2.7 Imaginary and Complex Numbers

Total Page:16

File Type:pdf, Size:1020Kb

2.7 Imaginary and Complex Numbers www.ck12.org Chapter 2. Radical Equations and Radical Functions 2.7 Imaginary and Complex Numbers Learning Objectives • Write square roots with negative radicands in terms of i • Recognize and write complex numbers in standard form • Describe the relationship between the sets of integers, rational numbers, real numbers and complex numbers • Plot z = a + bi in the complex number plane Introduction When working with quadratic equations, some quadratic equations have solutions that are not real. For example, an equation such as: (x − 1)2 + 4 = 0 does not have real solutions. No matter which method of solving quadratics we use, the solutions to that equation are not real numbers. (x − 1)2 + 4 = 0 (x − 1)2 = −4 √ (x − 1)=± −4 √ x − 1 = ± −4 √ x = 1 ± −4 x = 1 ± 2i √ √ Notice how the −4 simplified to 2i. This is because −1 = i. The solutions 1 + 2 i and 1 - 2i combine imaginary and real numbers, and are called complex numbers. The use of the word imaginary does not mean these numbers are useless. For a long period in the history of mathematics, it was thought that the square root of a negative number was in fact only within the mathematical imagination without real-world significance hence, imaginary. That has changed. Mathematicians now consider the imaginary number as another set of numbers that have real significance, but do not fit on what is called the number lineengineers, scientists, and others solve real world problems using complex numbers! √ Where did complex numbers come from? If you solve the equation x2 + 1 = 0, you get x = ± −1. But there is no real number that, when multiplied by itself, yields -1. To fix this problem, mathematicians defined the imaginary constant, i. by definition, √ i = −1 or squaring both sides, 119 2.7. Imaginary and Complex Numbers www.ck12.org i2 = −1 √ √ √Recall√ that you√ can simplify radicals by factoring out perfect squares in the radicand. For instance, 8 = 4 · 2 = 4 2 = 2 2. The same procedure√ works with i. If you have a negative number in the radicand, you can factor out the -1 and use the identity i = −1 to simplify. √ Example: Simplify −5 Solution: √ −5 = (−1) · (5) √ √ = −1 5 √ √ = i 5or 5i This also works in combination with the other method of factoring out perfect squares. See the following example. Notice the i can go before the square root or after the square root. However, the i can not go under the square root. √ Example: Simplify −72 Solution: √ −72 = (−1) · (72) √ √ = −1 72 √ = i 72 But, we’re not done yet. 72 = 36·2, so √ √ √ i 72 = i 36 2 √ = i(6) 2 √ √ = 6i 2or6 2i Standard Form of Complex Numbers ( Sometimes when you solve a quadratic equation, the solution has both a real part and an imaginary part. For example, if you want to solve (x + 2)2 + 8 = 0 then (x + 2)2 = −8 √ x + 2 = ± −8 √ √ x + 2 = ± −1 2 · 2 · 2 120 www.ck12.org Chapter 2. Radical Equations and Radical Functions √ x + 2 = ±2i 2 √ x = −2 ± 2i 2 √ √ x = −2 + 2i 2orx = −2 − 2i 2 √ Notice that these two solutions involve a real part, -2, and an imaginary part, ±2 2i z = a + bi is the standard or rectangular form of a complex number. A complex number is a number that has a real part (in this case a) and an imaginary part, that is, the imaginary number i with a coefficient b. Set of Complex Numbers (complex, real, irrational, rational, etc.) The complex numbers are a superset of the real numbers. Given z = a + bi,ifb = 0 then z is a real number. Every real number can be written as a complex number (just let it equal abwith b=0), but there are many more complex numbers than real numbers. Hence the complex numbers are a superset of the real numbers. When you were first introduced to mathematics, you probably begab by counting positive whole numbers, that is 1, 2, 3, 4,... Later, negative whole numbers were investigated. The set of all whole numbers, both positive and negative, including the number zero, is known as integers: ... - 2, - 1, 0, 1, 2,... Later, students were introduced to fractions. The set of all numbers that CAN be expressed as a quotient of two integers (where the denominator is not zero) is called the set of Rational Numbers. Rational Numbers can also − , 3 ,− 7 , . ,− . , be expressed as a terminating or repeating decimal. Some rational numbers are 1 5 3 0 6 5 27 0. Of