Latex Vector Notation

First put the following in your preamble (text copyable below the colored stuff):. Every command has a specific syntax to use. This will always be the case when we are dealing with the contours of a function as well as its gradient vector field. Usually you need to mark which Pawn is being captured, but ignore this for sake of simplicity. LaTeX tutorial using Texmaker. The indefinite integral is commonly applied in problems involving distance, velocity, and acceleration, each of which is a function of time. If the TeX Notation filter is activated, which set a LaTeX renderer, the same equation as above is obtained with the following control sequence:. It is important to keep them in a vector form so they can viewed or printed at any resolution without loss of quality. It also shows how to write a unit vector - the arrow that goes on the top of the letters representing the vector. or in abbreviated vector notation where u is the velocity vector and ∇. The main difference between the various types of matrix is the kind of delimeters that surround them. Basically, it is a $\LaTeX$ handwritten symbol recognition. Scroll down the page for examples and solutions. Compute answers using Wolfram's breakthrough technology & knowledgebase, relied on by millions of students & professionals. (1)Pisto des (1988): \Algebra. A matrix is a two-dimensional array of values that is often used to represent a linear transformation or a system of equations. Retrieved from "http://ufldl. The arrow vector notation is the standard in LaTeX, LaTeX Vectors With Hats. MATLAB recognizes vector operations if we use the vector notation. I am not 100% sure, but this is associated in my memory with J. Sum-class symbols, or accumulation symbols, are symbols whose sub- and superscripts appear directly below and above the symbol rather than beside it. Vector Paint is a drawing program, with a simple yet powerful interface, that lets you draw points, lines, curves and shapes. In L a T e X, subscripts and superscripts are written using the symbols ^ and _, in this case the x and y exponents where written using these codes. Using the Hadamard product, we can write the expression as. Unit vector: \vec { [1] } The documentation for Google Docs Equation editor (based on the Google Chat API) is not really extensive so most of the time the CodeCogs Online LaTex Equation Editor is the place to start and test if a LaTex expression from that editor also works in the Google Docs Equation Editor; they also have a link to better. One of the interesting things I noticed in the article is that the students prefer to type up their homework using LaTeX. Special Matrices and Matrix Properties. What is the usual way to do this? Of course, I think it would be best to state my notation somewhere in the beginning of paper. Stack Exchange network consists of 175 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers. For non-Greek stuff, see below. I have thought of \left( \stackrel{n}{r} \right), but the effect is obviously ugly. Special matrices are represented by special notation. linear algebra or $X,Y$ in e. Notation: We often represent a vector by some letter, just as we use a letter to denote a scalar (real number) in algebra. Vector notation is a commonly used mathematical notation for working with mathematical vectors, which may be geometric vectors or members of vector spaces. Used to be that all vectors would be bold-face and unit vectors would be denoted by the hat as well if you were to write it in LaTeX (for example) you used to have to write \hat{\vec{r}} with the \vec{r} giving you the boldface. Here, the vector notation, models the phenotypes of all the individuals in the dataset, denoted as a column vector. I have not been able to find a resource to tell me the standard notation for a normalized scalar value. Macros are. b) Given two points P; Q, the vector from P to Q is denoted PQ. This is not a comprehensive list. com Knowledge base dedicated to Linux and applied mathematics. Typesetting mathematics is one of LaTeX's greatest strengths. If one of the vector is identity vector, then the result is the sum of all the elements in the other one. Notation in which you express the x component as i and the y component as j, and you add them. The notation is sometimes more eﬃcient than the conventional mathematical notation we have been using. Denote by the probability of an event. LaTeX Source of Example 1. \end{document} (Note that this is not to be taken as an example of good mathematical exposition. The Wolfram Language has the world's largest collection of consistent multifont mathematical notation characters$LongDash]all fully integrated into both typesetting and symbolic expression construction. Basically if a number is bigger than 10^4 or smaller than 10^{-4} (applies to the absolute values of negative numbers too), it will be denoted in scientific notation. The main difference between the various types of matrix is the kind of delimeters that surround them. Special Matrices and Matrix Properties. Stack Exchange network consists of 175 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers. I am trying to understand the Matrix and Vector Notations on page 2 here: (the page is also pasted below, to make it easier to explain the problem). The Comprehensive LATEX Symbol List Scott Pakin ∗ 8 October 2002 Abstract This document lists 2590 symbols and the corresponding LATEX commands that produce them. * y, in numpy x*y), producing a new vector of same. since I am writing blog post that hosted by Github with Editor Atom , and use plugin markdown-preview-plus and mathjax-wrapper , and use mathjax Javascript display the math symbols on the web page. This week, we will go into some of the heavier. The Markdown parser included in the Jupyter Notebook is MathJax-aware. No - as the wikipedia article (see link above) points out there is no standard (i. How to Enter Answers in WeBWorK Be careful entering powers of trigonometric, and other, functions. php/Backpropagation_Algorithm". In certain cases it may be desirable to include “normal text” within an equation. So are 3*4 and 3 4 (3 space 4, not 34) but using an explicit multiplication symbol makes things clearer. LaTeX representation of hollow "R" symbol for real space? Post by clayton » Wed May 19, 2010 8:33 pm In LaTeX, how do I represent the hollow "R" symbol that designates the real number space?. \endgroup – KCd Apr 17 at 8:04. In the discussion of the applications of the derivative, note that the derivative of a distance function represents instantaneous velocity and that the. I have created a five video YouTube playlist Geometric Calculus, about 53 minutes in all, taken from the book. Again, the effect of the th variant on the phenotype is , the mean is , and the contribution of the environment on the phenotype is denoted by. I am not 100% sure, but this is associated in my memory with J. how the components of a tensor (in the above example, the tensor components would be the σij) transform under a change of reference frame. Tilde notation. In the following document, we will refer to special characters for all symbols other than the lowercase letters a-z, uppercase letters A-Z, figures 0-9, and English punctuation marks. The element-wise squaring makes the vector notation a bit more obscure in terms of everyday linear algebra operators. \left and \right can dynamically adjust the size, as shown by the next example:. This list of mathematical symbols by subject shows a selection of the most common symbols that are used in modern mathematical notation within formulas, grouped by mathematical topic. Notice that to insert the parentheses or brackets, the \left and \right commands are used. The derivative of a real function f may be indicated with differential notation or simply as f ′. If one of the vector is identity vector, then the result is the sum of all the elements in the other one. , integrals containing more than one integral sign) one finds that LaTeX puts too much space between the integral signs. If your document requires only a few simple mathematical formulas, plain LaTeX has most of the tools that you will need. \endgroup – Edward Xu Mar 14 at 18:14. De nition 1 (Naive De nition of a Set). 21 Sep 11 – fixed some typos and the vector notation 22 Sep 11 – fixed import of sklearn according to the new 0. Its done! Actually, the default interpreter in MATLAB for legend is 'tex', I guess. It all begins by writing the inner product. Scientific notation in LaTex? Hi, im engineering student therefore using allot of scientific notations. For example (1+2)* (3+4) and (1+2) (3+4) are both valid. functional analysis. This post highlights vector calculus and vector analysis. Software implementation Edit Plate notation has been implemented in various TeX / LaTeX drawing packages, but also as part of graphical user interfaces to Bayesian statistics programs such as BUGS and BayesiaLab. If A is a complex number a+bί, then A+1 returns a + 1 + bί. This supports my suspicion that you should check whether your LaTeX installation works. For the posted code, you can achieve this by using Part. Is a math Book. Tensor products of vector spaces are to Cartesian products of sets as direct sums of vectors spaces are to disjoint unions of sets. Specify vectors in Cartesian or polar coordinates, and see the magnitude, angle, and components of each vector. -10 -5 0 5 10 -10 -5 0 5 10 -0. But let me go over the different ways you could represent a vector. This leaﬂet provides information on symbols and notation commonly used in mathematics. In Unicode , it is the character at code point U+2207, or 8711 in decimal notation. We write $$\sim g(n)$$ to represent any quantity that, when divided by $$f(n)$$, approaches $$1$$ as $$n$$ grows. How to Enter Answers in WeBWorK Be careful entering powers of trigonometric, and other, functions. Our aim in this subsection is to give you a storehouse of examples to work with, to become comfortable with the ten vector space properties and to convince you that the multitude of examples justifies (at least initially) making such a broad definition as Definition VS. the weight of the car times the coefficient of kinetic friction. The Vector class implements vector quantities in $${\bf R}^n$$ for arbitrary $$n$$. VECTOR CALCULUS: USEFUL STUFF Revision of Basic Vectors A scalar is a physical quantity with magnitude only A vector is a physical quantity with magnitude and direction A unit vector has magnitude one. Symbols for Preference Relations Unicode Relation Hex Dec Name LAΤΕΧ ≻ U+227b 8827 SUCCEEDS \succ Strict Preference P U+0050 87 LATIN CAPITAL LETTER P P > U+003e 62 GREATER-THAN SIGN. Summary: Possibilities for the Solution Set of a System of Linear Equations Problem 288 In this post, we summarize theorems about the possibilities for the solution set of a system of linear equations and solve the following problems. Hi, What mathematical symbol is the one where it has a cross in a circle? Is it just multiplication? I've never seen it before. And, yes, it turns out that \curl \dlvf is equal to \nabla \times \dlvf. Thanks for contributing an answer to Stack Overflow! Please be sure to answer the question. Ask Question Asked 4 years, 1 month ago. It also shows how to write a unit vector - the arrow that goes on the top of the letters representing the vector. I realized that I didn't answer the question. Equation 30-3 gives Bself = µ 0I=2a and B other = µ 0Ia 2=2(a 2 +x2)3=2. How to write vector equation in Word Wide Spectrum. \begingroup Every time I have seen that notation used and unexplained, I've been confused by it. Some examples from the MathJax demos site are reproduced below, as well as the Markdown+TeX source. It is important to keep them in a vector form so they can viewed or printed at any resolution without loss of quality. However, instead of measuring this distance on the number line, a complex number's absolute value is measured on the complex number plane. Notice that the vectors of the vector field are all orthogonal (or perpendicular) to the contours. LaTeX is capable of displaying any mathematical notation. How can I pick up the argmin of a vector? I. lecture1_2 - Free download as PDF File (. But get G M_1 bold as well = bad. The way the curly bracket is facing indicates the direction of the hug. This one is a tricky one, you need to use the (surprisingly powerful) Microsoft Equation Editor which should be under the insert tab. In this section we will give a brief review of matrices and vectors. 