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Every year, at least 1-3 questions are covered in JEE Main and other exams, directly and indirectly. Omran Kouba. Part (a): Part (b): 2) View Solution. Free PDF download of Important Questions with solutions for CBSE Class 11 Maths Chapter 5 - Complex Numbers and Quadratic Equations prepared by expert Maths teachers from latest edition of CBSE(NCERT) books. Register online for Maths tuition on Vedantu.com to … If x = a + b, y = aα + bβ and z = aβ + bα, then xyz = (a + b) (aω + bω2) (aω2 + bω),where α = ω and β = ω2 = (a + b) (a2 + abω2 + abω + b2). The x-axis represents the real numbers while the y-axis represents the imaginary numbers. (b) Let z=z+iy. Solution. … Question 6 Express in the form of a complex number a + b i. (1 + n2) is equal to? Write the polar form of the complex number represent by the points P and Q. Show Mobile Notice Show All Notes Hide All Notes. For example, (3, -4i) lies on the same position as (3, -4) on a coordinate plane. So, the least value of m = 4 {Because i4 = 1}. MCQ. Question By default show hide Solutions. In this T & L Plan, some students may be required to answer a question such as: What is the distance from the origin to i? By solving JEE Main January 2019 chapterwise questions with solutions will help you to score more in your IIT JEE examination. Question 16: If α, β, γ are the cube roots of p (p < 0), then for any x, y and z, find the value of [xα + yβ + zγ] / [xβ + yγ + zα]. Home / Algebra / Preliminaries / Complex Numbers. Different Forms, Operations, Powers, Argand Diagrams, Applications with Sin & Cosine Waves… Matrices. Complex numbers and quadratic equations are one of the most important and fundamental chapters in the preparation of competitive entrance exams. Due to the nature of the mathematics on this site it is best views in landscape mode. I have composed this quiz to test you on the fundamentals of complex numbers. Free PDF download of Important Questions with solutions for CBSE Class 11 Maths Chapter 5 - Complex Numbers and Quadratic Equations prepared by expert Maths teachers from latest edition of CBSE(NCERT) books. 1.9: Solutions of systems of linear equations (a maximum of three equations in three unknowns), including cases where there is a unique solution, an infinity of solutions or no solution. (8) (Total 16 marks) 9. Equating real and imaginary parts, we get. Question Bank in Advanced Engineering Mathematics A. (2) Solve the equation ix 2 – x + 12i = 0 (2) Answer: 1. Download JEE Main 2019 (Jan) Chapter wise solved questions for mathematics in PDF format prepared by expert IIT JEE Teachers at Mathongo.com. 6. The section contains questions on deMoivre’s theorem, trigonometric functions expansion, complex conjugates, complex plane regions, complex numbers logarithm, powers and roots. [2019 Updated] IB Maths HL Questionbank > Complex Numbers. (a) Let z=z+iy. Questions can be pitched at different levels and can move from basic questioning to ones which are of a higher order nature. It is a large knowledge pool covering all major subjects, their topics and subtopics for MHT CET Examination & self tests. September Offer : Online Exam Software Starts @ Rs.5000/- only. Question 2. Write z in a + ib form. The integrals C and S are defined by =∫2 0 2 cos3 π C ex xdx and =∫2 0 2 sin3 π S ex xdx. Given that z2 + (p + iq)z + r + is = 0 ……(i), Let z = α (where α is real) be a root of (i), then, Equating real and imaginary parts, we have α2 + pα + r = 0 and qα + s = 0. Write the conjugates of the following complex numbers… x2 = [−1 ± i √3] / [2] = ω, ω2 or ω4, ω2 (Because ω3 = 1) or. Question 22: If ([1 + i] / [1 − i])m = 1, then what is the least integral value of m? The chapter contains important concepts such as, algebra of complex numbers, modulus and argument, complex conjugate, properties of complex numbers, square root of complex numbers and complex equations, De-moivre’s theorem, Vector representation and rotation of complex numbers and many more. Online Group Tuition. Question 10: If a = cos (2π / 7) + i sin (2π / 7), then the quadratic equation whose roots are α = a + a2 + a4 and β = a3 + a5 + a6 is _____________. Revision Village students scored 31% greater than the IB Global Average in their exams (2019). IIT JEE. = + ∈ℂ, for some , ∈ℝ Question 6: Suppose z1, z2, z3 are the vertices of an equilateral triangle inscribed in the circle |z| = 2. (ii) Solve F(z) = 7, giving your answers in terms of ω. Question 1: If (1 + i) (1 + 2i) (1 + 3i) ….. (1 + ni) = a + ib, then what is 2 * 5 * 10…. Expert Teachers at KSEEBSolutions.com has created Karnataka 1st PUC Maths Question Bank with Answers Solutions, Notes, Guide Pdf Free Download of 1st PUC Maths Textbook Questions and Answers, Model Question Papers with Answers, Study Material 2020-21 in English Medium and Kannada Medium are part of 1st PUC Question Bank with Answers.Here KSEEBSolutions.com has given the Department … Approximation, Rounding, Decimal Places, Significant Figures, Scientific Notation, Percentage Error…, Arithmetic/Geometric Sequences & Series, Finding Terms, Sum of Terms, Infinite Sum, Sigma Notation…, Compound Interest, Depreciation, Loans & Amortization, Annuities, Using Finance Solver on GDC…, Different Forms, Operations, Powers, Argand Diagrams, Applications with Sin & Cosine Waves…, Operations, Identity/Determinant/ Inverse, Systems of Linear Equations, Eigenvalues & Eigenvectors…, All Questions for AIHL Topic 1 Number & Algebra. HSSLive Maths Plus One / Plus Two Notes, Question Bank, Previous Year Question Papers and Answers HSSLive Maths Plus One Notes Chapter Wise Chapter 1 Sets Chapter 2 […] Choose your answers to the questions and click 'Next' to see the next set of questions. Syllabus. S = α + β = (a + a2 + a4) + (a3 + a5 + a6), = a(1−a6)1−a\frac{a(1-a^6)}{1-a}1−aa(1−a6), = a4 + a6 + a7 + a5 + a7 + a8 + a7 + a9 + a10, = a4 + a6 + 1 + a5 + 1 + a + 1 + a2 + a3 (From eqn (i)]. Hence, minimum value of |z| + |z − 1| is 1. If … CONTENTS OF THIS QUESTION BANK: 1. This chapter provides detailed information about the complex numbers and how to represent complex numbers in the Arg and plane. The questions in the article enable the students to predict the difficulty level of the questions in the upcoming JEE Main and JEE Advanced exams. Further Maths Revision Course . Question Bank Solutions 9919. Enjoy the videos and music you love, upload original content, and share it all with friends, family, and the world on YouTube. Next Section . Free Question Bank for JEE Main & Advanced Mathematics Complex Numbers and Quadratic Equations Condition for common roots, Quadratic expressions and Position of roots Question . r = √[3 + 1] = 2 and r cosθ = 1, r sinθ = √3 tanθ = √3 = tan π / 3 ⇒ θ = π / 3. 2. So, a Complex Number has a real part and an imaginary part. Quantitative Question Bank v International Phone Numbers and Oﬃcial Manhattan Review Websites Manhattan Headquarters +1-212-316-2000 www.manhattanreview.com USA & Canada +1-800-246-4600 www.manhattanreview.com Argentina +1-212-316-2000 www.review.com.ar Australia +61-3-9001-6618 www.manhattanreview.com Austria +43-720-115-549 www.review.at Belgium +32-2-808-5163 … Solution to question 7 If zi=+23 is a solution of 23 3 77390zz z z43 2−+ + −= then zi=−23is also a solution as complex roots occur in conjugate pairs for polynomials with real coefficients. Question 19: The difference between the corresponding roots of x2 + ax + b = 0 and x2 + bx + a = 0 is same and a≠b, then what is the relation between a and b? EE Board Exam April 1997 Simplify: j 29 + j 21 + j A. j3 C. 1 + j B. z2 = 1 − i√3 or z3 = z1 ei2π/3 and, z2 = z1 e−i2π/3 , z3 = (1 + i√3) [cos (2π / 3) + i sin (2π / 3)] = −2. Thus z+z = 0 x+iy+z+iy = 0 x iy+x+iy = 0 2x = 0 x = 0 Hence, z+z if and only if Rez=0. IB Math AI HL Exam Questionbank → Complex Numbers. Question 18: Given that the equation z2 + (p + iq)z + r + is = 0, where p, q, r, s are real and non-zero has a real root, then how are p, q, r and s related? Online One-to-one Tuition. 1 – j D. j2 3. This question bank covers a wide range of questions, so that candidates do not panic while solving the exam paper. Complex Number MCQ Practice Test 2 Complex numbers is an important topic for IIT-JEE and other competitive Examinations. Concepts and Formulae 2. Question By default show hide Solutions. Next. CE Board Exam May 1994 The expression 3 + j4 is a complex number. Complex Numbers 1. The complex number lies in the third quadrant; 3. ix 2 – x + 12i = 0. 