course there are an infinite number of rational numbers between any two whole numbers, so listing all rational numbers neatly is difficult (but it is possible–can you think of a way to do it?). Notice, that all integers are in the set of rational numbers (for example, 5 CAN be written as the quotient of 10 and = 10 2 since 5 2 ), so the integers are a subset of the rational numbers. Finally, when working with circles students encounter a number that can be approximated as a quotient of two integers but cannot be expressed EXACTLY as π π 22 that quotient, that is the number . Recall that was often expressed as APPROXIMATELY 7 or 3.14, BUT NOT EXACTLY THOSE VALUES. When first√ exploring using the√ Pythagorean Theorem to find the length of a diagonal of a square whose side is 1, the number 2 was introduced.√ 2 often was approximated as 1.4 or 1.414, but again you can’t possibly write out all of the decimals in 2. These two numbers are examples of IRRATIONAL numbers, that is numbers that cannot be expressed as a quotient of two integers, and therefore CANNOT be expressed as a terminating or repeating decimal. The set of all rational and irrational numbers together is called REAL numbers. The set of all real numbers AND imaginary numbers is called the set of Complex numbers. 121 2.7. Imaginary and Complex Numbers www.ck12.org Complex Number Plane In standard form z = a + bi, a complex number can be graphed using rectangular coordinates (a, b). a represents the x - coordinate, while b represents the y - coordinate. Alternatively, the x - coordinate represents “real number” values, while the y - coordinate represents the “imaginary” values. For example, given the complex number in standard form: z =2+2i, you can graph this number in the coordinate plane To graph this point, the coordinate (2, 2) is graphed as shown below: Lesson Summary When graphing a complex number using rectangular coordinates, the x-axis plots the real number, while the y-axis plots the coefficient of the imaginary number. Review Questions Using rectangular coordinate system, graph 1. Simplify the following radicals 122 www.ck12.org Chapter 2. Radical Equations and Radical Functions √ a. √−9 b. √−12 c. √−17 d. 140 − 108 2. Solve each equation and express it as a complex number. (Note: If the imaginary part is 0, you can still express the solution as a + bi, but you will have b=0 a. x2 +24=0 b. 2x2 -4x +7=0 3. Plot each of the following complex numbers: a. (4 + 2i) b. (-3 + i) c. (3 - 4i) d. 3i Review Answers 1. a. 3i √ b. 2√i 3 c. i√ 17 √ d. 32 =√4 2 2. a. x = ±2 √6i √ = ± 10i 2± 10i b. x 1 2 or 2 3. 123.
Recommended publications
  • How to Show That Various Numbers Either Can Or Cannot Be Constructed Using Only a Straightedge and Compass
    How to show that various numbers either can or cannot be constructed using only a straightedge and compass Nick Janetos June 3, 2010 1 Introduction It has been found that a circular area is to the square on a line equal to the quadrant of the circumference, as the area of an equilateral rectangle is to the square on one side... -Indiana House Bill No. 246, 1897 Three problems of classical Greek geometry are to do the following using only a compass and a straightedge: 1. To "square the circle": Given a circle, to construct a square of the same area, 2. To "trisect an angle": Given an angle, to construct another angle 1/3 of the original angle, 3. To "double the cube": Given a cube, to construct a cube with twice the area. Unfortunately, it is not possible to complete any of these tasks until additional tools (such as a marked ruler) are provided. In section 2 we will examine the process of constructing numbers using a compass and straightedge. We will then express constructions in algebraic terms. In section 3 we will derive several results about transcendental numbers. There are two goals: One, to show that the numbers e and π are transcendental, and two, to show that the three classical geometry problems are unsolvable. The two goals, of course, will turn out to be related. 2 Constructions in the plane The discussion in this section comes from [8], with some parts expanded and others removed. The classical Greeks were clear on what constitutes a construction. Given some set of points, new points can be defined at the intersection of lines with other lines, or lines with circles, or circles with circles.