870 Ct VERY RARE UNHEATED NR' CEYLON BLUE SAPPHIRE ROSE CUT NO RESERVE PRICE. The curl, on the other hand, is a vector. , 1 \times 10^{-2} or 4 \times 10^{-35}. \begingroup Every time I have seen that notation used and unexplained, I've been confused by it. Thanks mates, But I could not get the response , how do I use vector notation, I am not more familiar with maths notations. @duffymo Regardless of how you thing of LaTeX, this is a pretty good question. We use tilde notation to develop simpler approximate expressions. No installation, real-time collaboration, version control, hundreds of LaTeX templates, and more. Notation in econometrics: a proposal for a standard 77 he means (5 + √ −15)(5 − √ −15)= 25 − (−15)= 40, see Kline (1972, p. Integral expression can be added using the \int_{lower}^{upper} command. use the following search parameters to narrow your results: subreddit:subreddit find submissions in "subreddit" author:username find submissions by "username" site:example. Asking for help, clarification, or responding to other answers. The frequently used left delimiters include (, [ and {, which are obtained by typing (, [and \{respectively. Basic LaTeX Julie Mitchell This resource was adapted from notes provided by Jerry Marsden 1 Basic Formatting 1. I dont think you are talking about f' for deeivative of f. Visit the Learning Center. Note: LaTeX is case sensitive. In conjunction with the previous tutorial on math, by the end of the session, you will be in pretty good shape to write almost anything that your work will require. 5 (d) 8 (e) 15. The notation is sometimes more eﬃcient than the conventional mathematical notation we have been using. Basics of LaTeX for scientiﬁc papers and reports1 Sneˇzana Stanimirovi´c, November 21, 2016 1. Vector notation is a written method for representing quantities that possess both direction and magnitude, such as acceleration. Also, I would like a discussion of how this was developed to be able to express physics/engineering equations independently of the coordinate system. the normal force of the road times the coefficient of static friction. For example, a standard integral in LaTeX looks like  \int_a^b \! f(x) \, \mathrm{d}x. But once you have it broken up into the x and y components, you can just separately add the x and y components. Matrix Calculator. Retrieved from "http://ufldl. inequality symbols in math equality symbols math equal sign vector icon equality sign mathematical sum symbol stock vector equality mathematical. or in abbreviated vector notation where u is the velocity vector and ∇. In this paper we attempt to harmonize the various practices in econometrics notation. It also shows how to write a unit vector - the arrow that goes on the top of the letters representing the vector. The 18th and 19th centuries saw the creation and standardization of mathematical notation as used today. uk 1994/09/16 Abstract This package provides a set of new commands for representing vectors in various ways. 3 posts • Page 1 of 1. The second vector arrow above the symbol is substantially larger and it does not look good, especially when compared to the TeX/LaTeX result. Every nonzero vector has a corresponding unit vector, which has the same direction as that vector but a magnitude of 1. Dot notation for time derivatives (e. Our aim in this subsection is to give you a storehouse of examples to work with, to become comfortable with the ten vector space properties and to convince you that the multitude of examples justifies (at least initially) making such a broad definition as Definition VS. Similarly, \R^2 is a two-dimensional vector, and \R^3 is a three-dimensional vector. 5 How do you look at any of these coordinate systems and determine. The French Talbot company was reorganised by Anthony Lago (1893-1960) and after that, the Talbot-Lago name was used but on the home market the cars bore a Talbot badge. Typically this is the LaTeX code for the symbol. Calculates transpose, determinant, trace, rank, inverse, pseudoinverse, eigenvalues and eigenvectors. That list also includes LaTeX and HTML markup, and Unicode code points for each symbol (note that this article doesn't have the latter two, but they could certainly be added). You write (sin(t))^2 for the square of sin(t), and never sin^2t. Latex Bold Vector. Three Styles for LaTeX Vector Notation LaTeX Vector With an Arrow. However, instead of measuring this distance on the number line, a complex number's absolute value is measured on the complex number plane. As you will see, these behave in a fairly predictable manner. In vector addition, the zero vector is the additive identity vector: v + O = v. You can get the same file with more text here. You can choose to load either of them: \usepackage{amsfonts} % or \usepackage{amssymb}. We've documented and categorized hundreds of macros!. In this chapter we will tackle matters related to input encoding, typesetting diacritics and special characters. For Greek letters, put \boldsymbol in front of the Greek letter and you'll get the bold vector notation. 5 How do you look at any of these coordinate systems and determine. Latex-Math-Snippets. The normalized vector of is a vector in the same direction but with norm (length) 1. On a mistake, update as follows:. Thus, for example, the multiple integral is obtained by typing. Using R Markdown for Class Reports. Topics include: limits, integrals, summation notation, and vectors. It further allows the same raising and lowering operators to be used for many different roles according to the term they apply to. since I am writing blog post that hosted by Github with Editor Atom , and use plugin markdown-preview-plus and mathjax-wrapper , and use mathjax Javascript display the math symbols on the web page. Thanks mates, But I could not get the response , how do I use vector notation, I am not more familiar with maths notations. This guide provides an overview of how to get started with LaTeX, as well as resources and exercise to help new users of the program. For example, if you type \sqrt2 and click this icon, you'll get 2 in your document without opening MathType. For math, science, nutrition, history. Here is how LaTeX typesets the above source file. Vectors are elements of a vectorspace, depending on context they are commonly called $V,W$ in e. I have read the vector examples page on Wolfram Alpha's homepage, but I didn't know which method to use. Easy Scientific Notation In LaTeX filed in LaTeX , Math , Physics on Mar. Notebook documents ¶ Notebook documents contains the inputs and outputs of a interactive session as well as additional text that accompanies the code but is not meant for execution. This is a repository of latex snippets for sublime. But MathJax can also be configured to use HTML-CSS (for legacy browsers), SVG, and native MathML rendering when available in a browser. We generally use the coordinates x, y, and z to locate a particle at point P(x, y, z) in three dimensions. There is a shorthand for the last two commands: if you are using colon notation to access elements of a vector, you can use the end command to denote the end of the vector. As you see, the way the equations are displayed depends on the delimiter, in this case \[$ and . Unit vector: \vec { [1] } The documentation for Google Docs Equation editor (based on the Google Chat API) is not really extensive so most of the time the CodeCogs Online LaTex Equation Editor is the place to start and test if a LaTex expression from that editor also works in the Google Docs Equation Editor; they also have a link to better. Recall the notation that $\R$ stands for the real numbers. As does the notation $\mathbf i, \mathbf j, \mathbf k$ for the standard basis vectors in $\mathbf R^3$. The Ipe extensible drawing editor is a free vector graphics editor for creating figures in PDF or EPS format. It's possible to typeset integrals, fractions and more. , integrals containing more than one integral sign) one finds that LaTeX puts too much space between the integral signs. function , latex. Sum-class symbols, or accumulation symbols, are symbols whose sub- and superscripts appear directly below and above the symbol rather than beside it. This is the 2nd post in the series. Equation 30-3 gives Bself = µ 0I=2a and B other = µ 0Ia 2=2(a 2 +x2)3=2. Vector notation is a commonly used mathematical notation for working with mathematical vectors, which may be geometric vectors or members of vector spaces. ˙, ¨) can be obtained by following a letter variable with "\dot" for a first derivative and "\ddot" for a second derivative. As it is virtually impossible to list all the symbols ever used in mathematics, only those symbols which occur often in mathematics or mathematics education are. Recall Ax = Acos2 and Ay = Asin2. 