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  • Math Homework Help: A Guide to the Best AI Math Solver of 2023
    by Casey Allen on October 24, 2023 at 4:20 am

    About a quarter of the average college student's courseload is general education requirements. While these are graduation requirements, they also are usually time-wasters. They're challenging and stressful... but luckily, help is available. If you're looking for quick math homework help, an online AI math solver can bring your grades up quickly and effectively. Read on to The post Math Homework Help: A Guide to the Best AI Math Solver of 2023 first appeared on SquareCirclez. Related posts: 5 Best Free Math Problem Solvers Math problems allow students to learn new concepts and strengthen... Curriculum Webs - more homework needed "Weaving the Web into Teaching and Learning" Cunningham, C and... Buyer’s Guide: TI-84 Graphing Calculator Math classes can be daunting. From a young age, I... My dilemma - ethical math help Is there a difference between paying someone to do...

  • 5 Best Free Math Problem Solvers
    by Casey Allen on June 6, 2023 at 3:43 am

    Math problems allow students to learn new concepts and strengthen problem-solving skills. But many learners feel confused or frustrated if they can’t find the correct solution. A math problem solver is a handy tool that helps students doublecheck their work and identify errors. However, not all math problem solvers are created equal. Here are the The post 5 Best Free Math Problem Solvers first appeared on SquareCirclez. Related posts: Microsoft Math 3.0 Review MS Math 3.0 is a well-designed computer-based math tool.... Free math software downloads Wanting to use some math software but find it’s too... GraphSketch.com - free online math grapher GraphSketch is a free offering that allows the user to... Context Free math-based art Context Free is software you can use to produce some...

  • Reviewing Six Online Math Tutoring Services - What’s the Best?
    by Hugo Pegley on June 22, 2022 at 4:00 am

    Math is an exciting field of study that can lead to a variety of exciting careers or research projects. But if you're a student having difficulty with the topic, you might be thinking about enrolling in an online math tutoring program.  This is a great way for you to get assistance in a format and The post Reviewing Six Online Math Tutoring Services - What’s the Best? first appeared on SquareCirclez. Related posts: How to Pick A Live Math Chat Tutoring Service If you’re looking for a live math tutor, you are... How Much Does an Online Math Tutor Cost? Across the world, math is the key to understanding many... Online Algebra Math Tutor Many private and public high schools and colleges require students... Best Online Calculus Math Tutor: How to Choose Calculus and math require tremendous background information, practice, and good...

  • Picking the Best Online Precalculus Math Tutor
    by Hugo Pegley on June 22, 2022 at 3:55 am

    Students who want to go on to study math, science, engineering, and other disciplines in college, usually find that their chosen college values some prior knowledge of calculus. An online precalculus math tutor could be the answer. High schools commonly offer precalculus courses in the 11th grade before introducing calculus in the 12th. Precalculus is The post Picking the Best Online Precalculus Math Tutor first appeared on SquareCirclez. Related posts: How Much Does an Online Math Tutor Cost? Across the world, math is the key to understanding many... Best Online Calculus Math Tutor: How to Choose Calculus and math require tremendous background information, practice, and good... Online Algebra Math Tutor Many private and public high schools and colleges require students... Reviewing Six Online Math Tutoring Services - What’s the Best? Math is an exciting field of study that can lead...

  • How Much Does an Online Math Tutor Cost?
    by Hugo Pegley on June 15, 2022 at 4:17 am

    Across the world, math is the key to understanding many complex subject matters. It is also imperative that a student does not fall behind, as math typically builds on previous concepts. So, it is no secret that many typical high school and college students struggle in math classes. Due to this fact, skilled math tutors The post How Much Does an Online Math Tutor Cost? first appeared on SquareCirclez. Related posts: Online Algebra Math Tutor Many private and public high schools and colleges require students... Best Online Calculus Math Tutor: How to Choose Calculus and math require tremendous background information, practice, and good... How to Choose a Math Tutor Are you in need of mathematics support, or do you... How to Pick A Live Math Chat Tutoring Service If you’re looking for a live math tutor, you are...


Recent Questions - Mathematics Stack Exchange most recent 30 from math.stackexchange.com

  • Seeking feedback on my $\epsilon-\delta$ proof of $\lim\limits_{x \to a} x^2 = a^2$.
    by ten_to_tenth on June 21, 2024 at 1:42 pm

