To find the reciprocal of a complex number z = r cis θ in polar form, the modulus of the reciprocal is 1/r and the argument is -θ. The reciprocal is equal to 1/r cis -θ, where r is the modulus and θ is the argument of the original complex number z.
Abstract: In this paper we prove some extension of the Eneström-Kakeya theorem by relaxing the hypothesis of this result in several ways and obtain zero-free regions for polynomials with restricted coefficients and there by present some interesting generalizations and extensions of the Enestrom-Kakeya Theorem.
The document discusses using different numerical methods to find the highest root of the function f(x) = 2x^3 - 11.7x^2 + 17.7x - 5. It provides the following key details:
1) The roots are graphically determined to be 0.365, 1.922, and 3.563.
2) Using a fixed-point iteration method with x0 = 3 converges to 2.322 after 3 iterations, which is not the desired root.
3) The Newton-Raphson method converges to 3.56324 after 3 iterations, providing a cleaner convergence to the desired root.
4) Using a secant method with x0 =
The document discusses applying the distributive property to simplify algebraic expressions. It provides examples of distributing positive and negative coefficients over terms with variables and constants. Key parts of expressions like terms, coefficients, and constant terms are defined. Examples show identifying these parts and simplifying expressions using the distributive property and combining like terms. Practice problems at the end ask the reader to simplify expressions.
This document discusses solving a system of two equations with two unknowns using substitution. It shows the steps of substituting the value of x from one equation into the other equation to get an equation with only one unknown. This reduces the system to a single variable equation that can be solved for the value of x, which is then substituted back into one of the original equations to find the value of y. The solution to the system of equations in this example is x=6 and y=47.
Concepts of Fuzzy morphism from George Klir book Fuzzy Sets and Fuzzy Logic Theory and Application are presented here which incudes fuzzy homomorphism, strong homomorphism, Isomorphism, endomorphism, automorphism.
This document provides an overview of key concepts in natural language processing and probability models, including:
- Unigram and bigram language models that calculate word probabilities based on frequency counts
- Conditional probability and how it is used in ngram models
- Maximum likelihood estimation and how it estimates joint word probabilities
- Perplexity as a metric to evaluate language models
- Smoothing techniques like Laplace and discount smoothing to address data sparsity issues
- Backoff and interpolation methods that combine multiple probability estimates
The document explains these concepts through examples and formulas. It also discusses how ngram models are used for tasks like word segmentation.
This document discusses techniques for evaluating definite integrals using u-substitution, including:
1) Changing the limits of integration to match the u-substitution, avoiding the need for back substitution.
2) Choosing substitutions that give the needed "du" term inside the integral.
3) Being aware that the limits of integration may end up out of order after substitution.
4) Recognizing odd functions where the area above and below the x-axis cancel out upon integration.
Abstract: In this paper we prove some extension of the Eneström-Kakeya theorem by relaxing the hypothesis of this result in several ways and obtain zero-free regions for polynomials with restricted coefficients and there by present some interesting generalizations and extensions of the Enestrom-Kakeya Theorem.
The document discusses using different numerical methods to find the highest root of the function f(x) = 2x^3 - 11.7x^2 + 17.7x - 5. It provides the following key details:
1) The roots are graphically determined to be 0.365, 1.922, and 3.563.
2) Using a fixed-point iteration method with x0 = 3 converges to 2.322 after 3 iterations, which is not the desired root.
3) The Newton-Raphson method converges to 3.56324 after 3 iterations, providing a cleaner convergence to the desired root.
4) Using a secant method with x0 =
The document discusses applying the distributive property to simplify algebraic expressions. It provides examples of distributing positive and negative coefficients over terms with variables and constants. Key parts of expressions like terms, coefficients, and constant terms are defined. Examples show identifying these parts and simplifying expressions using the distributive property and combining like terms. Practice problems at the end ask the reader to simplify expressions.
This document discusses solving a system of two equations with two unknowns using substitution. It shows the steps of substituting the value of x from one equation into the other equation to get an equation with only one unknown. This reduces the system to a single variable equation that can be solved for the value of x, which is then substituted back into one of the original equations to find the value of y. The solution to the system of equations in this example is x=6 and y=47.
Concepts of Fuzzy morphism from George Klir book Fuzzy Sets and Fuzzy Logic Theory and Application are presented here which incudes fuzzy homomorphism, strong homomorphism, Isomorphism, endomorphism, automorphism.
