# S Function Math

Functional notation: The argument of the function. “Function definition. For every trigonometry function such as cos, there is an inverse function that works in reverse. GSolve is a function that allows you to solve an equation or system of equations by simply entering the equation(s). “Dan’s work on supergravity has changed how scientists think about physics beyond the standard model, combining principles of supersymmetry and Einstein’s theory of general relativity,” says Michael Sipser, dean of the MIT School of Science and the Donner Professor of Mathematics. How to use function in a sentence. keep in mind if you want large factorials, you'll need to use some sort of arbitrary precision integer or perhaps the BCMath functions. Functions and equations Here is a list of all of the skills that cover functions and equations! These skills are organized by grade, and you can move your mouse over any skill name to preview the skill. Green's Function. If you're seeing this message, it means we're having trouble loading external resources on our website. The geometry of algebraic topology is so pretty, it would seem. Are you looking for free 8th grade math games that you can play online? On this website you can find a variety of fun online games for 8th grade students and teachers. The goal is to find all roots of the function (all values). Moreover, there is only one solution to this equation, because the function f is strictly increasing (for b > 1), or strictly decreasing (for 0 < b < 1). If you're behind a web filter, please make sure that the domains *. CALCULUS AND TRIGONOMETRY The Derivatives of Trigonometric Functions. B To evaluate z(u+ 1/2), we need to substitute everything inside the parentheses into q, i. We provide the exact math help you need with online test prep courses for over 100 standardized tests; tutoring and homework help for middle/high school and college math; and a complete homeschool math curriculum. The notation is y=f(x). Symmetry of the Green function 3 2. Fredholm theorems). We start with the definition of the Dirichlet function. Therefore, can be written as a one-to-one function from (since nothing maps on to ). It is also sometimes necessary to carry out the reverse process, decomposing a complicated function into two or more simple functions. It explores major themes of mathematics, from humankind's earliest study of prime numbers, to the cutting-edge mathematics used to reveal the shape of the universe. Foreign Function Interface. h header defines various mathematical functions and one macro. RELATIONS versus FUNCTIONS GENERAL MATHEMATICS Samar College Galina V. The math worksheets are randomly and dynamically generated by our math worksheet generators. SAT Math Function Quiz 6. Free math lessons and math homework help from basic math to algebra, geometry and beyond. It could also mean just explaining what the function does in words. Garvan, eds. A set of questions on the concepts of a function, in calculus, are presented along with their solutions. There is only one macro defined in this library −. The geometry of algebraic topology is so pretty, it would seem. Return (Math. If you're seeing this message, it means we're having trouble loading external resources on our website. h header defines various mathematical functions and one macro. com is the smart way to conquer math. Generally speaking, a Green's function is an integral kernel that can be used to solve differential equations from a large number of families including simpler examples such as ordinary differential equations with initial or boundary value conditions, as well as more difficult examples such as inhomogeneous partial differential equations (PDE) with boundary conditions. Math lessons and fun games for kindergarten to sixth grade, plus quizzes, brain teasers and more. Improve your math knowledge with free questions in "Complete a table and graph a linear function" and thousands of other math skills. 4 quite the opposite, we see a monotone, but unbounded function. Math has nothing to do with communication, creativity or intuition. Our study guides are available online and in book form at barnesandnoble. An arithmetic function f is called multiplicative if f(mn) = f(m)f(n) whenever m;n are relatively prime. This means provide an equation with just A in it and C in it, where A is the only thing on the left hand side and there are no variables other than C on the right side. Edwards's "Riemann's Zeta Function," for what little mathematics I have is self taught. mean(var1, var2, var3, var4) Square Root Function. By definition: A function is strictly increasing on an interval, if when x 1 < x 2, then f (x 1) < f (x 2). Find the right K-12 lesson plans - for free. Some problems are best solved using a function table. Parents use the math worksheets on this website to give their children extra practice with essential math skills. With a bit of manipulation we can get to the Elo equation from the standard Logistic function. Exp(angle) - Math. Find helpful math lessons, games, calculators, and more. ETS function Statistical: Returns a future value based on existing (historical) values by using the AAA version of the Exponential Smoothing (ETS) algorithm. Problem 1 A right triangle has one side x and a hypotenuse of 10 meters. There are six functions