14 \times 10^5$or 3. Vector notation was introduced to physics in the end of 19-th century. ISO standard) notation. You might want to look at a few careful expositions to get a sense of the normal range of notational variation. NET 3d advocacy AMD64 asdf billboard blogging Books cluster code dave winer essay example fix Flash fox and geese game glfw graphics haskell Humor KPAX LaTeX LaTeX vector notation life Linux Mathcad monads Neal Stephenson opengl OSS Papervision3D Physics Pizza Hut puppet repa SBCL scientific notation sci fi SLIME thesis vector whedon Wine wxCL. Calculations. Problem: For equation (2), I think it should b. When you then click on the blue downarrow at the lower right, five options appear. Vectors are a very important concept in science and math and appear very early in calculus and physics classes in college or in advanced high school classes. When the currents are in the same directions, the fields are parallel and oB s= B elf +B ther. Assuming w=f(x,y,z) and u=, we have Hence, the directional derivative is the dot product of the gradient and the vector u. If A is a complex number a+bί, then A+1 returns a + 1 + bί. As an undergraduate, I clearly remember learning and using "hat notation" to describe unit vectors. Latex-Math-Snippets. Viewed 10k times 7. This is an index-notation question rather then the NS one: For incompressible flow and Newtonian fluid, the continuity equation is denoted with: $$\frac{\partial u_i}{\partial x_i} = 0,$$ which. Sign up to join this community. How do you represent vectors. How to write vector equation in Word Wide Spectrum. With the x-axis due east and the y-axis due north, what is the displacement in unit vector notation for the bird?. 1/2 (c) 2 2 (d) 1 0 0. , integrals containing more than one integral sign) one finds that LaTeX puts too much space between the integral signs. “Save as new equation…” allows you to keep the equation as a building block, which makes it available from the “Insert” ribbon. Tilde notation. Thus, for example, the multiple integral is obtained by typing. 8 m/s2 and ay = -2. LaTeX code for combination? aside from the conventional${_n}C_r$, how can I input "n taken r" using the more common parenthesis notation. Putting a bar or a tilde over a letter in LaTeX. We write $$\sim g(n)$$ to represent any quantity that, when divided by $$f(n)$$, approaches $$1$$ as $$n$$ grows. the weight of the car times the coefficient of kinetic friction. \documentclass{article} \usepackage{amsmath} \begin{document} Let$\boldsymbol\alpha$be a vector. Subsection EVS Examples of Vector Spaces. A LaTeX command begins with the command name, which consists of a \ followed by either (a) a string of letters or (b) a single non-letter. Can we get something like this going on Stack Overflow? I think it'd be appropriate as I at least. \end{document} (Note that this is not to be taken as an example of good mathematical exposition. Problem: For equation (2), I think it should b. [email protected]. Windows applications that create vector graphics (such a graphing and diagramming tools) usually allow you to export graphics in either the Encapsulated Postscript (EPS) format or as a Windows metafile (EMF or WMF). To find the unit vector u of the vector you divide that vector by its magnitude as follows: Note that this formula uses scalar multiplication, because the numerator is a vector and the denominator […]. How do you represent vectors. Online WYSIWYG Mathematics Editor (Equation Editor, Latex Editor), fast and powerful Editing features, inputting Normal text, Math symbols, and drawing Graph/Diagram in one single editor, help writing Math Document much easier. The direction of a vctor V is the unit vector U parallel to V: U = V j V. The FontName, FontWeight, and FontAngle properties do not have an effect. There is no doubt that the development of good notation has been of great importance in the history of mathematics. – “Toggle between TeX/LaTeX and MathType equations” (not available on the Mac) – This is a huge shortcut if you know the TeX language. It allows you to specify a variable for the offset, but the width must be constant. When you reference a vector with a colon, such as v (:), all the components of the vector are listed. com Knowledge base dedicated to Linux and applied mathematics. Number sets such as natural numbers or complex numbers are not provided by default by LaTeX. news & information AoPS News AoPS Wiki. Home > Latex > FAQ > Latex - FAQ > How to write a vector in Latex ? \vec,\overrightarrow. A free-body diagram is a special example of the vector diagrams that. The result of adding 11 km, north plus 11 km, east is a vector with a magnitude of 15. As you will see, these behave in a fairly predictable manner. We use tilde notation to develop simpler approximate expressions. As it is virtually impossible to list all the symbols ever used in mathematics, only those symbols which occur often in mathematics or mathematics education are. Question: Q: Physics font-vector notation I'm using Keynote in my Grade 11 physics class. of design science. 1/2 (b) 1/2. How can I code in R to get such regular scientific notation format as the xtable output?. Latex-Math-Snippets. Is it possible to tell in index notation whether a vector is a row or column vector, or is that supposed to be clear based on context?. Equation 30-3 gives Bself = µ 0I=2a and B other = µ 0Ia 2=2(a 2 +x2)3=2. † as before, open WinEdt and write the following: \documentclass. In a vector such as a=[2 3 1 4 5], I want to pick up 1 by min(a) and also want to have 3 that is the Nth element of this vector with the minimum 1. It could be either a row vector or a column vector. Normalized vectors (i. Performs LU, Cholesky, QR, Singular value. , while the elements of a set will usually be given lower-case letters, like x, y, z, v, etc. The ability to easily include mathematical notation within markdown cells using LaTeX, and rendered natively by MathJax. Two arguments for addition taking precedence over (being done before) the max or min: In a max-plus algebra the addition plays the role of multiplication, and is denoted as such, while the$\max$plays the role of addition and is denoted as such thus since multiplication takes precendce over addition this translates to addition taking precedence over$\max$; see Max-plus algebra. Topics include: limits, integrals, summation notation, and vectors. Whether you've loved the book or not, if you give your honest and detailed thoughts then people will find new books that are right for them. By relieving the brain of all unnecessary work, a good notation sets it free to concentrate on more advanced problems, and, in effect, increases the mental power of the race. clude annotation such as In the history of mathematics, these symbols have denoted numbers, shapes, patterns, and change. Also let’s auto-matically scale all examples x to have (Euclidean) length 1, since this doesn’t aﬀect which side of the plane they are on. This is the 2nd post in the series. An online LaTeX editor that's easy to use. 0 m and b =-13. Our aim in this subsection is to give you a storehouse of examples to work with, to become comfortable with the ten vector space properties and to convince you that the multitude of examples justifies (at least initially) making such a broad definition as Definition VS. I realized that I didn't answer the question. The length is denoted j V. Three vector approach. Sign up to join this community. But get G M_1 bold as well = bad. So can be viewed as the imanginary part of. In unit-vector notation, what is the velocity of the cart when it reaches its greatest y coordinate? - Slader A cart is propelled over an xy plane with acceleration components ax = 4. \documentclass{article} \usepackage{amsmath} \begin{document} Let$\boldsymbol\alpha$be a vector. 1 Compute the lengths of the following vectors: (a) 0 0. An "overdot" is a raised dot appearing above a symbol most commonly used in mathematics to indicate a derivative taken with respect to time (e. Pos tentang vector notation latex yang ditulis oleh blogkarinagoo. But MathJax can also be configured to use HTML-CSS (for legacy browsers), SVG, and native MathML rendering when available in a browser. You cannot avoid mathematical notation when reading the descriptions of machine learning methods. Note, that integral expression may seems a little different in inline and display math mode - in inline mode the integral symbol and the limits are compressed. Normalized vectors (i. set margin latex latex matrices First the basic environments which could be used for a matrix, all of them are provide by usepackage amsmath. Once you're in equation mode, you can mess around with the GUI and try to look for it, or just type: (abc)/vec The / you might recognize from latex, it works usually the same here, it indicates a keyword is about to happen, and it will apply that operation to. Let $$X$$ be the stochastic process described by this chain (remember, a stochastic process is just a random variable that evolves through time). The matrix environments are matrix, bmatrix, Bmatrix, pmatrix, vmatrix, Vmatrix, and smallmatrix. In vector addition, the zero vector is the additive identity vector: v + O = v. It is a sequel to my Geometric Algebra playlist. Anyone who works with LaTeX knows how time-consuming it can be to find a symbol in symbols-a4. Associated with this quantity is the general relationship and when the Lagrangian is invariant with respect to time translation the energy will be a conserved quantity (and also the Hamiltonian). You have to use parentheses; it will not accept "<2,-3>" notation. A more complete primer can be found here. Currently to get for example 2x10 -3 i am typing as 2 \times 10 {-3}. We use tilde notation to develop simpler approximate expressions. ,$1 \times 10^{-2}$or$4 \times 10^{-35}$. inequality symbols in math equality symbols math equal sign vector icon equality sign mathematical sum symbol stock vector equality mathematical. The result of adding 11 km, north plus 11 km, east is a vector with a magnitude of 15. (4) Potential. Displacement Vector. A vector of length one is said to be a unit vector. 1 Vectors and vector notation 355B Guided practice worksheet 4 Using Resource sheet 18. This chapter has shown many examples using subscript notation and range variables, Chapter 10, "Vectors and Matrices," showed many examples using vector notation without subscripts. The derivative of a real function f may be indicated with differential notation or simply as f ′. Maybe you are talking about the del operator for gradient, divergence, curl kind of things in physics or differential geometry then it is \nalba in LaTeX math mode. The$ 's around a command mean that it has to be used in maths mode, and they will temporarily put you into maths mode if you use them. Notation This section includes the most commonlyused notation in this book. The Anglo-French STD (Sunbeam-Talbot-Darracq) combine collapsed in 1935. Lecture notes on topological insulators Ming-Che Chang Department of Physics, National Taiwan Normal University, Taipei, Taiwan (Dated: December 26, 2018). , with n columns), then the product Ax is defined for n×1 column vectors x. In L a T e X, subscripts and superscripts are written using the symbols ^ and _, in this case the x and y exponents where written using these codes. Even if you are using only one bracket, both commands are mandatory. So here's the deal: the inner product is very common in mathematical work (and there's accepted notation for it), and what you describe is not (and so there's not established notation). Note: Here we just want to analyse the commands and structure of a LaTeX file. Thanks for contributing an answer to Stack Overflow! Please be sure to answer the question. Calculates transpose, determinant, trace, rank, inverse, pseudoinverse, eigenvalues and eigenvectors. 2 The Simplex Method. MathOverflow has an awesome engine where you can embed LaTeX in questions, answers, and comments.


