## Differential Equation Eigenvalue Problem

Apply matrix techniques to solve systems of linear ordinary differential equations with constant coefficients. n first order equations that we can solve separately. Find the eigenvalues and Eigen functions for the boundary value problem, x^{2}{y} for Teachers for Schools for Working Scholars for College. One mathematical tool, which has applications not only for Linear Algebra but for differential equations, calculus, and many other areas, is the concept of eigenvalues and eigenvectors. Therefore, for each eigenfunction Xn with corresponding eigen-value ‚n, we have a solution Tn such that the function un(x;t) = Tn(t)Xn(x) is a solution. In other words, we have to ﬁnd all of the numbers λ such that there is a solution of the equation AX = λX for some function X (X 6= 0) that satisﬁes the boundary conditions at 0 and at l. This function specifies the probability density for observing the particle at a given position,. Just as you said, the general solution of $$y''+\lambda y=0$$ has the form $$y=A\cos\sqrt{\lambda}x+B\sin\sqrt{\lambda}x\,,$$ (we assumed here that $\lambda>0$). A very important problem in matrix algebra is called the eigenvalue problem. eigenvalue problem, with the quadratic two-parameter eigenvalue problem as important special case; (b) to show the relevance of this problem to determine critical delays for various types of DDEs; (c) to provide alternative derivations of existing matrix pencil methods using the context of polynomial two-parameter eigenvalue problems;. 4, Modeling with First Order Equations. An ode is an equation for a function of. Nonlinear eigenvalue problems (NEPs) arise in many fields of science and engineering. [12,13] are in perfect agreement, (3) in applications such. 1 Estimating the Diffusion Coefficient in the Heat Equation 703 9. Solve first order differential equations using standard methods, such as separation of variables, integrating factors, exact equations, and substitution methods; use these methods to solve analyze real-world problems in fields such as economics, engineering, and the sciences. Differential Equations with Differential Equation many of the problems are difficult to make up on the spur of Real Eigenvalues - Solving systems of. We will work quite a few examples illustrating how to find eigenvalues and eigenfunctions. differential equations. ) The equation λ2−(a+b)λ+ab−h2 = 0 is called the eigenvalue equation of the matrix A. A differential equation is an equation that relates the rate of change of some process to other processes that are changing in time. txt) or read online for free. Chapter Five - Eigenvalues , Eigenfunctions , and All That The partial differential equation methods described in the previous chapter is a special case of a more general setting in which we have an equation of the form L 1 ÝxÞuÝx,tÞ+L 2 ÝtÞuÝx,tÞ = F Ýx,tÞ. One of the stages of solutions of differential equations is integration of functions. From "Mémoire sur l'integration des équations linéaires" (1840); image courtesy of Archive. In this paper, we are concerned with the solution of a class of boundary value problems −y″+f(x)y=λy, y(0)=0, y(∞)=0, where f(x) monotonically increases to infinity as n increases to infinity. 2 The Method of Elimination, 229 4. An eigenvalue λ of an nxn matrix A means a scalar (perhaps a complex number) such that Av=λv has a solution v which is not the 0 vector. They're both hiding in the matrix. So Mathematica provides us only one eigenvector ξ = [ 1, 0, 0] corresponding to the eigenvalue λ = 1 (therefore, A is defective) and one eigenvector v = <-1,1,0> corresponding eigenvalue λ = 0. A = [ 1 1 0 0 0 1 0 0 1]. 2 Sturm-Liouville Boundary Value Problems 723. The problem is to find the numbers, called eigenvalues, and their matching vectors, called eigenvectors. Ordinary Diﬀerential Equations Igor Yanovsky, 2005 2 Disclaimer: This handbook is intended to assist graduate students with qualifying examination preparation. Meade prepared for UC Davis students. I will be in my office on Tuesday (12/15) 10-12, Wednesday (12/16) 10-4. It assumes some knowledge of calculus, and explains the tools and concepts for analysing models involving sets of either algebraic or 1st order differential equations. In this article, we study the global bifurcation from in nity of non-linear eigenvalue problems for ordinary di erential equations of fourth order. The simplest example and one that will play a role in neurobiology is the decay to rest of the membrane potential. An important special case is when is an aﬃne function of y. NONLINEAR EIGENVALUE PROBLEMS 489 Many people have worked on nonlinear eigenvalue problems; in particular, for ordinary differential equations and integral equations. The language of dynamic phenomena is differential equations. The ﬁrst one is easier, especially in the 2 × 2 case. For example, whenever a new type of problem is introduced (such as first-order equations, higher-order equations, systems of differential. - Minimization problems for variational integrals, existence and regularity theory for minimizers and critical points, geometric measure theory - Variational methods for partial differential equations, linear and nonlinear eigenvalue problems, bifurcation theory - Variational problems in differential and complex geometry. , Bayramoglu, Mamed, and Aslanov, Khalig M. differential-equations eigenvalues schrodinger produces a confluent Heun differential equation with solution Eigenvalue Problem on a Rectangle with Dirichlet. Questions concerning eigenvectors and eigenvalues are central to much of the theory of linear. Strauss, Partial Differential Equations: An Introduction, Wiley G. Complex Eigenvalues – Solving systems of differential equations with complex eigenvalues. Then the chaotic behavior of the logistic map. The roots of this differential equation are called eigenvalues, and the corresponding functional solutions are known as eigenfunctions. In this series, we will explore temperature, spring systems, circuits, population growth, biological cell motion, and much more to illustrate how differential equations can be used to model nearly everything. The authors have sought to combine a sound and accurate (but not abstract) exposition of the. In particular, for any scalar ‚, the solution of the ODE for T is given by T(t) = Ae¡k‚t for an arbitrary constant A. In natural sciences and engineering, are often used differential equations and systems of differential equations. It is perhaps not surprising that one of the primary examples involves the L-shaped membrane read more >>. Naylor, Differential Equations of Applied Mathematics, John Wiley & Sons I. Osborne† (Received 1 June 2001; revised 18 October 2002) Abstract Discretisations of diﬀerential eigenvalue problems have a sensitivity to perturbations which is asymptotically least as h →0 when the diﬀerential equation is in ﬁrst order sys-tem form. New algorithms are suggested and old algorithms, i. A = [ 1 1 0 0 0 1 0 0 1]. This approach is based on the extension of the previously reported differential transfer matrix method with modified basis functions. Partial Differential Equations and Boundary Value Problems with Maple, Second Edition, presents all of the material normally covered in a standard course on partial differential equations, while focusing on the natural union between this material and the powerful computational software, Maple. 2 Properties of Sturm-Liouville Eigenvalue Problems There are several properties that can be proven for the (regular) Sturm-Liouville eigenvalue problem. Elementary Differential Equations with Boundary Value Problems integrates the underlying theory, the solution procedures, and the numerical/computational aspects of differential equations in a seamless way. Variation of Parameters for Matrices. The eigenvalue problem of complex structures is often solved using finite element analysis, but neatly generalize the solution to scalar-valued vibration problems. Note that the eigenfunction is defined up to a multiplicative constant so you can just set $\epsilon$=1 and there is only one parameter $\epsilon'$ to vary in order to achieve a properly decaying solution at infinity. NONLINEAR EIGENVALUE PROBLEMS 489 Many people have worked on nonlinear eigenvalue problems; in particular, for ordinary differential equations and integral equations. The fractional differential equation (1. I am looking for a way to solve them in Python. Two Distinct Real Eigenvalue Case. An important special case is when is an aﬃne function of y. Repeated Eigenvalues – Solving systems of differential equations with repeated eigenvalues. DiPrima and D. 14 Chapter 1. First, find a matrix that diagonalizes The eigenvalues of are and with corresponding eigenvectors and Diagonalize using the matrix whose columns consist of and to obtain and The system has the following form. Introduction. In the last few years, fractional differential equations have gained attentions due to their numerous applications in various aspects of science and technology. WILKES Department of Chemical Engineering, The University of Michigan, Ann Arbor, Michigan 48109, U. , Bayramoglu, Mamed, and Aslanov, Khalig M. This longer text consists of the main text plus three additional chapters (Eigenvalue Problems and Sturm—Liouville Equations; Stability of Autonomous Systems. Suppose, I have an differential equation like this one: mu1 u1[x] - u1''[x] - 10 u1[x] == 0 where mu1 is the eigenvalue and u1 is the eigenfuntion. Solving an eigenvalue problem means finding all its eigenvalues and associated eigenfunctions. Fourier Series and Systems of Differential Equations and Eigenvalue Problems by Leif Mejlbro. See [I, 4 and 51 where several references are given. Functional Anal. During the studying of linear. y'' + y' +λy = 0, y(0) = 0, y(2) = 0. Here a brief overview of the required con-cepts is provided. As a special case of the proposed method, we find a closed form for a parameterization of the critical surface for the scalar case. This page describes how it can be used in the study of vibration problems for a simple lumped parameter systems by considering a very simple system in detail. This approach is based on the extension of the previously reported differential transfer matrix method with modified basis functions. This process is experimental and the keywords may be updated as the learning algorithm improves. Solutions of S-L Problems. AN EIGENVALUE PROBLEM INVOLVING A FUNCTIONAL DIFFERENTIAL EQUATION ARISING IN A CELL GROWTH MODEL BRUCE VAN BRUNT 1 and M. From "Mémoire sur l'integration des équations linéaires" (1840); image courtesy of Archive. 1 Matrices and Linear Systems 264 5. To do this we have to distinguish two cases, called complete and defective. Going through these examples, the. Running some numerics suggests that the mapping h ↦ λn(Hh) where λn denotes the nth eigenvalue of Hh, n = 0, 1, 2, …, is monotone non-decreasing. Sturm-Liouville Eigenvalue Problems and Generalized Fourier Series Examples of Regular Sturm-Liouville Eigenvalue Problems We will now look at examples of regular Sturm-Liouville differential equations with various combinations of the three types of boundary conditions Dirichlet, Neumann and Robin. Chapter 9 Diffusion Equations and Parabolic Problems Chapter 10 Advection Equations and Hyperbolic Systems Chapter 11 Mixed Equations Part III: Appendices. We show that there exist classes of explicit numerical integration methods that can handle very stiff problems if the eigenvalues are separated into two clusters, one containing the "stiff," or fast, components, and one containing the slow components. Observe that eigendirections vary if we change parameters. where the eigenvalues are repeated eigenvalues. To actually solve ODE systems having complex eigenvalues, imitate the procedure in the following example. Lecture Description. Applications of the method to boundary value and initial value problems, as well as several examples are illustrated. pdf), Text File (. Krylov Subspace Methods for the Eigenvalue problem Presented by: Sanjeev Kumar Applications We need only few eigen (singular) pairs, and matrices can be large and sparse Solving homogeneous system of linear equations A x = 0. Chapter 12 Measuring Errors Chapter 13 Polynomial Interpolation and Orthogonal Polynomials Chapter 14 Eigenvalues and inner product norms Chapter 15 Matrix powers and exponentials. I believe that there is a misprint in the problem statement. Of our three ordinary differential equations, only two will be eigenvalue problems. The eigenvalues and eigenfunctions are characterized in terms of the Mittag-Leffler functions. With equations conveniently specified symbolically, the Wolfram Language uses both its rich set. More of Cauchy's method for solving a system of linear, first-order differential equations with constant coefficients, equivalent to a modern-day eigenvalue problem. has a solution if the admissible sets are all contained in a bounded open set called design region (known as Buttazzo-Dal Maso Theorem). Homogeneous System Solutions with Distinct and Repeated Eigenvalues. Let's see some examples of first order, first degree DEs. In this paper, we describe new localization results for nonlinear eigenvalue problems that generalize Gershgorin’s theorem, pseudospectral inclusion theorems, and the Bauer-Fike theorem. The average is now 70/100. 