that are the core of trigonometry. Math content is a value determined by the amount of subject matter that is within the project. Real World Math Horror Stories from Real encounters Math Riddles. Library Macros. They're all free to watch! ↓. Consider the following function: f(x) = 3x + 1. Below is an animation, zooming into the graph at x=1. Designed for freshman math majors to introduce them to one another, their major, their college, and the University. Symmetry of the Green function 3 2. Works for PCs, Macs and Linux. Functions can be fun with these original curriculum-aligned games created in collaboration with math teachers and our partners. What is a Function in Math? - Definition & Examples Video Functions in Math. MATH provides support for rounding negative numbers (toward zero, away from. Define math. For example, the function A = s 2 giving the area of a square as a function of its side length is not linear because its graph contains the points (1,1), (2,4) and (3,9. Which half of the function you use depends on what the value of x is. Covered topics include special functions, linear algebra, probability models, random numbers, interpolation, integration, regression, optimization problems and more. The number e is a mathematical constant having an approximate value equal to 2. Using Math To Calculate Your Weight In Outer Space. Learn math with free interactive flashcards. Functions find their application in various fields like representation of the computational complexity of algorithms, counting objects, study of sequences and strings, to name a few. An S-function is a computer language description of a Simulink ® block written in MATLAB ®, C, C++, or Fortran. Mathematics. Relating effortful control, executive function, and false belief understanding to emerging math and literacy ability in kindergarten. Cos(angle) End Function Example. 4 quite the opposite, we see a monotone, but unbounded function. One can de ne a function fby saying that for each xthe value of f(x) is the solution y of the equation x2 + 2y 3 = 0:. Free, online math games and more at MathPlayground. No need to enter the variable(s) to solve for. Here is an example of one: It is not hard to show that this series converges for all x. Get math help in algebra, geometry, trig, calculus, or something else. But let's use "f": We say "f of x equals x squared" what goes into the function is put inside parentheses after the name of the function: So f(x) shows us the function is called "f", and "x" goes in. 30314, USA [email protected] For example the inverse of y = 2x is y = ½ x. These features include whether (and where) a function is increasing or decreasing, whether a function is. Welcome to algebra-online. Special function, any of a class of mathematical functions that arise in the solution of various classical problems of physics. We notice the curve is either on or above the horizontal axis. Excel Magic Trick 628: VLOOKUP Data Validation IF Function Date Math Total After Discount Template. One of the most important applications of differential calculus is to find extreme function values. This function is overloaded in and (see complex sqrt and valarray sqrt). The graph of a step function looks like a series of small steps. The URI math department offers both Bachelor of Arts (B. Draw Math Function Your Own: 4. ) exists (i. keep in mind if you want large factorials, you'll need to use some sort of arbitrary precision integer or perhaps the BCMath functions. Free online maths games for kids. Continuity and Uniform Continuity 521 May 12, 2010 1. , is finite) , and iii. Maxima and Minima from Calculus. Double-click a category to display the examples that it includes. What's a Function (Intro to Domain and Range) 1 - Cool Math has free online cool math lessons, cool math games and fun math activities. Synonym Discussion of function. Parents use the math worksheets on this website to give their children extra practice with essential math skills. 15) t can be rewritten as (1. when the link is initially processed. Actuarial Mathematics and Life-Table Statistics Eric V. What I said above holds whether functions are defined in terms of sets or not. Struggling with scatterplots? Can't quite wrap your head around circumference? Here are resources and tutorials for all the major functions, formulas, equations, and theories you'll encounter in math class. In fact, it is absolutely convergent. exponential series. Consider ‰ ¡∆u = f x 2 Ω ‰ Rn u = g x 2 @Ω:. 13 Slimey Inc. There is a further restriction, though. Additional overloads are provided in this header ( ) for the integral types : These overloads effectively cast x to a double before calculations (defined for T being any integral type ). Find the area of the triangle as a function of x. students continue to rank around the middle of the pack, and behind many other advanced industrial nations. Use for fast finishers, homework assignments, or for something different in your classroom. Combining two functions by substituting one function's formula in place of each x in the other function's formula. Intuitively, a function is a process that associates to each element of a set X a single element of a set Y. Straight Angle. Explore math with desmos. Return (Math. If the value of x is negative, Math. example, if we know the shape of the function f(x; y), can we have any idea of the shape of the function u(x;y)? The answers the the above questions lie in the theory ofGreen's functions. That's it for the introduction. expected value (expectation) experiment. 