# RPM required to simulate Earth's gravity (Centrifugal acceleration)

1. Mar 22, 2014

### jwlgrant

1. The problem statement, all variables and given/known data

Here's the problem: "On a journey to Mars, one design is to have a section of the spacecraft rotate to simulate gravity. If the radius of this section is 30 meters, how many RPMs must it rotate to simulate one Earth gravity (1 g = 9.8 meters/sec^2)?"

2. Relevant equations

$A = \frac{V^2}{R}$

3. The attempt at a solution

I completed this problem, getting an answer of 5.5 RPM. Apparently I was wrong, and the answer should have been 14 RPM. I asked my instructor about it, and he showed me the calculations used to get 14 RPM, but it still didn't quite make sense to me.

Here's the process my instructor used to solve the problem:
Circumference is C = 2∏(30) = 188 meters.

1 rpm is like rotating 1 full circular rotation every minute, so 188/60 = 3.1 meters/sec.

Then V = 3.1 meters/sec x RPM.

Then A = (3.1 RPM)^2/188 = 0.05 x rpm^2.

So: 9.8 = 0.05rpm^2

RPM = 14

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Here's my process:
A = 9.8 (m)/(s^2)
R = 30 m
V = √(AR)
V = 17.15 m/s

1 revolution = circumference = 2∏30 m
[1 revolution/(2∏30 m)] * [60s/1 min] (1 rev is equal to 2π30m and sixty seconds is equal to one minute)
V = (17.15 m/s)/∏ RPM
V = 5.5 RPM

--

I realize, of course, that we both used different methods to get the answer. I thought I did it wrong, so I solved the problem again, using his method, to try to get the correct answer. He used 188m for the radius component when solving for A=V^2/R, but when I used 30m I got an answer of 5.5RPM. The radius, 30m, should be used in that equation, not 188m, right?

I would greatly appreciate any help in understanding this problem. I can't tell where/how I went wrong in solving it.

2. Mar 22, 2014

### Andrew Mason

Why is he dividing by 188 (circumference)?? Should be 30.

You are right and the instructor is wrong.

It is easier to use Ac = ω2r; so $\omega = \sqrt{A/r}$ where ω is the angular speed in radians/sec.

Letting RPM = 60ω/2∏:

$RPM = \frac{60}{2\pi}*\sqrt{A/r} = \frac{60}{2\pi}\sqrt{9.8/30} = 5.5$

AM

Last edited: Mar 22, 2014
3. Mar 23, 2014

### mr_pavlo

I also agree, it must be divided by radius = 30 not circumference.


# Suppose that 3 men and 3 women will be randomly seated at a round table having 6 evenly-spaced seats

Suppose that $3$ men and $3$ women will be randomly seated at a round table having $6$ evenly-spaced seats (with each seat being directly opposite one other seat).

(a) What is the probability that all $3$ men will be seated next to one another?

(b) What is the probability that starting at the northernmost seat and going clockwise around the table, a 2nd man will be encountered before a 2nd woman?

(c) What is the probability that no two men are seated directly opposite of one another?

(d) Suppose that the $6$ people are named Ashley, Barney, Carly, David, Emma, and Fred. What is the probability that starting with Ashley and going clockwise, the 6 people are seated in alphabetical order?

Attempted Solution:

(a) There are $6!$ = $720$ ways to seat these $6$ people. There are 6 different ways you can seat all three men next to each other, if you consider them as a clump. Each clump of three has $3!$ = 6 different ways to be arranged. Thus, there is $6*3!*3!$ = $216$ different arrangements. This gives a probability of $216\over{720}$ = $.3$.

(b) I wasn't sure about this but I realized that in any situation, either a 2nd man is reached before a 2nd woman or a 2nd woman is reached before a 2nd man. This gives a probability of $.5$.

(c) After drawing a picture of the table, this appears to be the same situation as part (a), giving a probability of $.3.$ I just realized this assumption was incorrect. Now I am getting that there are 8 different arrangements when considering males and females as clumps, with $3!$ different ways of arranging each clump, giving $8*3!*3!$ = $288$. This gives a probability of $288\over{720}$ = $.4$.

(d) $6\over{6}$ * $1\over{5}$ * $1\over{4}$ * $1\over{3}$ * $1\over{2}$ * $1\over{1}$ = $.00833$.

Any corrections to my attempted solutions would be greatly appreciated.

All of your answers are correct. I will assume that only the relative order of the people matters, that the men are named Barney, David, and Fred, and that the women are named Ashley, Carly, and Emma.

What is the probability that all three men will be seated next to each other?

We seat Ashley. The remaining people can be seated in $5!$ ways as we proceed clockwise around the table.

For the favorable cases, we have two blocks of people to arrange. Since the blocks of men and women must be adjacent, this can be done in one way. The block of men can be arranged in $3!$ ways. The block of women can be arranged in $3!$ ways. Hence, the probability that all the men are seated next to each other is $$\frac{3!3!}{5!} = \frac{3}{10}$$

What is the probability that starting at the northernmost seat and going clockwise around the table, a second man will be encountered before a second woman?

You made good use of symmetry. Nice solution.

What is the probability that no two men are seated directly opposite each other?

We count arrangements in which two men are seated directly opposite each other. There are $\binom{3}{2}$ ways to select the men. Once they are seated, there are $4$ ways to seat the third man relative to the already seated man whose name appears first alphabetically and $3!$ ways to seat the women as we proceed clockwise around the table from the third man. Hence, there are $$\binom{3}{2}\binom{4}{1}3!$$ seating arrangements in which two men are opposite each other. Hence, the probability that no two men are opposite each other is $$1 - \frac{\binom{3}{2}\binom{4}{1}3!}{5!} = 1 - \frac{3}{5} = \frac{2}{5}$$

Suppose that the six people are named Ashley, Barney, Carly, David, Emma, and Fred. What is the probability that starting with Ashley and going clockwise, the six people are seated in alphabetical order.

There is only one permissible seating arrangement. Hence, the probability that they are seated in alphabetical order is $$\frac{1}{5!} = \frac{1}{120}$$


# Thread: Need hints to solve antiderivative

1. ## Need hints to solve antiderivative

OK I have a slightly new problem that seems much more difficult.

$\displaystyle X'(t) = \mu_{k}gt + \frac{A}{4m} \int X'(t)^2dt$

constants:
$\displaystyle \mu_{k}$ = kinetic friction coefficient
$\displaystyle g$ = gravity
$\displaystyle A$ = cross sectional area
$\displaystyle m$ = mass

I am trying to solve $\displaystyle X(t)$ and then solve the specific function as an initial value problem.

I just need help solving $\displaystyle X'(t)$ for now. I should be OK after that.

Thanks

This is for a hobby physics project, not for homework.

2. Originally Posted by guitardude
OK I have a slightly new problem that seems much more difficult.

$\displaystyle X'(t) = \mu_{k}gt + \frac{A}{4m} \int X'(t)^2dt$

constants:
$\displaystyle \mu_{k}$ = kinetic friction coefficient
$\displaystyle g$ = gravity
$\displaystyle A$ = cross sectional area
$\displaystyle m$ = mass

I am trying to solve $\displaystyle X(t)$ and then solve the specific function as an initial value problem.

I just need help solving $\displaystyle X'(t)$ for now. I should be OK after that.

Thanks

This is for a hobby physics project, not for homework.
Differentiate both sides with respect to t. Then make the substitution U = X'. Solve the resulting DE for U. Hence solve for X.

3. I am still confused. If I differentiate both sides I will get $\displaystyle X''(t)$ right? which is what I started with. This equation is my work thus far on taking the antiderivative of $\displaystyle X''(t)$.

If I substitute $\displaystyle u=X'$ then I get $\displaystyle \int u^2$ The antiderivative of this part isn't as simple as $\displaystyle \frac{1}{3}u^3$ is it? Even so I would end up with $\displaystyle X(t) = constant*t^2 - constant* X'(t)^3 + C$ and I don't know $\displaystyle X'(t)$.

