In Z: x + 3 = 2 3x = 2 x = 1 No solution In Q: 3x = 2 x 2 = 2. x = 2 No solution. In R: x 2 = 2 x = 0 x = ± 2 No solution Z Q.

Size: px
Start display at page:

Download "In Z: x + 3 = 2 3x = 2 x = 1 No solution In Q: 3x = 2 x 2 = 2. x = 2 No solution. In R: x 2 = 2 x = 0 x = ± 2 No solution Z Q."

Transcription

1 THE UNIVERSITY OF NEW SOUTH WALES SCHOOL OF MATHEMATICS AND STATISTICS MATH 1141 HIGHER MATHEMATICS 1A ALGEBRA. Section 1: - Complex Numbers. 1. The Number Systems. Let us begin by trying to solve various algebraic equations. Suppose we only know about the set of natural numbers (written as N). Then we can solve the equation x 3 = 2 and obtain the solution x = 5. On the other hand, if we try to solve the equation x + 3 = 2 then there is no solution! To solve this equation we need a larger set of numbers which includes the negative whole numbers as well as the positive ones. This set is called the set of integers and is denoted by Z. Continuing this idea: In Z: x + 3 = 2 3x = 2 x = 1 No solution In Q: 3x = 2 x 2 = 2 In R: x = 2 No solution 3 x 2 = 2 x = 0 x = ± 2 No solution i 2 C 2 i π Z 1 0 N Q R 2 1

2 At each stage in the above we are able to solve each new type of equation by extending the set of numbers in which we are working. Hence, to solve the equation x 2 = 1 we introduce a new symbol i (much as we introduced the symbol 2 to solve x 2 = 2.) We define i to be the (complex) number whose square is 1. i.e. i 2 = 1. Using this new symbol, we can now solve x 2 = 1 to obtain solutions x = ±i. Furthermore, we can define the complex numbers by: Definition: The set of all numbers of the form a + bi where a, b are real numbers and i 2 = 1 is called the set of all complex numbers and denoted by C. Summary of Basic Rules and Notation: Let z = a + ib, w = c + id be complex numbers. Then (i) z ± w = (a ± c) + i(b ± d) (ii) zw = (ac bd) + i(ad + bc). (iii) z = a+ib c id = (ac+bd)+i(bc ad) w c+id c id c 2 +d 2 (iv) Re(z) = a, Im(z) = b. Example: z = 2 4i, w = 3 + i, find z + w, z w, zw, z. w Ex: Simplify (1 + i) 8. CP: Let n be an integer. By considering cases, simplify i n + i n+1 + i n+2. 2

3 As with the set of real numbers and the set of rational numbers, if we add, subtract, multiply, or divide (with the exception of division by 0) any two complex numbers, we again obtain a complex number. This property is called closure. The complex numbers are closed under addition, subtraction, multiplication and division (not by 0). A set of objects which has these (and a number of other properties) is called a field in mathematics. The real numbers and the rational numbers also form fields, but the integers do not, since they are not closed under division. The natural numbers are not closed under subtraction. The above examples are all infinite fields. In addition to these, there are finite fields, which you will study in more detail in higher years. Here are some simple examples. Definition: Given two integers a, b, we can write a = bq + r, with 0 r < b. We will write a r mod b, which we read as a is congruent to r modulo b. In words, r is the remainder when we divide a by b. Hence 7 2 mod 5, and 18 2 mod 4. We can now form the set Z 5 of the possible remainders when we divide by 5. So Z 5 = {0, 1, 2, 3, 4}. We can define the operations of addition and multiplication to be the same as in ordinary arithmetic, but the answers are calculated modulo 5. Thus, addition and multiplication tables can be drawn up as follows: From the tables, we can see that subtraction and division (except by 0) are defined. For example 2 4 = 3 since and 3 2 = 4 since The set Z 5 is an example of a finite field. CP: Make up the addition and multiplication tables for Z 6. Explain why this is NOT a field. Can you complete (and prove) the following Theorem: Z n is a field if and only if n is... 3

4 Equality: Two complex numbers are equal iff they have the same real and imaginary parts, i.e. if z = a + bi = w = c + di then we can conclude that a = c and b = d. Proof: Roots of Unity: A complex number α 1 is called an n-th root of unity if α n = 1. For example, if ω 3 = 1, and ω 1 then we can write, 4

5 Example: To show just a little of the power of complex numbers, we seek to find a simple closed formula for ( n ( n ( n ( n ) 3) 6) n) where n is an integer divisible by 6. We begin by noting that if ω is a complex cube root of unity then 1+ω k +ω 2k can only take the values Now expand out (1 + ω) n and (1 + ω 2 ) n. Hence if l is the largest integer such that 3l n we have ( n ( n ( n ( n = 0) 3) 6) 3l) 1 ( 2 n + (1 + ω) n + (1 + ω 2 ) n). 3 Finally, if n is a multiple of 6, 5

6 CP: Suppose n > 1 is a multiple of 4. By expanding (1+i) n, (1 i) n, (1+1) n, and (1 1) n, find, in as simple a form as you can, the sum ( n ( n ( n ( n ) 4) 8) n) Polynomial Equations: We can now solve ALL quadratic equations. Ex: Solve 5x 2 4x + 1 = 0 and z 2 3z + (3 + i) = 0. Note also that we can find new solutions to old equations such as x 3 1 = 0. Both of these are examples of the following remarkable theorem: Theorem. (Fundamental theorem of algebra, FTA) Suppose p(x) = a n x n + a n 1 x n a 1 x + a 0 is a polynomial, whose co-efficients a n,, a 1, a 0 are all real (or complex) numbers, then the equation p(x) = 0 has at least one root in the complex numbers. Corollary: The equation p(x) = 0 has exactly n (complex) solutions in the complex numbers (counting multiplicity). (The last proviso counting multiplicity refers to polynomials which may, for example, have factors such as (x 2) 4 in which case the root x = 2 is counted four times.) The above result tells us (among other things) that we do not need to find any larger set of numbers if we want to solve polynomial equations. The complex numbers contain all the roots of every polynomial. 6

7 Solution of Cubics: The FTA tells about existence, but doesn t give us the machinery to actually find the roots of a polynomial. For quadratics we have the quadratic formula, what about cubics? The first thing to observe is that every cubic equation x 3 + px 2 + qx + r = 0 can be rewritten in the following form: x 3 + ax + b = 0. This is achieved by the change of variable, x = y p 3. Example: Remove the square term in: x 3 6x 2 + x + 3. Put x = y + 2 Cardano stole from Tartaglia, the secret of solving the cubic. He made the change of variable x = u v. It is technically easier to put x = u + v. Example: Find the real root of x 3 + 3x = 1. 7

8 Strange things can happen when we apply this method to cubics which have three real roots. Example: Solving x 3 6x + 4 = 0 which has x = 2 as one of it roots. Square Roots: Solve z 2 = 3 + 4i. 8