221, Springer, Cham, 2017, pp. This is useful when you don't want to immediately compute an answer, or when you have a math "formula" to work on but don't know how to "process" it. Newton's Method Sometimes we are presented with a problem which cannot be solved by simple algebraic means. In mathematics, a function is a mathematical object that produces an output, when given an input - it could be a number, a vector, or anything that can exist inside a set of things. From the problem, we know that c(32) = 0 and c(68) = 20. DadsWorksheets. Moreover, there is only one solution to this equation, because the function f is strictly increasing (for b > 1), or strictly decreasing (for 0 < b < 1). There are no worksheets associated with these. Domain of a relation is the set of all x-coordinates of the ordered pairs of that relation. A formal parameter is used like a variable within the function's body, but it is declared in the function's parameter list; it is not declared in the function's body. They depend only on the the dynamics. Provides constants and static methods for trigonometric, logarithmic, and other common mathematical functions. Math exercises on quadratic functions. Algebra Basics: What Are Functions? - Math Antics mathantics. Express the Area A as a function of the Circumference C. nearest integer using current rounding mode with exception if the result differs (function) Floating point manipulation functions. To use Excel to determine the sample standard deviation of a set of quantitative data, type these numbers into a group of adjacent cells in a spreadsheet. In mathematics, a function is a relation between sets that associates to every element of a first set exactly one element of the second set. 4 quite the opposite, we see a monotone, but unbounded function. These can be found by looking at where the graph of a function crosses the x-axis, which is the horizontal axis in the xy-coordinate plane. How Can MathPapa Help You? We offer an algebra calculator to solve your algebra problems step by step, as well as lessons and practice to help you master algebra. Okay, with that out of the way let's get back to the definition of a function and let's look at some examples of equations that are functions and equations that aren't functions. What is a function?. Dirichlet function and its modifications. Welcome to It's Elemental - Element Math Game! Welcome to It's Elemental - Element Math Game! The computer will randomly pick an element and present you with that element's data from the Periodic Table of Elements. It is also a modification of Dirichlet function and sometimes called Riemann function. Strictly what I describe above is a 'relation', not a function. Primary or Elementary, Key Stage 1, ages 5-7 years, Numeracy, Math help activities and teacher resources to use in the classroom or for parents at home to help your child with maths. These problems generally involve the flow of electromagnetic, acoustic, or thermal energy. It could also mean just explaining what the function does in words. Kronecker delta function: is a function of two variables, usually integers, which is 1 if they are equal, and 0 otherwise. Covers the Vertical Line Test, along with how to know if a formula is a function even without the graph. To fully understand function tables and their purpose, you need to understand functions, and how they relate to variables. Calculus on the Web was developed with the support of the National Science Foundation COW is a project of Gerardo Mendoza and Dan Reich Temple University. Math 456 Lecture Notes: Bessel Functions and their Applications to To nd asymptotic behavior of Gamma-function as s!1, we use so called "Laplace s(z) is the. 2 CONDENSED In this lesson you will model a real-word situation with a quadratic function identify the x-intercepts, vertex, and line of symmetry of a parabola rewrite a quadratic function in vertex form You have looked at quadratic functions that model projectile motion. Compute exponential function (function ) frexp Get significand and exponent (function ) ldexp Generate value from significand and exponent (function ) log Compute natural logarithm (function ) log10 Compute common logarithm (function ) modf Break into fractional and integral parts (function ) exp2 Compute binary exponential function (function. The Function Iterator. GUTIERREZ´ NOVEMBER 5, 2013 Contents 1. If you are still confused, you might consider posting your question on our message board , or reading another website's lesson on domain and range to get another point of view. 7 of the Reference Manual. Improve your math knowledge with free questions in "Complete a table and graph a linear function" and thousands of other math skills. Green's functions are widely used in electrodynamics and quantum field theory, where the relevant differential operators are often difficult or impossible to solve exactly but can be solved perturbatively using Green's. Questions and Answers on Functions. Example 1b. RELATIONS versus FUNCTIONS GENERAL MATHEMATICS Samar College Galina V. The Barnes' G-Function, and related higher functions It is Free Math Help Boards We are an online community that gives free mathematics help any time of the day. You feed the machine an input, it does some calculations on it, and then gives you back another value - the result of the calculations. Many times one might have come across problems that relate to the prime factorization of a number, to the divisors of a number, to the multiples of a number and so on. Designed for freshman math majors to introduce them to one another, their major, their college, and the University. @achillehiu gave a good example. 