2) a)Find the eigenvalues and eigenfunctions of the boundary-value problem. In the nonlinear context a separatrix plays the role of an eigenfunction and the initial conditions that give rise to the separatrix play the role of eigenvalues. My name is Will Murray and I thank you very much for watching, bye bye. In this section we will define eigenvalues and eigenfunctions for boundary value problems. Two Coupled Oscillators Let's consider the diagram shown below, which is nothing more than 2 copies of an harmonic oscillator, the system that we discussed last time. Durán, Ariel L. Find the particular solution given that y(0)=3. differential-equations eigenvalues schrodinger produces a confluent Heun differential equation with solution Eigenvalue Problem on a Rectangle with Dirichlet. Eigenvalue problem. 2 The Method of Elimination 239 4. In this blog post — inspired by Strogatz (1988, 2015) — I will introduce linear differential equations as a means to study the types of love affairs two people might. Eigenvalue Problems for Odes 1 - Free download as PDF File (. 1: Eigenvalue Problems for y'' + λy = 0 - Mathematics LibreTexts Skip to main content. Additionally, the input data of the problem depend on a numerical parameter. This is an additional adjustable parameter we. Perfect for undergraduate and graduate studies. Partial Differential Equations with at Least Three Independent Variables. There are homogeneous boundary conditions in x and y. They are often introduced in an introductory linear algebra class, and when introduced there alone, it is hard to appreciate their importance. Ambrosetti and P. 8) Each class individually goes deeper into the subject, but we will cover the basic tools needed to handle problems arising in physics, materials sciences, and the life sciences. Differential Equations Calculator. The case a = b and h = 0 needs no investigation, as it gives an equation of a circle. can make the numerical solution of eigenvalue problems a difficult and challenging task due to possible crossing of eigenvalues. I believe that there is a misprint in the problem statement. It is perhaps not surprising that one of the primary examples involves the L-shaped membrane read more >>. • Ordinary Differential Equation: Function has 1 independent variable. Two Distinct Real Eigenvalue Case. Introductory Differential Equations with Boundary Value Problems Third Edition Martha L. The task is to compute the fourth eigenvalue of Mathieu's equation. These pages offer an introduction to the mathematics of such problems for students of quantum chemistry or quantum physics. fairly its not. this equation, and we end up with the central equation for eigenvalues and eigenvectors: x = Ax De nitions A nonzero vector x is an eigenvector if there is a number such that Ax = x: The scalar value is called the eigenvalue. Multiplying through by this, we get y0ex2 +2xex2y = xex2. Nonstiff ODE system due to van der Pol (assuming a small coefficient). where P and Q are functions of x. In this paper the authors develop a general framework for calculating the eigenvalues of a symmetric matrix using ordinary differential equations. In this set of equations, $$E$$ is an eigenvalue, which means there are only non-trivial solutions for certain values of $$E$$. 1 Eigenvalues and eigenvectors • Diagonalization Section 7. - Chapter 3: New Problem 35 on determination of radii of convergence of power series solutions of differential equations; new Example 3 just before the subsection on logarithmic cases in the method of Frobenius, to illustrate first the reduction-of-order formula with a simple non-series problem. EigenNDSolve uses a spectral expansion in Chebyshev polynomials and solves systems of linear homogenous ordinary differential eigenvalue equations with general (homogenous) boundary conditions. •This first- order homogeneous system of linear differential equations is already in matrix form. 18) g(0) = 0 and g(H) = 0. Eigenvalues! Eigenvalues! This page is a collection of online resources that might come in handy to anyone interested in learning about differential equations (on an introductory level), and also students who are taking their first diffeq course in college. Problems 1-5 are called eigenvalue problems. Ecker, andW. He is the author of numerous technical papers in boundary value problems and random differential equations and their applications. Let's say we have a linear differential operator $\hat A = \sum_k a_k \frac {d^k} {dx^k}$ and we want to solve the eigenvalue equation $\hat A f(x) = a f(x)$ which is. Partial Differential Equations and Boundary Value Problems with Maple, Second Edition, presents all of the material normally covered in a standard course on partial differential equations, while focusing on the natural union between this material and the powerful computational software, Maple. Knowledge beyond the boundaries. 3 The Geometry of First-Order Differential Equations 1. The Concept of Eigenvalues and Eigenvectors. In natural sciences and engineering, are often used differential equations and systems of differential equations. 1 Vector spaces and linear. There are various methods by which the continuous eigenvalue problem may be. The eigenvalue problem for such an A (with boundary conditions) is to ﬁnd all the possible eigenvalues of A. If there are two distinct, , real eigenvalues, with corresponding eigenvectors and , then the two solutions. My name is Will Murray and I thank you very much for watching, bye bye. Such values of λ, when they exist, are called the eigenvalues of the problem, and the corresponding solutions are the eigenfunctions associated to each λ. WILKES Department of Chemical Engineering, The University of Michigan, Ann Arbor, Michigan 48109, U. Lecture 18: Complex eigenvalues and spirals. This is all part of a larger lecture series on differential equations here on educator. , v(x,y,z,t). Eigenvalue range, specified as a two-element real vector. Differential equations are very common in physics and mathematics. Love, on the other hand, is humanity's perennial topic; some even claim it is all you need. The average is now 70/100. Repeated Eigenvalues – Solving systems of differential equations with repeated eigenvalues. differential-equations eigenvalues schrodinger produces a confluent Heun differential equation with solution Eigenvalue Problem on a Rectangle with Dirichlet. These limitations are appropriate for most quantum mechanics problems as well as many classical problems. Eigenvalues, Eigenvectors, and Di erential Equations William Cherry April 2009 (with a typo correction in November 2015) The concepts of eigenvalue and eigenvector occur throughout advanced mathematics. Eigenvalues of a Sturm Liouville differential equation tutorial of Differential equations I course by Prof Chris Tisdell of Online Tutorials. Diﬀerential equations are called partial diﬀerential equations (pde) or or-dinary diﬀerential equations (ode) according to whether or not they contain partial derivatives. 2 Properties of Sturm-Liouville Eigenvalue Problems There are several properties that can be proven for the (regular) Sturm-Liouville eigenvalue problem. Knowledge beyond the boundaries. So, it is natural to use the analytic multiplicity to count eigenvalues, and the analytic multiplicity plays an important role in the study of the dependence of the eigenvalues of a spectral problem on the differential equation boundary value problem (see, for example, [5]. 2 The Eigenvalue Method for Homogeneous. Solution to linear constant coeﬃcient ODE systems 90 7. A University Level Introductory Course in Differential Equations. In this section we will solve systems of two linear differential equations in which the eigenvalues are distinct real numbers. We consider an eigenvalue problem for stochastic ordinary differential operators with stochastic boundary conditions. Homogeneous Linear Equations; Forcing; Sinusoidal Forcing; Forcing and Resonance; Projects for Second-Order Differential Equations; 5 Nonlinear Systems. This book is aimed at students who encounter mathematical models in other disciplines. Introduction to Systems of Differential Equations 228 4. For the following system of linear differential equations:\frac{\mathrm{d} x}{\mathrm{d} Question: For the following system of linear differential equations:. Ambrosetti and P. Braselton AMSTERDAM •BOSTON HEIDELBERG • LONDON NEW YORK •OXFORD • PARIS SAN DIEGO SAN FRANCISCO •SINGAPORE SYDNEY • TOKYO Academic Press is an imprint of Elsevier. 2 The Eigenvalue Method for Homogeneous Systems 304. 1 Linear differential equations All linear equations involve a linear operator L. Questions concerning eigenvectors and eigenvalues are central to much of the theory of linear. The ball's acceleration towards the ground is the acceleration due to gravity minus the acceleration due to air resistance. Typical problems considered were elliptic partial differential equations of the form U xx + U yy = f(x'y)' (1) or U xx + U yy + X2U = 0 (2) where appropriate boundary conditions are specified so that the problem is self-adjoint, The four methods are relaxation, Galerkin, Rayleigh-. Worked Example. A very important problem in matrix algebra is called the eigenvalue problem. Differential equations are the language of the models that we use to describe the world around us. The Handbook of Ordinary Differential Equations: Exact Solutions, Methods, and Problems, is an exceptional and complete reference for scientists and engineers as it contains over 7,000 ordinary. In this section we will define eigenvalues and eigenfunctions for boundary value problems. KEYWORDS: Worksheets, Introduction to Matrices, Definitions, Matrix arithmetic, Identity matrices, Inverse matrices, Finding eigenvalues and eigenvectors, Using eigenvalues and eigenvectors to solve differential equations and discrete systems. is an eigenvalue if it is a pole of , where denotes the identity operator. Phase Plane. were developed and used for. , Schrödinger. Notice that this is exactly the same equation as in the first (both ends kept at 0 degree) heat conduction problem, due to the fact that both problems have the same set of eigenvalues (but with different eigenfunctions). Find the solution of y0 +2xy= x,withy(0) = −2. Here a brief overview of the required con-cepts is provided. The critical delays of a delay‐differential equation can be computed by solving a nonlinear two‐parameter eigenvalue problem. using eigenvalues and eigenvectors to solve boundary value problems for Laplace’s equation and other partial differential equations, analytically and via finite differences. Partial differential equations, 171–181, Lecture Notes in Pure and Appl. The problem is nonlinear with respect to the spectral parameter and involves generally nonlocal additional conditions specified by a Stieltjes integral. You will need to find one of your fellow class mates to see if there