    [Show full text]
  • Operations with Complex Numbers Adding & Subtracting: Combine Like Terms (풂 + 풃풊) + (풄 + 풅풊) = (풂 + 풄) + (풃 + 풅)풊 Examples: 1
    Name: __________________________________________________________ Date: _________________________ Period: _________ Chapter 2: Polynomial and Rational Functions Topic 1: Complex Numbers What is an imaginary number? What is a complex number? The imaginary unit is defined as 풊 = √−ퟏ A complex number is defined as the set of all numbers in the form of 푎 + 푏푖, where 푎 is the real component and 푏 is the coefficient of the imaginary component. An imaginary number is when the real component (푎) is zero. Checkpoint: Since 풊 = √−ퟏ Then 풊ퟐ = Operations with Complex Numbers Adding & Subtracting: Combine like terms (풂 + 풃풊) + (풄 + 풅풊) = (풂 + 풄) + (풃 + 풅)풊 Examples: 1. (5 − 11푖) + (7 + 4푖) 2. (−5 + 7푖) − (−11 − 6푖) 3. (5 − 2푖) + (3 + 3푖) 4. (2 + 6푖) − (12 − 4푖) Multiplying: Just like polynomials, use the distributive property. Then, combine like terms and simplify powers of 푖. Remember! Multiplication does not require like terms. Every term gets distributed to every term. Examples: 1. 4푖(3 − 5푖) 2. (7 − 3푖)(−2 − 5푖) 3. 7푖(2 − 9푖) 4. (5 + 4푖)(6 − 7푖) 5. (3 + 5푖)(3 − 5푖) A note about conjugates: Recall that when multiplying conjugates, the middle terms will cancel out. With complex numbers, this becomes even simpler: (풂 + 풃풊)(풂 − 풃풊) = 풂ퟐ + 풃ퟐ Try again with the shortcut: (3 + 5푖)(3 − 5푖) Dividing: Just like polynomials and rational expressions, the denominator must be a rational number. Since complex numbers include imaginary components, these are not rational numbers. To remove a complex number from the denominator, we multiply numerator and denominator by the conjugate of the Remember! You can simplify first IF factors can be canceled.
    [Show full text]
  • Theory of Trigintaduonion Emanation and Origins of Α and Π
    Theory of Trigintaduonion Emanation and Origins of and Stephen Winters-Hilt Meta Logos Systems Aurora, Colorado Abstract In this paper a specific form of maximal information propagation is explored, from which the origins of , , and fractal reality are revealed. Introduction The new unification approach described here gives a precise derivation for the mysterious physics constant (the fine-structure constant) from the mathematical physics formalism providing maximal information propagation, with being the maximal perturbation amount. Furthermore, the new unification provides that the structure of the space of initial ‘propagation’ (with initial propagation being referred to as ‘emanation’) has a precise derivation, with a unit-norm perturbative limit that leads to an iterative-map-like computed (a limit that is precisely related to the Feigenbaum bifurcation constant and thus fractal). The computed can also, by a maximal information propagation argument, provide a derivation for the mathematical constant . The ideal constructs of planar geometry, and related such via complex analysis, give methods for calculation of to incredibly high precision (trillions of digits), thereby providing an indirect derivation of to similar precision. Propagation in 10 dimensions (chiral, fermionic) and 26 dimensions (bosonic) is indicated [1-3], in agreement with string theory. Furthermore a preliminary result showing a relation between the Feigenbaum bifurcation constant and , consistent with the hypercomplex algebras indicated in the Emanator Theory, suggest an individual object trajectory with 36=10+26 degrees of freedom (indicative of heterotic strings). The overall (any/all chirality) propagation degrees of freedom, 78, are also in agreement with the number of generators in string gauge symmetries [4].