9 Conjugates: When solving a quadratic equation (with real roots) over the complex numbers, you will have observed that the solutions occur in pairs, in the form a + bi and a bi. There are called conjugate pairs. We say that a bi is the conjugate of a+bi (and vise-versa). To represent this, we use the notation z = a + bi, z = a bi. This conjugate operation has the following properties: i. z ± w = z ± w ii. zw = z.w iii. ( ) z w = z. w iv. z = z if and only if z is real. v. z + z = 2Re(z). You will prove these and similar results in the tutorial exercises. Also note that repeated application of (ii) gives (a + bi) n = (a + bi) n. The Argand Plane: Complex numbers can be represented using the Argand plane, which consists of Cartesian axes similar to that which you used to represent points in the plane. The horizontal axis is used to represent the real part and the vertical axis, (sometimes called the imaginary axis), is used to represent the imaginary part. For example, the following points have been plotted: 3, 2i, 3 + 2i, 4 + i i 2i 4 + i 3 Complex numbers then are 2-dimensional, in that we require two axes to represent them. Observe that a complex number z and its conjugate are simply reflections of each other in the real axis. 9

10 We lose the notion of comparison in the complex plane. That is, we cannot say whether one complex number is greater or lesser than another. You have already seen that complex numbers can be expressed either in Cartesian Form, a + ib, a, b R. We can also specify a complex number z by specifing the distance of z from the origin and the angle it makes with the positive real axis. This distance is called the modulus and written as z while the angle is called the argument and written as Arg(z). We insist, to remove ambiguity, that π < Arg(z) π. Pythagoras theorem gives: If z = a + bi then z = a 2 + b 2. Care must be taken to find the correct argument. It is easiest to find the related angle ˆθ such that tan ˆθ = b a and then use this to find the argument in the correct quadrant recalling that we use negative angles in the third and fourth quadrant. Ex: Find the modulus and argument of z = 1 + i 3 and w = 1 2i Properties of Modulus: The modulus function has the following properties: (i) zw = z w (ii) z = z, provided w 0. w w (iii) z n = z n (iv) z = 0 z = 0. 10

11 Example: (Sums of squares). We can write the integer 5 as = 2 + i 2. We can also write the integer 13 as = 2 + 3i 2. Hence Try doing the same for 17 and 29. CP: Use the idea above (not expansion) to prove that (a 2 +b 2 )(c 2 +d 2 ) = (ac bd) 2 +(ad+bc) 2 and conclude that, in general, the product of any two numbers which are each the sum of two integer squares, is itself the sum of the two integer squares. This begs the question: What numbers can be written as the sum of two integer squares? This is a hard problem. Experiment with prime numbers and make a conjecture. CP: Suppose A and B are two points in the complex plane corresponding to the complex numbers α = a + ib and β = c + id respectively. Explain why the triangle OAB is right-angled if and only if α β 2 = α 2 + β 2. B α β β α A O Show that if triangle OAB is right-angled then ac = bd. Deduce that if triangle OAB is right-angled then Re(αβ) = 0 11

12 Properties of the Argument: We can distinguish between the principal argument of z, written Arg(z), which is uniquely defined and takes values between π and π (excluding π), and the more general argument, written arg(z), which is a set of values. We have Arg(z) = arg(z) mod 2π, which means that we can recover Arg(z) from arg(z) by adding or subtracting the appropriate multiple of 2π. The Argument function has the following properties: (i) Arg(zw) = Arg(z)+Arg(w) mod 2π. (ii) Arg(z/w) = Arg(z) Arg(w) mod 2π. The proofs of these will become apparent later. Ex: Suppose α < 1, use a diagram to explain why Arg ( ) 1+α 1 α < π. 2 12

13 Polar Form: r sin θ r z = (r(cosθ + i sin θ) θ r cos θ From the diagram, we can see that the complex number z can be written in the form z = r(cosθ + i sin θ), where r is the modulus of z and θ is the argument of z. For example, the complex number 1 i can be written as 1 i = 2(cos( π 4 ) + i sin( π 4 )) = 2(cos π 4 i sin π 4 ). This is sometimes called the polar form of z. You will need to be able to convert a complex number from cartesian form, (a + bi), into polar form and vice-versa. For example, 3(cos π + i sin π ) = i3. 2 You may have seen the abbreviation cisθ to represent cosθ + i sin θ. You should not use that here, since your tutor may not know what it is. This form is NOT generally used in books beyond High School. Moreover, as we shall see, this polar form, is really a stepping stone to a much better form which involves e. One important fact about the polar form is a remarkable result called: De Moivre s Theorem: For any real number θ, and any integer n, we have (cosθ + i sin θ) n = cosnθ + i sin nθ. The proof of this, looking at the various cases of n is given in the algebra notes. The method of proof by induction is used. Note that the result also holds for n rational which we will find useful later than finding roots. Let us see how useful this result can be: Ex: Let z = 1 i 3. Find z

14 Let us write de Moivre s theorem as follows: Let f(θ) = cosθ + i sin θ, then (f(θ)) n = f(nθ). Also f(0) = 1. Euler did the following: He supposed we can differentiate the function, treating i just like a real number. If we momentarily ignore the logical difficulties involved then f (θ) = i(cos θ + i sin θ). Comparing this with d dθ (eiθ ) = ie iθ, we can see then that this function seems to have properties that are very similar to the exponential function e iθ. We will therefore define: Definition: e iθ = cosθ + i sin θ (and hence e iθ = cosθ i sin θ). This formula is sometimes called Euler s formula. We shall take it to be a definition of the complex exponential. Thus, any complex number z can be expressed in the polar form z = re iθ where r is the modulus and θ the argument of z. For example, z = 1 i = and z = 1 This last formula is quite remarkable since it links together the four fundamental constants of mathematics. In a very important sense, this is the best way to write complex numbers. We have used the term polar form in two different senses. From now on, when I say polar form, I will (generally) mean this new exponential form. You ought to be able to convert a complex number from cartesian form to polar form and vice-versa. Note the following important facts: (i) The conjugate of the complex number z = e iθ is given by z = e iθ. (ii) e iθ = e i(θ+2kπ) where k is an integer. We can write cosθ and sin θ in terms of the complex exponential as follows: cosθ = eiθ + e iθ 2 and sin θ = eiθ e iθ. 2i 14

15 From the polar form, we can deduce the properties of modulus and argument which we listed earlier. Let z = r 1 e iθ 1 and w = r 2 e iθ 2 then zw = r 1 r 2 e i(θ 1+θ 2 ) from which it follows that zw = z w and Arg(zw) = Arg(z) + Arg(w) mod 2π. Ex: Convert z = 2e i5π 6, w = 3e i π 3 to Cartesian form: Ex: Evaluate the product (1 + i)(1 i 3) in two ways to show that cos π 12 = The rules for multiplication of complex numbers in polar form tell us that when we multiply two complex numbers together, rotation and stretching are involved. In particular, 15

16 since i = e i π 2, multiplying a complex number by i has the effect of rotating z anti-clockwise about the origin, through an angle of 90. iz z Ex: Find the complex number obtained by rotating (4 + 2i) anti-clockwise about the origin through π 2. More generally, to rotate complex number anticlockwise around 0 through an angle θ, we multiply it by e iθ. Ex: Rotate 3 i anticlockwise about 0 through an angle of π 4. CP: Suppose w 1, w 2 are two complex numbers such that 0 < Arg(w 1 ) < Arg(w 2 ). Show that the triangle in the complex plane whose vertices are given by the origin, w 1 and w 2 is equilateral if and only if w w 2 2 = w 1 w 2. (Hint: Try to write w 2 in terms of w 1 using the rotation idea.) 16