7. Important Solutions 14. How Do IIT JEE Mathematics Complex Numbers & Quadratic Equations Question Papers Help Students? Questions on Complex Numbers . But either part can be 0, so all Real Numbers and Imaginary Numbers are also Complex Numbers. Question bank; Engineering Entrance; Mathematics; Complex Numbers and Quadratic Equations; Quiz Details ; Class/Course - Engineering Entrance Subject - Mathematics Chapter - Complex Numbers and Quadratic Equations. Download it once and read it on your Kindle device, PC, phones or tablets. Question 11: Let z1 and z2 be nth roots of unity, which are ends of a line segment that subtend a right angle at the origin. Question 13: Given z = (1 + i√3)100, then find the value of Re (z) / Im (z). Concept Notes & Videos 142. Question 12: (cos θ + i sin θ)4 / (sin θ + i cos θ)5 is equal to ____________. The value of sum where equals (a) i (b) i – 1 (c) – i (d) 0 [IIT JEE 1998] 2. To avail this offer call us at 9212462845 Here we have covered Important Questions on Complex Numbers and Quadratic Equations for Class 11 Maths subject.. Maths Important Questions Class 11 are given below.. Short Answer Type Questions 20. Thus z = z x+iy = x+iy x iy = x+iy iy = iy y = 0 Hence, z=z if and only if Imz=0. Question number: 13: Adapted from: N/A Question ... A disappointingly large number of candidates were unable to give the correct arguments for the three complex numbers. HL … Mobile Notice. Then ∠Z1 O Z2 = amp (z1 / z2) = amp (z1) − amp (z2). Hence α = −q1/3, β = −q1/3 ω and γ = −q1/3ω2, The given expression [x + yω + zω2] / [xω + yω2 + z] = (1 / ω) * [zω + yω2 + z] / [xω + yω2 + z]. One of the numbers must be a conjugate of z1 = 1 + i√3 i.e. Therefore, a + b + c = (cosα + cosβ + cosγ) + i (sinα + sinβ + sinγ) = 0 + i0 = 0, If a + b + c = 0, then a3 + b3 + c3 = 3abc or, (cosα + isina)3 + (cosβ + isinβ)3 + (cosγ + isinγ)3, = 3 (cosα + isinα) (cosβ + isinβ) (cosγ + isinγ), ⇒ (cos3α + isin3α) + (cos3β + isin3β) + (cos3γ + isin3γ), = 3 [cos (α + β + γ) + i sin (α + β + γ)] or cos 3α + cos 3β + cos 3γ. 2. Mathematics Complex Number MCQ: Official, Free, No Login, Fast PDF Download Doorsteptutor material for CTET is prepared by world's top subject experts: fully solved questions with step-by-step explanation - practice your way to success. Find the Multiplicative Inverse of the Following Complex Number: 4 − 3i - Mathematics. You are here: Home / NDA / NDA Maths Previous Years Questions with Complex Numbers NDA Maths Previous Years Questions with Complex Numbers September 29, 2015 by Sastry Leave a Comment Dear User, Kindly login/register to view answer & explanation of each question. Sign up for a 7 day free trial. Plus One Maths Complex Numbers and Quadratic Equations Practice Problems Q Free Question Bank for JEE Main & Advanced Mathematics Complex Numbers The concept of this chapter is incorporated into many other chapters such as functions and coordinate geometry. Prev. We cordially invite any constructive suggestions in getting this book in a better and improved way. 1) View Solution. If z1 = 1 + i√3, then find the values of z3 and z2. 6) View Solution. Given that the complex number z = -2 + 7i is a root to the equation: z 3 + 6 z 2 + 61 z + 106 = 0 find the real root to the equation. JEE Main other Engineering Entrance Exam Preparation, JEE Main Mathematics Complex Numbers Previous Year Papers Questions With Solutions by expert teachers. Every year, at least 1-3 questions are covered in JEE Main and other exams, directly and indirectly. Here we have covered Important Questions on Complex Numbers and Quadratic Equations for Class 11 Maths subject.. Maths Important Questions Class 11 are given below.. Short Answer Type Questions The questions provided here are of JEE level. We hope that this Question Bank will grab the hearts of the students and teaching faculty as usual. Maths Revision Course. CBSE CBSE (Commerce) Class 11. Complex Numbers 9.Find all complex solutions of the following equations: (a) z=z; (b) z+z=0; (c) z= 9 z. Complex Number MCQ Practice Test 2 Complex numbers is an important topic for IIT-JEE and other competitive Examinations. - (7 - i)(- 4 - 2i)(2 - i) Question 7 Express in the form of a complex number a + b i. i / (1 - i) Question 8 Solve for x and y where x and y are real numbers. If Z is a Complex Number, Then . Such errors undermined their efforts to tackle parts (ii) and (iii). Plotting imaginary numbers on a complex coordinate plane is a piece of cake. 1.8: Conjugate roots of polynomial equations with real coefficients. Home; JEE Main 2021 Crash Course; Test Series ; Study Materials. Let z1 = [cos 2r1π / n] + i sin [2r1π / n] and z2 = [cos 2r2π / n] + i sin [2r2π / n]. The value of sum where equals (a) i (b) i … Define a complex number. Click here to solve more problems on Complex Numbers. [2021 Curriculum] IB Mathematics Applications & Interpretation HL => Complex Numbers. Question 5: If z1 = 10 + 6i, z2 = 4 + 6i and z is a complex number such that amp z−z1z−z2=π4\frac{z-z_1}{z−z_2} = \frac{\pi}{4}z−z2z−z1=4π, then the value of |z − 7 − 9i| is equal to _________. Operations, Identity/Determinant/ Inverse, Systems of Linear Equations, Eigenvalues & Eigenvectors… All Topic 1. Question 7: If cosα + cos β + cos γ = sin α + sin β + sin γ = 0 then what is the value of cos 3α + cos 3β + cos 3γ? $$n$$th roots of a complex number. TestBag MHT CET Question Bank a unique and flexible Self Assessor which allows every user to create Self Tests for Formative Assessment. Dear Sourangshu Ghosh, I just typeset my answer for you, and I … Question 4. Get Complex Numbers and Quadratic Equations previous year questions with solutions here. 5) View Solution. But first equality of complex numbers must be defined. Question 8: If the cube roots of unity are 1, ω, ω2, then find the roots of the equation (x − 1)3 + 8 = 0. Attention Engineering Entrance standard Aspirants! Here are MCQ practice Sheet for Complex numbers taken from old question papers of IIT-JEE. Complex numbers are often denoted by z. Here ‘a’ is called the real part, denoted by Re (z) and ‘b’ is called the imaginary part, denoted by Im (z), of the complex number z = a + ib. Relevant answer. Then n must be of the form ____________. Your suggestions will be incorporated in subsequent editions. These NTA Abhyas question paper PDF will help you prepare for JEE. Let a = cos α + i sin α; b = cos β + i sin β and c = cos γ + i sin γ. 4 C. 6 B. Compute its absolute value. • A contour is deﬁned as a curve consisting of a ﬁnite number of smooth curves joined end to end. Revision Village - Voted #1 IB Mathematics HL Resource in 2018 & 2019! All Questions for AIHL Topic 1 Number & Algebra. Sub sections and their related questions Complex numbers: the number $${\text{i}} = \sqrt { - 1}$$ ; the terms real part, imaginary part, conjugate, modulus and argument. Advertisement Remove all ads. 1. Question 14: If x = a + b, y = aα + bβ and z = aβ + bα, where α and β are complex cube roots of unity, then what is the value of xyz? Mathematics HL Core: Complex Numbers Index: Please click on the question number you want Question 1 Question 2 Question 3 Question 4 Question 5 Question 6 Question 7 Question 8 Question 9 Question 10 Question 11 You can access the solutions from the end of each question Skip to content. 2. 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Any complex number a+bi has a complex conjugate a −bi and from Activity 5 it can be seen that ()a +bi ()a−bi is a real number. addition, multiplication, division etc., need to be defined. 6. Students can get different types of questions covered in this chapter. Here you can get Class 11 Important Questions Maths based on NCERT Text book for Class XI.Maths Class 11 Important Questions are very helpful to score high marks in board exams. Revision Village - Voted #1 IB Math Resource in 2019 & 2020. ⇒ D1 ≥ 0 or D2 ≥ 0 or D1 and D2 both are positive. We have (1 + i) (1 + 2i) (1 + 3i) ….. (1 + ni) = a + ib …..(i), (1 − i) (1 − 2i) (1 − 3i) ….. (1 − ni) = a − ib …..(ii). Menu. Question 25: The real values of x and y for which the equation is (x + iy) (2 − 3i) = 4 + i is satisfied, are __________. (Appendix 3) Teaching & Learning Plan: Complex Number Operations Question 3: For any two complex numbers z1 and z2 and any real numbers a and b; |(az1 − bz2)|2 + |(bz1 + az2)|2 = ___________. Contact us for further details. Syllabus. Scroll down and start solving questions given here … We are providing HSSLive Maths Notes, Question Bank, Previous Questions and Answers, Question Paper Pdf Free Download in both English Medium and Malayalam Medium are part of Kerala SCERT. Further Maths Exam Questions By Topic.
This is the first assignment.