    [Show full text]
  • Truly Hypercomplex Numbers
    TRULY HYPERCOMPLEX NUMBERS: UNIFICATION OF NUMBERS AND VECTORS Redouane BOUHENNACHE (pronounce Redwan Boohennash) Independent Exploration Geophysical Engineer / Geophysicist 14, rue du 1er Novembre, Beni-Guecha Centre, 43019 Wilaya de Mila, Algeria E-mail: [email protected] First written: 21 July 2014 Revised: 17 May 2015 Abstract Since the beginning of the quest of hypercomplex numbers in the late eighteenth century, many hypercomplex number systems have been proposed but none of them succeeded in extending the concept of complex numbers to higher dimensions. This paper provides a definitive solution to this problem by defining the truly hypercomplex numbers of dimension N ≥ 3. The secret lies in the definition of the multiplicative law and its properties. This law is based on spherical and hyperspherical coordinates. These numbers which I call spherical and hyperspherical hypercomplex numbers define Abelian groups over addition and multiplication. Nevertheless, the multiplicative law generally does not distribute over addition, thus the set of these numbers equipped with addition and multiplication does not form a mathematical field. However, such numbers are expected to have a tremendous utility in mathematics and in science in general. Keywords Hypercomplex numbers; Spherical; Hyperspherical; Unification of numbers and vectors Note This paper (or say preprint or e-print) has been submitted, under the title “Spherical and Hyperspherical Hypercomplex Numbers: Merging Numbers and Vectors into Just One Mathematical Entity”, to the following journals: Bulletin of Mathematical Sciences on 08 August 2014, Hypercomplex Numbers in Geometry and Physics (HNGP) on 13 August 2014 and has been accepted for publication on 29 April 2015 in issue No.
    [Show full text]
  • Chapter I, the Real and Complex Number Systems
    CHAPTER I THE REAL AND COMPLEX NUMBERS DEFINITION OF THE NUMBERS 1, i; AND p2 In order to make precise sense out of the concepts we study in mathematical analysis, we must first come to terms with what the \real numbers" are. Everything in mathematical analysis is based on these numbers, and their very definition and existence is quite deep. We will, in fact, not attempt to demonstrate (prove) the existence of the real numbers in the body of this text, but will content ourselves with a careful delineation of their properties, referring the interested reader to an appendix for the existence and uniqueness proofs. Although people may always have had an intuitive idea of what these real num- bers were, it was not until the nineteenth century that mathematically precise definitions were given. The history of how mathematicians came to realize the necessity for such precision in their definitions is fascinating from a philosophical point of view as much as from a mathematical one. However, we will not pursue the philosophical aspects of the subject in this book, but will be content to con- centrate our attention just on the mathematical facts. These precise definitions are quite complicated, but the powerful possibilities within mathematical analysis rely heavily on this precision, so we must pursue them. Toward our primary goals, we will in this chapter give definitions of the symbols (numbers) 1; i; and p2: − The main points of this chapter are the following: (1) The notions of least upper bound (supremum) and greatest lower bound (infimum) of a set of numbers, (2) The definition of the real numbers R; (3) the formula for the sum of a geometric progression (Theorem 1.9), (4) the Binomial Theorem (Theorem 1.10), and (5) the triangle inequality for complex numbers (Theorem 1.15).