17 The Triangle Inequality: The modulus operation has a number of other useful properties, but two very important ones are: i. zz = z 2 and ii. (The Triangle inequality), z 1 + z 2 z 1 + z 2. I will leave you to prove the first of these and look at the second: Proof of (ii): Ex: Prove that every root of the polynomial p(z) = z 4 + z + 3 lies outside the unit circle in the complex plane. 17

18 CP: a. Find an upper bound on the maximum of the modulus of p(z) = 4z 3 2z + 1 over all complex numbers z which lie on the unit circle. b. Prove that z 1 + z 2 z 1 z 2. (Hint: Start with z 1 = z 1 + z 2 z 2 ). c. Hence find the minimum value of the modulus in (a). (Note that there are two things to prove here.) Get MAPLE to plot the real and imaginary parts (use trigonometric polar form) of p as z moves around the unit circle. Powers and Roots of Complex Numbers: Ex: Find (1 3i) 10. To find roots of complex numbers, we will use the polar form. Note that to find the nth root of a complex number α, we are really solving z n = α and so we will convert α into polar form. Such an equation will have n solutions! (by the Fundamental Theorem of Algebra.) To get all of these solutions we express α in polar form, using the general argument, not the principal one. An example will make this clear. Ex: Find the 7th roots of 1. 18

19 If we plot these complex numbers we see that they lie on a circle radius 1 and are equally spaced around that circle. z 3 z 2 z 1 z 4 z 5 z 7 z 6 Ex: Find the 5th roots of 4(1 i). 19

20 Applications to Trigonometry: Euler s formula gives a dramatic relationship between the exponential and trigonometric functions. We can exploit this to deduce useful relationships and identities in trigonometry. Ex: Find an expression for cos 5θ in terms of sines and cosines. Observe that one can also easily obtain corresponding formula for sin 5θ by taking the imaginary parts of both sides. Web Activity: Use Google to find some information about the Chebychev Polynomials (there are various spellings of Chebychev) and see how they are related to the above example. Ex: Taking the problem the other way round, express sin 5 θ in terms of sine and cosines of multiples of θ. 20

21 Such a formula is extremely useful in integration, where one might, for example, wish to integrate sin 5 θ. CP: Find a similar formula for sin 4 θ cos 6 θ. Example: Suppose 0 < θ < 2π and n is a positive integer. Show that ( ) 1 e i(n+1)θ Re = 1 1 e iθ 2 + sin(n + 1)θ 2. 2 sin θ 2 Use this to find a simple closed formula for 1 + cosθ + cos 2θ cos(nθ). (Try to find a similar formula for the sine sum.) 21

22 Regions in the Complex Plane: In this section we see how to represent regions in the argand plane algebraically. For example, the set A = {z C : z 2} represents the set of points whose distance from the origin is less or equal to 2. (N.B. z α measures the distance between z and α.) Hence this set represents a disc radius 2 centre the origin Similarly, the set B = {z C : z+1 < 2} represents the open disc centre ( 1, 0) radius

23 The set C = {x C : 0 Arg(z) π } represents a wedge vertex at the origin, and arms 3 separated by an angle of 60. o Note that the origin in NOT included since the argument of 0 is not defined. Similarly the set D = {z C : 0 Arg(z i) π } represents a wedge centre the 6 point i as shown. 1 o Here are some further examples: Ex: Sketch {z C : z i + 1 < 2} {z C : Re(z) 0} 23

24 Ex: Sketch {z C : z 3 < 2} {z C : Im(z 3i) > 0} More on Polynomials: The fundamental theorem of algebra, mentioned above, tells us that in the complex plane, all polynomials have all their roots. This is a very powerful theoretical tool, but it does not explicitly tell us how to find these roots for a given polynomial. Moreover, if we know the roots then we also know how to factor the polynomial. You will need to recall a number of basic facts about polynomials from High School, which are: Remainder Theorem: If p(x) is a polynomial then the remainder r when p(x) is divided by x α is given by p(α). Factor Theorem: If p(α) = 0 then (x α) is a factor of p(x). It is important to look at what the underlying set is when we are factoring, for example, x 2 2 does NOT factor over the rational numbers, but it does over the real numbers, (x + 2)(x 2). Similarly, x does not factor over the real numbers but does over the complex numbers. From the fundamental theorem of algebra, it is clear that over the complex numbers, all polynomials completely factor (at least in theory) into linear factors. Theorem: Every polynomial (with real or complex co-efficients) of degree n 1 has a factorisation into linear factors of the form: p(z) = a(z α 1 )(z α 2 ) (z α n ) where α 1, α 2,, α n are the (complex) roots of p(z). This result, still does not tell us how to factor. Nor does it tell us much about factoring over the real and rational numbers. For example, does the polynomial x factor over the real numbers or rational numbers? 24

25 The key to factoring over the real numbers, is to firstly factor over the complex numbers since in the complex plane the polynomial falls to pieces into linear factors. Ex: Factor x over the complex numbers and hence over the real numbers. Ex: Factor x over the complex and real numbers. Note that if the co-efficients of the polynomial are real, then the roots occur in conjugate pairs. Theorem: Suppose p(x) is a polynomial with real co-efficients, then if α is a complex (non-real) root, then so is α. 25

26 Proof: From this is follows that: Theorem: A polynomial with real co-efficients can be factored into a product of real linear and/or real quadratic factors. Proof: Factor p(x) over the complex numbers in the form p(x) = a(x b 1 ) (x b r )(x α 1 )(x α 1 ) (x α s )(x α s ) where the b i s are real and the α i s are complex (non-real). By the above theorem, these must occur in conjugate pairs. Now each such pair of factors containing the conjugate pairs, can be expanded, viz: (x α)(x α) = (x 2 (α + α)x + αα). Now α+α = 2Re(α) and so is REAL, and also αα = α 2 is also REAL. Hence the quadratic we obtain has REAL co-efficients. 26

27 Ex: Show that z = i is a root of p(z) = z 4 2z 3 + 6z 2 2z + 5 = 0 and hence factor p over R and C. CP: (Not so hard.) Factor x 9 16x 5 x over the real numbers. The story over the rational numbers is much more complicated. It is possible to have polynomials of arbitrary degree which cannot be factored over the rational numbers. For example, if p is a prime, then x p 1 +x p x+1 cannot be factored over the rationals. (Can you prove this?) Moreover, there is no simple test to tell whether a given polynomial can be factored over the rationals. (More on this in MATH2400 and Higher Algebra in 3rd year). 27

Chapter 3: Complex Numbers

Chapter 3: Complex Numbers Chapter 3: Complex Numbers Daniel Chan UNSW Semester 1 2018 Daniel Chan (UNSW) Chapter 3: Complex Numbers Semester 1 2018 1 / 48 Philosophical discussion about numbers Q In what sense is 1 a number? DISCUSS