is something in these. ,xn and the time t as shown below dx1 --- = a11 x1 + a12 x2 +. However, this only satisfies the differential equation, not boundary conditions. To find real valued solutions. and eigenvalue systems of the form. Problem Sets Use of Problem Sets. The eigenvalues and eigenfunctions are viewed at a fixed point as functions of the interval length. evr(2) specifies the upper limit of the range, and must be finite. Because of that, problem of eigenvalues occupies an important place in linear algebra. senting solutions of the partial differential equation. Let me show you the reason eigenvalues were created, invented, discovered was solving differential equations, which is our purpose. Since we are going to be working with systems in which A is a 2 x 2 matrix we will make that assumption from the start. Incidentally we note that an alternative, yet related approach to the ME formalism—in the form of delay differential equation model—has been developed which does not require small gain and. A first‐order differential equation is said to be linear if it can be expressed in the form. My name is Will Murray and I thank you very much for watching, bye bye. which is the same as (4. With the aid of the spectral theory of compact self-adjoint operators in Hilbert spaces, we show that the spectrum of this problem consists of only countable real eigenvalues with finite multiplicity and the corresponding eigenfunctions form a complete. The problem is to find the numbers, called eigenvalues, and their matching vectors, called eigenvectors. We will be concerned with finite difference techniques for the solution of eigenvalue and eigenvector problems for ordinary differential equations. where the eigenvalues are repeated eigenvalues. This volume is application-oriented and rich in examples. Automatically selecting between hundreds of powerful and in many cases original algorithms, the Wolfram Language provides both numerical and symbolic solving of differential equations (ODEs, PDEs, DAEs, DDEs, ). 4* Initial and Boundary Conditions 20 1. - Minimization problems for variational integrals, existence and regularity theory for minimizers and critical points, geometric measure theory - Variational methods for partial differential equations, linear and nonlinear eigenvalue problems, bifurcation theory - Variational problems in differential and complex geometry. A University Level Introductory Course in Differential Equations. The Linear System Solver is a Linear Systems calculator of linear equations and a matrix calcularor for square matrices. The presentation does not presume a deep knowledge of mathematical and functional analysis. Somebody say as follows. 1 Vector spaces and linear. Solve first order differential equations using standard methods, such as separation of variables, integrating factors, exact equations, and substitution methods; use these methods to solve analyze real-world problems in fields such as economics, engineering, and the sciences. 1 First-Order Systems and Applications 246 4. Consider a linear homogeneous system of $$n$$ differential equations with constant coefficients, which can be written in matrix form as. of Statistics. Additionally, the input data of the problem depend on a numerical parameter. Homogeneous Linear Equations; Forcing; Sinusoidal Forcing; Forcing and Resonance; Projects for Second-Order Differential Equations; 5 Nonlinear Systems. Hopefully, those solu-tions will form a useful basis in some function space. Harrar II∗ M. Reference [1] J. Eigenvalue problems for linear differential equations, such as time-independent Schrö dinger equations, can be generalized to eigenvalue problems for nonlinear differential equations. 2 Properties of Sturm-Liouville Eigenvalue Problems 189 6. A class of problems to which our previous examples belong are the Sturm-Liouville eigenvalue problems. Let's say we have a linear differential operator $\hat A = \sum_k a_k \frac {d^k} {dx^k}$ and we want to solve the eigenvalue equation $\hat A f(x) = a f(x)$ which is. 2) (The roots are distinct if a 6= b or h 6= 0. 3 Numerical Methods for Systems, 240 Linear Systems of Differential Equations 255 5. Because the differential equation and boundary conditions for X(x), form a Sturm-Liouville problem, we know that the solutions to this problem, Xn(x), are an infinite set of orthogonal eigenfunctions. Numerical Methods for Differential Equations Chapter 1: Initial value problems in ODEs Gustaf Soderlind and Carmen Ar¨ evalo´ Numerical Analysis, Lund University Textbooks: A First Course in the Numerical Analysis of Differential Equations, by Arieh Iserles and Introduction to Mathematical Modelling with Differential Equations, by Lennart Edsberg. Here a brief overview of the required con-cepts is provided. is an idealized heat source of. WILKES Department of Chemical Engineering, The University of Michigan, Ann Arbor, Michigan 48109, U. Math 285: Differential Equations (3 credit hours) Course Description This is an introduction to ordinary differential equations with an emphasis on applications. 1 First-Order Systems and Applications, 219 4. Solve u = Au, where A =. Partial Differential Equations with at Least Three Independent Variables. Since our last example and that wraps up our lecture on repeated eigenvalues so, this is the systems of differential equations where we had repeated eigenvalues. The integrating factor is e R. Hence we have which implies that an eigenvector is We leave it to the reader to show that for the eigenvalue , the eigenvector is Let us go back to the system with complex eigenvalues. Thinking about solving coupled linear differential equations by considering the problem of developing a solution to the following homogeneous version of Eq. It is suitable for a one-semester undergraduate level or two-semester graduate level course in PDEs or applied mathematics. This book is aimed at students who encounter mathematical models in other disciplines. During the studying of linear. Durán, Ariel L. We use our results to analyze three. 2 The Eigenvalue Method for Homogeneous Systems 304. To find real valued solutions. problem of linear algebra is the eigenvalue problem which is more sophisticated. They are often introduced in an introductory linear algebra class, and when introduced there alone, it is hard to appreciate their importance. In this series, we will explore temperature, spring systems, circuits, population growth, biological cell motion, and much more to illustrate how differential equations can be used to model nearly everything. For example, whenever a new type of problem is introduced (such as first-order equations, higher-order equations, systems of differential. 3 Numerical Methods for Systems, 240 Linear Systems of Differential Equations 255 5. The eigenvalue problem for a linear system of ordinary differential equations is considered. The Concept of Eigenvalues and Eigenvectors. COMPOUND MATRIX METHOD FOR EIGENVALUE PROBLEMS IN MULTIPLE CONNECTED DOMAINS N. The complete case. This is the complex eigenvalue example from [1], Section 3. y'' + y' +λy = 0, y(0) = 0, y(2) = 0. For uncoupled systems,. The solution to the system of differential equations can therefore be written out as a linear combination of the real and imaginary parts of the eigenvector associated with eigenvalue. It is suitable for a one-semester undergraduate level or two-semester graduate level course in PDEs or applied mathematics. Since the eigenvalues are real distinct roots and one eigenvalue is positive with the other eigenvalue negative, the equilibrium point is a saddle and the solutions are unstable. And the lambda, the multiple that it becomes-- this is the eigenvalue associated with that eigenvector. In this series, we will explore temperature, spring systems, circuits, population growth, biological cell motion, and much more to illustrate how differential equations can be used to model nearly everything. Eigenvalue Problems for Odes 1 - Free download as PDF File (. The case a = b and h = 0 needs no investigation, as it gives an equation of a circle. Section 6: Systems of Equations: Lecture 2 | 1:03:54 min. A differential equation is an equation that relates the rate of change of some process to other processes that are changing in time. 8) Each class individually goes deeper into the subject, but we will cover the basic tools needed to handle problems arising in physics, materials sciences, and the life sciences. Solve u = Au, where A =. Two Distinct Real Eigenvalue Case. Introduction to Systems of Differential Equations 246 4. Math 285: Differential Equations (3 credit hours) Course Description This is an introduction to ordinary differential equations with an emphasis on applications. Notes on Diffy Qs: Differential Equations for Engineers. 1) Find the eigenfunctions and the equations that defines the eigenvalues for the given boundary-value problem. This volume is an introductory level textbook for partial differential equations (PDE's) and suitable for a one-semester undergraduate level or two-semester graduate level course in PDE's or applied mathematics. In this course, we will only study two-point boundary value problems for scalar linear second order ordinary di erential equations. Variation of Parameters for Matrices. Since we are going to be working with systems in which A is a 2 x 2 matrix we will make that assumption from the start. DIFFERENTIAL EQUATIONS. Chapters One to Five are organized according to the equations and the basic PDE's are introduced in an easy to understand manner. A basis of the system consists of two solution vectors y1(t)and y2(t), which are linearly independent and each of which satisﬁes the corresponding homogeneous equations L(y1)=0 and L(y2)=0. Introduction. Of our three ordinary differential equations, only two will be eigenvalue problems. Additionally, the input data of the problem depend on a numerical parameter. Parabolic partial differential equations are differential equations which depend on space and time. Causality and the Wave Equation Integrating the Bell Curve Compressor Stalls and Mobius Transformations Dual Failures with General Densities Phase, Group, and Signal Velocity Series Solutions of the Wave Equation The Limit Paradox Proof That PI is Irrational Simple Proof that e is Irrational The Filter Of Observation Eigenvalue Problems and. find (manually, numerically, and graphically) and interpret solutions to differential equations, systems of differential equations, and initial value problem s (A, B, C) 3. Sturm Liouville Problem (SLP) SL equation A classical "'Sturm-Liouville equation"', is a real second-order linear differential equation of the form d dx p(x) dy dx +q(x)y= λr(x)y, (1) In the simplest of cases all coefﬁcients are continuous on the ﬁnite closed interval [a,b], and p(x) has continuous derivative. 1 First-Order Systems and Applications 228 4. Let T:Ω→Cn×n be a matrix-valued function that is analytic on some simply-connected domain Ω⊂C. 3) with λ = µ, has a non-trivial solution is called a Sturm-Liouville Eigen Value Problem (SL-EVP). 8 - Endpoint Problems and Eigenvalues 3. A very important problem in matrix algebra is called the eigenvalue problem. , thermal conduction) and quantum mechanics (e. is an eigenvalue if it is a pole of , where denotes the identity operator. Naylor, Differential Equations of Applied Mathematics, John Wiley & Sons I. I'll do an example in a minute. SUMMARY An algorithm based on a compound matrix method is presented for solving difficult eigenvalue problems. In this lesson, our instructor Will Murray gives an introduction on complex eigenvalues for systems of equations. Vibrating Strings and Membranes. First, find a matrix that diagonalizes The eigenvalues of are and with corresponding eigenvectors and Diagonalize using the matrix whose columns consist of and to obtain and The system has the following form. 7 Modeling Problems Using First-Order Linear. for λ>0, λ = 0, λ < 0. Sturm-Liouville problem, or eigenvalue problem, in mathematics, a certain class of partial differential equations (PDEs) subject to extra constraints, known as boundary values, on the solutions.