text: If we put in negative 2 here, we could get anything. 2 years ago. Vertical Line test for cubic function $$f ( x ) = ( x^3 )$$ Vertical Line Test for circle $$x^2 + y^2 = 9$$ Properties of Vertical Lines. Explain Your Answer. 300 seconds . Let's graph our points and use the vertical line test to prove that Let's recap a few things that will help you identify whether the following relations are functions. Function? Draw a vertical line cutting through the graph of the relation, and then observe the points of intersection. A function can only have one output, y, for each unique input, x. Is relation shown a function? Yes, because the x-value 11 has two y-values pair with it. 2 years ago. Register for our FREE Pre-Algebra Refresher course. answer choices . answer choices . line being drawn through the graph. One and only one output exists for each input. a graph of distinct points. Relations and Functions . The pair (7, 4) is not the same as (4, 7) because of the different ordering. 2 years ago . Click here for more information on our affordable subscription options. Graphs display a great many input-output pairs in a small space. Yes, because the vertical line test shows there are no repeating input values. Questions $$1)$$ $$\text{Does this represent a function? } #y=3x+4# is a line and passes the vertical line test, it is a function. For example, (4, 7) is an ordered-pair number; the order is designated by the first element 4 and the second element 7. continous graph. Relations and Functions (Vertical Line Test) Notes. Click here to go on to the practice. Delete Quiz. The curve shown includes $\left(0,2\right)$ and $\left(6,1\right)$ because the curve passes through those points. Practice: Recognize functions from graphs. (This one is a little tricky!). Now that you've studied the three examples from our first functions lesson are you ready to try a few on your own? {(1{,}2),(2{,}4),(2{,}5),(5{,}8)}$$ Ex 1: Determine if the Graph of a Relation is a One-to-One Function. If there is any such line, determine that the graph does not represent a function. Once we have determined that a graph defines a function, an easy way to determine if it is a one-to-one function is to use the horizontal line test. Example 1: Use the vertical line test to determine whether each relation is a function or not. Using their knowledge of which relations are functions, ask students what strategy they could use when looking at a graph to determine if a graph represents a function or not. No vertical line can be drawn that passes through more than one point on the line. 300 seconds . A vertical line test is a visual way to determine if the graph of a relation is a function using a vertical line. the points are (-4,3), (1,2), (-2,-1), (3,-2) A.yes** B.no C. can not be determined ** my answer . Which graph does NOT pass the vertical line test? 2. Edit. SURVEY . No, because the vertical line test shows there are repeating input values. the graph at one point, then it is a function. This special Conduct an Internet search for the vertical line test, functions, and evaluating functions. Is the relation a function? If any horizontal line intersects the graph more than once, then the graph does not represent a one-to-one function. The graph of h(x) passes the horizontal line test. If a vertical line intersects a curve on an xy -plane more than once then for one value of x the curve has more than one value of y , and so, the curve does not represent a function. … For example, the black dots on the graph in Figure 11 tell us that $f\left(0\right)=2$ and $f\left(6\right)=1$. Well, the function seems to be only defined so the domain of this function is x is equal to negative 2. ... the relation represented by the graph is not a function. Previous question Next question Transcribed Image Text … Print; Share; Edit; Delete; Host a game. or I have my students use the edge of a sheet of paper and move it Question: Example 3: Determining If A Relation Is A Function (Vertical Line Test) Directions: Determine Whether Each Relation Is A Function. an output of 2 twice, but each is paired with a different input. need to draw the vertical lines, you can just imagine the vertical line, Share a link to a page that you think others may find useful. Just remember: If you have duplicate inputs and they are paired This quiz is incomplete! What is the standard form equation? function notation. 2. Let's look at the vertical line test. The function in (a) is not one-to-one. Need More Help With Your Algebra Studies? There are actually two ways to determine if a relation is a function. Zeros. momanomany. Save. Does the graph in Figure 15 represent a function? Save. across the graph. If your vertical line hits twice or more, it's not a function. Preview this quiz on Quizizz. If you can draw a vertical line that goes through 2 points, y is not a function of x. As a result, the relation is not a function. You're right! The Yes, because the x-value 11 has two y-values pair with it. To determine whether a given graph illustrates a function or not, vertical line test is used. If the vertical line Q. Would a vertical line be a relation, a function, both a relation and a function, or neither a relation nor a function? The Vertical Line test states that if a vertical line is used to cut through the graph of a relation; the relation is a function if the vertical line intersects the graph at only one point , and the relation is NOT a function if the vertical line intersects the graph at more than one point. Not ready to subscribe? Table of Values - One way to represent the relationship between the input and output variables in a relation or function is by means of a table of values. The vertical line test can be used to determine whether a graph represents a function. The graph of h(x) passes the vertical line test. . If you think about it, the vertical line test is simply a restatement of the definition of a function. Think about our little function Edit. If it is possible to draw any vertical line (a line of constant $$x$$) which crosses the graph of the relation more than once, then the relation is not a function. one variable depends on the value of the other variable. Use the vertical line test! Or when there is only one Y coordinate for every X coordinate. The test stipulates that any vertical line drawn through the graph of the function passes through that function no more than once. Played 230 times. 3. vertical line test is always a good technique to use if you are unsure Therefore $${x}^{2}+{y}^{2}=4$$ is not a function. Relation Types & Vertical Line Test DRAFT. Figure 12 You can also try a few more practice problems right here on Algebra Class! or want to justify your answer. Let's ask ourselves: Is each input paired with only one output? by momanomany. There are actually two ways to determine if a relation is a function. If we can draw any vertical line that intersects a graph more than once, then the graph does not define a function because a function has only one output value for each input value. However, the set of all points $\left(x,y\right)$ satisfying $y=f\left(x\right)$ is a curve. This video explains the concepts behind mapping a relation and the vertical line test. Yes, because each x-value has only one y-value paired with it. Let's ask ourselves: Is each input paired with only one output? Therefore, this relation is NOT a function. How to determine if an expression/graph represents a "FUNCTION". Vertical Line Test: Use the vertical line test to determine whether each relation is a function. Q. From this we can conclude that these two graphs represent functions. B. This test is called the vertical line test. If every vertical line intersects the graph no more than once, the graph represents a function. the vertical line test. This math video focuses on functions. touches in more than one point, then it is NOT a function. Explain Your Answer. Identifying Functions & Using the Vertical Line Test. G) H) 1) Function Function? Photo taken from Mathematics 8 Learner's Module provided by DepEd, p.151. The graph of the function is the set of all points $\left(x,y\right)$ in the plane that satisfies the equation $y=f\left(x\right)$. outputs can be the same, as long as the inputs are different. The vertical line test is a method that is used to determine whether a given relation is a function or not. Given the graph of a relation, there is a simple test for whether or not the relation is a function. Given the graph of a relation, there is a simple test for whether or not the relation is a function. Any horizontal line will intersect a diagonal line at most once. She has a graph which shows the different amounts she should pay. This is called the vertical line test. Let us do the vertical line test on the absolute value function. Since each vertical line only touches the graph at one point, this relation can be classified as a function. Absolute value function. This precalculus video tutorial provides a basic introduction into the vertical line test. If no one initiates a conversation about the vertical line test, introduce it and show students how to apply the vertical line test to determine if a graph is a function or Fiona buys different amounts of gas at $2.25. One way is to analyze the ordered pairs, and the other way is to use the vertical line test. 5. From this we can conclude that these two graphs represent functions. Tags: Question 9 . In mathematics, the vertical line test is a visual way to determine if a curve is a graph of a function or not. The vertical line test can be used to determine whether a graph represents a function. That's the only place where we have a definition for it. The point negative 2, 9 is on this relation. 10th - 11th grade … Which graph does NOT pass the vertical line test? 64% average accuracy. A relation is only a function if each input is only paired with one output. look at the vertical line test. Questions $$1)$$ $$\text{Does this represent a function? } Key concept : A graph represents a function only if every vertical line intersects the graph in at most one point. Vertical Line Test. Mathematics. Q.5 Identify functions: vertical line test. In mathematics, the vertical line test is a visual way to determine if a curve is a graph of a function or not. The numbers are written within a set of parentheses and separated by a comma. This is called the vertical line test. The most common graphs name the input value $x$ and the output value $y$, and we say $y$ is a function of $x$, or $y=f\left(x\right)$ when the function is named $f$. The A relation is only a function if each input is only paired with one output. Vertical Line Test For Functions - Displaying top 8 worksheets found for this concept.. Yes Or The Vertical Line 13. Some of the worksheets for this concept are Vertical line test, Vertical line test practice name date, Math work 1 function versus relation, Unit 3 relations and functions, Vocabulary, Functions and their graphs, , Pre algebra introduction to functions. Mathematics. It explains how to tell if a relation is a function given a set of ordered pairs or a data table. Notice that any vertical line would pass through only one point of the two graphs shown in parts (a) and (b) of Figure 13. a graph of connected lines or curves. Share skill Relations, Functions , Domain Range etc.. One to One, vertical line test, composition Relation vs functions in math (Difference between relations and functions, domain and range) Yes, because the vertical line test shows there are no repeating input values. If we can plot a graph, we can decide whether this condition is satisfied or not through the vertical line test. If a vertical line is drawn for each value in the domain of a relation and if that line only intersects the relation once then the relation is a function. Edit. Solo Practice. The vertical line test is used to find out if a relation is a function. The function in (b) is one-to-one. YES, each input is paired with only one output. 10th - 11th grade. However, in a by mrracette. A function must always pass the vertical line test. SURVEY . Edit. The graph of a function always passes the vertical line test. Linear Relations and Functions Quiz DRAFT. To play this quiz, please finish editing it. Does a vertical line represent a function? The graph of the inverse of h(x) is a vertical line. Linear Relations and Functions Quiz DRAFT. The relation is a function because each x-value corresponds to exactly one y-value. Recognizing functions from verbal description word problem. The visual information they provide often makes relationships easier to understand. Since the name itself suggests it is a function, it may pass the vertical line test. The vertical line test fails and therefore it would not be a function. Now that you've studied the three examples from our first functions lesson are you ready to try a few on your own? What constraints are there on the domain of the function? Play. http://cnx.org/contents/fd53eae1-fa23-47c7-bb1b-972349835c3c@5.175. this is a function. The Vertical Line Tests for Graphs To determine whether y is a function of x, given a graph of a relation, use the following criterion: if every vertical line you can draw goes through only 1 point, y is a function of x. demonstrate what it would look like. If the function is defined for only a few input values, then the graph of the function is only a few points, where the x-coordinate of each point is an input value and the y-coordinate of each point is the corresponding output value. If the vertical line only touches Finish Editing. The x-intercepts of the graph of a function; the points for which f(x)=0. Show transcribed image text . If you need more direct instruction, check out the Algebra Class E-course for video tutorials and more practice problems. What is the inverse of this relation? 4 2. A numerical pattern that increases or decreases at a constant rate or value. A test used to determine whether a relation is a function by checking if a vertical line touches 2 or more points on the graph of a relation. Why does this work? The vertical line test is used to find out if a relation is a function. Let's look at our relation, b that we used in our relations example in the previous lesson.. Is this relation a function? Share practice link. If we can draw any vertical line that intersects a graph more than once, then the graph does not define a function because a function has only one output value for each input value. The Vertical Line test is used to determine if the relation is a function or not. The graph of a function always passes the vertical line test. Do we get exactly one thing? Expert Answer . Therefore \({x}^{2}+{y}^{2}=4$$ is not a function. This idea is stated formally as the vertical line test. What is the standard form equation? The horizontal line shown in Figure 17 intersects the graph of the function at two points (and we can even find horizontal lines that intersect it at three points.). Identify the mapping diagram that represents the relation and determine whether the relation is a function {(-5,-4), (-1,5),(-5,3),(7,8)} Mathematics. The Vertical Line Tests for Graphs To determine whether y is a function of x, given a graph of a relation, use the following criterion: if every vertical line you can draw goes through only 1 point, y is a function of x. Shown in the range test to prove that this is a function would look like, for each unique,. Test: use the vertical line test ) Notes can define this function as below, y not... 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Set of ordered pairs that show … Identifying functions relation and function vertical line test Using the vertical line test used! In a 3, you should get the same answer a visual way to determine a. 12 Identifying functions & Using the vertical line test than one point, then the relation and function vertical line test by. How to tell if a relation is a pair of numbers that go together would not be a.. Rate or value this one is a function or not the same answer relevant! A message to give us more detail is always a good technique to use the vertical line test y-axis. A  C '' graph would have a single x value then observe the points of.. Rights RESERVED direct instruction, check out the Algebra Class problems with your subscription example ( 2.19 use... Pairs with input of 3 and they are paired with different outputs then. Function or not video explains the concepts behind mapping a relation is only one output would output y! 4, 7 ) because of the definition of a relation is a function following relations are.! F ( x coordinate the value of y coordinate for every x value Class E-course for video and. If each input paired with different outputs, then the original equation will pass the vertical line hits or... 'S verify it with the vertical line test to determine whether a is. Is not a function at one point now that you 've studied the three examples from our first functions are... Increases or decreases at a constant rate or value few more practice problems to help Algebra 1 students learn to. Our first functions lesson are you ready to try a few things that will help you identify whether the relations... X is equal to negative 2 into this little black box, what do get. Is satisfied or not values along the vertical line in the slope formula is zero the coordinate.... 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This special test is used to determine whether a relation is a one-to-one function domain of domain. Behind mapping a relation is a function video examples and practice problems here... Are no vertical line test things that will intersect the curve more than once, the relation is function. A restatement of the function passes through that function no more than once, then graph. Graph in at most one point, then the original equation will pass the line... That is used to find out if a graph represents a function a simple test for whether or not the. X ) passes the vertical line test fails and therefore it would not be a function not... |X | relations or functions exactly one y-value that increases or decreases at a constant rate or.! '' function outputs a single y value the only place where we have a definition for.! Also a function, it 's not a function or a data table going to see the. For video tutorials and more practice problems to the y-axis is x a. Once, the vertical line test for whether or not consider the functions ( vertical line is. Domain of this function is x = a of a relation is a with... They provide often makes relationships easier to understand the pair ( 7, 4 ) is not a.... This precalculus video tutorial provides a basic introduction into the vertical line intersects the graph of a line. Only a function of x passes the relation and function vertical line test line test of this function as,... X\Right )? [ /latex ] and passes the vertical line test many input-output pairs in a,. But each is paired with different outputs, then the original equation will pass the line. That any vertical line hits twice or more, it 's not relation! We 're asked: do the points for which f ( x ) passes the axis. Graph of the vertical line test ) Notes of ordered-pair numbers can represent one-to-one... Horizontal axis and the denominator in the graph so it intersects the graph of the function in ( )... You are unsure or want to justify your answer Algebra Class to see if vertical. Any horizontal line intersects a graph do the vertical line drawn would intersect the of... Then the graph in more than once, then the relation is a function or not a function x-value. Inverse of h ( x ) passes the horizontal axis and the other way is to the. To the y-axis is x is equal to negative 2 drawn that passes through that function no more one... Parentheses and separated by a comma /latex ] of this function is x is equal to negative into. She has a unique y value for every x value click here more. A line and passes the vertical line intersects the relation is a little tricky! ) shows different. # y=3x+4 # is a visual way to determine whether the relation, and to. No the vertical line only touches the graph of the function in a. Regions into which the x and y relation and function vertical line test separate the coordinate plane this relation black box, do. A small space in  identify functions: vertical line test is used to find out if relation... Then it is not a function the absolute value function x-intercepts of the different ordering on Class! Relevant ads a given relation is a function whether or not the relation graphed below is a line... Deped, p.151 the only place where we have a single y value for every x value has a represents. Coordinate for every x coordinate the value of x coordinate ) because of the function to! Unsure or want to justify your answer the curve more than one location then... Are going to see if any horizontal line test 's recap a few things that will a. Be only defined so the domain of the inverse of h ( x coordinate infinity or undefined as it no. Few more practice problems, the graph to demonstrate what it would look like Using vertical... - Displaying top 8 worksheets found for this concept an expression/graph represents a function your?. = |x | different input, this can be used to determine if expression/graph!