More information

AH Complex Numbers.notebook October 12, 2016

AH Complex Numbers.notebook October 12, 2016 Complex Numbers Complex Numbers Complex Numbers were first introduced in the 16th century by an Italian mathematician called Cardano. He referred to them as ficticious numbers. Given an equation that does

More information

10.1 Complex Arithmetic Argand Diagrams and the Polar Form The Exponential Form of a Complex Number De Moivre s Theorem 29

10.1 Complex Arithmetic Argand Diagrams and the Polar Form The Exponential Form of a Complex Number De Moivre s Theorem 29 10 Contents Complex Numbers 10.1 Complex Arithmetic 2 10.2 Argand Diagrams and the Polar Form 12 10.3 The Exponential Form of a Complex Number 20 10.4 De Moivre s Theorem 29 Learning outcomes In this Workbook

More information

Complex Numbers. 1 Introduction. 2 Imaginary Number. December 11, Multiplication of Imaginary Number

Complex Numbers. 1 Introduction. 2 Imaginary Number. December 11, Multiplication of Imaginary Number Complex Numbers December, 206 Introduction 2 Imaginary Number In your study of mathematics, you may have noticed that some quadratic equations do not have any real number solutions. For example, try as

More information

) z r θ ( ) ( ) ( ) = then. Complete Solutions to Examination Questions Complete Solutions to Examination Questions 10.

) z r θ ( ) ( ) ( ) = then. Complete Solutions to Examination Questions Complete Solutions to Examination Questions 10. Complete Solutions to Examination Questions 0 Complete Solutions to Examination Questions 0. (i We need to determine + given + j, j: + + j + j (ii The product ( ( + j6 + 6 j 8 + j is given by ( + j( j

More information

Solutions to Tutorial for Week 3

Solutions to Tutorial for Week 3 The University of Sydney School of Mathematics and Statistics Solutions to Tutorial for Week 3 MATH9/93: Calculus of One Variable (Advanced) Semester, 08 Web Page: sydney.edu.au/science/maths/u/ug/jm/math9/

More information

1. COMPLEX NUMBERS. z 1 + z 2 := (a 1 + a 2 ) + i(b 1 + b 2 ); Multiplication by;

1. COMPLEX NUMBERS. z 1 + z 2 := (a 1 + a 2 ) + i(b 1 + b 2 ); Multiplication by; 1. COMPLEX NUMBERS Notations: N the set of the natural numbers, Z the set of the integers, R the set of real numbers, Q := the set of the rational numbers. Given a quadratic equation ax 2 + bx + c = 0,

More information

or i 2 = -1 i 4 = 1 Example : ( i ),, 7 i and 0 are complex numbers. and Imaginary part of z = b or img z = b

or i 2 = -1 i 4 = 1 Example : ( i ),, 7 i and 0 are complex numbers. and Imaginary part of z = b or img z = b 1 A- LEVEL MATHEMATICS P 3 Complex Numbers (NOTES) 1. Given a quadratic equation : x 2 + 1 = 0 or ( x 2 = -1 ) has no solution in the set of real numbers, as there does not exist any real number whose

More information

Complex Numbers. Vicky Neale. Michaelmas Term Introduction 1. 2 What is a complex number? 2. 3 Arithmetic of complex numbers 3

Complex Numbers. Vicky Neale. Michaelmas Term Introduction 1. 2 What is a complex number? 2. 3 Arithmetic of complex numbers 3 Complex Numbers Vicky Neale Michaelmas Term 2018 Contents 1 Introduction 1 2 What is a complex number? 2 3 Arithmetic of complex numbers 3 4 The Argand diagram 4 5 Complex conjugation 5 6 Modulus 6 7 Argument

More information

Quick Overview: Complex Numbers

Quick Overview: Complex Numbers Quick Overview: Complex Numbers February 23, 2012 1 Initial Definitions Definition 1 The complex number z is defined as: z = a + bi (1) where a, b are real numbers and i = 1. Remarks about the definition:

More information

Complex Numbers Introduction. Number Systems. Natural Numbers ℵ Integer Z Rational Q Real Complex C

Complex Numbers Introduction. Number Systems. Natural Numbers ℵ Integer Z Rational Q Real Complex C Number Systems Natural Numbers ℵ Integer Z Rational Q R Real Complex C Number Systems Natural Numbers ℵ Integer Z Rational Q R Real Complex C The Natural Number System All whole numbers greater then zero

More information

Math 632: Complex Analysis Chapter 1: Complex numbers

Math 632: Complex Analysis Chapter 1: Complex numbers Math 632: Complex Analysis Chapter 1: Complex numbers Spring 2019 Definition We define the set of complex numbers C to be the set of all ordered pairs (a, b), where a, b R, and such that addition and multiplication

More information

Overview of Complex Numbers

Overview of Complex Numbers Overview of Complex Numbers Definition 1 The complex number z is defined as: z = a+bi, where a, b are real numbers and i = 1. General notes about z = a + bi Engineers typically use j instead of i. Examples

More information

3 + 4i 2 + 3i. 3 4i Fig 1b

3 + 4i 2 + 3i. 3 4i Fig 1b The introduction of complex numbers in the 16th century was a natural step in a sequence of extensions of the positive integers, starting with the introduction of negative numbers (to solve equations of

More information

C. Complex Numbers. 1. Complex arithmetic.

C. Complex Numbers. 1. Complex arithmetic. C. Complex Numbers. Complex arithmetic. Most people think that complex numbers arose from attempts to solve quadratic equations, but actually it was in connection with cubic equations they first appeared.

More information

The modulus, or absolute value, of a complex number z a bi is its distance from the origin. From Figure 3 we see that if z a bi, then.

The modulus, or absolute value, of a complex number z a bi is its distance from the origin. From Figure 3 we see that if z a bi, then. COMPLEX NUMBERS _+i _-i FIGURE Complex numbers as points in the Arg plane i _i +i -i A complex number can be represented by an expression of the form a bi, where a b are real numbers i is a symbol with

More information

Re(z) = a, For example, 3 + 2i = = 13. The complex conjugate (or simply conjugate") of z = a + bi is the number z defined by

Re(z) = a, For example, 3 + 2i = = 13. The complex conjugate (or simply conjugate) of z = a + bi is the number z defined by F COMPLEX NUMBERS In this appendi, we review the basic properties of comple numbers. A comple number is a number z of the form z = a + bi where a,b are real numbers and i represents a number whose square

More information

ALGEBRAIC LONG DIVISION

ALGEBRAIC LONG DIVISION QUESTIONS: 2014; 2c 2013; 1c ALGEBRAIC LONG DIVISION x + n ax 3 + bx 2 + cx +d Used to find factors and remainders of functions for instance 2x 3 + 9x 2 + 8x + p This process is useful for finding factors

More information

1 Complex Numbers. 1.1 Sums and Products

1 Complex Numbers. 1.1 Sums and Products 1 Complex Numbers 1.1 Sums Products Definition: The complex plane, denoted C is the set of all ordered pairs (x, y) with x, y R, where Re z = x is called the real part Imz = y is called the imaginary part.

More information

(x 1, y 1 ) = (x 2, y 2 ) if and only if x 1 = x 2 and y 1 = y 2.