This document provides an overview of key concepts in natural language processing and probability models, including:
- Unigram and bigram language models that calculate word probabilities based on frequency counts
- Conditional probability and how it is used in ngram models
- Maximum likelihood estimation and how it estimates joint word probabilities
- Perplexity as a metric to evaluate language models
- Smoothing techniques like Laplace and discount smoothing to address data sparsity issues
- Backoff and interpolation methods that combine multiple probability estimates
The document explains these concepts through examples and formulas. It also discusses how ngram models are used for tasks like word segmentation.
This document discusses techniques for evaluating definite integrals using u-substitution, including:
1) Changing the limits of integration to match the u-substitution, avoiding the need for back substitution.
2) Choosing substitutions that give the needed "du" term inside the integral.
3) Being aware that the limits of integration may end up out of order after substitution.
4) Recognizing odd functions where the area above and below the x-axis cancel out upon integration.
Lesson 3bi proving euler’s equation without using power seriesjenniech
This document outlines a calculus proof of Euler's equation without using power series. It starts with the function f(θ) = e^(iθ)(cosθ + i sinθ) and takes the derivative of f'(θ) using the product rule. The derivative implies that f(θ) = k, where substituting θ = 0 gives k = 1. Multiplying both sides by e^(iθ) yields Euler's famous formula e^(iθ) = cosθ + i sinθ.
The document introduces complex numbers and different ways to represent them, including:
1) Imaginary numbers, represented by i, which allows for solutions to equations with "hidden roots". Complex numbers have both a real and imaginary part.
2) Polar form represents complex numbers using modulus (distance from origin) and argument (angle from positive x-axis).
3) Exponential or Euler's form uses modulus and an imaginary exponent to represent complex numbers, where the angle must be in radians.
4) Operations like addition, subtraction, multiplication and division are introduced for complex numbers, along with converting between rectangular, polar and exponential forms.
This document contains three sections about solving complex equations and deriving trigonometric identities using complex numbers. The first section explains how to find the cube root of 2+2i using polar form and de Moivre's Theorem. The second section prompts solving the cube root of 8 + 8i. The third section instructs deriving a formula for cos 4θ in terms of cos θ using binomial expansion and de Moivre's Theorem then equating real and imaginary parts.
This document discusses complex roots of unity and finding their values. It begins by explaining that roots of unity are complex numbers raised to a power that gives 1. It then provides an example of finding the cube roots of unity. There are 3 cube roots of unity, which are 1, cis(2π/3) and cis(4π/3). In general, the equation zn=1 has n solutions, with the first being 1 and the others being cis(2πk/n) for k=1,2,...n-1.
Lesson 6 complex roots of a polynomial equationjenniech
The document discusses complex roots of polynomial equations. It notes that any polynomial with complex roots will have roots that come in conjugate pairs. It asks questions about the relationship between the coefficients of a polynomial and the sum and product of its roots. Specifically, it asks the reader to find the values of a and b for a quartic polynomial given that two of its roots are 3i and 1-2i.
Lesson 4 operations with complex numbers in polar form p1 2jenniech
The document discusses operations with complex numbers in polar form. It provides the polar form representations of two complex numbers z1 and z2 as r1(cosθ1 + i sin θ1) and r2(cosθ2 + i sin θ2). It asks the reader to find the product and quotient of z1 and z2 using the polar form, and notes that the product can also be expressed as r1 cisθ1 × r2 cisθ2. It provides hints about using trigonometric addition formulas to simplify the expressions and concludes with two practice problems asking the reader to find the product and quotient of complex numbers given in polar form.
e^iπ = cos(π) + i sin(π) = -1, which is a special case of Euler's equation. When the angle θ is set to π in Euler's formula, it yields the unexpected but important result of e^iπ + 1 = 0, which connects the five most important mathematical constants: 0, 1, e, i, and π.
Lesson 7 de moivre theorem proof by inductionjenniech
This document outlines the proof by induction of DeMoivre's Theorem. It presents the three steps of an inductive proof: 1) Show the statement is true for the initial value, here n=1; 2) Assume the statement is true for an arbitrary value k; 3) Show that if true for k then it must also be true for k+1, completing the induction proof that the statement is true for all positive integers n.
Lesson 5 interesting properties of complex conjugates p1 2jenniech
This document discusses properties of complex conjugates. It lists six properties to prove about complex conjugates: 1) z+z* and zz* are always real. 2) If z=rcisθ then z*=r cis(-θ). 3) (z+w)*=z*+w* . 4) (zw)*=z*w* . 5) (z/w)*=(z*)/(w*) . 6) (zn)*=(z*)n. It also provides two practice problems: (i) find the complex conjugate of z=(1+iw)2 and (ii) find a complex number z such that z=3z+2z*=5+2i.