    [Show full text]
  • Doing Physics with Quaternions
    Doing Physics with Quaternions Douglas B. Sweetser ©2005 doug <[email protected]> All righs reserved. 1 INDEX Introduction 2 What are Quaternions? 3 Unifying Two Views of Events 4 A Brief History of Quaternions Mathematics 6 Multiplying Quaternions the Easy Way 7 Scalars, Vectors, Tensors and All That 11 Inner and Outer Products of Quaternions 13 Quaternion Analysis 23 Topological Properties of Quaternions 28 Quaternion Algebra Tool Set Classical Mechanics 32 Newton’s Second Law 35 Oscillators and Waves 37 Four Tests of for a Conservative Force Special Relativity 40 Rotations and Dilations Create the Lorentz Group 43 An Alternative Algebra for Lorentz Boosts Electromagnetism 48 Classical Electrodynamics 51 Electromagnetic Field Gauges 53 The Maxwell Equations in the Light Gauge: QED? 56 The Lorentz Force 58 The Stress Tensor of the Electromagnetic Field Quantum Mechanics 62 A Complete Inner Product Space with Dirac’s Bracket Notation 67 Multiplying quaternions in Polar Coordinate Form 69 Commutators and the Uncertainty Principle 74 Unifying the Representations of Integral and Half−Integral Spin 79 Deriving A Quaternion Analog to the Schrödinger Equation 83 Introduction to Relativistic Quantum Mechanics 86 Time Reversal Transformations for Intervals Gravity 89 Unified Field Theory by Analogy 101 Einstein’s vision I: Classical unified field equations for gravity and electromagnetism using Riemannian quaternions 115 Einstein’s vision II: A unified force equation with constant velocity profile solutions 123 Strings and Quantum Gravity 127 Answering Prima Facie Questions in Quantum Gravity Using Quaternions 134 Length in Curved Spacetime 136 A New Idea for Metrics 138 The Gravitational Redshift 140 A Brief Summary of Important Laws in Physics Written as Quaternions 155 Conclusions 2 What Are Quaternions? Quaternions are numbers like the real numbers: they can be added, subtracted, multiplied, and divided.
    [Show full text]
  • 50 Mathematical Ideas You Really Need to Know
    50 mathematical ideas you really need to know Tony Crilly 2 Contents Introduction 01 Zero 02 Number systems 03 Fractions 04 Squares and square roots 05 π 06 e 07 Infinity 08 Imaginary numbers 09 Primes 10 Perfect numbers 11 Fibonacci numbers 12 Golden rectangles 13 Pascal’s triangle 14 Algebra 15 Euclid’s algorithm 16 Logic 17 Proof 3 18 Sets 19 Calculus 20 Constructions 21 Triangles 22 Curves 23 Topology 24 Dimension 25 Fractals 26 Chaos 27 The parallel postulate 28 Discrete geometry 29 Graphs 30 The four-colour problem 31 Probability 32 Bayes’s theory 33 The birthday problem 34 Distributions 35 The normal curve 36 Connecting data 37 Genetics 38 Groups 4 39 Matrices 40 Codes 41 Advanced counting 42 Magic squares 43 Latin squares 44 Money mathematics 45 The diet problem 46 The travelling salesperson 47 Game theory 48 Relativity 49 Fermat’s last theorem 50 The Riemann hypothesis Glossary Index 5 Introduction Mathematics is a vast subject and no one can possibly know it all. What one can do is explore and find an individual pathway. The possibilities open to us here will lead to other times and different cultures and to ideas that have intrigued mathematicians for centuries. Mathematics is both ancient and modern and is built up from widespread cultural and political influences. From India and Arabia we derive our modern numbering system but it is one tempered with historical barnacles. The ‘base 60’ of the Babylonians of two or three millennia BC shows up in our own culture – we have 60 seconds in a minute and 60 minutes in an hour; a right angle is still 90 degrees and not 100 grads as revolutionary France adopted in a first move towards decimalization.