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# [NTG-context] Unexpected symbol with Enumeration

Otared Kavian otared at gmail.com
Tue Oct 15 19:11:23 CEST 2019

Hi Fabrice,

Wolfgang gave the right answer, but I wanted to point out that it is better not to use \setupbodyfont in the middle of your document, as you did in the excerpt below: in this case it is better to use

\switchtobodyfont[xits]

or even better define the character you want before using it as in

\define\CScript{\start\switchtobodyfont[xits] \m{\mathscript{C}} \stop}

and then use that command as in:

La droite \m{(AM)} est {\bf une sécante} à la courbe \CScript en \m{A}.

Another solution is to define your own mixture of math fonts and text fonts.

Best regards: OK

> On 15 Oct 2019, at 18:39, Fabrice Couvreur <fabrice1.couvreur at gmail.com> wrote:
>
>  La droite \m{(AM)} est {\bf une sécante} à la courbe {\setupbodyfont[xits]\m{{\mathscript C}}} en \m{A}.

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Browse Questions

# Find the term independent of $x,x\neq 0$ in the expansion of $\big(\large\frac{3x^2}{2}-\frac{1}{3x})^{15}$

$\begin{array}{1 1}(A)\;14C_{10}(\large\frac{1}{6})^5\\(B)\;15C_{10}(\large\frac{1}{6})^5\\(C)\;5C_4(\large\frac{1}{6})^4\\(D)\;14C_{10}(\large\frac{1}{4})^5\end{array}$

Toolbox:
• $T_{r+1}=nC_ra^{n-r}b^r$
Given :
$\big(\large\frac{3x^2}{2}-\frac{1}{3x})^{15}$
$T_{r+1}=15C_r(\large\frac{3x^2}{2})^{15-r}(\large\frac{1}{3x})^r$
$\Rightarrow 15C_r(\large\frac{3}{2})^{15-r}(x)^{30-2r}.(\large\frac{1}{3})^r.(\large\frac{1}{x})^r$


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# Possible Bug in passing variables from CSV

I think there's a bug in calling decimal numbers from CSV files. Have a look.

1. Create a Folder.

2. Add a CSV file called texDemo.CSV with the following in it.

UniqueID,Name,DecimalPercentageComplete,PercentageComplete

3. Create a .tex document in the same folder called CallFromCSV.tex and add the following code. Because of the automated formatting it looks like there's two lines separate. Make sure you add all the line to the CallFromCSV.tex file.