(x 1, y 1 ) = (x 2, y 2 ) if and only if x 1 = x 2 and y 1 = y 2. 1. Complex numbers A complex number z is defined as an ordered pair z = (x, y), where x and y are a pair of real numbers. In usual notation, we write z = x + iy, where i is a symbol. The operations of

More information

MTH 362: Advanced Engineering Mathematics

MTH 362: Advanced Engineering Mathematics MTH 362: Advanced Engineering Mathematics Lecture 1 Jonathan A. Chávez Casillas 1 1 University of Rhode Island Department of Mathematics September 7, 2017 Course Name and number: MTH 362: Advanced Engineering

More information

1 Review of complex numbers

1 Review of complex numbers 1 Review of complex numbers 1.1 Complex numbers: algebra The set C of complex numbers is formed by adding a square root i of 1 to the set of real numbers: i = 1. Every complex number can be written uniquely

More information

Review Sheet for the Final Exam of MATH Fall 2009

Review Sheet for the Final Exam of MATH Fall 2009 Review Sheet for the Final Exam of MATH 1600 - Fall 2009 All of Chapter 1. 1. Sets and Proofs Elements and subsets of a set. The notion of implication and the way you can use it to build a proof. Logical

More information

MTH4101 CALCULUS II REVISION NOTES. 1. COMPLEX NUMBERS (Thomas Appendix 7 + lecture notes) ax 2 + bx + c = 0. x = b ± b 2 4ac 2a. i = 1.

MTH4101 CALCULUS II REVISION NOTES. 1. COMPLEX NUMBERS (Thomas Appendix 7 + lecture notes) ax 2 + bx + c = 0. x = b ± b 2 4ac 2a. i = 1. MTH4101 CALCULUS II REVISION NOTES 1. COMPLEX NUMBERS (Thomas Appendix 7 + lecture notes) 1.1 Introduction Types of numbers (natural, integers, rationals, reals) The need to solve quadratic equations:

More information

Complex numbers, the exponential function, and factorization over C

Complex numbers, the exponential function, and factorization over C Complex numbers, the exponential function, and factorization over C 1 Complex Numbers Recall that for every non-zero real number x, its square x 2 = x x is always positive. Consequently, R does not contain

More information

MATH 135: COMPLEX NUMBERS

MATH 135: COMPLEX NUMBERS MATH 135: COMPLEX NUMBERS (WINTER, 010) The complex numbers C are important in just about every branch of mathematics. These notes 1 present some basic facts about them. 1. The Complex Plane A complex

More information

Introduction to Complex Analysis

Introduction to Complex Analysis Introduction to Complex Analysis George Voutsadakis 1 1 Mathematics and Computer Science Lake Superior State University LSSU Math 413 George Voutsadakis (LSSU) Complex Analysis October 2014 1 / 67 Outline

More information

Find the common ratio of the geometric sequence. (2) 1 + 2

Find the common ratio of the geometric sequence. (2) 1 + 2 . Given that z z 2 = 2 i, z, find z in the form a + ib. (Total 4 marks) 2. A geometric sequence u, u 2, u 3,... has u = 27 and a sum to infinity of 8. 2 Find the common ratio of the geometric sequence.

More information

COMPLEX NUMBERS II L. MARIZZA A. BAILEY

COMPLEX NUMBERS II L. MARIZZA A. BAILEY COMPLEX NUMBERS II L. MARIZZA A. BAILEY 1. Roots of Unity Previously, we learned that the product of z = z cis(θ), w = w cis(α) C is represented geometrically by the rotation of z by the angle α = arg

More information

Complex Numbers. Introduction

Complex Numbers. Introduction 10 Assessment statements 1.5 Complex numbers: the number i 5 1 ; the term s real part, imaginary part, conjugate, modulus and argument. Cartesian form z 5 a 1 ib. Sums, products and quotients of complex

More information

Here are brief notes about topics covered in class on complex numbers, focusing on what is not covered in the textbook.

Here are brief notes about topics covered in class on complex numbers, focusing on what is not covered in the textbook. Phys374, Spring 2008, Prof. Ted Jacobson Department of Physics, University of Maryland Complex numbers version 5/21/08 Here are brief notes about topics covered in class on complex numbers, focusing on

More information

PURE MATHEMATICS AM 27

PURE MATHEMATICS AM 27 AM Syllabus (014): Pure Mathematics AM SYLLABUS (014) PURE MATHEMATICS AM 7 SYLLABUS 1 AM Syllabus (014): Pure Mathematics Pure Mathematics AM 7 Syllabus (Available in September) Paper I(3hrs)+Paper II(3hrs)

More information

MATH Fundamental Concepts of Algebra

MATH Fundamental Concepts of Algebra MATH 4001 Fundamental Concepts of Algebra Instructor: Darci L. Kracht Kent State University April, 015 0 Introduction We will begin our study of mathematics this semester with the familiar notion of even

More information

COMPLEX NUMBERS AND QUADRATIC EQUATIONS

COMPLEX NUMBERS AND QUADRATIC EQUATIONS Chapter 5 COMPLEX NUMBERS AND QUADRATIC EQUATIONS 5. Overview We know that the square of a real number is always non-negative e.g. (4) 6 and ( 4) 6. Therefore, square root of 6 is ± 4. What about the square

More information

Complex Analysis Homework 1: Solutions

Complex Analysis Homework 1: Solutions Complex Analysis Fall 007 Homework 1: Solutions 1.1.. a) + i)4 + i) 8 ) + 1 + )i 5 + 14i b) 8 + 6i) 64 6) + 48 + 48)i 8 + 96i c) 1 + ) 1 + i 1 + 1 i) 1 + i)1 i) 1 + i ) 5 ) i 5 4 9 ) + 4 4 15 i ) 15 4

More information

Discrete mathematics I - Complex numbers

Discrete mathematics I - Complex numbers Discrete mathematics I - Emil Vatai (based on hungarian slides by László Mérai) 1 January 31, 018 1 Financed from the financial support ELTE won from the Higher Education Restructuring

More information

PURE MATHEMATICS AM 27

PURE MATHEMATICS AM 27 AM SYLLABUS (2020) PURE MATHEMATICS AM 27 SYLLABUS 1 Pure Mathematics AM 27 (Available in September ) Syllabus Paper I(3hrs)+Paper II(3hrs) 1. AIMS To prepare students for further studies in Mathematics

More information

University of Regina. Lecture Notes. Michael Kozdron

University of Regina. Lecture Notes. Michael Kozdron University of Regina Mathematics 32 omplex Analysis I Lecture Notes Fall 22 Michael Kozdron kozdron@stat.math.uregina.ca http://stat.math.uregina.ca/ kozdron List of Lectures Lecture #: Introduction to

More information

Complex Numbers. Integers, Rationals, and Reals. The natural numbers: The integers:

Complex Numbers. Integers, Rationals, and Reals. The natural numbers: The integers: Complex Numbers Integers, Rationals, and Reals The natural numbers: N {... 3, 2,, 0,, 2, 3...} The integers: Z {... 3, 2,, 0,, 2, 3...} Note that any two integers added, subtracted, or multiplied together

More information

Algebraic. techniques1

Algebraic. techniques1 techniques Algebraic An electrician, a bank worker, a plumber and so on all have tools of their trade. Without these tools, and a good working knowledge of how to use them, it would be impossible for them

More information

Topic 4 Notes Jeremy Orloff

Topic 4 Notes Jeremy Orloff Topic 4 Notes Jeremy Orloff 4 Complex numbers and exponentials 4.1 Goals 1. Do arithmetic with complex numbers.. Define and compute: magnitude, argument and complex conjugate of a complex number. 3. Euler

More information

1 Complex numbers and the complex plane

1 Complex numbers and the complex plane L1: Complex numbers and complex-valued functions. Contents: The field of complex numbers. Real and imaginary part. Conjugation and modulus or absolute valued. Inequalities: The triangular and the Cauchy.