I forgot two negative signs, not that it matters much here but it should be:

$\displaystyle X'(t) = -\mu_{k}gt - \frac{A}{4m} \int X'(t)^2dt$

I previously solved the initial value problem for constant acceleration as a differential equation:

$\displaystyle X''(t) = B$
so $\displaystyle X'(t) = Bt + C$
so $\displaystyle X(t) = Bt^2 + Ct + D$
which is our familiar physics equation $\displaystyle S = S_{i} + V_{i}t + \frac{1}{2} a(t^2)$

You can the solve the specific function and find B,C,D given 3 X(t) samples.

Now, instead of assuming a constant acceleration, I am trying to solve this with $\displaystyle B = -\mu_{k}g - \frac{1}{4m}Av^2$ and that is how I got $\displaystyle X''(t) = -\mu_{k}g - \frac{1}{4m}Av^2 = -\mu_{k}g - \frac{1}{4m}AX'(t)^2$

It is not obvious to me yet how many X(t) samples I will need to solve this as an initial value problem?

4. Originally Posted by mr fantastic
Differentiate both sides with respect to t. Then make the substitution U = X'. Solve the resulting DE for U. Hence solve for X.
Differentiate both sides wrt t: $\displaystyle X'' = \mu_k g + \frac{A}{4m} \, (X')^2$.

Substitute U = X': $\displaystyle U' = \mu_k g + \frac{A}{4m} \, U^2 \Rightarrow \frac{dU}{dt} - \frac{A}{4m} \, U^2 = \mu_k g$.

Solve this DE for U: I'd suggest using the integrating method.

Then solve $\displaystyle \frac{dX}{dt} = U$ for X.

5. I worked through it with my differential equations prof and got to:

$\displaystyle X'(t) = \frac{-\sqrt{\mu_{k}g}}{\sqrt{\frac{A}{4m}}} * \tan((\sqrt{\mu_{k}g})(\sqrt{\frac{A}{4m}})((t-C_{1})))$

I verified the previous equation against simulations as well as with mathematica. Next I used matlab and then simplified to get back to :

$\displaystyle X(t) = \frac{-4m}{A} * \ln(\sec^2((\sqrt{\mu_{k}g})(\sqrt{\frac{A}{4m}})( t-C_{1}))) + C_{2}$

With the two unknowns, $\displaystyle C_{1}, C_{2}$

I was expecting it to have a single $\displaystyle \sec$ not $\displaystyle sec^2$ ?

This will be messy to solve as an initial value problem but possible.

Is there any way to make this function continuous? Tan() and Sec() make these hard to use and I don't see why it is necessary.

6. Originally Posted by guitardude
I worked through it with my differential equations prof and got to:

$\displaystyle X'(t) = \frac{-\sqrt{\mu_{k}g}}{\sqrt{\frac{A}{4m}}} * \tan((\sqrt{\mu_{k}g})(\sqrt{\frac{A}{4m}})((t-C_{1})))$

I verified the previous equation against simulations as well as with mathematica. Next I used matlab and then simplified to get back to :

$\displaystyle X(t) = \frac{-4m}{A} * \ln(\sec^2((\sqrt{\mu_{k}g})(\sqrt{\frac{A}{4m}})( t-C_{1}))) + C_{2}$

With the two unknowns, $\displaystyle C_{1}, C_{2}$

I was expecting it to have a single $\displaystyle \sec$ not $\displaystyle sec^2$ ?

This will be messy to solve as an initial value problem but possible.

Is there any way to make this function continuous? Tan() and Sec() make these hard to use and I don't see why it is necessary.
I can't quite understand how you've written the solution for X(t) but it looks like you have a log of a product. So use the usual log rule to break this up into a sum of logs.

Then it looks like one of those logs has the form $\displaystyle \log \sec^2 \theta$. Using the usual log rule you get $\displaystyle \log \sec^2 \theta = \log \cos^{-2} \theta = -2 \log \cos \theta$ .....

7. Originally Posted by mr fantastic
I can't quite understand how you've written the solution for X(t) but it looks like you have a log of a product. So use the usual log rule to break this up into a sum of logs.

Then it looks like one of those logs has the form $\displaystyle \log \sec^2 \theta$. Using the usual log rule you get $\displaystyle \log \sec^2 \theta = \log \cos^{-2} \theta = -2 \log \cos \theta$ .....

Hopefully this helps.... It is of form:
$\displaystyle X(t) = D * \ln( \sec^2(E(t-C_{1})) ) + C_{2}$

For the known constants $\displaystyle D,E$

I will have to solve $\displaystyle C_{1}$ which I can do given two pairs of corresponding X,t

Does your suggestion of splitting $\displaystyle \ln(\sec^2(\theta))$ up still apply?

8. Originally Posted by guitardude
Hopefully this helps.... It is of form:
$\displaystyle X(t) = D * \ln( \sec^2(E(t-C_{1})) ) + C_{2}$

For the known constants $\displaystyle D,E$

I will have to solve $\displaystyle C_{1}$ which I can do given two pairs of corresponding X,t

Does your suggestion of splitting $\displaystyle \ln(\sec^2(\theta))$ up still apply?
Yes.


# Math Help - A French Week

1. ## A French Week

For about 10 years after the French Revolution, the French government attempted to base measures of time on multiple of ten: One week consisted of 10 days, one day consisted of 10 hours, one hour consisted of 100 minutes, and one minute consisted of 100 seconds. What are the ratios of (a) the French decimal week to the standard week and (b) the French decimal second to the standard second?

(a) $1 \mbox{ French Week} = 1 \mbox{ French Week} \left(\frac{10 \mbox{ days}}{1 \mbox{ French Week}}\right)\left(\frac{10 \mbox{ hrs}}{1 \mbox{ day}}\right)$ $\left(\frac{100 \mbox{ min}}{1 \mbox{ hr}}\right)\left(\frac{100 \mbox{ s}}{1 \mbox{ min}}\right)$

$=10^6 \mbox{ seconds}$

$1 \mbox{ Normal Week} = 1 \mbox{ Normal Week} \left(\frac{7 \mbox{ days}}{1 \mbox{ Normal Week}}\right)\left(\frac{24 \mbox{ hrs}}{1 \mbox{ day}}\right)$ $\left(\frac{60 \mbox{ min}}{1 \mbox{ hr}}\right)\left(\frac{60 \mbox{ s}}{1 \mbox{ min}}\right)$

$=604800 \mbox{ seconds}$

So the ratio is 1.65, but the answers say that it is 1.43 ?

Also, in the first question I assumed that the 'French' second was the same as the 'normal' second, so wouldn't my answer to the second question make the answer to my first question invalid? Thanks.

2. Originally Posted by DivideBy0
For about 10 years after the French Revolution, the French government attempted to base measures of time on multiple of ten: One week consisted of 10 days, one day consisted of 10 hours, one hour consisted of 100 minutes, and one minute consisted of 100 seconds. What are the ratios of (a) the French decimal week to the standard week and (b) the French decimal second to the standard second?

(a) $1 \mbox{ French Week} = 1 \mbox{ French Week} \left(\frac{10 \mbox{ days}}{1 \mbox{ French Week}}\right)\left(\frac{10 \mbox{ hrs}}{1 \mbox{ day}}\right)$ $\left(\frac{100 \mbox{ min}}{1 \mbox{ hr}}\right)\left(\frac{100 \mbox{ s}}{1 \mbox{ min}}\right)$

$=10^6 \mbox{ seconds}$

$1 \mbox{ Normal Week} = 1 \mbox{ Normal Week} \left(\frac{7 \mbox{ days}}{1 \mbox{ Normal Week}}\right)\left(\frac{24 \mbox{ hrs}}{1 \mbox{ day}}\right)$ $\left(\frac{60 \mbox{ min}}{1 \mbox{ hr}}\right)\left(\frac{60 \mbox{ s}}{1 \mbox{ min}}\right)$

$=604800 \mbox{ seconds}$

So the ratio is 1.65, but the answers say that it is 1.43 ?

Also, in the first question I assumed that the 'French' second was the same as the 'normal' second, so wouldn't my answer to the second question make the answer to my first question invalid? Thanks.
Hello,

the ratio between the weeks is:

$\frac{\text{french week}}{\text{normal week}}=\frac{10}7 \approx 1.42857...$

to b):
$1 \text{ french day}= 10 \ h$ I assume that a new french day lasts as long as a normal day. Then you'll get the ratio:

$\frac{10 \ h \cdot 100\ min \cdot 100\ sec}{24\ h \cdot 60\ min \cdot 60\ sec} \approx 11.574...$ That means one of the new french seconds is a very short time interval. So better check my calculations.

3. Originally Posted by earboth
Hello,

the ratio between the weeks is:

$\frac{\text{french week}}{\text{normal week}}=\frac{10}7 \approx 1.42857...$

to b):
$1 \text{ french day}= 10 \ h$ I assume that a new french day lasts as long as a normal day. Then you'll get the ratio:

$\frac{10 \ h \cdot 100\ min \cdot 100\ sec}{24\ h \cdot 60\ min \cdot 60\ sec} \approx 11.574...$ That means one of the new french seconds is a very short time interval. So better check my calculations.
Why wouldn't DivideByZero's calculations be correct? When comparing two terms of measurement, you must relate them somehow

4. Hello, DivideBy0!

A fascinating problem . . . took me a few moments . . .