221, Springer, Cham, 2017, pp. The function is to add 3. Bing helps you turn information into action, making it faster and easier to go from searching to doing. Find the area of the triangle as a function of x. I have modeled a dynamic system( 5 inputs and 10 outputs) using S-Function Level-1 and Level-2. This example uses the Exp method of the Math class to return e raised to a power. Beyond simple math and grouping (like "(x+2)(x-4)"), there are some functions you can use as well. If the number is negative, NaN is returned. One example of a function is the relation between x (any real number) and twice that number, or 2x. In this tutorial we develop a number of important properties of the Barne's G-function - a close relative of the Double Gamma function (N. Slud Mathematics Department University of Maryland, College Park °c 2001. The advantage is that ﬁnding the Green's function G depends only on the area D and curve C, not on F and f. The Mathematica Trajectory It's Come a Long Way in Three Decades. It is often written as f(x) where x is the input. The second function is similar. Thomae's function: is a function that is continuous at all irrational numbers and discontinuous at all rational numbers. Each is described, with an example, on this page. The math library is documented in section 6. u(x,y) of the BVP (4). Green's functions are widely used in electrodynamics and quantum field theory, where the relevant differential operators are often difficult or impossible to solve exactly but can be solved perturbatively using Green's. For instance, if we needed to find the roots of the polynomial , we would find that the tried and true techniques just wouldn't work. In mathematics, a function is a mathematical object that produces an output, when given an input - it could be a number, a vector, or anything that can exist inside a set of things. CTypes - A package for calling the functions of dlls/shared libraries. The x value is referred to as the independent variable and the y as the dependent variable. You feed the machine an input, it does some calculations on it, and then gives you back another value - the result of the calculations. 85 and 112; Atkin and Morain 1993; Berndt 1994, p. Students will learn language and concepts used throughout mathematics, and will learn how to write mathematical proofs, which makes Math 19 excellent preparation for upper level courses. W HEN ONE THING DEPENDS on another, as for example the area of a circle depends on the radius -- in the sense that when the radius changes, the area will change -- then we say that the first is a "function" of the other. Generate an unlimited number of custom math worksheets instantly. These problems generally involve the flow of electromagnetic, acoustic, or thermal energy. x is called the argument of the function, and y is the value of the function. Function y = f(x) is continuous at point x=a if the following three conditions are satisfied : i. Really clear math lessons (pre-algebra, algebra, precalculus), cool math games, online graphing calculators, geometry art, fractals, polyhedra, parents and teachers areas too. Mathematical methods in quantum field theory, quantum mechanics, statistical mechanics, condensed matter, nuclear and atomic physics. Let's see what these definitions mean using our vending machine example. Unlike some of the numeric methods of class StrictMath, all implementations of the equivalent functions of class Math are not defined to return the bit-for-bit same results. What Is an S-Function? S-functions (system-functions) provide a powerful mechanism for extending the capabilities of the Simulink ® environment. The Dedekind eta function is implemented in the Wolfram Language as DedekindEta[tau]. Passing too few parameters, which can be avoided by checking IsFunc() 's return value (which is the number of mandatory parameters plus one). For many functions the rule which tells you how to compute it is not an explicit formula, but instead an equation which you still must solve. The use of the IF function with numeric values is based on using different comparison operators to express your conditions. STUDY GUIDES. It translates Python code to fast C code and supports calling external C and C++ code natively. com for more Free math videos and additional subscription based content!. The square root of the given number. One example of a function is the relation between x (any real number) and twice that number, or 2x. A debt of gratitude is owed to the dedicated staff who created and maintained the top math education content and community forums that made up the Math Forum since its inception. Then the value of x for which the derivative of f(x) with respect to x is equal to zero corresponds to a maximum, a minimum or