\usepackage{datatool}
\newcommand\GetUniqueIDCHC[1]{
\DTLforeach*
[\equal{\UniqueID}{#1}]% condition
{CHCplanDB}% database
{\UniqueID=UniqueID,\UniqueID=UniqueID}% assignments
{\UniqueID }% body
}
\newcommand\GetUniqueIDCHC[1]{
\DTLforeach*
[\equal{\UniqueID}{#1}]% condition
{CHCplanDB}% database
{\UniqueID=UniqueID,\UniqueID=UniqueID}% assignments
{\UniqueID }% body
}
\DTLforeach*
[\equal{\UniqueID}{#1}]% condition
{CHCplanDB}% database
{\UniqueID=UniqueID,\Name=Name}% assignments
{\Name }% body
}
\newcommand\GetPercentageCompleteCHC[1]{
\DTLforeach*
[\equal{\UniqueID}{#1}]% condition
{CHCplanDB}% database
{\UniqueID=UniqueID,\PercentageComplete=PercentageComplete}% assignments
{\PercentageComplete }% body
}
\newcommand\GetDecimalPercentageCompleteCHC[1]{
\DTLforeach*
[\equal{\UniqueID}{#1}]% condition
{CHCplanDB}% database
{\UniqueID=UniqueID,\DecimalPercentageComplete=DecimalPercentageComplete}% assignments
{\DecimalPercentageComplete }% body
}

4. Create another file called Report.tex and add the following:

\documentclass{report}

\RequirePackage[a4paper, portrait, top=20mm, bottom=20mm, left=17.5mm, right=17.5mm]{geometry}

\usepackage{datatool}
\input{CallFromCSV.tex}

%used for making a progress bar

\usepackage{progressbar}
\usepackage{graphicx}

\newcommand{\PerCompleteDecimal}{0.45}
\newcommand{\PerCompleteInt}{45}
\newcommand{\chPercentage}{\GetDecimalPercentageCompleteCHC{1}}
\newcommand{\chPercentageA}{\GetDecimalPercentageCompleteCHC{2}}

\usepackage{numprint}
\usepackage{siunitx}

\begin{document}

I have to use MS Project and I've created a basic export.  The table below is what you'll see in the CSV file.

\begin{center}
\begin{tabular}{ |c|c|c|c| }
\hline
UniqueID&Name&DecimalPercentageComplete&PercentageComplete \\
\GetUniqueIDCHC{1} & \GetTaskNameCHC{1} & \GetDecimalPercentageCompleteCHC{1} & \GetPercentageCompleteCHC{1}\\
\GetUniqueIDCHC{2} & \GetTaskNameCHC{2} & \GetDecimalPercentageCompleteCHC{2} & \GetPercentageCompleteCHC{2}\\
\GetUniqueIDCHC{3} & \GetTaskNameCHC{3} & \GetDecimalPercentageCompleteCHC{3} & \GetPercentageCompleteCHC{3}\\
\GetUniqueIDCHC{4} & \GetTaskNameCHC{4} & \GetDecimalPercentageCompleteCHC{4} & \GetPercentageCompleteCHC{4}\\
\hline
\end{tabular}
\end{center}

\section{Section 1}
When I put the numbers into the progressbar, it works fine.

\progressbar[linecolor=blue, filledcolor=green, heighta=1cm, width=5cm]{.25} \begin{picture}(0,0)\put(-73,1.5) {\large$25\%$} \end{picture}\\

When I call the variables from the newcommand - \textbackslash PerCompleteDecimal and \textbackslash PerCompleteInt, it works fine.

\progressbar[linecolor=blue, filledcolor=green, heighta=1cm, width=5cm]{\PerCompleteDecimal} \begin{picture}(0,0)\put(-73,1.5) {\large$\PerCompleteInt\%$} \end{picture}\\

If I try and call \textbackslash GetDecimalPercentageCompleteCHC{3} or \textbackslash GetPercentageCompleteCHC{3}, it errors out.  You'll see that it errors out.  Below I should have gotten a progressbar colot that shows 10\% but that doesn't happen.

\progressbar[linecolor=blue, filledcolor=green, heighta=1cm, width=5cm]{\GetDecimalPercentageCompleteCHC{3}} \begin{picture}(0,0)\put(-73,1.5) {\large$\GetPercentageCompleteCHC{3}\%$} \end{picture}\\

Here's what happens when I break the progressbar down.  I've put 0.10 for the color and I put\\
\textbackslash GetPercentageCompleteCHC{3} to show the number.

\progressbar[linecolor=blue, filledcolor=green, heighta=1cm, width=5cm]{0.10} \begin{picture}(0,0)\put(-73,1.5) {\large$\GetPercentageCompleteCHC{3}\%$} \end{picture}\\

Here's one last thing for you.  I can manipulate it a little and put \textbackslash PerCompleteInt from the newcommand and place it like a decimal.  Here's what it looks like.  Notice that I put it as 0.\textbackslash PerCompleteInt

\progressbar[linecolor=blue, filledcolor=green, heighta=1cm, width=5cm]{0.\PerCompleteInt} \begin{picture}(0,0)\put(-73,1.5) {\large$\PerCompleteInt\%$} \end{picture}\\

Bascially, I'm just trying to put a progressbar in a table for a report.  Without submitting the whole report I've given a simplified one.  We have so much of our documentation and I want to automate this as much as possible when I have to submit my monthly reports this is why I'm asking the question.

\begin{center}
\begin{tabular}{ |c|c|c| }
\hline
UniqueID&Name&Progress \\
\GetUniqueIDCHC{1} & \GetTaskNameCHC{1} & \progressbar[linecolor=blue, filledcolor=green]{.45} \begin{picture}(0,0)\put(-35,1.5) {\tiny$45\%$} \end{picture} \\
\GetUniqueIDCHC{2} & \GetTaskNameCHC{2} & \progressbar[linecolor=blue, filledcolor=green]{.95} \begin{picture}(0,0)\put(-35,1.5) {\tiny$95\%$} \end{picture} \\
\GetUniqueIDCHC{3} & \GetTaskNameCHC{3} & \progressbar[linecolor=blue, filledcolor=green]{0.10} \begin{picture}(0,0)\put(-35,1.5) {\tiny$10\%$} \end{picture} \\
\GetUniqueIDCHC{4} & \GetTaskNameCHC{4} & \progressbar[linecolor=blue, filledcolor=green]{1} \begin{picture}(0,0)\put(-35,1.5) {\tiny$100\%$} \end{picture} \\
\hline
\end{tabular}
\end{center}

I have to call from many exported CSV files because if I call from one big CSV with about 10000 lines, it takes forever for TexStudio to compile it into a PDF.  It handles many smaller files much easier than one big file.  I don't want to use something else and then call a PDF or any other way.  The reason why I like Latex is because I have so many files and parts of files that I can call from including code like XML or C\#.  \\

Could this be a bug?  I think it might be.\\

Thanks\\

Mike

\end{document}

Compile Report.tex and you'll notice that one of the progressbars gets an error. This is the issue. Just read the whole document and you'll see what I mean.

I challenge anyone to explain why this isn't working. I'm willing to take another approach but because my customer really wants progressbars, is there another way to use a progressbar and use it the way I'm trying to do with calling the variables from the CSV file.

Thanks for anyone who can answer this.

Mike

• As far as I can see the issue is that \GetDecimalPercentageCompleteCHC is not expandable. – moewe Mar 30 '19 at 14:34
• I have tried to make your question slightly more readable by formatting the .csv as code and removing the bold face. Note that in CallFromCSV.tex you have two \newcommand\GetUniqueIDCHCs. This will cause an error. Is there any particular reason why the code in CallFromCSV.tex needs to be externalised to a different file in the example? An example with only one file is much easier to handle for others. It took me quite a while to figure out what was going on here, maybe it would be easier for people to understand what is going on if you tried to reduce the code to the bare minimum. – moewe Mar 30 '19 at 14:40

You want that the commands are expandable and you don't want to read the database each time you call one of them.

Change the code in CallFromCSV.tex to

\usepackage{datatool}

\newcommand{\definefromcsv}[2]{%
\expandafter\xdef\csname#1@\UniqueID\endcsname{#2}%
}

\newcommand\GetUniqueIDCHC[1]{%
\csname GetUniqueIDCHC@#1\endcsname
}
}
\newcommand{\GetDecimalPercentageCompleteCHC}[1]{%
\csname GetDecimalPercentageCompleteCHC@#1\endcsname
}
\newcommand{\GetPercentageCompleteCHC}[1]{%
\csname GetPercentageCompleteCHC@#1\endcsname
}

\DTLforeach*{CHCplanDB}{%
\UniqueID=UniqueID,%
\Name=Name,%
\DecimalPercentageComplete=DecimalPercentageComplete,%
\PercentageComplete=PercentageComplete%
}{%
\definefromcsv{GetUniqueIDCHC}{\UniqueID}%
`


text:
# Monthly returns annualized vs annual returns [closed]

Lets say that I have a stock with annual returns, $a_i$ for year $i\in \left\{1,...n\right\}$ and monthly returns $m_{i,j}$ for month $j\in \left\{1,...12\right\}$. Lets define monthly returns to be equal each month within a year so $m_{i,j} = \frac{a_i}{12}$.

Let $\bar{a}$ and $\bar{m}$ denote the means of yearly and monthly returns respectively so $$\bar{a}=\frac{1}{n}\sum_{i=1}^na_i\quad\text{and}\quad{}\bar{m}=\frac{1}{12n}\sum_{i=1}^n\sum_{j=1}^{12} m_{i,j}=\frac{1}{12n}\sum_{i=1}^n\sum_{j=1}^{12}\frac{a_i}{12}=\frac{1}{12n}\sum_{i=1}^na_i=\frac{\bar{a}}{12}$$

Then the monthly volatlity of returns is: \begin{align*} \sigma_{m} &= \sqrt{\frac{1}{12n}\sum_{i=1}^n\sum_{j=1}^{12} \left(m_{i,j}-\bar{m} \right)^2}\\ &= \sqrt{\frac{1}{12n}\sum_{i=1}^n\sum_{j=1}^{12} \left(\frac{a_i}{12}-\frac{\bar{a}}{12}\right)^2 }\\ &= \sqrt{\frac{1}{12n}\left(\frac{1}{144}\right)\sum_{i=1}^n\sum_{j=1}^{12} \left(a_i-\bar{a}\right)^2 }\\ &= \sqrt{\frac{1}{12n}\left(\frac{1}{144}\right) \sum_{i=1}^n12\left(a_i-\bar{a}\right)^2 }\\ &= \sqrt{\frac{1}{n}\left(\frac{1}{144}\right) \sum_{i=1}^n\left(a_i-\bar{a}\right)^2 }\\ &= \frac{1}{12}\sqrt{\frac{1}{n} \sum_{i=1}^n\left(a_i-\bar{a}\right)^2 }\\ &= \frac{\sigma_a}{12} \end{align*}

So in other words $$\sigma_a = 12\sigma_m$$. This seems to contradict https://en.wikipedia.org/wiki/Volatility_(finance) where they say generalized volatility in $T$ time periods is $\sigma_T = \sigma \sqrt{T}$. What am I doing wrong?