This document provides instructions for solving equations using the form z = x + iy. It gives the example equations (i) z^2 = -4i and (ii) z^2 = 9i to solve using the hint that if two complex numbers are equal, their real and imaginary parts must be equal. It also lists the example equations (i) z^2 = 2 + 2√3i and (ii) z^2 = 5 + i.
Lesson 3 argument polar form of a complex numberjenniech
This document discusses representing complex numbers in polar form using modulus (r) and argument (θ) instead of Cartesian coordinates (x, y). It shows how to convert between polar and Cartesian forms, defines the notation cis(θ) for representing complex numbers in polar form, and lists some properties of cis(θ) including rules for addition, multiplication, and equivalence under modulo 2π.
Lesson 1 imaginary and complex numbers p1 8jenniech
1. Imaginary and complex numbers can be used to solve quadratic equations that have no real solutions. The square root of -1 is defined as i, and complex numbers have both a real and imaginary part.
2. An Argand diagram represents complex numbers graphically by plotting the real part on the x-axis and the imaginary part on a perpendicular y-axis. Common complex number operations like addition, subtraction, multiplication, and division can be performed by treating complex numbers as vectors.
3. The modulus of a complex number z = a + bi is the distance from the point (a, b) to the origin on the Argand diagram, and represents the absolute value of z. The argument of z,
Lesson 1b two interesting questions for youjenniech
This document poses two interesting questions and asks what is wrong with two proofs presented. The first proof is not shown, while the second proof claims to prove something but does not actually include any logical steps, reasoning, or evidence to support its claim.
This document provides an overview of key concepts in probability for the IB Standard Level course, including Venn diagrams, conditional probability, and probability tree diagrams. It includes links to examples and practice problems involving each of these topics. Students are intended to understand how to use strategies like Venn and tree diagrams to solve real-life probability problems.
Your rubric multimedia project how tall is the sports block_jenniech
This rubric is for evaluating a multimedia project on how to calculate the height of a sports block using trigonometry. The rubric assesses students on content, oral presentation, organization, sources, and presentation. For content, it evaluates the depth of subject knowledge and inclusion of essential trigonometry information. For oral presentation, it considers whether the delivery is interesting, rehearsed and holds audience attention. The organization category assesses whether the content is logically structured. Sources examines whether source information is collected and documented properly. Presentation assesses rehearsal, delivery and maintaining audience interest.
Finding the magnitude of a vector in 2 d and 3djenniech
To find the magnitude (length) of a vector in 2D or 3D, one uses the formula that takes the square root of the sum of the squares of the vector's components. For a vector a with components a1, a2, and a3, the magnitude is calculated as the square root of a1^2 + a2^2 + a3^2. The magnitude of a vector is denoted by |a|.
This document contains a student worksheet for investigating vectors in 2D and 3D using Autograph software. It includes tasks on expressing vectors in terms of other vectors, finding vector equations of lines given conditions, calculating lengths and angles of vectors, and using vectors to solve geometric problems in 2D and 3D. The worksheet guides students through using Autograph's vector tools to explore and check their work.
The document discusses scalar products of vectors and different types of vector relationships. It asks to define the scalar product using notation, calculate products of perpendicular and parallel vectors, and explain the difference between parallel and coincident vectors with a diagram. Perpendicular vectors have a scalar product of 0, parallel vectors maintain the same direction but their scalar product depends on length and angle, and coincident vectors overlap completely and have the greatest possible scalar product.
Converting vector equation of a line to cartesian formjenniech
The document describes how to convert a vector equation of a line to Cartesian form. A vector equation defines a line using a point (a,b,c) on the line and a direction vector (k,m,n). It explains how to write the x, y, z coordinates in terms of λ and rearrange to make λ the subject, then equate the expressions to find the Cartesian equation of the line. The Cartesian equation defines the line using the standard x, y, z variables.
The scalar product form for the equation of a planejenniech
The document discusses deriving the scalar product form of the equation of a plane. It defines two points A and P on the plane and their position vectors a and r. It explains that the vector from A to P, AP, is equal to r - a. Using the scalar product of the position vector r and the normal vector n to the plane, along with the position vector a, allows obtaining the scalar product equation of the plane, r·n = a·n. This equation can then be written in Cartesian form.
Lesson 3bi proving euler’s equation without using power seriesjenniech
This document outlines a calculus proof of Euler's equation without using power series. It starts with the function f(θ) = e^(iθ)(cosθ + i sinθ) and takes the derivative of f'(θ) using the product rule. The derivative implies that f(θ) = k, where substituting θ = 0 gives k = 1. Multiplying both sides by e^(iθ) yields Euler's famous formula e^(iθ) = cosθ + i sinθ.