an inflexion point of the function f(x). One of the biggest cross-national. Log in Sign up. Briefly, a function is continuous at a point x 0 if and only if Definition of Continuity. ) exists (i. Sets, Functions, Relations 2. Panela RELATIONS FUNCTIONS A relation is also a set of ordered pairs (x, y). Paul Garrett: Riemann’s Explicit/Exact formula (October 2, 2010) The main trick (known to Riemann and before) is to use Jacobi’s functional equation for (z) to convert the other part of the integral, from 0 to 1, into a similar integral from 1 to +1, up to more elementary terms admitting direct analysis. Math problem of the week A says B&C tell the truth B says A tell the truth C says A lies and B tells the truth A's. All the functions available in this library take double as an argument and return double as the result. It would be written as f(x)=2x. exclusive (of events) expand (in algebra) expanded form. The natural display shows fractions, roots and exponents as you would expect it from mathematics. The range of a function is the set of numbers that the function can produce. With a bit of manipulation we can get to the Elo equation from the standard Logistic function. Functional notation: The argument of the function. Write expressions in equivalent forms to solve problems. Strictly what I describe above is a 'relation', not a function. The logistic function is the inverse of the natural logit function and so can be used to convert the logarithm of odds into a probability. A function is an operation on numbers of some set that gives (calculates) one number for every number from the domain. The CEILING function rounds a number to the nearest integer or multiple of significance. Public Function Asinh(ByVal value As Double) As Double ' Calculate inverse hyperbolic sine, in radians. For example, click C-files. SQRT1_2 // returns the square root of 1/2 Math. Garvan, eds. There are no separate math functions in PowerShell. The function machine should write his function rule on a piece of paper and slip it in his pocket. Let be a one-to-one function as above but not onto. 2 CONDENSED In this lesson you will model a real-word situation with a quadratic function identify the x-intercepts, vertex, and line of symmetry of a parabola rewrite a quadratic function in vertex form You have looked at quadratic functions that model projectile motion. As mentioned above, functions may have one, zero, or even many x-intercepts. Specifically, each input into a function has exactly one corresponding and correct output. Math by Kahoot! - Functions. 0 End Function Example. When a function's output is always boolean, the function can be called a predicate, which plays a major role in logic. A formal parameter is used like a variable within the function's body, but it is declared in the function's parameter list; it is not declared in the function's body. Step Function. This site was created with Mathematica and developed. Math lessons and fun games for kindergarten to sixth grade, plus quizzes, brain teasers and more. Math has little to do with "me" (except in terms of money). Cos(angle) End Function Example. In order to determine whether a rounding operation involves a midpoint value, the Round method multiplies the original value to be rounded by 10 n, where n is the desired number of fractional digits in the return value, and then determines whether the remaining fractional portion of the value is greater than or equal to. In other words, φ( n ) is the number of integers m coprime to n such that 1 ≤ m ≤ n. ” From Math. But let's use "f": We say "f of x equals x squared" what goes into the function is put inside parentheses after the name of the function: So f(x) shows us the function is called "f", and "x" goes in. Parents use the math worksheets on this website to give their children extra practice with essential math skills. It is often written as f(x) where x is the input. To use Excel to determine the sample standard deviation of a set of quantitative data, type these numbers into a group of adjacent cells in a spreadsheet. Return (Math. One can de ne a function fby saying that for each xthe value of f(x) is the solution y of the equation x2 + 2y 3 = 0:. Understanding basic input output function prepare students for pre algebra. Dim MyVar2 As Double = Math. Math 3210-3 HW 6 Solutions (0,∞). You can see steps and explanations of your algebra homework problems covering:. 30314, USA [email protected] For example, given the function f(x) = 3(x), solving for x = -1. Math 259: Introduction to Analytic Number Theory The Riemann zeta function and its functional equation (and a review of the Gamma function and Poisson summation) Recall Euler’s identity: [ (s) :=] X1 n=1 n @s= Y pprime 0 X1 c p=1 p c ps 1 A= Y pprime 1 1 p s: (1) We showed that this holds as an identity between absolutely convergent sums and. W HEN ONE THING DEPENDS on another, as for example the area of a circle depends on the radius -- in the sense that when the radius changes, the area will change -- then we say that the first is a "function" of the other. This says that the Green's function is the solution to the differential equation with a forcing term given by a point source. The nature of the gaussian gives a probability of 0. REDISH Department of Physics, University of Maryland College Park, MD, 20742-4111 