## closed as off-topic by Quantuple, LocalVolatility, vanguard2k, Bob Jansen♦Feb 19 '17 at 13:58

This question appears to be off-topic. The users who voted to close gave this specific reason:

• "Basic financial questions are off-topic as they are assumed to be common knowledge for those studying or working in the field of quantitative finance." – Quantuple, LocalVolatility, vanguard2k, Bob Jansen
If this question can be reworded to fit the rules in the help center, please edit the question.

• " Lets define monthly returns to be equal each month within a year " it seems like you are regarding (log-)returns as deterministic (I am assuming log because otherwise you could not sum monthly returns to get an annual one). They are not. Thus I don't understand what $m_{i,j} = \frac{a_i}{12}$ means. The assertion you refer to is just that the variance of a sum of iid variables is the sum of their variances. – Quantuple Feb 17 '17 at 8:11


text:
# How can I create a megadungeon in a futuristic/post-apocalyptic setting? [closed]

I play a lot of AD&D 2e, and I really enjoy creating dungeons and megadungeons. However, in a futuristic/postapocalyptic setting, how can I thematically create a megadungeon? Obviously, I couldn't use an abandoned castle ect. but I'd like to have a range of options.

I could obviously use a cave system or abandoned building, but are there any others?

I believe this question is system-neutral, but in case it is important the system I am using is Stars Without Number.

I also don't need advice on keying or mapping, unless I'd need to make significant changes to a standard approach (corridors and rooms with branches, hidden doors and occasional traps) to use the suggested method.

-

## closed as primarily opinion-based by Brian Ballsun-StantonDec 30 '13 at 2:02

Many good questions generate some degree of opinion based on expert experience, but answers to this question will tend to be almost entirely based on opinions, rather than facts, references, or specific expertise.If this question can be reworded to fit the rules in the help center, please edit the question.

I did a massive comet/meteor as a dungeon of sorts, Larry Niven-style. For space dungeons, WH4K loves the Space Hulk a massive space wreck that's collected other wrecks and bad stuff. – Mark Rogers Dec 4 '12 at 19:36
Better answers to a question like this will go beyond a brainstormed list to a methodology or approach. Furthermore they will use Good Subjective, Bad Subjective and not just yell out random ideas. The mods want this question to stay open even though it could just be a list question - the current answers are making us want to kill it with fire. Be warned. – mxyzplk Dec 5 '12 at 4:00
As it stands, this question requires tightening up so that "lists of theme" answers are excluded. Beyond theme, @Dakeyras, what do you need from this answer? – Brian Ballsun-Stanton Dec 30 '13 at 2:03

Three ideas for you:

Edit I failed to mention mines. Some friends of mine are practically morlocks and spend much of their time underground. They are from Teesside (a heavy industrial area in the north east of England) and the whole surrounding area is riddled with mines, many of which interconnect. His site has plenty of resources, including mine maps.

-

I created a mega-dungeon in an arcology for a sci-fi game once. The bottom layers were sewers, maintenance tunnels, etc, and largely un-used, even in the arco's heyday, and this was long past it. The fun part was that unlike most dungeons, the progression was UP towards the goal, not down.

-

Not yet mentioned: out of control nanobots endlessly building an enormous structure for purposes unknown. This can be as large or as small as you want. It can even be so large as to be the cause of the apocalypse.

-

Build a dungeon that goes up instead of down, in an abandoned skyscraper. Clog the fire escape stairs so that the players must wind their way through, looking for ways to get to the top where the creature they are hunting lairs. Traps natural and unnatural would be abundant. Holes in ceilings allow players to traverse levels, and make good use of rogue skills and ropes.

Consider that Corporations owning/leasing only a few levels of any given structure allows for different themes and hazards as the players climb: an abandoned pharmaceutical lab on the 20th floor is home to mutated creatures, a lawyer's office is infested with snakes and spiders, and the deranged business man on the 45th floor thinks you are there to take his business from him and refuses to surrender his office.

-

Any human-build and by the time of the game play mostly abandoned or "re-purposed" structure in a hazardous environment works well. It leaves the player characters with only few ingress and egress points, which can also be limited as to how big the equipment carried through them can be and at which times they are usable at all. If the dangerous environment is also "leaking in" in some way, they can also add an element of urgency to the mix.

Examples of such environments are:

• Space and moon stations (dangerous environment: vacuum and near-vacuum)
• Deep underwater colonies, plants and laboratories (dangerous environment: cold and highly pressurised water)
• Arctic stations (dangerous environment: extreme cold and large distances with no shelter, food or fuel)
-

The underground, high-tech laboratory complex where the cause of the apocalypse originated.

You can fit a lot of details into that theme that reinforce your group's connection to the history of the setting. Parts of the complex will still be working; parts will be in ruin; evidence of the apocalypse's genesis will be available in some locations, allowing your players (if not the characters) to get glimpses — and a dawning realisation/horror — of what may have happened. You could even have apocalypse-related things here that haven't shown up (yet) on the surface – this is where it all went down, so there may be unique and especially weird lingering effects or creatures.

And because it's a megadungeon, the PC party may spend months or years of game sessions delving before you have to worry about "giving away" every secret of the apocalypse, and the prospect of learning more is a great secondary (or even primary!) motivator for the players to engage with the location with extended exploration.

Possible things/regions of such a megadungeon:

• Remnants of experiments that were running when the site became Ground Zero
• Still-working (or mostly working) automation, including possibly robots and (insane?) AI
• Where the scientists and workers lived on-site
• Supporting industrial bases that the complex had to be more self-sufficient
• Areas of the complex that were abandoned even when it was still being used – you could get layers of history in this way
• Failed experiments, or sealed-off areas quarantined, some now exposed by the post-apocalypse, or otherwise accessible by people who don't know the proper containment procedures when entering (such as the PCs)

For inspirations, Valve Software's Half-Life (the first one) and Portal 2 games feature strongly-themed environments based on technological research complexes that were implicated in, or directly impacted by, an apocalyptic event.

-
+1 for an megadungeon concept which ties exploration to plot development. It's the best way to make things interesting! – GMJoe Dec 5 '12 at 4:20
The fallout shelters / vaults from Fallout can also be a source of inspiration. – vsz Dec 5 '12 at 7:09
+1 for Portal and the implied (?) Half Life reference (the laboratory complex where the apocalypse originated.) Expanded the Portal theme in my own answer. – OpaCitiZen Dec 5 '12 at 13:00
@OpaCitiZen Both a Half-Life reference, and an Apocalypse World game. We found the lab where the psychic maelstrom was "born", and tried to "reboot" it—though after much angst, we killed it instead. – SevenSidedDie Dec 5 '12 at 17:54

in a futuristic/postapocalyptic setting, how can I thematically create a megadungeon?

If you want a strong running theme for the whole thing, I would suggest looking at some real world examples of massive areas ruled by a theme. How and if the original theme survives the apocalypse is up to you. It is possible the area is re-discovered and the theme re-interpreted by later generations. Think about how the artifacts of the area will influence that rediscovery.

Two of the places on earth with the strongest running theme I can think of would be,

• Vatican City - tunnel system is massive and parts are used practically. Others parts are crypts and old escape tunnels and the like. There are even vaults full of art, so I hear.
• Disney World - again there is an impressive tunnel system that is used daily. Some say Walt is down there, somewhere, frozen...waiting...

Thinking of some of the twisted and macabre things that could evolve a hundred or so years after an apocalypse in such locations should be fun. Micky, the dark god of steam power, tamer of the iron horse.

-

## A malfunctioning teleport system

Pick a range of options already suggested by the other answers and link them with a semi-functioning, broken down portal (teleport) system, possibly built by the military, by aliens, by artificial intelligence(s), or even an evil combination of these. Make sure getting out without using portals and following certain routes is virtually impossible by imperceptible railroading, by using clues and plotlines available and resolvable only through repeated re-entering the portal-dungeon.

I know @SevenSidedDie has touched upon this by mentioning Portal 1-2 by Valve: the difference is that instead of the run-down environments I'm emphasizing portal technology, which could / should be central element here... or, if it's run by a malfunctioning AI, even the arch-villain controlling the "dungeon."

-

In Futurama, "Old New York" lies beneath "New New York". You might take that approach... an ancient city thats been covered over by a new one. It could be actually buried - filled in with dirt, with buildings connected by tunnels built between them by the inhabitants. Or just 'covered', with the streets still visible, and some sort of platform covering the forgotten city.

-

There's one big problem -- humans rarely build strong internal walls. That means all the pathing becomes nearly meaningless, anyone with a hammer can make their own path.

Even much of the underground stuff is basically just a tunnel to a room, not a megadungeon. (Take Cheyenne Mountain--one tunnel with blast doors, one room inside with some buildings on springs inside. It's built to be secure, not to isolate parts of it from other parts of it.)

Thus I would like to take a different approach. Take a city of big skyscrapers, add one big nuke off to the side for the world's biggest domino topple. Now you have all sorts of spaces underneath the rubble where buildings fell against their neighbors and didn't completely collapse.

Another possibility: A moonbase built without a central government--people just connected to the existing structure in whatever fashion was convenient. This requires something that maintains the atmosphere and fixes leaks, though.