### CSIR JUNE 2011 PART C QUESTION 63 SOLUTION (When $d(x,y) = |f(x) - f(y)|$ defines a metric)

Which of the following is/are metric on $\Bbb R$?
1)$d(x,y) = min\{x,y\}$,
2)$d(x,y) = |x-y|$,
3) $d(x,y) = |x^2 - y^2|$,
4)$d(x,y) = |x^3 - y^3|$.
Solution

option (1): NO. $d(-1,0) = -1$ which is negative.
option (2): YES. Given metric is the usual metric.
option (3): NO. $d(-1,1) = 0$. In a metric space $d(x,y) =0$ if and only if $x=y$.
option (4):YES.  We will verify all the axioms.
axiom 1: Clearly $d(x,y) \ge 0$,
axiom 2: Let $x,y \in \Bbb R$, then $d(x,y) = 0$ if and only if $|x^3-y^3|=0$ if and only if x^3 = y^3 if and only if $x=y$. [Since f(x)=x^3 is injective]
axiom 3: $d(x,y) = d(y,x)$ follows.
axiom 4: (Triangle inequality)
Let $x,y,z \in \Bbb R$. $d(x,y) = |x^3-y^3| = |x^3-z^3+z^3-y^3| \le |x^3-z^3| + |z^3-y^3|$ and this is equal to $d(x,z) + d(z,y)$. This proves that $d$ is a metric.
Bonus: Define a function $d$ on $\Bbb R$ by $d(x,y) = |f(x) - f(y)|$where $f: \Bbb R \to \Bbb R$ is a function. What is the necessary and sufficient condition on $f$ to make $d$ a metric. The answer is it is enough $f$ to be injective.
Proof: Proof of option (4) work for this general case!!!
In option (3), $f(x) =x^2$ which is not injective so not a metric.
In option (4), $f(x) = x^3$ is injective, so it is a metric.
Exercises:
Which of the following define metric on $\Bbb R$?
1) $d(x,y) = |sin\,x - \sin y|$.  ($f(x) = sin\,x$).
2) $d(x,y) = |e^x - e^y|$. ($f(x) = e^x$).
3) $d(x,y) = |log \,x - log \,y|$ on $\Bbb R^+$. ($f(x) = log \,x$).
4) $d(x,y) = |\sqrt x - \sqrt y|$ on $\Bbb R^+$. ($f(x) = \sqrt x$).
5) $d(x,y) = |tan\,x - tan\,y|$ on $(-\frac{\pi}{2},\frac{\pi}{2})$. ($f(x) = tan\,x$).
6) $d(x,y) = |\frac{1}{x} - \frac{1}{y}|$ on $\Bbb R^+$. ($f(x) =\frac{1}{x}$).
You can ask this question for any function and you know the answer!!!
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### NBHM 2020 PART A Question 4 Solution $$\int_{-\infty}^{\infty}(1+2x^4)e^{-x^2} dx$$
Evaluate : $$\int_{-\infty}^{\infty}(1+2x^4)e^{-x^2} dx$$ Solution : \int_{-\infty}^{\infty}(1+2x^4)e^{-x^2} dx = \int_{-\infty}^{\inft...


# How many ways are there to choose strings of length $3$ from $S=\{1,2,3,4\}$ for $3$ people if each each digit from $S$ has to appear at least once?

Let $S=\{1,2,3,4\}$. There're three persons: Jack, Mary, Ann which choose $3$ digits from $S$ to compose a string of $3$ digits.

For example, a) Jack chose $112$, Mary chose $222$, Ann chose $123$; b) Jack chose $123$, Mary chose $122$, Ann chose $433$.

In how many ways can they form the strings? In how many ways can they form the strings if each digit from $S$ has to appear at least in one string?

For the second subquestion the example a) would not be a valid choice because $4$ doesn't appear in any string, while example b) is valid.

The first subquestion is easy for each string there're $4^3$ possibilities and there're $3$ different people so $3!\cdot 4^3$.

The second subquestion is the harder one. I think this can be solved using inclusion/exclusion. We can count the ways where a given digit doesn't appear in any string, then two digits don't appear in any string, then 3 digits.

The case when one digit doesn't appear in any string: $${4\choose 3}3^3\cdot 3!$$ that is we choose only $3$ digits from $4$ available. The in each string there're $3^3$ options to build the string. Lastly there're $3$ strings in total. By this logic the final answer is: $${4\choose 3}3^3\cdot 3!-{4\choose 2}2^3!\cdot 3+{4\choose 1}1^3\cdot 3!$$

The problem is that when I permute the persons (the $3!$ in the calculation) it works if not two strings are the same. But if some strings are the same I will count too many if I use permutation.

• For the first question, your answer seems to be based on the assumption that no two people can choose the same string. Each person can choose a string in $4^3$ ways, so the number of ways three people could independently choose a string is $(4^3)^3 = 4^9$. – N. F. Taussig Jun 12 '18 at 11:22

Let $S = \{1, 2, 3, 4\}$. Jack, Mary, and Ann each choose three digits from $S$ to compose a string of $3$ digits. For example, Jack chose $112$, Mary chose $222$, Ann chose $123$. In how many ways can they form the strings?

Since the question makes clear that repetition is permitted, each person can choose his or her string in $4^3$ ways. Hence, there are $(4^3)^3 = 4^9$ ways for Jack, Mary, and Ann to choose their strings.

Another way to see this is to observe that there are four choices for each of the nine entries, so the three strings can be selected in $4^9$ ways.

In how many ways can Jack, Mary, and Ann form the strings if each digit in $S$ has to appear in at least one string?

Suppose Jack chooses the string $(j_1, j_2, j_3)$, Mary chooses the string $(m_1, m_2, m_3)$, and Ann chooses the string $(a_1, a_2, a_3)$. Notice that we could concatenate the three strings of length $3$ into a single string of length $9$: $(j_1, j_2, j_3, m_1, m_2, m_3, a_1, a_2, a_3)$.

As stated above, we have four choices in $S$ for each of the nine entries, so there are $4^9$ ways to choose the strings. If we exclude $k$ of the four elements in $S$, the nine entries in the concatenated string can each be filled in $(4 - k)^9$ ways. Hence, the number of strings from which $k$ elements of $S$ have been excluded is $$\binom{4}{k}(4 - k)^9$$ Therefore, by the Inclusion-Exclusion Principle, the number of admissible strings is $$\sum_{k = 0}^{4} (-1)^k\binom{4}{k}(4 - k)^9 = \binom{4}{0}4^9 - \binom{4}{1}3^9 + \binom{4}{2}2^9 - \binom{4}{3}1^9 + \binom{4}{4}0^9$$

Where did you make your error?

You seem to have made the assumption that no two of the three people could choose the same string. However, Jack, Mary, and Ann can choose their strings independently.

• I really like your suggestion to think of the problem as concatenating the strings into one big string. – user123429842 Jun 12 '18 at 12:31


# Proving $x$ is a given quotient of logarithms

I'm practicing some questions on logarithms at the moment in order that I'm up to speed with the problem solving aspect before I embark on my PHD in chemical engineering at Boston college next year.

I've been studying the laws of logarithms and what I am to do when it is necessary to add and subtract logs. So for example, for the first part of the question that is causing me trouble, I would utilize the first law of logarithms.

My issue comes with the "Prove part", and ascertaining a value for $x$ when it is already involved in the first part of the question.