More information

18.03 LECTURE NOTES, SPRING 2014

18.03 LECTURE NOTES, SPRING 2014 18.03 LECTURE NOTES, SPRING 2014 BJORN POONEN 7. Complex numbers Complex numbers are expressions of the form x + yi, where x and y are real numbers, and i is a new symbol. Multiplication of complex numbers

More information

Revision Problems for Examination 1 in Algebra 1

Revision Problems for Examination 1 in Algebra 1 Centre for Mathematical Sciences Mathematics, Faculty of Science Revision Problems for Examination 1 in Algebra 1 Arithmetics 1 Determine a greatest common divisor to the integers a) 5431 and 1345, b)

More information

COMPLEX NUMBERS AND SERIES

COMPLEX NUMBERS AND SERIES COMPLEX NUMBERS AND SERIES MIKE BOYLE Contents 1. Complex Numbers 1 2. The Complex Plane 2 3. Addition and Multiplication of Complex Numbers 2 4. Why Complex Numbers Were Invented 3 5. The Fundamental

More information

Notes on Complex Numbers

Notes on Complex Numbers Notes on Complex Numbers Math 70: Ideas in Mathematics (Section 00) Imaginary Numbers University of Pennsylvania. October 7, 04. Instructor: Subhrajit Bhattacharya The set of real algebraic numbers, A,

More information

Topic 1 Part 3 [483 marks]

Topic 1 Part 3 [483 marks] Topic Part 3 [483 marks] The complex numbers z = i and z = 3 i are represented by the points A and B respectively on an Argand diagram Given that O is the origin, a Find AB, giving your answer in the form

More information

CM2202: Scientific Computing and Multimedia Applications General Maths: 3. Complex Numbers

CM2202: Scientific Computing and Multimedia Applications General Maths: 3. Complex Numbers CM2202: Scientific Computing and Multimedia Applications General Maths: 3. Complex Numbers Prof. David Marshall School of Computer Science & Informatics A problem when solving some equations There are

More information

z = a + ib (4.1) i 2 = 1. (4.2) <(z) = a, (4.3) =(z) = b. (4.4)

z = a + ib (4.1) i 2 = 1. (4.2) <(z) = a, (4.3) =(z) = b. (4.4) Chapter 4 Complex Numbers 4.1 Definition of Complex Numbers A complex number is a number of the form where a and b are real numbers and i has the property that z a + ib (4.1) i 2 1. (4.2) a is called the

More information

Complex numbers in polar form

Complex numbers in polar form remember remember Chapter Complex s 19 1. The magnitude (or modulus or absolute value) of z = x + yi is the length of the line segment from (0, 0) to z and is denoted by z, x + yi or mod z.. z = x + y

More information

Exercises for Part 1

Exercises for Part 1 MATH200 Complex Analysis. Exercises for Part Exercises for Part The following exercises are provided for you to revise complex numbers. Exercise. Write the following expressions in the form x+iy, x,y R:

More information

MAT01A1: Complex Numbers (Appendix H)

MAT01A1: Complex Numbers (Appendix H) MAT01A1: Complex Numbers (Appendix H) Dr Craig 14 February 2018 Announcements: e-quiz 1 is live. Deadline is Wed 21 Feb at 23h59. e-quiz 2 (App. A, D, E, H) opens tonight at 19h00. Deadline is Thu 22 Feb

More information

B Elements of Complex Analysis

B Elements of Complex Analysis Fourier Transform Methods in Finance By Umberto Cherubini Giovanni Della Lunga Sabrina Mulinacci Pietro Rossi Copyright 21 John Wiley & Sons Ltd B Elements of Complex Analysis B.1 COMPLEX NUMBERS The purpose

More information

With this expanded version of what we mean by a solution to an equation we can solve equations that previously had no solution.

With this expanded version of what we mean by a solution to an equation we can solve equations that previously had no solution. M 74 An introduction to Complex Numbers. 1 Solving equations Throughout the calculus sequence we have limited our discussion to real valued solutions to equations. We know the equation x 1 = 0 has distinct

More information

Exercises for Part 1

Exercises for Part 1 MATH200 Complex Analysis. Exercises for Part Exercises for Part The following exercises are provided for you to revise complex numbers. Exercise. Write the following expressions in the form x + iy, x,y

More information

Complex Numbers. Rich Schwartz. September 25, 2014

Complex Numbers. Rich Schwartz. September 25, 2014 Complex Numbers Rich Schwartz September 25, 2014 1 From Natural Numbers to Reals You can think of each successive number system as arising so as to fill some deficits associated with the previous one.

More information

Complex Numbers and Polar Coordinates

Complex Numbers and Polar Coordinates Chapter 5 Complex Numbers and Polar Coordinates One of the goals of algebra is to find solutions to polynomial equations. You have probably done this many times in the past, solving equations like x 1

More information

Introduction. The first chapter of FP1 introduces you to imaginary and complex numbers

Introduction. The first chapter of FP1 introduces you to imaginary and complex numbers Introduction The first chapter of FP1 introduces you to imaginary and complex numbers You will have seen at GCSE level that some quadratic equations cannot be solved Imaginary and complex numbers will

More information

1MA1 Introduction to the Maths Course

1MA1 Introduction to the Maths Course 1MA1/-1 1MA1 Introduction to the Maths Course Preamble Throughout your time as an engineering student at Oxford you will receive lectures and tuition in the range of applied mathematical tools that today

More information

JUST THE MATHS UNIT NUMBER 6.2. COMPLEX NUMBERS 2 (The Argand Diagram) A.J.Hobson

JUST THE MATHS UNIT NUMBER 6.2. COMPLEX NUMBERS 2 (The Argand Diagram) A.J.Hobson JUST THE MATHS UNIT NUMBER 6.2 COMPLEX NUMBERS 2 (The Argand Diagram) by A.J.Hobson 6.2.1 Introduction 6.2.2 Graphical addition and subtraction 6.2.3 Multiplication by j 6.2.4 Modulus and argument 6.2.5

More information

COMPLEX NUMBERS

COMPLEX NUMBERS COMPLEX NUMBERS 1. Any number of the form x+iy where x, y R and i -1 is called a Complex Number.. In the complex number x+iy, x is called the real part and y is called the imaginary part of the complex

More information

Day 6: 6.4 Solving Polynomial Equations Warm Up: Factor. 1. x 2-2x x 2-9x x 2 + 6x + 5

