbers to be found: x x x i Imaginary numbers take the form bi, where br, i. Page of 9
Complex numbers Typically the variable used for real numbers is x. For complex numbers we often use the variable z, where z a bi, where a b R i and i,,. A complex number is written in this way:. a is called the real part, and is written as Re. bi is called the imaginary part, and is written as Im The set of all complex numbers is C, a bi a, br, i C a bi a, br, i Addition & Subtraction of Complex numbers When adding or subtracting complex numbers, add or subtract the real parts, then add or subtract the imaginary parts. a bi c di a c b d i a bi c di a c b d i And aa bi bi cc di di a c b d i z 3i and w 5i Q) Calculate z + w Q) Calculate z w ( 3 i) ( 5 i) ( 3 i) ( 5 i) 3i 5i 3i 5i 3i 5i 3i 8i Page of 9
Multiplication of Complex numbers Use the same approach as you d use when multiplying polynomials together in algebra; multiply every term in one bracket by every term in the other. a bic di ac adi bci bdi ac ad bci bd ac bd ad bci z 3 i and w 5i Q) Calculate zw 3i 5i 0i 3i 5i 7i 5( ) 7i 5 7 7i i Complex Conjugate In order to divide complex numbers we must first define the conjugate of a complex number. If z a bi, then the complex conjugate of z, written as z, is defined by: zz. a bia bi aa ab ba i bb a b Q. Calculate z.z, where z 3i. 3i 3i 4 6i 6i 4 9( ) 49 3 9i real number z a bi To find the complex conjugate just change the sign of the imaginary part. The conjugate of a complex number is a reflection/ image through the x-axis (y-value of point changes sign) If a complex number is added to or multiplied by its conjugate, the answer will always be a real number. If complex conjugates are subtracted, the answer is always an imaginary number. Page 3 of 9
Division of Complex numbers As we cannot divide by a complex number, we must first multiply the fraction above & below by the conjugate of the denominator. a bi c di a bi c di c di c di a bic di c d ac bd ad bci c d ac bd ad bc i c d c d Q. Express z w in the form a bi, where a, br, i 3i 5i by conjugate of bottom 5i 5i 0i 3i 5i 5i 5i 5i 3i 5( ) 5( ) 3i 5 5 33i 6 i top and bottom by 3 Page 4 of 9
Example. Basic Operations with Complex numbers Q Simplify and write your answer in the form a bi : (i) 3i 4 5i (ii) 4 i 3 i (iii) 5 i 6 3i (iv) 3i 5i (v) 3 4i5 6i (vi) 8 3i i 7 4i (vii) 4 i3 i (viii) 35i 7i 3i (ix) 3 4i i5 6i (x) 4 i i3 5i Q Simplify and write your answer in the form a bi : (i) 3i i4 5i (ii) 7 i i 9i (iii) 3 5i i3 i (iv) 4i 3i 7 4i (v) 34 i i 5i (vi) 4i i3 i (vii) 7 i 5 6i (viii) 3i 5 i (ix) 3 i 5i (x) 3i i 3i Example. Basic Operations with Complex numbers Q Express in the form a bi : (i) i (vi) 4 i i (ii) (vii) i 3 i 7 3 5i (iii) (viii) 3i 3 4i 3 3 i (iv) 4 i 3 i (ix) 3 i 4i 68i (v) 4 3i (x) 3i Q Express in the form a bi : (i) (vi) 5 i 6 (ii) 3i i i (vii) i 5 4i (iii) 5 4i i 5i (viii) i i (iv) 3i6i (ix) i3 4i i 3 6i (v) 36i3i 5 (x) i i Page 5 of 9
Argand Diagram We represent all real numbers on a onedimenional number line. We use two-dimensional plane to represent complex numbers. We use the horizontal axis to plot the real part (Re) and a vertical axes to plot the imaginary part (Im). Plot complex numbers exactly like you plot points, where the x co-ordinate is the real part (Re) and the y co-ordinate is the imaginary part (Im). Q w 3 i. Plot w and w on an Argand diagram.. Im 3 w3i 4 3 0 3 Re. 3 w3i Any multiple of z is collinear with z and the origin Page 6 of 9
a Jean Kelly Modulus The modulus of a complex number, z a bi, is the distance, from the origin on an Argand diagram to the point. Im 3 z 3i ab. c 4 3 0 b 3 Re 3 The modulus of z is written as z, where z a b This formula comes from Pythagoras theorem: c a b z 3 i, w 5i z 3 i, w 5i Q) Calculate z Q. Calculate z 3 i 3 3 4 94 3 9 3 Q) Calculate z Q. Calculate z 3 i 3 3 4 94 93 3 Q3) Calculate w Q.3 Calculate w 5i 5 5 6 5i 5 5 6 Page 7 of 9