I know I'll most likely to be shut down for lack of evidence and research for this, but I would like to know what the most logical first step would be and which rule I need to follow. It doesn't follow on from the other questions I've been solving and so I can't problem solve it as easily as I can the rest.

Here is my question:

If $4^x\cdot 5^{3x+1}=10^{2x+1}$, prove that $x=\dfrac{\log(2)}{\log(5)}$.

Thanks, not a problem if I get shut down, I know this isn't in the spirit of the website and would normally never ask a question in such a manner.

• For the sake of typesetting, it is recommended to use MathJax and $\LaTeX$, or at the very least use parenthesis so that there is no confusion as to what is meant to go where. Visit this page for information on how to typeset mathematics here. Is the equation meant to be $4^x 5^{3x+1} = 10^{2x+1}$? Is it meant to be $4^{x5^{3x}+1}=10^{2x}+1$ is it meant to be $4^x5^3x+1 = 10^2x+1$ or something else entirely? – JMoravitz May 2 '16 at 1:27
• Seems like your question is perfectly in line with the spirit of the website. – M47145 May 2 '16 at 1:28
• Thanks for the link, I must learnt the code as it'll prove invaluable to me. The equation is the first you mentioned, with everything being a power apart from the 4, 5 and 10. Thanks, M47145. I'm sat here with an inordinate amount of maths textbooks open trying to get my head around this one! – New Zealand's finest May 2 '16 at 1:31

$$4^x\cdot 5^{3x+1} = 10^{2x+1}$$

$~~$Let us begin by moving everything to one side by dividing both sides by $10^{2x+1}$

$$\dfrac{4^x\cdot 5^{3x+1}}{10^{2x+1}} = 1$$

$~~$Let us separate the denominator using the fact that $10=2\cdot 5$

$$\dfrac{4^x\cdot 5^{3x+1}}{(2\cdot 5)^{2x+1}} = 1$$

$~~$Now, $(ab)^c = a^c\cdot b^c$

$$\dfrac{4^x\cdot 5^{3x+1}}{2^{2x+1}\cdot 5^{2x+1}} = 1$$

$~~$Let us factor out a factor of two from the denominator using $a^{b+c} = a^b\cdot a^c$

$$\dfrac{4^x\cdot 5^{3x+1}}{2^{2x}\cdot 2\cdot 5^{2x+1}}=1$$

$~~$Now, using $a^{bc} = (a^b)^c$ simplify $2^{2x}$ in terms of a power of four

$$\dfrac{4^x\cdot 5^{3x+1}}{4^x\cdot 2\cdot 5^{2x+1}}=1$$

$~~$Cancel like terms and use $\frac{a^b}{a^c} = a^{b-c}$

$$\frac{5^x}{2}=1$$

$~~$Multiply both sides by two

$$5^x=2$$

$~~$Take the logarithm of each side

$$\log(5^x)=\log(2)$$

$~~$Use the property of logarithms that $\log(a^b)=b\log(a)$

$$x\log(5)=\log(2)$$

$~~$Divide each side by $\log(5)$ to arrive at the desired result

$$x=\frac{\log(2)}{\log(5)}$$

• It is not recommended to use quotes for highlighting. – Ian Miller May 2 '16 at 2:04
• @IanMiller given the amount of text inbetween each step, I find it harder to quickly read without, but have edited it according to your suggestion. Hopefully this is a fine compromise. – JMoravitz May 2 '16 at 2:20
• I replaced your single \$equations with double \$\$equations to make it look more like your original. – Ian Miller May 2 '16 at 2:30 • JMoravitz, thank you for this excellent breakdown. It was invaluable to my learning, honestly. I really appreciate it. – New Zealand's finest May 5 '16 at 4:29 Alternative Solution: Recall the laws of logarithms: 1.$\log a^b = b \log a$. 2.$\log ab = \log a + \log b$3.$\log \frac{a}{b} = \log a - \log bBy taking the logarithm on both sides of your equation and applying rule (1), you will get: $$x \log 4 + (3x + 1) \log 5 = (2x + 1) \log (10)$$ which can be simplified to $$(\log 4 + 3 \log 5 – 2 \log 10) x = \log(10) - \log(5)$$ Using the mentioned rules, we get $$\log\left(\frac{4 \times 5^3}{10^2}\right) x = \log\left(\frac{10}{5}\right)$$ which establish your result. • Thank you FALAM for helping me to understand this utlizing the laws of logarithms, I really appreciate it. – New Zealand's finest May 5 '16 at 4:29 \begin{align} 4^x \cdot 5^{3x+1} & = 2^{2x} \cdot 5^{3x+1} \\ & = 2^{2x} \cdot 5^{2x} \cdot 5^{x+1} \\ & = 10^{2x} \cdot 5^{x+1} \end{align} If10^{2x} \cdot 5^{x+1} = 10^{2x+1}$(the hypothesis of the problem), then $$5^{x+1} = 10$$ $$5^x \cdot 5^1 = 10$$ $$5^x = 2$$ which gives us, by definition, $$x = \log_5 2$$ which can be rewritten, using$\log_b x = \frac{\log x}{\log b}\$, as

$$x = \frac{\log 2}{\log 5}$$

• Thanks for this version Brian, it was very helpful to me. – New Zealand's finest May 5 '16 at 4:30


# Largest positive integer $n$ for which $n^3+100$ is divisible by $n+10$

Problem: What is that largest positive integer $$n$$ for which $$n^3+100$$ is divisible by $$n+10$$?

The solution from Art of Problem Solving: If $$n+10 \mid n^3+100$$, $$\gcd(n^3+100,n+10)=n+10$$. Using the Euclidean algorithm, we have $$\gcd(n^3+100,n+10)= \gcd(-10n^2+100,n+10)$$ $$= \gcd(100n+100,n+10)$$ $$= \gcd(-900,n+10)$$, so $$n+10$$ must divide $$900$$. The greatest integer $$n$$ for which $$n+10$$ divides $$900$$ is $$\boxed{890}$$; we can double-check manually and we find that indeed $$900\mid 890^3+100$$.

Question: How can $$\gcd(n^3+100,n+10)= \gcd(-10n^2+100,n+10)$$ $$= \gcd(100n+100,n+10)$$?

• Are you aware $\gcd(A,B) = \gcd(A \pm kB, B)$ for any integer $k$? $n^3+100 = n^3 + 10n^2 - 10n^2 + 100= n^2(n+10) -10n^2 + 100$ so $\gcd(n^3 + 100, n+10) = \gcd(n^2(n+10)-10n^2 + 100, n+10) = \gcd(-10n^2 + 100, n+100)$. Sep 19 at 4:39

You know that $$n$$ 'acts as' $$-10$$, since

$$n \equiv -10 \quad \text { mod } (n + 10)$$ like so:

$$n^3 + 100 = (n + 10 - 10)n^2 + 100 = \underbrace{(n+10)n^2}_\text{multiple of (n+10)} - 10n^2 + 100$$ You could 'cheat' and plug in $$-10$$ immediately:

$$\gcd(n^3 + 100, n+ 10) = \gcd(-900, n+ 10)$$

• gcd(100n+100,n+10) here 100n+100 how I derived? Sep 19 at 1:04
• @ABIDB11152 like so: $$-10n^2 + 100 = -10(n+10-10)n + 100 = -10n(n+10) + 100n + 100$$ Sep 19 at 9:27

The Euklidean Algorithm uses the fact that $$\gcd(a,b)=\gcd(a+kb,b)$$ We choose $$k=-n^2$$ so that the $$n^3$$ term will be removed. We get $$\gcd(n^3+100,n+10)= \gcd((n^3+100)-n^2(n+10),n+10)$$ So we have $$=\gcd(-10n^2+100,n+10)$$ Now we remove $$n^2$$ by choosing $$k=10n$$ $$=\gcd((-10n^2+100)+10n(n+10),n+10)$$ $$=\gcd(100n+100,n+10)$$ and now for $$k=-100$$ we get $$=\gcd((100n+100)-100(n+10),n+10)$$ $$=\gcd(-900,n+10)$$

You can perform long division to obtain the following result, $$n^3+100=(n^2-10n+100)(n+10)-900$$ $$\implies n+10\mid 900$$ Therefore the largest value of $$n=890$$.

$$\gcd(A,B) = \gcd(A \pm kB, B)$$ for any integer $$k$$.

For example $$\gcd(58, 16) = \gcd(3\times 16 + 10, 16) = \gcd(3\times 16+ 10-3\times 16, 16) = \gcd(10, 16)$$.

......

So $$\gcd(n^3 + 100, n+10) = \gcd(n^2(n+10) - 10n^2 + 100, n+10)=$$

$$\gcd(n^2(n+10) - 10n^2 + 100- n^2(n+10), n+10)=\gcd(-10n^2 + 100, n+10)=$$

$$\gcd(-10n(n + 10) +100n + 100, n+10) = \gcd(-10n(n+10) + 10n + 100 + 10n(n+10), n+10)=$$

$$\gcd(10n + 100, n+10)$$


# Q: Is 0.9999… repeating really equal to 1?

Mathematician: Usually, in math, there are lots of ways of writing the same thing. For instance:

$\frac{1}{4}$ = $0.25$ = $\frac{1}{\frac{1}{1/4}}$ = $\frac{73}{292}$ = $(\int_{0}^{\infty} \frac{\sin(x)}{\pi x} dx)^2$

As it so happens, 0.9999… repeating is just another way of writing one. A slick way to see this is to use:

$0.9999... = (9*0.9999...) / 9 = ((10-1) 0.9999...) / 9$

$= (10*0.9999... - 0.9999...) / 9$

$= (9.9999... - 0.9999...) / 9$

$= (9 + 0.9999... - 0.9999...) / 9 = 9 / 9 = 1$

One.