Day 6: 6.4 Solving Polynomial Equations Warm Up: Factor. 1. x 2-2x x 2-9x x 2 + 6x + 5 Day 6: 6.4 Solving Polynomial Equations Warm Up: Factor. 1. x 2-2x - 15 2. x 2-9x + 14 3. x 2 + 6x + 5 Solving Equations by Factoring Recall the factoring pattern: Difference of Squares:...... Note: There

More information

MATHS (O) NOTES. SUBJECT: Maths LEVEL: Ordinary Level TEACHER: Jean Kelly. The Institute of Education Topics Covered: Complex Numbers

MATHS (O) NOTES. SUBJECT: Maths LEVEL: Ordinary Level TEACHER: Jean Kelly. The Institute of Education Topics Covered: Complex Numbers 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

More information

ACCESS TO SCIENCE, ENGINEERING AND AGRICULTURE: MATHEMATICS 2 MATH00040 SEMESTER /2018

ACCESS TO SCIENCE, ENGINEERING AND AGRICULTURE: MATHEMATICS 2 MATH00040 SEMESTER /2018 ACCESS TO SCIENCE, ENGINEERING AND AGRICULTURE: MATHEMATICS 2 MATH00040 SEMESTER 2 2017/2018 DR. ANTHONY BROWN 2. Complex Numbers 2.1. Introduction to Complex Numbers. The first thing that it is important

More information

Secondary Honors Algebra II Objectives

Secondary Honors Algebra II Objectives Secondary Honors Algebra II Objectives Chapter 1 Equations and Inequalities Students will learn to evaluate and simplify numerical and algebraic expressions, to solve linear and absolute value equations

More information

CHAPTER 1 COMPLEX NUMBER

CHAPTER 1 COMPLEX NUMBER 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

More information

Complex Numbers. Basic algebra. Definitions. part of the complex number a+ib. ffl Addition: Notation: We i write for 1; that is we

Complex Numbers. Basic algebra. Definitions. part of the complex number a+ib. ffl Addition: Notation: We i write for 1; that is we Complex Numbers Definitions Notation We i write for 1; that is we define p to be p 1 so i 2 = 1. i Basic algebra Equality a + ib = c + id when a = c b = and d. Addition A complex number is any expression

More information

Complex Numbers. Math 3410 (University of Lethbridge) Spring / 8

Complex Numbers. Math 3410 (University of Lethbridge) Spring / 8 Complex Numbers Consider two real numbers x, y. What is 2 + x? What is x + y? What is (2 + x)(3 + y)? What is (2x + 3y)(3x + 5y)? What is the inverse of 3 + x? What one fact do I know for sure about x

More information

The Complex Numbers c ). (1.1)

The Complex Numbers c ). (1.1) The Complex Numbers In this chapter, we will study the basic properties of the field of complex numbers. We will begin with a brief historic sketch of how the study of complex numbers came to be and then

More information

SPECIALIST MATHEMATICS

SPECIALIST MATHEMATICS SPECIALIST MATHEMATICS (Year 11 and 12) UNIT A A1: Combinatorics Permutations: problems involving permutations use the multiplication principle and factorial notation permutations and restrictions with

More information

Algebra 2 Khan Academy Video Correlations By SpringBoard Activity

Algebra 2 Khan Academy Video Correlations By SpringBoard Activity SB Activity Activity 1 Creating Equations 1-1 Learning Targets: Create an equation in one variable from a real-world context. Solve an equation in one variable. 1-2 Learning Targets: Create equations in

More information

Algebra 2 Khan Academy Video Correlations By SpringBoard Activity

Algebra 2 Khan Academy Video Correlations By SpringBoard Activity SB Activity Activity 1 Creating Equations 1-1 Learning Targets: Create an equation in one variable from a real-world context. Solve an equation in one variable. 1-2 Learning Targets: Create equations in

More information

Unit 3 Specialist Maths

Unit 3 Specialist Maths Unit 3 Specialist Maths succeeding in the vce, 017 extract from the master class teaching materials Our Master Classes form a component of a highly specialised weekly program, which is designed to ensure

More information

COMPLEX ANALYSIS TOPIC V: HISTORICAL REMARKS

COMPLEX ANALYSIS TOPIC V: HISTORICAL REMARKS COMPLEX ANALYSIS TOPIC V: HISTORICAL REMARKS PAUL L. BAILEY Historical Background Reference: http://math.fullerton.edu/mathews/n2003/complexnumberorigin.html Rafael Bombelli (Italian 1526-1572) Recall

More information

This leaflet describes how complex numbers are added, subtracted, multiplied and divided.

This leaflet describes how complex numbers are added, subtracted, multiplied and divided. 7. Introduction. Complex arithmetic This leaflet describes how complex numbers are added, subtracted, multiplied and divided. 1. Addition and subtraction of complex numbers. Given two complex numbers we

More information

Lecture 1 Complex Numbers. 1 The field of complex numbers. 1.1 Arithmetic operations. 1.2 Field structure of C. MATH-GA Complex Variables

Lecture 1 Complex Numbers. 1 The field of complex numbers. 1.1 Arithmetic operations. 1.2 Field structure of C. MATH-GA Complex Variables Lecture Complex Numbers MATH-GA 245.00 Complex Variables The field of complex numbers. Arithmetic operations The field C of complex numbers is obtained by adjoining the imaginary unit i to the field R

More information

Complex Numbers. James K. Peterson. September 19, Department of Biological Sciences and Department of Mathematical Sciences Clemson University

Complex Numbers. James K. Peterson. September 19, Department of Biological Sciences and Department of Mathematical Sciences Clemson University Complex Numbers James K. Peterson Department of Biological Sciences and Department of Mathematical Sciences Clemson University September 19, 2013 Outline 1 Complex Numbers 2 Complex Number Calculations

More information

Complex Numbers. Outline. James K. Peterson. September 19, Complex Numbers. Complex Number Calculations. Complex Functions

Complex Numbers. Outline. James K. Peterson. September 19, Complex Numbers. Complex Number Calculations. Complex Functions Complex Numbers James K. Peterson Department of Biological Sciences and Department of Mathematical Sciences Clemson University September 19, 2013 Outline Complex Numbers Complex Number Calculations Complex

More information

College Trigonometry

College Trigonometry College Trigonometry George Voutsadakis 1 1 Mathematics and Computer Science Lake Superior State University LSSU Math 131 George Voutsadakis (LSSU) Trigonometry January 2015 1 / 25 Outline 1 Functions

More information

Chapter 3: Polynomial and Rational Functions

Chapter 3: Polynomial and Rational Functions Chapter 3: Polynomial and Rational Functions 3.1 Polynomial Functions A polynomial on degree n is a function of the form P(x) = a n x n + a n 1 x n 1 + + a 1 x 1 + a 0, where n is a nonnegative integer

More information

Solutions to Exercises 1.1

Solutions to Exercises 1.1 Section 1.1 Complex Numbers 1 Solutions to Exercises 1.1 1. We have So a 0 and b 1. 5. We have So a 3 and b 4. 9. We have i 0+ 1i. i +i because i i +i 1 {}}{ 4+4i + i 3+4i. 1 + i 3 7 i 1 3 3 + i 14 1 1

More information

Lesson 73 Polar Form of Complex Numbers. Relationships Among x, y, r, and. Polar Form of a Complex Number. x r cos y r sin. r x 2 y 2.