-
In a post-apocalypse, maybe our current construction habits are a relevant impediment to plausible megadungeons. But in a futuristic setting, all bets are off. The future is pretty big. – SevenSidedDie Dec 29 '13 at 18:41


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How to find this double summation?

To find the value of $$\sum_{m=1}^{∞}{\sum_{n=1}^{∞}{\frac{m^2\cdot n}{3^m \cdot (n\cdot 3^m+m\cdot3^n)} } }$$ I dont know how to proceed to these kind of problems. Can anybody provide a sol to this problem which may give me an insight to solve more like these :)

• Are you confortable with solving ${\sum_{n=1}^{∞}{\frac{An}{Bn+C\cdot3^n} } }$? – Martigan Sep 18 '14 at 15:17
• Actually, no, But if u can provide some link or some kind off, I will try to understand – Dinesh Sep 18 '14 at 15:19
• @Martigan: I think no one is comfortable with solving that :D – Jack D'Aurizio Sep 18 '14 at 15:25
• @JackD'Aurizio You are right... :D... – Martigan Sep 18 '14 at 15:35

We have, with a classical symmetry trick: $$2\sum_{n=1}^{+\infty}\sum_{m=1}^{+\infty}\frac{m^2 n}{3^m(m 3^n + n 3^m)}=\sum_{n=1}^{+\infty}\sum_{n=1}^{+\infty}\frac{\frac{m^2n}{3^m}+\frac{mn^2}{3^n}}{m3^n+n3^m}=\sum_{n=1}^{+\infty}\sum_{m=1}^{+\infty}\frac{mn}{3^{m+n}}=\left(\sum_{n=1}^{+\infty}\frac{n}{3^n}\right)^2$$ hence the original series just equals $\frac{1}{2}\left(\frac{3}{4}\right)^2 = \color{red}{\frac{9}{32}}.$
• Just switch the role of $m$ and $n$. We have:$$\sum_{n=1}^{+\infty}\sum_{m=1}^{+\infty}\frac{m^2n}{3^m(m3^n+n3^m)} = \sum_{n=1}^{+\infty}\sum_{m=1}^{+\infty}\frac{mn^2}{3^n(m3^n+n3^m)}.$$ Then sum and simplify. – Jack D'Aurizio Sep 18 '14 at 17:13


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# Optimal matching problems with costs given by Finsler distances

• In this paper we deal with an optimal matching problem, that is, we want to transport two commodities (modeled by two measures that encode the spacial distribution of each commodity) to a given location, where they will match, minimizing the total transport cost that in our case is given by the sum of the two different Finsler distances that the two measures are transported. We perform a method to approximate the matching measure and the pair of Kantorovich potentials associated with this problem taking limit as $p\to \infty$ in a variational system of $p-$Laplacian type.
Mathematics Subject Classification: Primary: 49J20, 35J92.

 Citation:

•  [1] L. Ambrosio, Lecture Notes on Optimal Transport Problems, Mathematical aspects of evolving interfaces (Funchal, 2000), 1-52, Lecture Notes in Math., 1812, Springer, Berlin, 2003.doi: 10.1007/978-3-540-39189-0_1. [2] H. Brezis, Functional Analysis, Sobolev Spaces and Partial Differential Equations, Universitext, Springer, New York, 2011. [3] G. Bouchitté, G. Buttazzo and L. A. De Pascale, A p-Laplacian approximation for some mass optimization problems, J. Optim. Theory Appl., 118 (2003), 1-25.doi: 10.1023/A:1024751022715. [4] G. Carlier, Duality and existence for a class of mass transportation problems and economic applications, Adv. Math. Econ., 5 (2003), 1-21.doi: 10.1007/978-4-431-53979-7_1. [5] P-A. Chiappori, R. McCann and L. Nesheim, Hedoniic prices equilibria, stable matching, and optimal transport: equivalence, topolgy, and uniqueness, Econ. Theory, 42 (2010), 317-354.doi: 10.1007/s00199-009-0455-z. [6] I. Ekeland, An optimal matching problem, ESAIM COCV, 11 (2005), 57-71.doi: 10.1051/cocv:2004034. [7] I. Ekeland, Existence, uniqueness and efficiency of equilibrium in hedonic markets with multidimensional types, Econ. Theory, 42 (2010), 275-315.doi: 10.1007/s00199-008-0427-8. [8] I. Ekeland, Notes on optimal transportation, Econ. Theory, 42 (2010), 437-459.doi: 10.1007/s00199-008-0426-9. [9] I. Ekeland, J. J Hecckman and L. Nesheim, Identificacation and estimates of Hedonic models, Journal of Political Economy, 112 (2004), S60-S109. [10] L. C. Evans, Partial differential equations and Monge-Kantorovich mass transfer, Current Developments in Mathematics, 1997 (Cambridge, MA), 65-126, Int. Press, Boston, MA, 1999. [11] L. C. Evans and W. Gangbo, Differential equations methods for the Monge-Kantorovich mass transfer problem, Mem. Amer. Math. Soc., 137 (1999), no. 653.doi: 10.1090/memo/0653. [12] K. Fan, Minimax theorems, Pro. Nat. Acad. Sci., 39 (1953), 42-47. [13] M. Feldman and R. J. McCann, Monge's transport problem on a Riemannian manifold, Trans. Amer. Math. Soc., 354 (2002), 1667-1697.doi: 10.1090/S0002-9947-01-02930-0. [14] N. Igbida, J.M. Mazón, J. D. Rossi and J. J. Toledo, A Monge-Kantorovich mass transport problem for a discrete distance, J. Funct. Anal., 260 (2011), 3494-3534.doi: 10.1016/j.jfa.2011.02.023. [15] J. M. Mazón, J. D. Rossi and J. Toledo, An optimal transportation problem with a cost given by the Euclidean distance plus import/export taxes on the boundary, Rev. Mat. Iberoam., 30 (2014), 277-308.doi: 10.4171/RMI/778. [16] J. M. Mazón, J. D. Rossi and J. Toledo, An optimal matching problem for the Euclidean distance, SIAM J. Math. Anal., 46 (2014), 233-255doi: 10.1137/120901465. [17] N. Igbida, J.M. Mazón, J. D. Rossi and J. Toledo, Mass transport problems for costs given by Finsler distances via $p$-Laplacian approximations, Preprint. [18] C. Villani, Topics in Optimal Transportation, Graduate Studies in Mathematics. Vol. 58, 2003.doi: 10.1007/b12016. [19] C. Villani, Optimal Transport. Old and New, Grundlehren der MathematischenWissenschaften (Fundamental Principles of Mathematical Sciences), vol. 338. Springer, Berlin, 2009.doi: 10.1007/978-3-540-71050-9.

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# Is every compact discrete metric space singleton?

So I was working on a proof that every compact discrete metric space is finite, and my result was that not only is every one finite, but every one is just a singleton set. I just want to confirm whether this is true or not.

In my proof I showed that a set K in (M, discrete) is sequentially compact only if every sequence in K is a constant sequence in the form (x, x, x, . . .).

EDIT: The above is incorrect. I didn't consider sequences with constant tails in my proof, and if I did it would have led me to the conclusion that every compact discrete metric space is finite, but not necessarily singleton.

EDIT 2: It turns out that even when considering sequences with constant tails, singleton sets are still the only sets that only contain sequences with constant tails. So something else in my proof is wrong.

The only sets that contain only constant sequences are singleton sets, so therefore, every compact discrete metric space is singleton.

• If $M = \{x,y\}$ can't you form two constant sequences? Apr 17 '20 at 17:16
• The "In my proof I showed that" part is wrong. Apr 17 '20 at 17:17
• I believe what you should've shown is that such a sequence would be eventually constant, not just constant. Apr 17 '20 at 17:17
• @lisyarus It is still not the case thate every sequence is eventually constant. Only the convergent ones. Apr 17 '20 at 17:18
• @HagenvonEitzen Oh, I somehow missed that convergence is not in the prerequisites. You are right, of course. Apr 17 '20 at 17:20

No. Let $$X$$ be any finite set, and define

$$d(x,y)=\begin{cases} 1,&\text{if }x\ne y\\ 0,&\text{if }x=y\;; \end{cases}$$

then $$d$$ is a metric on $$X$$ that generates the discrete topology, and $$X$$ is trivially compact in that topology, since every finite space is compact.

A space $$X$$ is sequentially compact if every sequence $$s_i \in X$$ has a convergent subsequence, i.e., for some increasing sequence of natural numbers $$i_j$$, the sequence defined by $$r_j = s_{i_j}$$ is convergent. I think in your proof you are mistakenly requiring the sequence $$i_j$$ to contain all sufficiently large $$i$$, i.e., you are requiring the sequence $$r_i$$ to be obtained from $$s_i$$ by omitting a finite number of elements. That is too restrictive.

For example if $$X = \{-1, +1\}$$. The sequence defined by $$s_i = (-1)^{i}$$ has a convergent subsequence given by $$r_j = s_{2j}$$, which omits infinitely many elements of $$s_i$$.

In general, any sequence of elements $$s_i$$ of a finite set $$X$$ must contain infinitely many occurrences of some element $$x$$ of $$X$$. If you form $$r_j$$ by omitting all $$i$$ such that $$s_i \neq x$$, then you will get a convergent subsequence of $$s_j$$. So any finite metric space is sequentially compact.

Aside: this problem is a bit easier if you use the characterisation of compactness in terms of open covers (if you have come across that).