For about 10 years after the French Revolution, the French government
attempted to base measures of time on multiple of ten:
One week consisted of 10 days, one day consisted of 10 hours,
one hour consisted of 100 minutes, and one minute consisted of 100 seconds.

What are the ratios of (a) the French decimal week to the standard week
and (b) the French decimal second to the standard second?

$\text{1 French Week }\:=\:\frac{\text{1 French Week}}{1} \times \frac{\text{10 days}}{\text{1 French Week}} \times \frac{\text{10 hrs}}{\text{1 day}}$ $\times \frac{\text{100 min}}{\text{1 hr}} \times \frac{\text{100 s}}{\text{1 min}}$
. . $=\;10^6 \text{ seconds}$

$\text{1 Normal Week} \;= \;\frac{\text{1 Normal Week}}{1} \times \frac{\text{7 days}}{\text{1 Normal Week}} \times \frac{\text{24 hrs}}{\text{1 day}} \times \frac{\text{60 min}}{\text{1 hr}} \times \frac{\text{60 s}}{\text{1 min}}$
. . $=\;604800 \text{ seconds}$

All of this is absolutely correct!
Note that (nearly) all the units are unequal . . . weeks, hours, minutes, seconds.

The only unit that both systems agree upon is the length of one day.
. . (Perhaps measured from sunrise to sunrise.)

$a)\;\;\frac{\text{1 Frech week}}{\text{1 Normal week}} \;=\;\frac{\text{10 days}}{\text{7 days}} \;\approx\; 1.43$

$b)\;\;\frac{\text{1 French day}}{\text{1 Normal day}} \; = \; \frac{10^5\text{ French sec}}{86,400\text{ Normal sec}} \;=\;1.1574
$

. . The French second is about 1½ jiffys longer than a Normal second.

5. Thanks, I don't know why I thought days could be of different lengths.


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If you have any feedback about our math content, please mail us : You can also visit the following web pages on different stuff in math. In other words, to write a complex number in rectangular form means to express the number as a+bi (where a is the real part of the complex number and bi is the imaginary part of the complex number). Use rectangular coordinates when the number is given in rectangular form and polar coordinates when polar form is used. In polar form, the multiplying and dividing of complex numbers is made easier once the formulae have been developed. Find powers of complex numbers in polar form. Figure 5. In other words, there are two ways to describe a complex number written in the form a+bi: To write a complex number in rectangular form you just put it into the standard form of a complex number by writing it as a+bi. Dividing complex numbers: polar & exponential form. The major difference is that we work with the real and imaginary parts separately. 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In the complex number a + bi, a is called the real part and b is called the imaginary part. Plot each point in the complex plane. Multiplying complex numbers is much like multiplying binomials. To write a complex number in rectangular form you just put it into the standard form of a complex number by writing it as a+bi. If z = x + iy , find the following in rectangular form. But then why are there two terms for the form a+bi? For Example, we know that equation x 2 + 1 = 0 has no solution, with number i, we can define the number as the solution of the equation. Addition of Complex Numbers . Find (3e 4j)(2e 1.7j), where j=sqrt(-1). Answer. Multiplication and division of complex numbers is easy in polar form. Label the x-axis as the real axis and the y-axis as the imaginary axis. (5 + j2) + (2 - j7) = (5 + 2) + j(2 - 7) = 7 - j5 (2 + j4) - (5 + j2) = (2 - 5) + j(4 - 2) = -3 + j2; Multiplying is slightly harder than addition or subtraction. 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Sum of all three four digit numbers formed using 0, 1, 2, 3. A complex number in rectangular form means it can be represented as a point on the complex plane. Active 1 year, 6 months ago. After having gone through the stuff given above, we hope that the students would have understood, "How to Write the Given Complex Number in Rectangular Form". We start with an example using exponential form, and then generalise it for polar and rectangular forms. Convert a complex number from polar to rectangular form. This can be a helpful reminder that if you know how to plot (x, y) points on the Cartesian Plane, then you know how to plot (a, b) points on the Complex Plane. Recall that FOIL is an acronym for multiplying First, Outer, Inner, and Last terms together. The reciprocal of zero is undefined (as with the rectangular form of the complex number) When a complex number is on the unit circle r = 1/r = 1), its reciprocal equals its complex conjugate. For the rest of this section, we will work with formulas developed by French mathematician Abraham de Moivre (1667-1754). bi+a instead of a+bi). 18 times root 2 over 2 again the 18, and 2 cancel leaving a 9. Notice the rectangle that is formed between the two axes and the move across and then up? Using either the distributive property or the FOIL method, we get I get -9 root 2. The correct answer is therefore (2). Multiplying by the conjugate . This video shows how to multiply complex number in trigonometric form. This calculator does basic arithmetic on complex numbers and evaluates expressions in the set of complex numbers. ; The absolute value of a complex number is the same as its magnitude. The Number i is defined as i = √-1. To divide, divide the magnitudes and … Hence the Re (1/z) is (x/(x2 + y2)) - i (y/(x2 + y2)). The following development uses trig.formulae you will meet in Topic 43. Rectangular Form of a Complex Number. 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As imaginary unit use i or j (in electrical engineering), which satisfies basic equation i 2 = −1 or j 2 = −1.The calculator also converts a complex number into angle notation (phasor notation), exponential, or polar coordinates (magnitude and angle). When you’re dividing complex numbers, or numbers written in the form z = a plus b times i, write the 2 complex numbers as a fraction. Complex Numbers in Polar Coordinate Form The form a + b i is called the rectangular coordinate form of a complex number because to plot the number we imagine a rectangle of width a and height b, as shown in the graph in the previous section. The imaginary unit i with the property i 2 = − 1 , is combined with two real numbers x and y by the process of addition and multiplication, we obtain a complex number x + iy. Multiplication . $\text{Complex Conjugate Examples}$ $\\(3 \red + 2i)(3 \red - 2i) \\(5 \red + 12i)(5 \red - 12i) \\(7 \red + 33i)(5 \red - 33i) \\(99 \red + i)(99 \red - i)$ Now that we can convert complex numbers to polar form we will learn how to perform operations on complex numbers in polar form. Well, rectangular form relates to the complex plane and it describes the ability to plot a complex number on the complex plane once it is in rectangular form. We start with an example using exponential form, and then generalise it for polar and rectangular forms. Show Instructions. Example 1 – Determine which of the following is the rectangular form of a complex number. To convert from polar form. Google custom search here method of notation is valid for numbers. 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# Math Help - [SOLVED] probability question(digits)

1. ## [SOLVED] probability question(digits)

Question:
A four-digit number, in the range 0000 to 9999 inclusive, is formed. Find the probability that
(a) the number begins or ends with 0,
(b) the number contains exactly two non-zero digits.

Can you help me solve this question? I have tried but didn't manage to get the real answer(but I strongly support my own answer).

Thanks

2. For part A, here are my thoughts:

For the first 0000 to 0999, there are 1000 numbers. Then you have to add in all the numbers ending in 0. These are a little trickier to see. So we are looking for 1000, 1010, 1020, ... , 1090. That's a total of 10 numbers between 1000 and 1090 (inclusive). We also have to remember 1100, 1110, 1120, ... , 1190. So there are 10 numbers between each increment of a hundred. And between each thousand there are 10 increments of 100. Finally we are starting at 999 and proceeding to 9999, in which there are 9 increments of a thousand. So my total for part A comes to:

$\frac {1000 + 10 * 10 * 9}{10000} = \frac{19}{100}$

3. Originally Posted by stpmmaths
Question:
A four-digit number, in the range 0000 to 9999 inclusive, is formed. Find the probability that
(a) the number begins or ends with 0,
(b) the number contains exactly two non-zero digits.

Can you help me solve this question? I have tried but didn't manage to get the real answer(but I strongly support my own answer).

Thanks
(a) You want the combinations 0 X X X or X X X 0. Using the pigeon hole principle:

(1)(10)(10)(10) + (10)(10)(10)(1) = 2000.

But you have to subtract numbers of the form 0 X X 0 otherwise they get counted twice in the above: (1)(10)(10)(1) = 100.

So the total number of numbers satisfying the restriction is 1900.

The total number of numbers withut restriction is (10)(10)(10)(10) = 10000.

So the probability is 1900/10000 = 19/100.

You can take a similar approach for (b).

4. A more systematic way of doing part A is to count first the numbers beginning with 0, and then the numbers ending with 0 and then subtracting the number of numbers beginning and ending with 0 to get the total number of numbers beginning or ending with 0. We get $10^3+10^3-10^2$, giving the same probability as eXist calculated.
For part B,there are 4 choose 2 =6 ways to pick the two zero digits and 9*9 ways to choose the two non-zero digits. It is worth thinking about how many 4 digit numbers contain respectively exactly 0,1,2,3 and 4 0s. You should see a connection with the binomial theorem (expand $(9+1)^4$ to see it more clearly).

5. Hello stpmmaths

Welcome to Math Help Forum!
Originally Posted by stpmmaths
Question:
A four-digit number, in the range 0000 to 9999 inclusive, is formed. Find the probability that
(a) the number begins or ends with 0,
(b) the number contains exactly two non-zero digits.

Can you help me solve this question? I have tried but didn't manage to get the real answer(but I strongly support my own answer).