Another approach, that makes it a bit clearer what is going on, is to consider limits. Let’s define:

$p_{1} = 0.9$
$p_{2} = 0.99$
$p_{3} = 0.999$
$p_{4} = 0.9999$

and so on.

Now, our number $0.9999...$ is bigger than $p_{n}$ for every n, since our number has an infinite number of 9’s, whereas $p_{n}$ always has a finite number, so we can write:

$p_{n} < 0.9999... \le 1$ for all n.

Taking 1 and subtracting all parts of the equation from it gives:

$1-p_{n} > 1-0.9999... \ge 0$

Then, we observe that:

$1 - p_{n} = 1 - 0.99...999 = 0.00...001 = \frac{1}{10^n}$
and hence

$\frac{1}{10^n} > 1-0.9999... \ge 0$.

But we can make the left hand side into as small a positive number as we like by making n sufficiently large. That implies that 1-0.9999… must be smaller than every positive number. At the same time though, it must also be at least as big as zero, since 0.9999… is clearly not bigger than 1. Hence, the only possibility is that

$1-0.9999... = 0$

and therefore that

$0.9999... = 1$.

What we see here is that 0.9999… is closer to 1 than any real number (since we showed that 1-0.9999… must be smaller than every positive number). This is intuitive given the infinitely repeating 9’s. But since there aren’t any numbers “between” 1 and all the real numbers less than 1, that means that 0.9999… can’t help but being exactly 1.

Update: As one commenter pointed out, I am assuming in this article certain things about 0.9999…. In particular, I am assuming that you already believe that it is a real number (or, if you like, that it has a few properties that we generally assume that numbers have). If you don’t believe this about 0.9999… or would like to see a discussion of these issues, you can check this out.

This entry was posted in -- By the Mathematician, Equations, Math. Bookmark the permalink.

### 58 Responses to Q: Is 0.9999… repeating really equal to 1?

1. mathman says:

lol. You need to know what each “number” is before you can take the difference. Of course the difference is zero, because you say that they are the same before you subtract, assuming you are subtracting “numbers”

2. George says:

While the mathematical argument presented here is correct, I have to say that limit and/or mathematical infinity introduced created confusions for many. What I wanted to say is that math infinity is really a math trick. It cannot and should not be real in the real physical world. Today this infinity plagues physics and its calculations. Maybe we need something new…

3. EMINEM says:

go to this https://www.youtube.com/watch?v=wsOXvQn3JuE site and get a mathematical proof that 0.999… doesnot equal to 1….

4. . says:

EMINEM, the video was published on April fool’s day for a reason!

5. Josh says:

The simplest explanation I’ve seen for this is:

1/3 = 0.333…
(1/3)3=1 = (0.333…)3=0.999…
1 = 0.999…

I don’t see how anyone can come to any conclusion other than 1=0.999… after understanding this.

6. mathman says:

1 / 3 = ??
Use long division. There are 2 parts: the quotient and the remainder. For every iteration.
First iteration is 0 R 1/3
Second is 0.3 R 1/30
Third is 0.33 R 1/300
Etc etc.

Where does your remainder go, Josh?
You don’t account for it anywhere.

The remainder is never zero.

How many 9’s are there in your 0.999… ?

7. Josh says:

Hey mathman,

There are infinitely many 9s in my “0.999…”. If it were anything other than infinitely many, then it would be less than 1.

And as far as 1/3, your “etc. etc.” is an infinitely repeating “0.333…”. I’m failing to see how anything you’ve presented has disproved what I said.

8. Angel says:

@mathman:

The remainder does not need to be accounted for. The division 1/3 is never formally complete if there is a remainder. Therefore, one must keep iterating the operations in long division. As the number of iterations n approaches infinity, the remainder does approach zero. That is what matters here. The number of 3s in the decimal expansion of 1/3 is infinite, and so is the number of 9s in the figure 0.999…


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# Thread: Evaluate the line integral

1. ## Evaluate the line integral

Hello,
please try to solve this question.
Thanks

2. $\oint_C x^3 dx - 2xy dy = \iint_D \left( \frac{\partial (-2xy)}{\partial x} - \frac{\partial x^3}{\partial y} \right) \ dA = -2\iint_D y \ dA =$ $- 2\int_0^2 \int_0^{2-x} y \ dy \ dx$

3. Evaluate the line integral $\int_C x^3 dx - 2xy dy$ where C comprises the three sides of a triangle joining O (0, 0) A (2, 0) and B (0, 2)
From O to A we have that y does not change so dy = 0.
$\int_{O to A} x^3 dx - 2xy dy = \int_0^2 x^3 dx$

$= \frac{1}{4}x^4|_0^2 = \frac{1}{4}2^4 = 4$

From A to B the path is the line from (2, 0) to (0, 2). This is a segment of the line y = -x + 2 as x goes from 2 to 0, so
$dy = -dx$

Thus
$\int_{(2,0) to (0,2)} x^3 dx - 2xy dy$

$= \int_2^0 x^3 dx + 2x(-x + 2)(-dx)$

$= \int_2^0 (x^3 + 2x^2 - 4x)dx$

$= \left ( \frac{1}{4}x^4 + \frac{2}{3}x^3 - 2x^2 \right ) _2^0$

$= - \left ( \frac{1}{4}2^4 + \frac{2}{3}2^3 - 2 \cdot 2^2 \right ) = - \left ( 4 + \frac{16}{3} - 8 \right )$

$= 4 - \frac{16}{3}$

So the overall integral will be:
$4 + 4 - \frac{16}{3} = 8 - \frac{16}{3} = \frac{8}{3}$

-Dan

4. Originally Posted by topsquark
From O to A we have that y does not change so dy = 0.
$\int_{O to A} x^3 dx - 2xy dy = \int_0^2 x^3 dx$

$= \frac{1}{4}x^4|_0^2 = \frac{1}{4}2^4 = 4$

From A to B the path is the line from (2, 0) to (0, 2). This is a segment of the line y = -x + 2 as x goes from 2 to 0, so
$dy = -dx$

Thus
$\int_{(2,0) to (0,2)} x^3 dx - 2xy dy$

$= \int_2^0 x^3 dx + 2x(-x + 2)(-dx)$

$= \int_2^0 (x^3 + 2x^2 - 4x)dx$

$= \left ( \frac{1}{4}x^4 + \frac{2}{3}x^3 - 2x^2 \right ) _2^0$

$= - \left ( \frac{1}{4}2^4 + \frac{2}{3}2^3 - 2 \cdot 2^2 \right ) = - \left ( 4 + \frac{16}{3} - 8 \right )$

$= 4 - \frac{16}{3}$

So the overall integral will be:
$4 + 4 - \frac{16}{3} = 8 - \frac{16}{3} = \frac{8}{3}$

-Dan
My answer is exactly negative of your answer if you do it out by Green's theorem! Maybe you did it clockwise.

5. Originally Posted by ThePerfectHacker
My answer is exactly negative of your answer if you do it out by Green's theorem! Maybe you did it clockwise.
I suppose I should have suspected this, but I didn't realize it was an integral over a closed area. I only did the integral from O to A to B. I never connected B to O.

So from B (0, 2) to O (0, 0) dx = 0, with x = 0. Thus
$\int_{(0, 2) to (0, 0)} x^3 dx - 2xy dy = \int_2^0 (-2 \cdot 0 \cdot y) dy = \int_2^0 0 dy = 0$

So my final integral is STILL $\frac{8}{3}$.

And if you look at the succession of points I did the integral in the counterclockwise sense. I can't argue with your result (though I was never very good at implementing Green's theorem anyway) but I can't find a mistake in my own work?

-Dan


# Integration with limits and options.

I found this exercise in an old exam but I don't know how to attack it because is a limit of an integration and I don't know if the limit affects the process of the integral or it makes it easier. The exercise is this:

The value of the limit $$\lim_{h\to 0}\frac{1}{h}\int^{2+h}_2\sqrt{1+t^3}dt$$ Is:

a) $0$

b) $1$

c) $\sqrt{2}$

d) $2$

e) $3$

I want to know two things:

1) If there is a method of how to solve limit-integral problems.

2) An explained solution of this example to at least try to understand how to solve exercises of the same style.

As $h \rightarrow 0$, we have $\displaystyle \int_2^{2+h} \sqrt{1+t^3} \ dt \rightarrow 0$. Thus, we arrive at a $0/0$ indeterminant form when trying to evaluate the limit, so we can apply L'Hopital's rule.

The hardest part about doing this is evaluating $\displaystyle \frac{d}{dh}\int_2^{2+h} \sqrt{1 + t^3} \ dt$. For a continuous function $f,$ the fundamental theorem tells us that $\displaystyle \frac{d}{dx} \int_a^x f(t) \ dt = f(x)$. To get our integral into this form, we can simply shift the function to the left $2$ units and adjust the bounds of integration accordingly: $$\displaystyle \frac{d}{dh}\int_2^{2+h} \sqrt{1 + t^3} \ dt = \frac{d}{dh} \int_0^h \sqrt{1 + (t+2)^3} \ dt = \sqrt{1+(h+2)^3}$$

• Yes, my answer as is in the exam was 3 because of that, but I don't know how affects L'Hopital to the integral. It lets the fuction inside the integrate just like it's now? (I mean $\sqrt{1+t^3}$) – MonsieurGalois Jun 24 '16 at 4:26
• @MonsieurGalois, added some detail – Kaj Hansen Jun 24 '16 at 4:32

Use L'Hopital's rule as follows: $$\lim_{h \to 0} \frac{\int_{2}^{2 + h} \sqrt{1 + t^3}dt}{h}$$ $$= \lim_{h \to 0} \frac{\sqrt{1 + (2 + h)^3}}{1}$$ $$= \boxed{3}.$$

And we are finished. Hope this helped!