    [Show full text]
  • Complex Numbers -.: Mathematical Sciences : UTEP
    2.4 Complex Numbers For centuries mathematics has been an ever-expanding field because of one particular “trick.” Whenever a notable mathematician gets stuck on a problem that seems to have no solution, they make up something new. This is how complex numbers were “invented.” A simple quadratic equation would be x2 10. However, in trying to solve this it was found that x2 1 and that was confusing. How could a quantity multiplied by itself equal a negative number? This is where the genius came in. A guy named Cardano developed complex numbers off the base of the imaginary number i 1 , the solution to our “easy” equation x2 1 . The system didn’t really get rolling until Euler and Gauss started using it, but if you want to blame someone it should be Cardano. My Definition – The imaginary unit i is a number such that i2 1 . That is, i 1 . Definition of a Complex Number – If a and b are real numbers, the number a + bi is a complex number, and it is said to be written in standard form. If b = 0, the number a + bi = a is a real number. If b≠ 0, the number a + bi is called an imaginary number. A number of the form bi, where b ≠ 0, is called a pure imaginary number. Equality of Complex Numbers – Two complex numbers a + bi and c + di, written in standard form, are equal to each other a bi c di if and only if a = c and b = d. Addition and Subtraction of Complex Numbers – If a + bi and c + di are two complex numbers written in standard form, their sum and difference are defined as follows.
    [Show full text]
  • Quaternions by Wasinee Siewrichol
    Quaternions Wasinee Siewsrichol Historical Context Complex numbers was a very popular subject in the early eighteenth hundreds. People knew how to multiply two numbers, but they did not know how to multiply three numbers. This question puzzled William Rowan Hamilton for a very long time. Hamilton wrote to his son: "Every morning in the early part of the above-cited month [Oct. 1843] on my coming down to breakfast, your brother William Edwin and yourself used to ask me, 'Well, Papa, can you multiply triplets?' Whereto I was always obliged to reply, with a sad shake of the head, 'No, I can only add and subtract them.'" Hamilton eventually found the solution, but in the fourth dimension. The concept of quaternions was first invented by the Irish mathematician William Rowan Hamilton on Monday, October 16th 1843 in Dublin, Ireland. Hamilton was on his way to the Royal Irish Academy with his wife and as he was passing over the Royal Canal on the Brougham Bridge. He made a dramatic realization that he immediately carved into the stone on the bridge. It is hard to tell, but on the bottom it writes i2 = j2 = k2 = ijk = −1 The first thing that Hamilton did was set the fourth dimension equal to zero. Then he spent the rest of his life trying to find a practical use for quaternions, but he did not find anything. By the early nineteenth century, Professor Josiah Willard Gibbs of Yale came up with the vector dot products and the cross product. Vectors are huge in physics for finding the velocity, acceleration, and force of an object.
    [Show full text]
  • A Brief History of Mathematics for Dynamic Systems
    A brief history of mathematics for dynamic systems Advances in mathematics and technology are usually made at a painstakingly slow pace, with small sparks of individual brilliance that are accompanied by good luck or divine inspiration. Over the past millennia, various cultures have produced groups of gifted individuals that have had a significant impact on modern mathematics, engineering, and technology, including: Egypt (3000 BC), Babylon (2000 BC), Greece and China (500 BC), India (500 AD), Africa (800 AD), Europe and Asia (1500 AD), United States and Soviet Union (1900 AD), and Worldwide Web (2000+ AD). 2.1.1 Real number systems • Egyptians (3000 BC): Egyptians used hieroglyphics (picture writing) for numerals. The system was based on 10, but did not include a zero or the principle of place value. The (approximate) hieroglyphic symbols shown below combine to depict the number 1,326 as | @@@ ^^ //////. 1 / Stroke 10 ^ Arch 100 @ Coiled Rope 1,000 | Lotus Flower 10,000 > Finger 100,000 ~ Tadpole • Babylonians (2000 BC): Developed a number system based on 60. Their number system was more consistent and structured than the Egyptian system and was simpler for mathematical calculations. The number of seconds in a minute (60) and number of minutes in an hour (60) are a consequence of the Babylonian number system. • Hittites in Mesopotamia (2000-700 BC): Developed the precursor to the Roman numeral system e.g., their first four numbers were I, II, III,andIIII (note how I looks like a finger or a stick). • Romans (500 BC): The early Roman system was based on the Hittite system, e.g., its first four numbers were I, II, III,andIIII.