USA Mathematics is an essential element of physics problem solving, but experts often fail to appreciate exactly how they use it. 7 of the Reference Manual. Composition. Think of a black box with an inlet chute and an outlet chute - drop in one number, and a number comes out. Math has little to do with "me" (except in terms of money). Simple to create equations and formulae for your documents! Math is Apache OpenOffice's component for mathematical equations. Loading Unsubscribe from mathantics? Cancel Unsubscribe. Example 3 Determine which of the following equations are functions and which are not functions. The poles of the transfer function are the eigenvalues of the system ma-trix Ain the state space model. All the functions available in this library take double as an argument and return double as the result. The range of a function is the set of numbers that the function can produce. Explore the concept of functions by putting values into this machine and observing its output. I have modeled a dynamic system( 5 inputs and 10 outputs) using S-Function Level-1 and Level-2. The Vertical Line Test 1 - Cool Math has free online cool math lessons, cool math games and fun math activities. HTML math follows general practice in mathematical typesetting by rendering functions, numbers and other constants in an upright font, while variables are rendered in an italic font. It’s about unlocking the joy of discovery when an idea finally makes sense. Real World Math Horror Stories from Real encounters Math Riddles. Generate an unlimited number of custom math worksheets instantly. Note: this method can be generalized to 3D domains. Matlab allows you to create symbolic math expressions. Sets, Functions, Relations 2. It gives access to the underlying C library functions. It is a description. The “ceiling” function in mathematics is defined differently than the CEILING function in SketchUp DC (which is “The CEILING function rounds a number to the nearest integer or multiple of significance”). 7 of the Reference Manual. Return (Math. Included: * function machines for students to find the missing numbers * function machines for students to identify the rule being used * blank, fillable (editable) function machines you can customize to meet your students' needs * flexible materials that can be used to introduce, practice, interven. Struggling with scatterplots? Can't quite wrap your head around circumference? Here are resources and tutorials for all the major functions, formulas, equations, and theories you'll encounter in math class. The math functions can handle values within the range of integer and float types. This activity will show students how to set up a function table and find all the needed answers. A beautiful, free Scientific Calculator from Desmos. This says that the Green's function is the solution to the differential equation with a forcing term given by a point source. It translates Python code to fast C code and supports calling external C and C++ code natively. For example,. Explore math with desmos. A function is a mathematical operator that takes in one number and outputs another number. Specifically, each input into a function has exactly one corresponding and correct output. The Structure Silicone™ Confetti pans are so pretty that they will open your appetite even before you start cooking! Discover this 24-count Mini Muffin Pan that is so easy to use!. There's not much of a difference. Mathematics Standards of Learning for Virginia Public Schools - February 2009 5 Algebra, Functions, and Data Analysis The following standards outline the content for a one-year course in Algebra, Functions, and Data Analysis. If this approach resonates with you, welcome aboard. manufactures skin moisturizer. Let S and T be sets. has two options, general or applied. Math 103 CSUN, Spring 2009 6 Write an expression for revenue as a function of the weekly demand for avocados. For example, the function A = s 2 giving the area of a square as a function of its side length is not linear because its graph contains the points (1,1), (2,4) and (3,9. S(" Following this type the location of the cells where the data is and then close the parentheses with" )". In these examples, physical constraints force the independent variables to be positive numbers. Double-click a category to display the examples that it includes. The fact-checkers, whose work is more and more important for those who prefer facts over lies, police the line between fact and falsehood on a day-to-day basis, and do a great job. Today, my small contribution is to pass along a very good overview that reflects on one of Trump’s favorite overarching falsehoods. Namely: Trump describes an America in which everything was going down the tubes under Obama, which is why we needed Trump to make America great again. And he claims that this project has come to fruition, with America setting records for prosperity under his leadership and guidance. “Obama bad; Trump good” is pretty much his analysis in all areas and measurement of U.S. activity, especially economically. Even if this were true, it would reflect poorly on Trump’s character, but it has the added problem of being false, a big lie made up of many small ones. Personally, I don’t assume that all economic measurements directly reflect the leadership of whoever