The document introduces complex numbers and different ways to represent them, including:
1) Imaginary numbers, represented by i, which allows for solutions to equations with "hidden roots". Complex numbers have both a real and imaginary part.
2) Polar form represents complex numbers using modulus (distance from origin) and argument (angle from positive x-axis).
3) Exponential or Euler's form uses modulus and an imaginary exponent to represent complex numbers, where the angle must be in radians.
4) Operations like addition, subtraction, multiplication and division are introduced for complex numbers, along with converting between rectangular, polar and exponential forms.
This document contains three sections about solving complex equations and deriving trigonometric identities using complex numbers. The first section explains how to find the cube root of 2+2i using polar form and de Moivre's Theorem. The second section prompts solving the cube root of 8 + 8i. The third section instructs deriving a formula for cos 4θ in terms of cos θ using binomial expansion and de Moivre's Theorem then equating real and imaginary parts.
This document discusses complex roots of unity and finding their values. It begins by explaining that roots of unity are complex numbers raised to a power that gives 1. It then provides an example of finding the cube roots of unity. There are 3 cube roots of unity, which are 1, cis(2π/3) and cis(4π/3). In general, the equation zn=1 has n solutions, with the first being 1 and the others being cis(2πk/n) for k=1,2,...n-1.
Lesson 6 complex roots of a polynomial equationjenniech
The document discusses complex roots of polynomial equations. It notes that any polynomial with complex roots will have roots that come in conjugate pairs. It asks questions about the relationship between the coefficients of a polynomial and the sum and product of its roots. Specifically, it asks the reader to find the values of a and b for a quartic polynomial given that two of its roots are 3i and 1-2i.
Lesson 4 operations with complex numbers in polar form p1 2jenniech
The document discusses operations with complex numbers in polar form. It provides the polar form representations of two complex numbers z1 and z2 as r1(cosθ1 + i sin θ1) and r2(cosθ2 + i sin θ2). It asks the reader to find the product and quotient of z1 and z2 using the polar form, and notes that the product can also be expressed as r1 cisθ1 × r2 cisθ2. It provides hints about using trigonometric addition formulas to simplify the expressions and concludes with two practice problems asking the reader to find the product and quotient of complex numbers given in polar form.
e^iπ = cos(π) + i sin(π) = -1, which is a special case of Euler's equation. When the angle θ is set to π in Euler's formula, it yields the unexpected but important result of e^iπ + 1 = 0, which connects the five most important mathematical constants: 0, 1, e, i, and π.
Lesson 7 de moivre theorem proof by inductionjenniech
This document outlines the proof by induction of DeMoivre's Theorem. It presents the three steps of an inductive proof: 1) Show the statement is true for the initial value, here n=1; 2) Assume the statement is true for an arbitrary value k; 3) Show that if true for k then it must also be true for k+1, completing the induction proof that the statement is true for all positive integers n.
Lesson 5 interesting properties of complex conjugates p1 2jenniech
This document discusses properties of complex conjugates. It lists six properties to prove about complex conjugates: 1) z+z* and zz* are always real. 2) If z=rcisθ then z*=r cis(-θ). 3) (z+w)*=z*+w* . 4) (zw)*=z*w* . 5) (z/w)*=(z*)/(w*) . 6) (zn)*=(z*)n. It also provides two practice problems: (i) find the complex conjugate of z=(1+iw)2 and (ii) find a complex number z such that z=3z+2z*=5+2i.
This document provides instructions for solving equations using the form z = x + iy. It gives the example equations (i) z^2 = -4i and (ii) z^2 = 9i to solve using the hint that if two complex numbers are equal, their real and imaginary parts must be equal. It also lists the example equations (i) z^2 = 2 + 2√3i and (ii) z^2 = 5 + i.
Lesson 3 argument polar form of a complex numberjenniech
This document discusses representing complex numbers in polar form using modulus (r) and argument (θ) instead of Cartesian coordinates (x, y). It shows how to convert between polar and Cartesian forms, defines the notation cis(θ) for representing complex numbers in polar form, and lists some properties of cis(θ) including rules for addition, multiplication, and equivalence under modulo 2π.
Lesson 1 imaginary and complex numbers p1 8jenniech
1. Imaginary and complex numbers can be used to solve quadratic equations that have no real solutions. The square root of -1 is defined as i, and complex numbers have both a real and imaginary part.
2. An Argand diagram represents complex numbers graphically by plotting the real part on the x-axis and the imaginary part on a perpendicular y-axis. Common complex number operations like addition, subtraction, multiplication, and division can be performed by treating complex numbers as vectors.