    [Show full text]
  • Structurally Hyperbolic Algebras Dual to the Cayley-Dickson and Clifford
    Unspecified Journal Volume 00, Number 0, Pages 000–000 S ????-????(XX)0000-0 STRUCTURALLY-HYPERBOLIC ALGEBRAS DUAL TO THE CAYLEY-DICKSON AND CLIFFORD ALGEBRAS OR NESTED SNAKES BITE THEIR TAILS DIANE G. DEMERS For Elaine Yaw in honor of friendship Abstract. The imaginary unit i of C, the complex numbers, squares to −1; while the imaginary unit j of D, the double numbers (also called dual or split complex numbers), squares to +1. L.H. Kauffman expresses the double num- ber product in terms of the complex number product and vice-versa with two, formally identical, dualizing formulas. The usual sequence of (structurally- elliptic) Cayley-Dickson algebras is R, C, H,..., of which Hamilton’s quater- nions H generalize to the split quaternions H. Kauffman’s expressions are the key to recursively defining the dual sequence of structurally-hyperbolic Cayley- Dickson algebras, R, D, M,..., of which Macfarlane’s hyperbolic quaternions M generalize to the split hyperbolic quaternions M. Previously, the structurally- hyperbolic Cayley-Dickson algebras were defined by simply inverting the signs of the squares of the imaginary units of the structurally-elliptic Cayley-Dickson algebras from −1 to +1. Using the dual algebras C, D, H, H, M, M, and their further generalizations, we classify the Clifford algebras and their dual orienta- tion congruent algebras (Clifford-like, noncommutative Jordan algebras with physical applications) by their representations as tensor products of algebras. Received by the editors July 15, 2008. 2000 Mathematics Subject Classification. Primary 17D99; Secondary 06D30, 15A66, 15A78, 15A99, 17A15, 17A120, 20N05. For some relief from my duties at the East Lansing Food Coop, I thank my coworkers Lind- say Demaray, Liz Kersjes, and Connie Perkins, nee Summers.
    [Show full text]
  • Abstract Algebra Lecture Notes
    Abstract Algebra Lecture Notes David M. McClendon Department of Mathematics Ferris State University Fall 2018 edition 1 Contents Contents 2 0 Review of Math 324 4 0.1 Sets . 4 0.2 Relations . 12 0.3 Functions . 14 0.4 Cardinality . 19 0.5 Summary of proof techniques . 21 0.6 Outlines for common types of proofs . 23 1 Major problems of abstract algebra 33 1.1 Straightedge and compass constructions . 33 1.2 Polynomial equations . 44 1.3 Summary: the major questions we want to tackle . 56 2 Integers and rational numbers 57 2.1 N: the natural numbers . 57 2.2 Z: the integers . 64 2.3 Q: the rational numbers . 69 2.4 Divisibility . 73 2.5 The Euclidean algorithm and applications . 79 2.6 Congruence classes modulo n ....................... 86 3 Real and complex numbers 98 3.1 Theorems of Hippasus and Theatitus . 98 3.2 R: the real numbers . 101 3.3 C: complex numbers . 107 3.4 Fundamental Theorem of Algebra . 115 3.5 Complex roots, cubic equations and regular polygons . 117 2 Contents 4 Polynomial rings 126 4.1 Definition and basic properties . 126 4.2 Irreducibility tests . 135 5 Fields 141 5.1 Field extensions . 142 5.2 Algebraic extensions . 143 5.3 Linear algebra and field extensions . 146 5.4 Classical construction problems, revisited . 152 6 Morphisms 156 6.1 What is a homomorphism? . 156 6.2 Isomorphisms and invariants . 160 6.3 Ring homomorphisms . 165 6.4 Automorphisms . 173 7 Groups 174 7.1 An update on the big picture . 174 7.2 What is a group? .
    [Show full text]