Example 3. Argand Diagrams & Modulus Q If z 3i and z 3 i, plot the following complex numbers on an Argand diagram: (i) z (ii) z (iii) z 3z (iv) zz Q Let u i, where i. Plot on an Argand diagram: (i) u (ii) u 3. Q3 Let u 3 4 i, where i. Plot on an Argand diagram: (i) u (ii) u 5. i Q4 Let z 5 3 i. Plot z and z on an Argand diagram. Q5 Let u 4 i, where i. Plot on an Argand diagram: (i) u (ii) u 4. Q6 Let w i. Plot w and w on an Argand diagram. Q7 Let z 3 i and z 5 i. Plot z, z and z z on an Argand diagram. Q8 Let Q9 w 3 i, where i. Plot on an Argand diagram: (i) w (ii) iw. z (i) (ii) Q0 If z 4 i, where i. Q If z 3 Q If z 5 Q3 z 8 Q4 z 3 5 3 4 Plot z, z, z and z on an Argand diagram. Make one observation about the pattern of the points on the diagram. i calculate z 4z. i calculate z 4z 4 i. i and w i, investigate if z w z w. ki, where k R. If z 0, find the possible values of k. i. If z ki 58, find the possible values of k R. Q5 u 4 3 i and w 6 8 i. Find the value of the real number k such, that u k w. Q6 w 3 i. Investigate whether iw w iw w. Q7 z i where i,. (i) Plot 3 z, z and z on an Argand diagram. (ii) Comment on the pattern of the points on the diagram. Page 8 of 9
0 Ordinary Level Paper : Q3 (5 Marks) The complex number z 4i, where i. (a) Plot z and z on the Argand diagram. (b) Show that z z. (c) What does part (b) tell you about the points you plotted in part (a)? (d) Let k be a real number such that zk 5. Find the two possible values of k. 0 SEC Ordinary Level Sample P: Q3 (5 Marks) Two complex numbers are and, where. (a) Given that, evaluate. (b) Plot u, v, and w on the Argand diagram given. (c) Find. Page 9 of 9
Solving/Finding Roots of Quadratic Equations Use the Quadratic formula: (pg. 0) log tables The two roots/solutions are always conjugates Q Solve z 6z 3 0 and write your answers in the form a bi, where a, b R. z 6z3 0 b b 4ac z sub in : a, b 6 & c 3 a 6 6 4 3 6 36 5 6 6 64i... 6 6 4i 4i 3 i Conjugates N.B. If you solve a quadratic equation Let y 0 to find the roots/solutions/ x values, you are trying to find where the Happy/Sad face curve cuts the x axis. If the solutions are complex numbers, then the curve does not touch the x-axis! Try and draw a graph of the function x 6x3 0, where 6 x 0 N.B. When trying to find the points of intersection between a line and a curve/circle in algebra, you may end up with a quadratic equation that results in solutions that are complex numbers and therefore the line does not intersect the curve or circle! Try and solve the equations x y 6 0and x y 0 for the points of intersection Page 0 of 9
Verifying Roots of Quadratic Equations To prove that a complex number is a root of a quadratic equation, substitute the complex number into the equation for the variable and the answer should be equal to zero. Q. Verify that 3i is a root of the equation down the other root. Let z 3i, Sub in: z 4z3 0 i i i i i 3 4 3 3 0 3 3 4 3 3 0 4 6i 6i 9i 8 i 3 0 4 i 9 8 i 3 0 4 9 8 3 0 5 5 0 0 0 Other root conjugate z 4z3 0 and write z 3i Q. z 4 i is one root of the equation k and write down the other root. z z k 8 0, find the value of Let z 4 i, Sub in: z z k 8 0 i i k 4 8 4 0 4 i 4 i 8 4 i k 0 6 4i 4i i 3 8i k 0 6 8i 3 8i k 0 6 3 k 0 7 k 0 k 7 Other root conjugate z 4 i Page of 9
If you know the roots of a quadratic equation, use the formula: z z sum of roots product of roots 0 to form the Quadratic Equation. Q3. If z 4 5 i is a root of the equation and the value of c. z bz c 0. Find the value of b z 4 5 i and z 4 5 i are the two roots Sub in : z z sum of roots product of roots 0 z z i i i i 4 5 4 5 4 5 4 5 0 6 0i z z 4 4 5i z 5i z 8 6 5 0 0i 5i 0 z z b c 8 4 0 8, 4 Example 4. Solving Quadratic Equations and Verifying roots Q Solve each equation and write the answers in the form a bi. (i) z 4z 0 0 (ii) z 0z 6 0 (iii) z (iv) z 6z34 0 (v) z 0z 9 0 4z 9 0 Q Verify that each complex number is a root of the equation and write down the other root: (i) 4 3i, z 8z 5 0 (ii) i, z z5 0 (iii) 5 4i, z 0z 4 0 (iv) 7 i, z 4z 50 0, z z 37 0 (v) 6 i Q3 Form a quadratic equation with the roots 5 3 i. Q4 If 3 3i is one of the roots of the equation of a and b. z az b 0, find the value Page of 9