3. The modulus of a complex number z = a + bi is the distance from the point (a, b) to the origin on the Argand diagram, and represents the absolute value of z. The argument of z,
Lesson 1b two interesting questions for youjenniech
This document poses two interesting questions and asks what is wrong with two proofs presented. The first proof is not shown, while the second proof claims to prove something but does not actually include any logical steps, reasoning, or evidence to support its claim.
This document provides an overview of key concepts in probability for the IB Standard Level course, including Venn diagrams, conditional probability, and probability tree diagrams. It includes links to examples and practice problems involving each of these topics. Students are intended to understand how to use strategies like Venn and tree diagrams to solve real-life probability problems.
Your rubric multimedia project how tall is the sports block_jenniech
This rubric is for evaluating a multimedia project on how to calculate the height of a sports block using trigonometry. The rubric assesses students on content, oral presentation, organization, sources, and presentation. For content, it evaluates the depth of subject knowledge and inclusion of essential trigonometry information. For oral presentation, it considers whether the delivery is interesting, rehearsed and holds audience attention. The organization category assesses whether the content is logically structured. Sources examines whether source information is collected and documented properly. Presentation assesses rehearsal, delivery and maintaining audience interest.
Finding the magnitude of a vector in 2 d and 3djenniech
To find the magnitude (length) of a vector in 2D or 3D, one uses the formula that takes the square root of the sum of the squares of the vector's components. For a vector a with components a1, a2, and a3, the magnitude is calculated as the square root of a1^2 + a2^2 + a3^2. The magnitude of a vector is denoted by |a|.
This document contains a student worksheet for investigating vectors in 2D and 3D using Autograph software. It includes tasks on expressing vectors in terms of other vectors, finding vector equations of lines given conditions, calculating lengths and angles of vectors, and using vectors to solve geometric problems in 2D and 3D. The worksheet guides students through using Autograph's vector tools to explore and check their work.
The document discusses scalar products of vectors and different types of vector relationships. It asks to define the scalar product using notation, calculate products of perpendicular and parallel vectors, and explain the difference between parallel and coincident vectors with a diagram. Perpendicular vectors have a scalar product of 0, parallel vectors maintain the same direction but their scalar product depends on length and angle, and coincident vectors overlap completely and have the greatest possible scalar product.
Converting vector equation of a line to cartesian formjenniech
The document describes how to convert a vector equation of a line to Cartesian form. A vector equation defines a line using a point (a,b,c) on the line and a direction vector (k,m,n). It explains how to write the x, y, z coordinates in terms of λ and rearrange to make λ the subject, then equate the expressions to find the Cartesian equation of the line. The Cartesian equation defines the line using the standard x, y, z variables.
The scalar product form for the equation of a planejenniech
The document discusses deriving the scalar product form of the equation of a plane. It defines two points A and P on the plane and their position vectors a and r. It explains that the vector from A to P, AP, is equal to r - a. Using the scalar product of the position vector r and the normal vector n to the plane, along with the position vector a, allows obtaining the scalar product equation of the plane, r·n = a·n. This equation can then be written in Cartesian form.
This slide is special for master students (MIBS & MIFB) in UUM. Also useful for readers who are interested in the topic of contemporary Islamic banking.
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The simplified electron and muon model, Oscillating Spacetime: The Foundation...RitikBhardwaj56
Discover the Simplified Electron and Muon Model: A New Wave-Based Approach to Understanding Particles delves into a groundbreaking theory that presents electrons and muons as rotating soliton waves within oscillating spacetime. Geared towards students, researchers, and science buffs, this book breaks down complex ideas into simple explanations. It covers topics such as electron waves, temporal dynamics, and the implications of this model on particle physics. With clear illustrations and easy-to-follow explanations, readers will gain a new outlook on the universe's fundamental nature.
How to Fix the Import Error in the Odoo 17Celine George
An import error occurs when a program fails to import a module or library, disrupting its execution. In languages like Python, this issue arises when the specified module cannot be found or accessed, hindering the program's functionality. Resolving import errors is crucial for maintaining smooth software operation and uninterrupted development processes.
Lesson 3 c reciprocal of a complex number is polar form
1. Reciprocal of a Complex Number in Polar form
Let z = r cis 𝜃 and 𝑧.
1
𝑧
=1
Suppose that
1
𝑧
= 𝛽𝑐𝑖𝑠𝛾
so
𝑧.
1
𝑧
=1 is
remember that 1cis 0 = 1
𝑟𝛽𝑐𝑖𝑠( 𝜃 + 𝛾) = 1 𝑐𝑖𝑠 0
What is an expression for the modulus?
What is an expression for the arguments?
What can you say about
1
𝑧
=