occupies the Oval Office, nor am I smart enough to figure out what causes what in the economy. But the idea that presidents get the credit or the blame for the economy during their tenure is a political fact of life. Trump, in his adorable, immodest mendacity, not only claims credit for everything good that happens in the economy, but tells people, literally and specifically, that they have to vote for him even if they hate him, because without his guidance, their 401(k) accounts “will go down the tubes.” That would be offensive even if it were true, but it is utterly false. The stock market has been on a 10-year run of steady gains that began in 2009, the year Barack Obama was inaugurated. But why would anyone care about that? It’s only an unarguable, stubborn fact. Still, speaking of facts, there are so many measurements and indicators of how the economy is doing, that those not committed to an honest investigation can find evidence for whatever they want to believe. Trump and his most committed followers want to believe that everything was terrible under Barack Obama and great under Trump. That’s baloney. Anyone who believes that believes something false. And a series of charts and graphs published Monday in the Washington Post and explained by Economics Correspondent Heather Long provides the data that tells the tale. The details are complicated. Click through to the link above and you’ll learn much. But the overview is pretty simply this: The U.S. economy had a major meltdown in the last year of the George W. Bush presidency. Again, I’m not smart enough to know how much of this was Bush’s “fault.” But he had been in office for six years when the trouble started. So, if it’s ever reasonable to hold a president accountable for the performance of the economy, the timeline is bad for Bush. GDP growth went negative. Job growth fell sharply and then went negative. Median household income shrank. The Dow Jones Industrial Average dropped by more than 5,000 points! U.S. manufacturing output plunged, as did average home values, as did average hourly wages, as did measures of consumer confidence and most other indicators of economic health. (Backup for that is contained in the Post piece I linked to above.) Barack Obama inherited that mess of falling numbers, which continued during his first year in office, 2009, as he put in place policies designed to turn it around. By 2010, Obama’s second year, pretty much all of the negative numbers had turned positive. By the time Obama was up for reelection in 2012, all of them were headed in the right direction, which is certainly among the reasons voters gave him a second term by a solid (not landslide) margin. Basically, all of those good numbers continued throughout the second Obama term. The U.S. GDP, probably the single best measure of how the economy is doing, grew by 2.9 percent in 2015, which was Obama’s seventh year in office and was the best GDP growth number since before the crash of the late Bush years. GDP growth slowed to 1.6 percent in 2016, which may have been among the indicators that supported Trump’s campaign-year argument that everything was going to hell and only he could fix it. During the first year of Trump, GDP growth grew to 2.4 percent, which is decent but not great and anyway, a reasonable person would acknowledge that — to the degree that economic performance is to the credit or blame of the president — the performance in the first year of a new president is a mixture of the old and new policies. In Trump’s second year, 2018, the GDP grew 2.9 percent, equaling Obama’s best year, and so far in 2019, the growth rate has fallen to 2.1 percent, a mediocre number and a decline for which Trump presumably accepts no responsibility and blames either Nancy Pelosi, Ilhan Omar or, if he can swing it, Barack Obama. I suppose it’s natural for a president to want to take credit for everything good that happens on his (or someday her) watch, but not the blame for anything bad. Trump is more blatant about this than most. If we judge by his bad but remarkably steady approval ratings (today, according to the average maintained by 538.com, it’s 41.9 approval/ 53.7 disapproval) the pretty-good economy is not winning him new supporters, nor is his constant exaggeration of his accomplishments costing him many old ones). I already offered it above, but the full Washington Post workup of these numbers, and commentary/explanation by economics correspondent Heather Long, are here. On a related matter, if you care about what used to be called fiscal conservatism, which is the belief that federal debt and deficit matter, here’s a New York Times analysis, based on Congressional Budget Office data, suggesting that the annual budget deficit (that’s the amount the government borrows every year reflecting that amount by which federal spending exceeds revenues) which fell steadily during the Obama years, from a peak of $1.4 trillion at the beginning of the Obama administration, to $585 billion in 2016 (Obama’s last year in office), will be back up to $960 billion this fiscal year, and back over $1 trillion in 2020. (Here’s the New York Times piece detailing those numbers.) Trump is currently floating various tax cuts for the rich and the poor that will presumably worsen those projections, if passed. As the Times piece reported: