1 Complex Numbers Learning Outcomes Investigate arithmetic with complex numbers in rectangular form . Calculate the conjugate of a complex number. Solve quadratic equations with complex roots. Solve problems about fractions of complex numbers using complex conjugates. Illustrate and read complex numbers on an Argand diagram. Solve problems about the modulus of a complex number. Discuss and illustrate symmetries and rotations on an Argand diagram. 2

Investigate Arithmetic Recall the quadratic formula: Previously, the square root was not allowed to contain a negative number. Now it can! Theyre called imaginary numbers. pg 1 3 Investigate Arithmetic The imaginary number, is usually written for short. Using rules for surds, we can write any imaginary number as a multiple of . e.g.

e.g. pg 1 4 Investigate Arithmetic e.g. Simplify each of the following to a product with : a) b) c) d) e) f) g) pg 2-3

5 Investigate Arithmetic - and A complex number is made up of a real part and an imaginary part added or subtracted together. e.g. in , the real part is , the imaginary part is . e.g. in , the real part is , the imaginary part is . In general, a complex number, , is written as either: or is the name of the complex number. , the real part of , and , the imaginary part of .

pg 2-3 6 Investigate Arithmetic - and Write down the real and imaginary parts of each of the following complex numbers: a) b) c) d) e) f) g)

pg 3-4 7 Investigate Arithmetic Add / Subtract Works just like regular algebra operate on like terms, do not operate on unlike terms: e.g. Let and , pg 3-4 8 Investigate Arithmetic Add / Subtract Given , , , find: a)

b) c) d) e) f) g) h) pg 3-4 9 Investigate Arithmetic Multiply Real Distribution works the same as normal algebra get extra copies of terms inside the bracket. e.g.

e.g. e.g. pg 3-4 10 Investigate Arithmetic Multiply Real Given , , , find: a) b) c) d) e) f) g)

h) pg 4-5 11 is a rule that pops up all over complex numbers recognise it well! Investigate Arithmetic Multiply Imaginary Almost like regular algebra. Recall definition of

Thus, e.g. pg 4-5 12 Investigate Arithmetic Multiply Imaginary Simplify the following and express in the form : a) b) c) d) e) f)

g) h) pg 4-5 13 Investigate Arithmetic Multiply Imaginary Given , , , find: a) b) c) d) e) f)

g) h) pg 4-5 2004 OL P1 Q4 Investigate Arithmetic Multiply Imaginary Given that , simplify: 2007 OL P1 Q4 and write your answer in the form , where , . Given that , simplify:

2010 OL P1 Q4 and write your answer in the form , where , . Given that , simplify: 14 and write your answer in the form , where , . pg 4-5 15 2003 OL P1 Q4 Investigate Arithmetic Multiply Imaginary

Given that , find the value of: i. 2006 OL P1 Q4 ii. Given that , write in its simplest form: 2011 OL P1 Q4 is the complex number , where . a) Find and . b) Verify that

c) Using or otherwise, find , , and . d) Based on (c) or otherwise, state whether is positive or negative. pg 5-6 16 Formerly was used to mean complex conjugate, but this is older notation and quite rare nowadays.

Calculate the Complex Conjugate For a complex number , its complex conjugate is . i.e. its real part is the same, but its imaginary part has opposite sign. e.g. e.g. e.g. e.g. pg 5-6 17 Calculate the Complex Conjugate e.g. Find the complex conjugate of the following: a)

h) b) i) c) j) d) k) e)

l) f) m) g) n) pg 5-6 18 Calculate the Complex Conjugate

2005 OL P1 Q4 Let a) Write down , the complex conjugate of . b) Find real numbers and such that 2011 OL P1 Q4 Let a) Find in the form , where , . b) Show that c) Show that , where is the complex conjugate of . pg 12-17 19

Solve Quadratics with Complex Roots Recall that the roots of a quadratic equation are the -co-ordinates where it cuts the -axis. Recall also that polynomials have a number of roots equal to their highest power. Thus quadratics must have two roots even if they dont cross the axis! It turns out the roots are complex.

pg 12-17 20 Solve Quadratics with Complex Roots e.g. Solve . Using the quadratic formula: We get: pg 12-17 Solve Quadratics with Complex Roots 21 2

4 Solve the following quadratic equations: = 2 a) b) c) d) e) f) g) h) pg 12-17

22 Solve Quadratics with Complex Roots If two shapes do not intersect, they will give complex value when solving simultaneously. As a shortcut, just check if is negative as that is where the comes from in complex roots: e.g. Prove that the line and the circle do not intersect. 1. 2. 1. So they do not intersect. pg 12-17

23 Solve Quadratics with Complex Roots Prove that the following shapes do not intersect: a) b) c) pg 12-17 24

Solve Quadratics with Complex Roots 2004 OL P1 Q4 Solve . Write your answers in the form , where , . 2006 OL P1 Q4 Solve . Write your answers in the form , where , . 2009 OL P1 Q4 Let . i.

Show that is a solution of . pg 7-8 25 SP about Fractions of Complex Numbers Given , , , , find: a) b) c) d) Note that none of the results have an imaginary part. In general, adding or multiplying a complex number to/by its conjugate results in a real number.

pg 7-8 26 SP about Fractions of Complex Numbers Dividing complex numbers by real numbers is trivial: e.g. Dividing complex number by complex numbers is not: e.g. So we make it easier by making the denominator real. pg 7-8 27

SP about Fractions of Complex Numbers Multiply top and bottom by the conjugate of the denominator: pg 7-8 28 SP about Fractions of Complex Numbers Distribute the brackets. pg 7-8 29 SP about Fractions of Complex Numbers

Recall that and simplify the resulting fraction. pg 7-8 30 SP about Fractions of Complex Numbers Simplify the following complex numbers. Write your answer in the form , where , : a) e) b) f)

c) d) pg 7-8 31 SP about Fractions of Complex Numbers 2013 OL P1 Q1 Let and , where . Find in the form , where , . 2014 OL P1 Q2

Let and , where Find a complex number such that . Give your answer in the form , where , . pg 7-8 32 SP about Fractions of Complex Numbers 2006 OL P1 Q4 a) Express in the form . Remember

identically equal polynomials! 2003 OL P1 Q4 b) Hence, or otherwise, find the values of the real numbers and such that: Let a) Simplify . b) and are real numbers such that: Find the value of and .

pg 17-19 33 Illustrate on an Argand Diagram An Argand diagram is a co-ordinate plane with on the horizontal () axis and on the vertical () axis. For a complex number , it can be plotted as on an Argand diagram. pg 17-19

34 Illustrate on an Argand Diagram Illustrate each of the following complex numbers on a single Argand diagram. a) b) c) d) e) f) pg 17-19 35

Illustrate on an Argand Diagram 2005 OL P1 Q4 Let , where . Plot the following on an Argand diagram. i. ii. 2008 OL P1 Q4 Let , where . Plot the following on an Argand diagram. i. ii.

pg 17-19 36 2011(S) OL P1 Q3 Illustrate on an Argand Diagram Two complex numbers are and , where . a) Given that , evaluate . b) Plot , , and on an Argand diagram. 2011 OL P1 Q5 is the complex number , where .

i. Find and . ii. Verify that . iii. Show , , , and on an Argand diagram. iv. Make one observation about the pattern of points on the diagram. pg 19-23 37 Solve Problems about Modulus The modulus of a complex number, , is its distance from the origin. e.g. Distance formula:

e.g. Pythagoras: pg 19-23 38 Solve Problems about Modulus In general, for , Formula: . Find the modulus of each of the following: a) b) c) d) e)

f) g) pg 19-23 39 Solve Problems about Modulus 2003 OL P1 Q4 Let and . i. Plot , , and on the same Argand diagram.

ii. Investigate whether . 2005 OL P1 Q4 Let . i. Express in the form , where , . ii. Investigate whether . 2006 OL P1 Q4 Let where . Calculate . pg 19-23

40 Solve Problems about Modulus 2012 OL P1 Q3 The complex number , where . a) Plot and on an Argand diagram. b) Show that . c) What does part (b) tell you about the points you plotted in (a)? d) Let be a real number such that . Find the two possible values of . pg 24-28

41 Discuss and Illustrate Symmetries Recall axial symmetries and rotations from transformation geometry. Given a complex number , plot and on an Argand diagram: What transformation do you apply to to get ? What is the angle formed by joining and to the origin? pg 24-28

42 Discuss and Illustrate Symmetries As and are symmetric about the Real axis, the line segments each make an angle with the axis. We can form right-angled triangles with: pg 24-28 43 Discuss and Illustrate Symmetries Given , find and plot and on an Argand diagram. Describe the transformation of to . It may help to join and to the origin. In general, multiplying a complex number by rotates

the complex number by 90o anti-clockwise in the complex plane. Multiplying by rotates by 90o clockwise. We can prove for this example using slopes: pg 24-28 44 Discuss and Illustrate Symmetries Converting complex numbers to co-ordinates: These slopes are reciprocal negatives, so the angle between these lines is 90o.