You should recognize that for well behaved $f(x)$ and fixed $x$: $$\frac{1}{h}\int^{x+h}_x f(t)dt=\frac{1}{h}\left[f(x)h+R_h(x) \right]$$ where $\lim_{h\to 0}R_h(x)/h=0$. So: $$\lim_{h\to 0}\frac{1}{h}\int^{x+h}_x f(t)dt=f(x)$$

An alternate way of showing this is to observe that for a well behaved function on an interval containing $[x,x+h]$: $$h \times \min_{t\in[x,x+h]}f(t)\le\int^{x+h}_x f(t)dt\le h \times \max_{t\in[x,x+h]}f(t)$$ and when $h\to 0$ we have $\min_{t\in[x,x+h]}f(t)\to \max_{t\in[x,x+h]}f(t)\to f(x)$.

Hence as $f(x)=\sqrt{1+t^3}$ is well behaved around $t=2$: $$\lim_{h\to 0}\frac{1}{h}\int^{2+h}_2\sqrt{1+t^3}dt=\left.\sqrt{1+t^3}\right|_{t=2}=3$$

• Why $R_h(x)$ tends to 0? – MonsieurGalois Jun 24 '16 at 4:47
• There are a number of ways of showing this, as I have specified that $f(x)$ be well behaved we may take that to mean that $f(x)$ is differentiable on an interval containing $[x,x+h]$ when the given property follows from the Lagrange form of the remainder in the Taylor series up to the zeroth order term. – Conrad Turner Jun 24 '16 at 5:57

By the Mean Value Theorem for Definite Integrals, https://en.wikipedia.org/wiki/Mean_value_theorem#First_Mean_Value_Theorem_for_Definite_Integrals,

$\displaystyle\int_2^{2+h}\sqrt{1+t^3}dt=\left(\sqrt{1+c^3}\right)h$ for some c between 2 and $2+h$,

so $\displaystyle\lim_{h\to 0}\frac{1}{h}\int^{2+h}_2\sqrt{1+t^3}dt=\lim_{c\to 2}\sqrt{1+c^3}=\sqrt{9}=3$.

$\textbf{Alternate solution:}$

Let $\displaystyle G(x)=\int_2^x \sqrt{1+t^3}dt,\;$ so $G^{\prime}(x)=\sqrt{1+x^3}$ by the Fundamental Theorem of Calculus.

Then $\displaystyle\lim_{h\to 0}\frac{1}{h}\int^{2+h}_2\sqrt{1+t^3}dt=\lim_{h\to 0}\frac{G(2+h)-G(2)}{h}=G^{\prime}(2)=\sqrt{1+2^3}=3$

• How you get the $\sqrt{1+c^3}h$? – MonsieurGalois Jun 24 '16 at 4:45
• I have added some explanation to my answer. – user84413 Jun 24 '16 at 20:29

You need to use Fundamental Theorem of Calculus. Let $$F(x) = \int_{2}^{x}\sqrt{1 + t^{3}}\,dt = \int_{2}^{x}f(t)\,dt$$ where $f(x) = \sqrt{1 + x^{3}}$ and we need to calculate the limit $$\lim_{h \to 0}\frac{F(2 + h)}{h}$$ Note that $F(2) = 0$ and hence $$\lim_{h \to 0}\frac{F(2 + h)}{h} = \lim_{h \to 0}\frac{F(2 + h) - F(2)}{h} = F'(2)$$ provided the limit exists.

By Fundamental Theorem of Calculus we know that $F'(x)$ exists if the function $f(x) = \sqrt{1 + x^{3}}$ is continuous at $x$ and then $F'(x) = f(x)$. Thus $F'(2) = f(2) = 3$.

The use of L'Hospital's Rule is unnecessary and is a roundabout way to evaluate this limit.


But they are not a linear pair because they are not connected, and they do not share a common side. The line through points A, B and C is a straight line. Since the non-adjacent sides of a linear pair form a line, a linear pair of angles is always supplementary. Linear Pair: Definition, Theorem & Example ... we can say that angle NSI and angle ISD also form a pair of adjacent angles. In the adjoining figure, ∠AOC and ∠BOC are two adjacent angles whose non-common arms OA and OB are two opposite rays, i.e., BOA is a line. True, if they are adjacent and share a vertex and one side. Definition: Two angles that add up to 180°, Adjacent Angle – Definition, Examples & More. The vertex of an angle is the endpoint of the rays that form the sides of the angle. Example: Find the pairs of adjacent angles in the following figure. In Other Words, It Is Between A Right Angle And A Straight Angle. Start studying M54.2c. E. Both angles … In this article, we are going to discuss the definition of adjacent angles and vertical angles in detail. (ii) Angles in a linear pair which are (iii) Complementary angles that do not form a not supplementary. When two lines intersect, the angles in a linear pair have sides that are opposite rays; but the vertical angles formed by the two intersecting lines also have sides that are opposite rays. Solution: ∠ 1 and ∠ 2 are adjacent to each other∠ 2 and ∠ 3 are adjacent to each other, Solution: ∠ APD and ∠ DPC are adjacent to each other∠ DPC and ∠ CPB are adjacent to each other. If Two Angles Form A Linear Pair, The Angles Are Supplementary. In the figure above, the two angles ∠BAC and ∠CAD share a common side (the blue line segment AC). In the diagram below, ∠ABC and ∠DBE are supplementary since 30°+150°=180°, but they do not form a linear pair since they are not adjacent. Common side. Two Obtuse Angles Can Be Adjacent Angles. v. Angles which are neither complementary nor adjacent. At times, in geometry, the pair of angles are used. The two angles will change so that they always add to 180°, In the figure above, the two angles ∠PQR and ∠JKL are supplementary because they always add to 180°. If the two supplementary are adjacent to each other then they are called linear pair. A. D. The angles are congruent and are right angles. Consider the following figure in which a ray $$\overrightarrow{OP}$$ stand on the line segment $$\overline{AB}$$ as shown: The angles ∠POB and ∠POA are formed at O. Draw a linear pair of angles. A linear pair is two angles that add up to be 180o.A linear pair is two adjacent, supplementary angles.Adjacent means they share ONE ray.Supplementary means add … Two angles are said to be linear if they are adjacent angles formed by two intersecting lines. Non-common side makes a straight line or Sum of angles is 180°. The endpoints of the ray from the side of an angle are called the vertex of an angle. The angles in a linear pair are supplementary. Yes. O. Additionally, the ray OB is the common arm between these two angles. Adjacent Angles Are Two Angles That Share A Common Vertex, A Common Side, And No Common Interior Points. Vertically Opposite Angles: When two lines intersect, then the angles that are opposite one another at the intersection are called Vertically Opposite Angles. Two angles that overlap, one inside the other sharing a side and vertex in the figure on the right, the two angles ∠PSQ and ∠PSR overlap. Both sets (top and bottom) are supplementary but only the top ones are linear pairs because these ones are also adjacent. Example : Logical Equivalence Lateral Area . Pair of adjacent angles whose measures add up to form a straight angle is known as a linear pair. iv. See the first picture below. Related Questions to study. Another way of defining them is: “two angles that share a side and a vertex, but do not share any interior points”. Smenevacuundacy and 26 more users found this answer helpful Adjacent angles can be a complementary angle or supplementary angle when they share the common vertex and side. Example: If following angles make a linear pair, find the value of q. The measure of rotation of a ray, when it is rotated about its endpoint is known as the angle formed by the ray between its initial and final position. Two Adjacent Angle Can Be Complementary Too If They Add Up To 90°. In this picture there is a flatscreen Toshiba television. They share a common side, but not a common vertex. On the other hand, two right angles will always make a linear pair as their sum is equal to 180°. Two angles form a linear pair. Before you know all these pairs of angles there is another important concept which is called ‘angles on a straight line’. A Linear Pair Forms A Straight Angle Which Contains 180º, So You Have 2 Angles Whose Measures Add To 180, Which Means They Are Supplementary. Linear pair forms two supplementary angles. This television has a pair of supplementary angles that are not a linear pair (one green, one blue). Two adjacent angles forming a linear pair are in the ratio 7 : 5 find the angles. Solution: 130° + 50° = 180°Since the sum of these angles is equal to two right angles, so they can make a linear pair. This statement is correct. It can also be said that angles of the linear pair are always supplementary to each other. A: Angles that are next to each other are known as adjacent angles, i.e., two angles with one common arm. Two angles are said to be supplementary if the sum of both the angles is 180 degrees. Adjacent angles which are not in linear pair. Adjacent angles, Linear pair, Vertically opposite angles. In A Right Triangle, The Altitude From A Right-angled Vertex Will Split The Right Angle Into Two Adjacent Angle; 30°+60°, 40°+50°, Etc. The measure of a straight angle is 180 degrees, so a linear pair of angles must add up to 180 degrees. The two angles are said to be adjacent angles when they share the common vertex and side. Draw the pairs of angles as described below. Fill in the blanks: If two adjacent angles are supplementary, they form a _____. | Definition & Examples - Tutors.com In the figure, ∠1 and ∠3 are non-adjacent angles. If that is not possible, say why. linear pair. Let us take example of the angles, shown in following figure: ∠ COB and ∠ BOA have a common vertex, i.e. … A linear pair of angles has two defining characteristics: 1) the angles must be supplmentary; 2) The angles must be adjacent ; In the picture below, you can see two sets of angles. Although they share a common side (PS) and a common vertex (S), they are not considered adjacent angles when they overlap like this. 1 answer. Linear Pair of Angles. Two Adjacent Angle Can Be Supplementary Too, If They Add Upto 180°. (i v) When two lines intersect opposite angles are supplementary. Adjacent Angles Are Angles That Come Out Of The Same Vertex. Adjacent angles are angles that are side by side. Supplementary angles a and b do not form linear pair. Vertical Angles Are Always Congruent, Which Means That They Are Equal. Solution: 110° + 70° = 180°Since the sum of these angles is equal to two right angles, so they can make a linear pair. Can Two Obtuse Angles Be Adjacent? If two angles form a linear pair, the angles are supplementary. Note: ∠ APD and ∠ CPB are not adjacent to each other, because they don’t have a common arm in spite of having a common vertex. A pair of adjacent angles formed by intersecting lines is called a Linear Pair of Angles. Obviously, the larger angle ∠BAD is the sum of the two adjacent angles. If two angles are not adjacent, then they do not form a linear pair. They are supplementary because each angle is 90 degrees so they add up to 180 degrees. A linear pair of angles is formed when two lines intersect. To identify whether the angles are adjacent or not, we must remember its basic properties that are … How to Find Adjacent Angles. Draw a pair of vertically opposite angles. They also share a common vertex (point A). , ∠1 and ∠3 are non-adjacent angles that angles of the linear which. And a straight angle the two lines intersect opposite angles are not adjacent:! Angle adjacent angles which are not in a linear pair is the common vertex and edge but do not form a linear pair, there are two do... Adjacent angles whose measures add up to form a straight line ( with their )... Inverses statement Contrapositive statement conditional statement - edu … two angles do Overlap! 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# Exact division and geometric sequences