Thanks
Part (b). If the number contains exactly two non-zero digits, then it contains 2 non-zero and 2 zero digits.

So: imagine you have 4 empty boxes in a line in front of you. You have to (i) place a zero into two of the boxes, and then (ii) place a digit from 1 to 9 in the remaining two boxes.

For stage (i), there are $\binom42 = 6$ ways of choosing which boxes shall contain the zeros.

For stage (ii), there are 9 choices of non-zero digit to go in the remaining left-hand empty box, and 9 to go in the right-hand box. Total number of choices = $9 \times 9 = 81$.

So the number of different numbers with exactly two non-zero digits is $6 \times 81 = 486$. So the probability is $\frac{486}{10000}=0.0486$.

6. Originally Posted by mr fantastic
(a) You want the combinations 0 X X X or X X X 0. Using the pigeon hole principle:

(1)(10)(10)(10) + (10)(10)(10)(1) = 2000.

But you have to subtract numbers of the form 0 X X 0 otherwise they get counted twice in the above: (1)(10)(10)(1) = 100.

So the total number of numbers satisfying the restriction is 1900.

The total number of numbers withut restriction is (10)(10)(10)(10) = 10000.

So the probability is 1900/10000 = 19/100.

You can take a similar approach for (b).
For part A,
As you said, (1)(10)(10)(10) + (10)(10)(10)(1) - (1)(10)(10)(1) = 1900. If we do this we still consider 0XX0 in the probability, right? But the question stated begins or ends with 0 not with both

7. Originally Posted by stpmmaths
For part A,
As you said, (1)(10)(10)(10) + (10)(10)(10)(1) - (1)(10)(10)(1) = 1900. If we do this we still consider 0XX0 in the probability, right? But the question stated begins or ends with 0 not with both
If the statement is "begins or ends with a zero digit" it means begins with a zero digit or ends with a zero digit or both begins and ends with a zero.
That is the logical nature of the connective or.

Hello stpmmaths

Welcome to Math Help Forum!
Part (b). If the number contains exactly two non-zero digits, then it contains 2 non-zero and 2 zero digits.

So: imagine you have 4 empty boxes in a line in front of you. You have to (i) place a zero into two of the boxes, and then (ii) place a digit from 1 to 9 in the remaining two boxes.

For stage (i), there are $\binom42 = 6$ ways of choosing which boxes shall contain the zeros.

For stage (ii), there are 9 choices of non-zero digit to go in the remaining left-hand empty box, and 9 to go in the right-hand box. Total number of choices = $9 \times 9 = 81$.

So the number of different numbers with exactly two non-zero digits is $6 \times 81 = 486$. So the probability is $\frac{486}{10000}=0.0486$.

I thought we should consider the position(placement) of the digits? If we do it by we are not considering the position. According to your statement "
imagine you have 4 empty boxes in a line in front of you. You have to (i) place a zero into two of the boxes, and then (ii) place a digit from 1 to 9 in the remaining two boxes.

For stage (i), there are $\binom42 = 6$ ways of choosing which boxes shall contain the zeros.

"

This makes me confused between and .
Lets say we change the question to:-
A four-digit number, in the range 0000 to 9999 without repetitions for all digits except digit 0, is formed. Find the probability that
(a) ....
(b) the number contains exactly two non-zero digits.

9. ## Permutations and Combinations

Hello stpmmaths
Originally Posted by stpmmaths

I thought we should consider the position(placement) of the digits? If we do it by we are not considering the position. According to your statement "
imagine you have 4 empty boxes in a line in front of you. You have to (i) place a zero into two of the boxes, and then (ii) place a digit from 1 to 9 in the remaining two boxes.

For stage (i), there are $\binom42 = 6$ ways of choosing which boxes shall contain the zeros.

"

This makes me confused between and .
Lets say we change the question to:-
A four-digit number, in the range 0000 to 9999 without repetitions for all digits except digit 0, is formed. Find the probability that
(a) ....
(b) the number contains exactly two non-zero digits.
You are raising two different points here. First, the difference between $^nP_r$ and $^nC_r$.

• $^nP_r$ gives the number of ways of choosing $r$ objects from $n$, and then arranging them in order. So, for instance, if we want to choose 2 from 4, and the objects are A, B, C and D, then the choice A, C will be counted as a different choice from C, A. There are therefore $^4P_2=4 \times 3 = 12$ possible choices.

• $^nC_r$ gives the number of ways of choosing $r$ objects from $n$, where the order doesn't matter. In the above example, A, C and C, A would be counted as the same choice, and there are therefore $^4C_2=\frac{4\times 3}{2}= 6$ possible choices.

As an example, suppose that a committee chairman and secretary have to be chosen from a short-list of 4 people. Here, A, C is a different selection from C, A because A and C are fulfilling different roles. So you use $^4P_2$. But if you have to choose 2 delegates to attend a conference, from 4 people, you'll use $^4C_2$. Why? Because A, C and C, A are now the same choice - there's no difference between the roles of A and C.

So, in question (b), the number of different pairs of positions that can be occupied by a zero is $^4C_2=6$, because one zero looks just like the other - the order in which you place them in the 'empty boxes' doesn't matter.

In each of the remaining boxes (containing non-zeros) the order in which we fill the boxes does matter: 0709 will be different from 0907, for instance. Each box can then be filled in 9 ways, so - using the 'r-s Principle' - the total number of ways in which these two boxes can be filled, one after the other, is $9^2 =81$.

Your second question is: how is this affected if we're not allowed to choose the same non-zero digit twice? The answer is this:

• The first stage - choosing which 'boxes' are to be filled with zeros - is unaltered. It's still 6 choices.

• In the second stage, suppose we fill the left-hand of the two remaining boxes first. There are 9 ways of doing this. Since we're not allowed to choose the same digit again, there are then 8 ways of filling the last box. Total number of choices: $9\times 8 = 72$. Notice that this is the number of ways of choosing 2 items from 9, where the order matters; in other words, it's $^9P_2$.

So the overall answer to your modified question would be $6 \times 72 = 432$ choices, or a probability of $0.0432$.

Does that clear things up?

10. Thanks

So if i modified the question to:-
A five-digit number, in the range 00000 to 99999 inclusive, is formed. Find the probability that
(a) ....
(b) the number contains exactly two non-zero digits.

Then the answer would be 6 x 729 = 4374. Am I right?

11. Hello stpmmaths
Originally Posted by stpmmaths
Thanks

So if i modified the question to:-
A five-digit number, in the range 00000 to 99999 inclusive, is formed. Find the probability that
(a) ....
(b) the number contains exactly two non-zero digits.

Then the answer would be 6 x 729 = 4374. Am I right?
Correct!

And if repetition is not allowed in the non-zero digits, it would be $6 \times ^9P_3 = 6 \times9\times8\times7 = 3024$.

13. ## probability question

So the term C^2_4 relates to choosing the positions for (non) zero digits whereas terms 9^2 or P^2_9 are the numbers of ways to choose non-zero digits for a given positions.

14. Hello kobylkinks
Originally Posted by kobylkinks
So the term C^2_4 relates to choosing the positions for (non) zero digits whereas terms 9^2 or P^2_9 are the numbers of ways to choose non-zero digits for a given positions.
If this is a question, and not just a statement, then yes, you are right in your analysis!

You're also right to write '(non) zero digits' because it doesn't matter whether you think of this part of the 'box-filling' process as choosing the boxes that will contain zeros, or the boxes that will contain non-zeros; the result is the same: there are $^4C_2= 6$ possible choices.


# How to put a matrix in Reduced Echelon Form

5. (6 pts) Find a basis for the following subspace: $$\left\{ \begin{bmatrix} r - s - 2t - u \\ -r + 2s + 5t + 2u \\ s + 3t + u \\ 3r + s + 5u \end{bmatrix} \in \mathbb{R}^4 : \text{r, s, t and u are scalars} \right\}.$$

Original image here.

So I have this matrix: $$\begin{pmatrix} 1 &-1 &-2 &-1 \\ -1 &2& 5 &2 \\ 0 &1 &3 &1 \\ 3 &1 & 0 & 5 \end{pmatrix}$$ If I compute the the matrix in reduced echelon form, I get: $$\begin{pmatrix} 1&0&1&0\\ 0&1&0&3\\ 0&0&1&-2/3\\ 0 & 0 & 0 & 0 \end{pmatrix}$$

Please note that i want to put the matrix in reduced echelon form , and not row reduced echelon form.

I know that the three rules of reduced echelon form are:

1) The first non zero number of a row is 1 (leading entry).

2) An all zero row is placed at the end of the matrix.

3)A leading entry of a row is placed to the right of a leading entry of the previous row.

If it were to be in row reduced echelon form , then another property would hold, or else that each column that contains a leading entry, all other entries in the same column are zeros.

So is my matrix correctly put in a row echelon form ?