Geometrical properties of Complex numbers (Transformations). Rotations A rotation turns a point through an angle about a fixed point If a point (complex number) is multiplied by i, the number is rotated by 90 anti-clockwise about the origin. This is a positive rotation. If a point (complex number) is multiplied by i, the number is rotated by 90 clockwise about the origin. This is a negative rotation On an Argand diagram: Multiplication by i rotates a complex number by 90 anti-clockwise Multiplication by i rotates a complex number by 80 anti-clockwise Multiplication by i 3 rotates a complex number by 70 anti-clockwise Multiplication by 4 i rotates a complex number by 360 anti-clockwise Multiplication by 3 4 i, i, i and i reverses the direction of the rotation to clockwise N.B. 3 4 i, i i and i Q. z 3 i. i) Represent ii) iii) 3 4 z, iz, i z, i z and i z on an Argand diagram. Using the origin as the centre point, draw a circle through the 3 4 complex numbers z, iz, i z, i z and i z. What do you notice? Verify that 3 4 z iz i z i z i z, i.e., prove that all the points are the same distance from the origin. ( Modulus Radius ) Page 3 of 9
Ƶ ( i 4 Ƶ) i Ƶ 90 360 80 70 3 i Ƶ i Ƶ z 3i z 3i 3 4 9 3 iz i( 3 i) i 3i i 3( ) 3 i i z ( 3 i) 3i i z iz 3 i 3 9 4 3 3 i z i( 3 i) i 3i i 3( ) 3 i 3 i z 3i 3 4 9 3 3 i 3 9 4 3 4 i z ( 3 i) 3i z 4 iz z i 3 3 4 9 3 Modulus Radius Length 3 Page 4 of 9
. Translations A translation is the image of an object by moving every point of the object in the same direction and same distance away, without rotating or resizing the object; simply changing the location of the object. If you add a given complex number to each complex number that makes up an object you will translate that object and move it to a different location to create an image. the addition of complex numbers means that you are translating / moving them on an Argand diagram. Q The four complex numbers the vertices of a square. A i, B( 3 i), C(3 3 i) and D(3 i) form i) Plot the complex numbers on an Argand diagram (complex plane). ii) If z 3 i, evaluate and plot the points on an Argand diagram: P A z, Q B z, R C z and S D z. iii) Describe the transformation that is the addition of z. Q R B C P S A D Page 5 of 9
ii) P A z P ( i) (3 i) P 33i Q B z Q ( 3 i) (3 i) Q 45i R C z R (3 3 i) (3 i) R 65i S D z S (3 i) (3 i) S 63i iii) The transformation that maps the square ABCD onto the quadrilateral PQRS is a translation, where all the points A, B, C and D are all moved the same distance and in the same direction on the complex plane. Page 6 of 9
3. Dilations A dilation is the resizing of an object, making it larger or smaller. If a complex number is multiplied by a real number (scalar), then its modulus (distance from the origin) will be multiplied by this scalar. If real number, then the dilation is referred to as a stretching and the object is enlarged. If real number, then the dilation is referred to as a contracting and the object is reduced. Q. The three points A i, B( 3 i) and C(3 i) are vertices of a triangle. i) Plot the complex numbers on an Argand diagram (complex plane). ii) If k 3, evaluate and plot the points on an Argand diagram: P ka, Q kb and R kc. iii) Describe the transformation that is the multiplication by k. P B P R A C Page 7 of 9
ii) P ka P 3( i) P 33i Q kb Q 3( 3 i) Q 39i R kc R 3(3 i) R 93i iii) From the diagram we see that all the points, A B and C are moved further from the origin by a factor of k 3. We call the transformation that maps the triangle ABC onto the triangle PQR a dilation by a factor of 3. The points A, B and C are said to be stretching the complex plane and the triangle PQR is an enlargement of the triangle ABC. N.B. If you multiply a complex number by i, its modulus will be doubled and it will be rotated by 90. N.B. If z is a complex number, then kz k z, where k is a real number. Page 8 of 9
Example 5. Transformations Q 3 z 4 i, z 3 i, z i and w i. (i) Plot the points z, z and z 3 on an Argand diagram. (ii) Evaluate z w, z w and z3 w and plot the answers on the Argand diagram (iii) Describe the transformation that is the addition of w. Q 3 z 4 i, z 3 i, z i and k. (i) Plot the points z, z and z 3 on an Argand diagram. (ii) Evaluate kz, kz and kz 3 and plot the answers on the Argand diagram. (iii) Describe the transformation that is the multiplication of k. Q3 z 3 i. (a) Show on an Argand diagram: (i) z (ii) z (iii) 3z (iv) 4z (v) 5z (b) Describe the transformation of z in parts (ii) to (v) above. Q4 z 3 i. (a) Show on an Argand diagram: (i) z (ii) z (iii) 3 z (iv) 4 z (v) 5 z (b) Describe the transformation of z in parts (ii) to (v) above. Q5 z i. (i) Find z, if z iz. (ii) Plot z and z on an Argand diagram. (iii) Describe the transformation that maps z onto z. Q5 z 3 4 i. (ii) Find z, if z iz. (ii) Plot z and z on an Argand diagram. (iii) Describe the transformation that maps z onto z. Page 9 of 9