I imagine this problem has a name that I don't know; it's probably some sort of exact division problem. Here goes: imagine you have to divide 4 cookies among 11 people. Divide the cookies equally among the 11 people with the constraint that you are a World Champion at halving cookies. :-)

So naturally, you just halve all the cookies until there are more pieces than people, then repeat for each new size. With the (4, 11) case above, there are 4 pieces to start. Halve them all until there are 16 pieces and everyone eats one piece. There are five pieces left over, halve them all until there are 20, everyone eats one...you get the idea.

It's lovely that each person will get 1/4 + 1/16 + 1/32 + 1/64 + 1/256 + 1/4096 + 1/16384 + ... = 4/11 of the total.

But the pretty part -- for me! -- is that the sequence above is not geometric; it's actually five different geometric sequences. (That's how I see it, anyway.) It's 1/4 + 1/4096 + ... and 1/16 + 1/16384 + ... and so on; each of the five sequences has a common ratio of 1/1024, so it's straightforward to show the sum is exactly 4/11.

Here's the Mathematica part. With pen & paper, it took a little playing around to realize that the denominators of the first five terms above are 4, 16, 32, 64, 256, and then the structure of the problem repeats. At that point, the doubling process gives the same number of "remaining" pieces, so those five denominators are the foundation of the sum. Just to check, I computed

Total[
Sum[1/2^# (1/2^10)^n, {n, 0, Infinity}] & /@ {2, 4, 5, 6, 8}
]

The result is, in fact, 4/11. Sweet. Similarly, I tried two other cases: 3 cookies and 5 people as well as 5 cookies and 9 people. The pattern so far is this, where the formatting is [cookies, people] --> listOfPortions.

[3, 5] --> {1/2, 1/16, 1/32, 1/256, ...}

[5, 9] --> {1/2, 1/32, 1/64, 1/128, 1/2048, ...}

Each case "works," and each person would end up with c/p as the total amount. But despite having three examples, I don't see an explicit pattern, and I think there are few ways to describe the pattern. I could describe it in terms of the actual portions, or I could describe it with the exponents on each denominator. So at this point, I have three questions:

• Does this problem have a name?!

• Do you have hints or suggestions for a function like portions[c, p, n] that gives the first n terms of the sequence based on c cookies and p people?

• How would you present this problem to a group of students? What are your thoughts? What other functions or computations would you show them?

The logic is straightforward: double the current number of pieces until it exceeds the number of people, subtract the number of people from that doubled number, and repeat. But I'm not sure how to translate that into the terms of a sequence that will sum to c/p. This feels like a NestList[] or NestWhileList[] situation, but I don't have it yet.

• Tell them it's worms instead of cookies. Nobody will want them, and you won't spend now to infinity cutting things in half. – Daniel Lichtblau Sep 21 at 14:26
• @DanielLichtblau I figure I'm the one holding the knife...I can keep them as cookies and have them all myself. :-) – mathalicious Sep 22 at 0:46
• Holding the knife...yes, good point. – Daniel Lichtblau Sep 22 at 13:19

The following function gives the full solution to the problem:

cookieHalves[c_Integer, p_Integer] /;
CoprimeQ[2, p](* Euler's theorem only holds for 2 and p comprime *):=

Module[
{
ratio = 2^EulerPhi[p]
},
{1/ratio, DeleteCases[0]@NumberExpand[(ratio - 1) c/p, 2]/ratio}
]

(* {1/1024, {1/4, 1/16, 1/32, 1/64, 1/256}} *)

(* {1/16, {1/2, 1/16}} *)

(* {1/64, {1/2, 1/32, 1/64}} *)

The first number is the common ratio of the sequences, the list gives the starting numbers.

To get the first $$n$$ terms by "brute force", you can use the following:

cookieHalves[c_, p_, n_] := Most@NumberExpand[c/p*2^n, 2, n + 1]/2^n

(* {1/4, 0, 1/16, 1/32, 1/64, 0, 1/256, 0, 0, 0, 1/4096, 0} *)

(* {1/2, 0, 0, 1/16, 1/32, 0, 0, 1/256, 1/512, 0, 0, 1/4096} *)

(* {1/2, 0, 0, 0, 1/32, 1/64, 1/128, 0, 0, 0, 1/2048, 1/4096} *)

This is essentially using NumberExpand to get the list of fractions. Since the function is designed for integers, we expand $$\frac{c}{p}2^n$$ and divide the terms by $$2^n$$ again. The last term is the fractional remainder, which is why we drop it.

### Derivation of full solution

Here's how I approached this: For your 4/11 example, you have found that

\begin{align} \frac{4}{11}&=\left(\frac{1}{4}\frac{1}{1024}+\frac{1}{4}\frac{1}{1024^2}+\cdots\right)\\ &+\left(\frac{1}{16}\frac{1}{1024}+\frac{1}{16}\frac{1}{1024^2}+\cdots\right)\\ &+\left(\frac{1}{32}\frac{1}{1024}+\frac{1}{32}\frac{1}{1024^2}+\cdots\right)\\ &+\left(\frac{1}{64}\frac{1}{1024}+\frac{1}{64}\frac{1}{1024^2}+\cdots\right)\\ &+\left(\frac{1}{256}\frac{1}{1024}+\frac{1}{256}\frac{1}{1024^2}+\cdots\right) \end{align}

Sum[Sum[1/n 1/1024^i, {i, 0, ∞}], {n, {4, 16, 32, 64, 256}}]
(* 4/11 *)

This can be rewritten as a single geometric series with ratio $$1024$$:

\begin{align} \frac{4}{11}&=\left(\frac{1}{4}+\frac{1}{16}+\frac{1}{32}+\frac{1}{64}+\frac{1}{256}\right)\left(\frac{1}{1024}+\frac{1}{1024^2}+\cdots\right)\\ &=\frac{93}{256}\left(\frac{1}{1024}+\frac{1}{1024^2}+\cdots\right)\\ &=4\frac{93}{1024}\left(\frac{1}{1024}+\frac{1}{1024^2}+\cdots\right) \end{align}

1/4 + 1/16 + 1/32 + 1/64 + 1/256
(* 93/256 *)

So effectively, you have found a way to write $$\frac{1}{11}$$ as a geometric series with ratio $$2^{-n}$$. Knowing the formula for the geometric series, we can rewrite the above as

\begin{align} \frac{4}{11}&=4\frac{93}{1024}\frac{1}{1-\frac{1}{1024}}\\ &=4\frac{93}{1024}\frac{1024}{1024-1}\\ &=4\frac{93}{1024}\frac{1024}{1023}\\ &=4\frac{93}{1024}\frac{1024}{93\cdot11}\\ \end{align}

As you can see, the trick is to write $$11$$ as $$\frac{1023}{93}$$, or equivalently,

$$2^{10}-1=11\cdot 93$$

This means that we need to find $$n$$ such that

$$2^n-1\equiv0\ \mod p$$

where $$p$$ is the number of people. Luckily, this problem is already solved by Euler's theorem, which states that

$$2^{\varphi(p)}-1\equiv0\ \mod p$$

where $$\varphi(n)$$ is the totient function (EulerPhi).

The last remaining step is then to find the write the fraction (here $$\frac{4\cdot 93}{1024}$$) as sum of terms with the form $$\frac{1}{2^n}$$. The numerator of the fraction is in general given by

$$2^{\varphi(p)}\frac{c}{p}$$

(where $$c$$ is the number of cookies and $$p$$ the number of people), and this is an integer thanks to Euler's theorem. Rewriting this as a sum of powers of $$2$$ (using NumberExpand), we get the final result, which is implemented with the code at the top.

• This is really thorough, Lukas, and the link to Euler's theorem and EulerPhi[] is perfect. Thank you. – mathalicious Sep 21 at 13:37

It all boils down to a binary representation of 4/11, or n/m if you have n cookies and m people. This representation can have a finite number or infinite number of digits. You get e.g. 16 digits (starting from the first non-zero digit) of this representation by:

RealDigits[4/11, 2, 16]
(*{{1, 0, 1, 1, 1, 0, 1, 0, 0, 0, 1, 0, 1, 1, 1, 0}, -1}*)

meaning the first few digits of the binary representation are: 0.01011101000101110... This tells you, each person gets:

1/4 + 1/16 + 1/32 + 1/64 + ... cookies

• Oh jeez...yes, of course. It is precisely the binary representation. Thanks, Daniel. – mathalicious Sep 21 at 13:38