I think you are confused on terminology. Your matrix is correctly put into one of its row echelon forms. Note that these are not unique, as we could add the second row to the first row and the result would still be in row echelon form, that is, \begin{pmatrix} 1&1&1&3\\ 0&1&0&3\\ 0&0&1&-2/3\\ 0 & 0 & 0 & 0 \end{pmatrix} Is still in row echelon form. In order to get to the reduced row echelon form (sometimes called reduced echelon form), we must, as you said, make $0$'s in every column which contains a leading $1$. In your example, we can subtract the third row from the first \begin{pmatrix} 1&0&1&0\\ 0&1&0&3\\ 0&0&1&-2/3\\ 0 & 0 & 0 & 0 \end{pmatrix} to obtain \begin{pmatrix} 1&0&0&2/3\\ 0&1&0&3\\ 0&0&1&-2/3\\ 0 & 0 & 0 & 0 \end{pmatrix} Which is now in reduced row echelon form. Hopefully that clears up any confusion you had.

• In order to find a basis for a subspace, should i use row echelon form or row reduced echelon form ? Moreover, would the basis be composed of non zero rows or columns with pivot entries ? – Sara Martini Jun 16 '18 at 20:32
• And what is the advantage of dealing with a NON reduced row echelon form ? – Rene Schipperus Jun 16 '18 at 20:37

With reference to the original problem we need to find a basis for

$$\begin{bmatrix} r-s-2t-u \\ -r+2s+5t+2u \\ s+3t+u \\ 3r+s+5u \end{bmatrix}=\begin{pmatrix} 1 &-1 &-2 &-1 \\ -1 &2& 5 &2 \\ 0 &1 &3 &1 \\ 3 &1 & 0 & 5 \end{pmatrix}\begin{pmatrix} r \\ s \\ t \\ u \end{pmatrix}$$

note that the LHS is a vector $\in \mathbb{R^4}$ and in the RHS the given coefficients are arranged by columns, therefore we need to find a basis for the column space of the matrix in on the RHS.

Then we have

$$\begin{pmatrix} 1 &-1 &-2 &-1 \\ -1 &2& 5 &2 \\ 0 &1 &3 &1 \\ 3 &1 & 0 & 5 \end{pmatrix}\to \begin{pmatrix} 1 &-1 &-2 &-1 \\ 0 &1& 3 &1 \\ 0 &1 &3 &1 \\ 3 &1 & 0 & 5 \end{pmatrix}\to \begin{pmatrix} 1 &-1 &-2 &-1 \\ 0 &1& 3 &1 \\ 0 &1 &3 &1 \\ 0 &4 & 6 & 8 \end{pmatrix}\to \begin{pmatrix} 1 &-1 &-2 &-1 \\ 0 &1& 3 &1 \\ 0 &0 &0 &0 \\ 0 &0 & -6 & 4 \end{pmatrix}\to \begin{pmatrix} 1 &-1 &-2 &-1 \\ 0 &1& 3 &1 \\ 0 &0 & -6 & 4\\ 0 &0 &0 &0 \end{pmatrix}$$

which is the RREF. From here we can conclude that a basis is given by the first three vectors of the original matrix that is

$$\{(1,-1,0,3),(-1,2,1,1),(-2,5,3,0)\}$$

Note that we can also continue and obtain

$$\to \begin{pmatrix} 1 &0 &1 &0 \\ 0 &1& 3 &1 \\ 0 &0 & 1 & -\frac23\\ 0 &0 &0 &0 \end{pmatrix}\to \begin{pmatrix} 1 &0 &0 &\frac23 \\ 0 &1& 0 &3 \\ 0 &0 & 1 & -\frac23\\ 0 &0 &0 &0 \end{pmatrix}$$

the latter is the Gauss-Jordan form of the RREF but it is not needed to establish a basis for the given subspace.

• Is the row echelon form sufficient in order to find the basis for a subspace? – Sara Martini Jun 16 '18 at 19:58
• @SaraMartini Yes and you can stop to the first step I've indicated, if you are looking to the columns a basis is given by the first three columns in the original matrix, if you are looking for the rows you can choose the first the second and the fourth rows in the originl matrix an a basis or the first, the second and the third row in the RREF. – gimusi Jun 16 '18 at 20:03
• Im sorry but i don't really understand what you wrote... sorry :( I am willing to find a basis for a subspace, and i am not sure whether to use the row echelon form or reduced row echelon form ... or if i can use both interchangeably. If i used the row echelon form , then the basis are the first three columns of the original matrix. But what do you mean when you say that they can be rows too? I am kinds of confused – Sara Martini Jun 16 '18 at 20:09
• @SaraMartini It depends upon which subspace you are looking for and how you arrange the vector in the matrix. – gimusi Jun 16 '18 at 20:10
• This is the subspace.... (r-s-2t-u) , (-r+2s+5t+2u), (s+3t+u), (3r+s+5u) – Sara Martini Jun 16 '18 at 20:13


# Identifying Types of Converging Series

1. Aug 3, 2015

### DameLight

Hello, I'm looking for some help with this problem for my Calculus 2 class. Since it's a summer class my professor wasn't able to explain everything fully so if you can help me that would be great : )

1. The problem statement, all variables and given/known data

Select the FIRST correct reason why the given series converges.

A. Convergent geometric series
B. Convergent p series
C. Comparison (or Limit Comparison) with a geometric or p series
D. Alternating Series Test
E. None of the above

1. Σn=1 6(4)n/72n
2. Σn=1 sin2(6n)/n2
3. Σn=1 cos(nπ)/ln(2n)
4. Σn=1 (−1)n/(6n+5)
5. Σn=1 (n+1)(24)n/52n
6. Σn=1 (−1)n * √(n)/(n+9)

2. Relevant equations
Geometric Series:
Σn=1 arn-1 will converge if -1 < r < 1

P Series:
Σn=1 1/np converges if p > 1

Alternating Series Test:
1) bn+1 </= bn for all n and
2) limn->∞ bn = 0

3. The attempt at a solution
1. A
2. B
3. E
4. D
5. C
6. D

Last edited by a moderator: Aug 3, 2015
2. Aug 3, 2015

### Ray Vickson

I get (in order 1-6) A, C, D, D, E, D.

3. Aug 4, 2015

### DameLight

My professor told me that I got numbers 2 and 3 correct

4. Aug 4, 2015

### Ray Vickson

I disagree. Here are the details.
1. $\sum 6 \;4^n/7^{2n} = 6 \sum (4/49)^n$, so geometric (A).
2. In $\sum \sin^2 (6n) /n^2$ the terms are (i) not geometric; (ii) not alternating; (iii) not in the simple form $1/n^p$. However, $0 \leq \sin^2(6n)/n^2 \leq 1/n^2$, so a comparison with the series $1/n^2$ works (C).
3. $\sum \cos(n \pi)/ \ln(n)$ is an alternating series, since $\cos(n \pi) = (-1)^n$. So, D.
4. $\sum (-1)^n / (6n+5)$ is alternating, so D.
5. $\sum (n+1) \; 24^n / 5^{2n} = \sum (n+1) r^n, \;\; r = 24/25$. Here, A,B,C,D do not apply, and that leaves E.
6. $\sum (-1)^n \sqrt{n}/(n+9)$ (or $\sum (-1)^n \sqrt{ n/(n+9)}$---can't tell which you mean) is alternating in either interpretation, so D.

5. Aug 4, 2015

### DameLight

The correct answer ended up being:

1. A
2. C
3. D
4. D
5. C
6. D

Thank you for your help : )

6. Aug 4, 2015

### Ray Vickson

How do you get 5 C? Did you read my detailed explanation?

7. Aug 4, 2015

### micromass

Why wouldn't the comparison test apply in $5$? You can do $n+1\leq \alpha^n$ for some small enough $\alpha>1$.

8. Aug 4, 2015

### Ray Vickson

OK. That would be an answer to my question (why?) that the OP did not answer. For all I know the OP may have used false reasoning.

9. Aug 4, 2015

### DameLight

I don't want to assume, but that sounded a bit rude.

I am assuming that going off of your explanation for 5 that we can compare that to a geometric function ∑n=0 anrn and since the absolute value of r is less than 1 we can go ahead and reason that the limit would be equivalent to a/(1-r)

10. Aug 5, 2015

### Ray Vickson

No: a geometric series would be $\sum r^n$ (or $\sum c r^n = c \sum r^n$ for constant $c$), but $\sum a_n r^n$ is NOT a geometric series if $a_n$ varies with $n$. However, if $a_n \to c = \text{constant}$ as $n \to \infty$, then comparison with the series $\sum c r^n$ can be made to work without too much effort (although some extra effort is needed), However, in your case you have $a_n = n+1 \to +\infty$ as $n \to \infty$, so that type of argument does not work automatically. If you thought it did, that is what what I would call "false reasoning".

However, as 'micromass' has pointed out, if you choose a small enough $\epsilon > 0$ so that $(1+\epsilon) r < 1$, then for some finite $N> 0$ we have $n+1 \leq (1+\epsilon)^n$ for all $n \geq N$, so for $n \geq N$ we have $0 \leq (n+1) r^n \leq [(1+\epsilon) r]^n$ and we can compare with the geometric series $\sum [(1+\epsilon) r]^n$. Note, however, that some such argument as this must be acknowledged and recognized in order for C to be a valid and well-founded answer. Without this realization, just saying it would, again, be false reasoning.

Last edited: Aug 5, 2015