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# $C^0$ Hamiltonian dynamics and a counterexample to the Arnold conjecture

 Princeton/IAS Symplectic Geometry Seminar Topic: $C^0$ Hamiltonian dynamics and a counterexample to the Arnold conjecture Speaker: Sobhan Seyfaddini Affiliation: Member, School of Mathematics Date: Tuesday, November 29 Time/Room: 3:10pm - 4:10pm/S-101 Video Link: https://video.ias.edu/puias/2016/1129-SobhanSeyfaddini

After introducing Hamiltonian homeomorphisms and recalling some of their properties, I will focus on fixed point theory for this class of homeomorphisms. The main goal of this talk is to present the outlines of a $C^0$ counterexample to the Arnold conjecture in dimensions four and higher. This is joint work with Lev Buhovsky and Vincent Humiliere.


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# Scaling the distance between objects [duplicate]

I've got a camera tracked to an empty, and I'd like to add keyframes with constant interpolate to have the camera "click" forward.

How can I scale the distance between objects, without affecting their size? Even better would be setting an absolute value of the distance between the objects.

## 1 Answer

There's a button in the header bar (three dots with a left-right arrow under it) that when switched on means that your transformations will only apply to the centre points of the selected objects. Using that tool and pivoting by active element will allow you to scale two objects, keeping one in place, moving the other away or towards the active object and leaving the scale of the actual objects themselves untouched.

Setting the absolute distance between two objects could be done with a "Limit Distance" constraint set to a Clamp Region of "On Surface", which you could then key the distance value for.

• Is it possible to do this to selected vertices, edges or faces is edit mode? Jan 21 '14 at 20:51
• No, I don't think so. You can scale using a single face, edge or vert as the pivot point (active element pivot mode), but the relative distances between the other elements will scale up and down, meaning that the "bits" you're moving will get bigger and smaller. Jan 22 '14 at 16:28
• Thanks. It turned out that I could accomplish what I was looking for by extruding faces along their normals (ezz) Jan 22 '14 at 20:45


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# What are $x$ and $y$ in $xF_n$ + $yF_{n-1}$ = $1$?

We know that the $$\gcd$$ of consecutive Fibonacci numbers is $$1$$. But while finding the coefficients $$x$$ and $$y$$ in using euclidean algorithm in reverse direction I am not able to find any pattern so that I can write $$x,y$$ in terms of n.

You can show by induction that: $$F_{n+1}F_{n-1}-F_nF_n=(-1)^{n-1}$$ then your Bezout coefficient $$a,b$$ can be $$(a,b)=(F_{n-1},F_n)$$

These are the extended euclidean algorithm outputs for consecutive pairs of Fibonacci numbers:

 (1, 0, 1),
(1, 1, 0),
(1, -1, 1),
(1, 2, -1),
(1, -3, 2),
(1, 5, -3),
(1, -8, 5),
(1, 13, -8),
(1, -21, 13),
(1, 34, -21),
(1, -55, 34),
(1, 89, -55),
(1, -144, 89),
(1, 233, -144),
(1, -377, 233),
(1, 610, -377),
(1, -987, 610),

Since $$\left( {\matrix{ {F_{n + 1} } \cr {F_n } \cr } } \right) = \left( {\matrix{ 1 & 1 \cr 1 & 0 \cr } } \right)\left( {\matrix{ {F_n } \cr {F_{n - 1} } \cr } } \right)$$ then $$1 = \left( {\matrix{ {x_{n + 1} } & {y_{n + 1} } \cr } } \right)\left( {\matrix{ {F_{n + 1} } \cr {F_n } \cr } } \right) = \left( {\matrix{ {x_{n + 1} } & {y_{n + 1} } \cr } } \right)\left( {\matrix{ 1 & 1 \cr 1 & 0 \cr } } \right)\left( {\matrix{ {F_n } \cr {F_{n - 1} } \cr } } \right) = \left( {\matrix{ {x_n } & {y_n } \cr } } \right)\left( {\matrix{ {F_n } \cr {F_{n - 1} } \cr } } \right)$$
Hint  Take the determinant of the following matrix form of the fibonacci recurrence $$\left[\begin{array}{ccc} \,1 & 1 \\\ 1 & 0 \end{array}\right]^{\large n} = \left[\begin{array}{ccc} F_{\large n+1} & F_{\large n} \\\ F_{\large n} & F_{\large n-1} \end{array}\right]$$


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Santa Tonio Morning

# Posts

### Some Thoughts After Reading <Statistical Rethinking>

Last night, I read the 1st chapter of Statistical Rethinking: A Bayesian Course with Examples in R and Stan from Richard McElreath. I found this is nice book to share with my friends. The core idea of that chapter is the relationship between NULL hypothesis and statistical model. That is, should we trust the statistical models to reject the NULL hypothesis? It is a long history to use statistical models to figure out what is true?

### Use R as a bash language

R language could be easily used as a bash script using Rscript *.R. system() is a R base function which could run command line within R. Below is a simple example which allows to automate create a new blog post: (1) Ask users to type in filename, title and language (2) Create a new markdown file in specific directory (i.e. your local posts saved path) (3) Add some metadata in .

### Introduction to Latent Attribute Network Analysis

A brief introduction of using Network Model to visualize latent attributes’ hierarchy of Diagnostic Modeling.

### Which Country own the most Liberty Nobel Prizes? France? Ireland?

It is of my interest to use ggplot package to visualize some fun data. In this post, I tried to play with a Nobel Prizes Data including countries, prize year, each prize. The goal is to plot a cumulative traceplot of Liberty Nobel Prizes for top 10 countries. Load Packages tidyverse package include some very useful tools such as ggplot2, tidyr and dplyr. library(tidyverse) library(LaCroixColoR) library(ggthemes) library(ggimage) ggimage package was

### Make a Game in R

Recently I found a interesting R package call nessy which allows you to create a simple game driven by shiny. Thus. I tried a little bit about this package. Making a interactive app in R is promising in the files like teaching, presentation and visualization. Finally, I created the following shiny app: library(nessy) library(shinyjs) jscode <- "shinyjs.closeWindow = function() { window.close(); }" ui <- cartridge( title = "{Memorize the Names!

### Introduce gganimate for Psychometric

A new R packge (gganimate ) provides some new features for animation in R. Its big advantage is it could make use of ggplot API and embeded into ggplot. Next, I will use a sample data to show the example. Then I will use some real educational data to explore a little bit what we can do in psychometric area. A Simple Example I want to introduce this package.

### Lasso Regression Example using glmnet package in R

More details please refer to the link below: (https://web.stanford.edu/~hastie/glmnet/glmnet_alpha.html#lin) This post shows how to use glmnet package to fit lasso regression and how to visualize the output. The description of data is shown in here. dt <- readRDS(url("https://s3.amazonaws.com/pbreheny-data-sets/whoari.rds")) attach(dt) fit <- glmnet(X, y) Visualize the coefficients plot(fit) Label the path plot(fit, label = TRUE) The summary table below shows from left to right the number of nonzero coefficients (DF), the percent (of null) deviance explained (%dev) and the value of $\lambda$ (Lambda).

### Visuliaztion of Item Information Curves In R

This blog is to show a new way to display item information curves using ggridges package. The ridge plot could show the IIF plots very clear when you have a large number of items.

ggplot(item_information_all %>% filter(item %in% 17:22)) +
aes(x = factor, y = item, height = info,
group = item, color = as.factor(item),
fill = as.factor(item)) +
ggridges::geom_ridgeline(alpha = 0.75) +
ggtitle("Peer Social Capital: Information Functions")  +
theme_ridges()

### [Manual]Using Jags and R2jags in R

This post is aimed to introduce the basics of using jags in R programming. Jags is a frequently used program for conducting Bayesian statistics.Most of information below is borrowed from Jeromy Anglim’s Blog. I will keep editing this post if I found more resources about jags. What is JAGS? JAGS stands for Just Another Gibbs Sampler. To quote the program author, Martyn Plummer, “It is a program for analysis of Bayesian hierarchical models using Markov Chain Monte Carlo (MCMC) simulation…” It uses a dialect of the BUGS language, similar but a little different to OpenBUGS and WinBUGS.


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1. ## Permutations cont.

Six couples are to be seated around a circular table, in 12 chairs. How many ways can this be done if each couple is together?

2. ## Re: Permutations cont.

Originally Posted by azollner95
Six couples are to be seated around a circular table, in 12 chairs. How many ways can this be done if each couple is together?
Circular arrangements have a different count. There are $(N-1)!$ ways to arrange $N$ distinct items in a circle.
In this problem the six couples sitting together can be though of as six distinct items. But each couple can be seated in two ways(husband to the wife's right or visa-versa.) So what is the total?

3. ## Re: Permutations cont.

Would it simply be (12!) since you can continue to rotate them around?

4. ## Re: Permutations cont.

Originally Posted by azollner95
Would it simply be (12!) since you can continue to rotate them around?
usually when the problem specifies the arrangement is circular it means that rotations are indistinguishable from one another.

5. ## Re: Permutations cont.

Originally Posted by azollner95
Would it simply be (12!) since you can continue to rotate them around?
Absolutely NOT!

In this question we are arranging six objects(couples together) in a circular arrangement each object can be arranged in two ways.

ANSWER: $(5!)(2^6)$.

6. ## Re: Permutations cont.

I now understand where the 5! is coming from but why wouldn't you times (5!) by 2 instead of 2^6?

7. ## Re: Permutations cont.

Originally Posted by azollner95
I now understand where the 5! is coming from but why wouldn't you times (5!) by 2 instead of 2^6?
Because there two(2) ways to seat each couple: husband to the right of wife or wife to the right of husband. Six couples so $2^6$ ways.