Lesson 73 Polar Form of Complex Numbers. Relationships Among x, y, r, and. Polar Form of a Complex Number. x r cos y r sin. r x 2 y 2. Lesson 7 Polar Form of Complex Numbers HL Math - Santowski Relationships Among x, y, r, and x r cos y r sin r x y tan y x, if x 0 Polar Form of a Complex Number The expression r(cos isin ) is called the

More information

Complex Numbers CK-12. Say Thanks to the Authors Click (No sign in required)

Complex Numbers CK-12. Say Thanks to the Authors Click  (No sign in required) Complex Numbers CK-12 Say Thanks to the Authors Click http://www.ck12.org/saythanks No sign in required) To access a customizable version of this book, as well as other interactive content, visit www.ck12.org

More information

Topic 1 Notes Jeremy Orloff

Topic 1 Notes Jeremy Orloff Topic 1 Notes Jeremy Orloff 1 Complex algebra and the complex plane We will start with a review of the basic algebra and geometry of complex numbers. Most likely you have encountered this previously in

More information

Lecture 5. Complex Numbers and Euler s Formula

Lecture 5. Complex Numbers and Euler s Formula Lecture 5. Complex Numbers and Euler s Formula University of British Columbia, Vancouver Yue-Xian Li March 017 1 Main purpose: To introduce some basic knowledge of complex numbers to students so that they

More information

8. Complex Numbers. sums and products. basic algebraic properties. complex conjugates. exponential form. principal arguments. roots of complex numbers

8. Complex Numbers. sums and products. basic algebraic properties. complex conjugates. exponential form. principal arguments. roots of complex numbers EE202 - EE MATH II 8. Complex Numbers Jitkomut Songsiri sums and products basic algebraic properties complex conjugates exponential form principal arguments roots of complex numbers regions in the complex

More information

EDEXCEL NATIONAL CERTIFICATE UNIT 28 FURTHER MATHEMATICS FOR TECHNICIANS OUTCOME 2- ALGEBRAIC TECHNIQUES TUTORIAL 2 - COMPLEX NUMBERS

EDEXCEL NATIONAL CERTIFICATE UNIT 28 FURTHER MATHEMATICS FOR TECHNICIANS OUTCOME 2- ALGEBRAIC TECHNIQUES TUTORIAL 2 - COMPLEX NUMBERS EDEXCEL NATIONAL CERTIFICATE UNIT 8 FURTHER MATHEMATICS FOR TECHNICIANS OUTCOME - ALGEBRAIC TECHNIQUES TUTORIAL - COMPLEX NUMBERS CONTENTS Be able to apply algebraic techniques Arithmetic progression (AP):

More information

Some commonly encountered sets and their notations

Some commonly encountered sets and their notations NATIONAL UNIVERSITY OF SINGAPORE DEPARTMENT OF MATHEMATICS (This notes are based on the book Introductory Mathematics by Ng Wee Seng ) LECTURE SETS & FUNCTIONS Some commonly encountered sets and their

More information

An angle in the Cartesian plane is in standard position if its vertex lies at the origin and its initial arm lies on the positive x-axis.

An angle in the Cartesian plane is in standard position if its vertex lies at the origin and its initial arm lies on the positive x-axis. Learning Goals 1. To understand what standard position represents. 2. To understand what a principal and related acute angle are. 3. To understand that positive angles are measured by a counter-clockwise

More information

1. Complex Numbers. John Douglas Moore. July 1, 2011

1. Complex Numbers. John Douglas Moore. July 1, 2011 1. Complex Numbers John Douglas Moore July 1, 2011 These notes are intended to supplement the text, Fundamentals of complex analysis, by Saff and Snider [5]. Other often-used references for the theory

More information

Course 2BA1: Hilary Term 2006 Section 7: Trigonometric and Exponential Functions

Course 2BA1: Hilary Term 2006 Section 7: Trigonometric and Exponential Functions Course 2BA1: Hilary Term 2006 Section 7: Trigonometric and Exponential Functions David R. Wilkins Copyright c David R. Wilkins 2005 Contents 7 Trigonometric and Exponential Functions 1 7.1 Basic Trigonometric

More information

Mathematics Extension 2 HSC Examination Topic: Polynomials

Mathematics Extension 2 HSC Examination Topic: Polynomials by Topic 995 to 006 Polynomials Page Mathematics Etension Eamination Topic: Polynomials 06 06 05 05 c Two of the zeros of P() = + 59 8 + 0 are a + ib and a + ib, where a and b are real and b > 0. Find

More information

CHAPTER 8. COMPLEX NUMBERS

CHAPTER 8. COMPLEX NUMBERS CHAPTER 8. COMPLEX NUMBERS Why do we need complex numbers? First of all, a simple algebraic equation like x = 1 may not have a real solution. Introducing complex numbers validates the so called fundamental

More information

Chapter 8B - Trigonometric Functions (the first part)

Chapter 8B - Trigonometric Functions (the first part) Fry Texas A&M University! Spring 2016! Math 150 Notes! Section 8B-I! Page 79 Chapter 8B - Trigonometric Functions (the first part) Recall from geometry that if 2 corresponding triangles have 2 angles of

More information

MT3503 Complex Analysis MRQ

MT3503 Complex Analysis MRQ MT353 Complex Analysis MRQ November 22, 26 Contents Introduction 3 Structure of the lecture course............................... 5 Prerequisites......................................... 5 Recommended

More information

Chapter 7 PHASORS ALGEBRA

Chapter 7 PHASORS ALGEBRA 164 Chapter 7 PHASORS ALGEBRA Vectors, in general, may be located anywhere in space. We have restricted ourselves thus for to vectors which are all located in one plane (co planar vectors), but they may

More information

P3.C8.COMPLEX NUMBERS

P3.C8.COMPLEX NUMBERS Recall: Within the real number system, we can solve equation of the form and b 2 4ac 0. ax 2 + bx + c =0, where a, b, c R What is R? They are real numbers on the number line e.g: 2, 4, π, 3.167, 2 3 Therefore,

More information

Notice that these numbers march out along a spiral. This continues for all powers of 1+i, even negative ones.

Notice that these numbers march out along a spiral. This continues for all powers of 1+i, even negative ones. 18.03 Class 6, Feb 16, 2010 Complex exponential [1] Complex roots [2] Complex exponential, Euler's formula [3] Euler's formula continued [1] More practice with complex numbers: Magnitudes Multiply: zw

More information

Chapter 9: Complex Numbers

Chapter 9: Complex Numbers Chapter 9: Comple Numbers 9.1 Imaginary Number 9. Comple Number - definition - argand diagram - equality of comple number 9.3 Algebraic operations on comple number - addition and subtraction - multiplication

More information

Pure Further Mathematics 2. Revision Notes

Pure Further Mathematics 2. Revision Notes Pure Further Mathematics Revision Notes October 016 FP OCT 016 SDB Further Pure 1 Inequalities... 3 Algebraic solutions... 3 Graphical solutions... 4 Series Method of Differences... 5 3 Comple Numbers...

More information