    I'm seeking feedback on my understanding of the $\epsilon-\delta$ limit proof for quadratic functions, specifically for $\lim\limits_{x \to a} x^2 = a^2$. After studying multiple proofs, I've noticed that while the general approach is similar, the justifications for certain steps often vary. Some explanations, such as 'because $\delta$ is typically small, we suppose $\delta \leq 1$', don't seem entirely rigorous or satisfying. Nevertheless, these various proofs have helped me grasp the overall idea. I've attempted to synthesize my understanding into the following proof. As I'm still new to this topic, I would greatly appreciate feedback on my work. Prove that $\lim\limits_{x \to a} x^2 = a^2$. Let $\epsilon > 0$. We wish to show that there exists $\delta$ such that $0 < |x-a| < \delta$ implies $|x^2 - a^2| < \epsilon$. Working backwards, note that $|x^2 - a^2| < \epsilon \iff |x-a||x+a| < \epsilon$. If $\delta$ exists at all, there must also exist a value of $\delta \leq 1$ that works. Thus, it is sufficient to consider if it's possible to find $\delta \leq 1$. When $|x - a| < \delta \leq 1$, $$|x + 1| = |(x -a) + 2a| \leq |x-a| + 2|a| < 1 + 2|a| \implies |x - a||x + a| < |x - a|(1+2|a|)$$Thus, as long as we also have $|x-a|(1+2|a|) < \epsilon$ or $|x-a| < \dfrac{\epsilon}{1+2|a|}$, we will have $|x-a||x+a| < \epsilon$. To sum up, if it's possible to find a working $\delta$, it's also possible to find a $\delta \leq 1$. For $\delta \leq 1$, we will have a working $\delta$ if it also satisfies $\delta < \dfrac{\epsilon}{1+2|a|}$. By taking $\delta = \min\left\{1, \dfrac{\epsilon}{1+2|a|}\right\}$, we will have $$|x - a| < \delta \implies |x-a| < \dfrac{\epsilon}{1+2|a|} < \dfrac{\epsilon}{|x+a|} \implies |x^2 - a^2| < \epsilon$$ It follows that $\lim\limits_{x \to a} f(x) = a^2$.

  • Galois group of $\mathbb{C}(t)$ over $\mathbb{C}(t-t^{-1})$
    by riescharlison on June 21, 2024 at 1:28 pm

    I was asked to determine the Galois group for the following extensions: first $\mathbb{C}(t+t^{-1})\subset \mathbb{C}(t)$ and then $\mathbb{C}(t^n+t^{-n})\subset \mathbb{C}(t)$ for a certain $n \in \mathbb{N}$. Now I can see the Galois group is not trivial since the automorphism that sends $t$ to $t^{-1}$ is the identity for $\mathbb{C}(t+t^{-1})$ and as such is in the Galois group. Now for the first extension, we can find the minimal polynomial for the element $t$ which is $x^2-(t+t^{-1})x+1$. I am not entirely sure wether this is enough to prove that Galois group is just $\mathbb{Z}/2\mathbb{Z}$ but I'm mainly having trouble with the second extension. We also have another automorphism which is the identity on $\mathbb{C}(t+t^{-1})$ namely the one that sends $t$ to $\zeta_n t$ with $\zeta_n$ the root of unity. However I don't really see how to use this to determine the Galois group similarly we can also find a polynomial (I'm not sure whether it is the minimal one) for which $t$ is a root, namely, $x^2n-(t^n+t^{-n})x^n+1$. Any tips on how to proceed?

  • Understanding the implication in linear algebra regarding vectors
    by Aljaz Brodar on June 21, 2024 at 1:22 pm

    Let $V$ be a subspace of $\mathbb{R}^n$ with the usual dot product, and let $\mathbf{z}, \mathbf{w} \in V$ be fixed vectors. If for every $\mathbf{v} \in V$ it holds that $\mathbf{z} \cdot \mathbf{v} = \mathbf{w} \cdot \mathbf{v}$, then $\mathbf{z} = \mathbf{w}$. I'm trying to understand why inserting the $\mathbf{v} = 0$ is not the correct approach. This is considered an implication right, so if I take $\mathbf{z}, \mathbf{w}$ as different vectors, then it holds that $\mathbf{z} \cdot \mathbf{v} = \mathbf{w} \cdot \mathbf{v}$ is 0 = 0 and the consequent $\mathbf{z} = \mathbf{w}$ is false when $\mathbf{v} = 0$? Why is this the wrong approach?

  • Sum of two random variables has same distribution as sum of two uniform
    by JWM on June 21, 2024 at 1:21 pm

    Let $U$, $V$ be two independant random variables with uniform distribution on $\{1, \dots, n \}$. Let $X$ and $Y$ be two independant random variables, each with values in $\{1, \dots, n \}$ such that $X+Y$ has the same probability distribution as $U+V$. Is it true that $X$ and $Y$ have both uniform distributions on $\{1, \dots, n \}$ ?

  • Intuition of weak and weak* Topology on dual space
    by TeX_User on June 21, 2024 at 1:10 pm

    If we take any normed space $X$ and observe their dual space $X^*$ we can compare different topologies on the space: That is the norm topology on $X^*$ ($\tau^*$), weak topology ($\tau_w$) and weak$*$ topology ($\tau_{w*}$). I am trying to find an intuition to "show" that $$\tau_w \subseteq \tau^*.$$ I know that the weak topology on the dual is the coarsest topology such that any function $x^*: X^* \to \mathbb{K}$ is continous (where $\mathbb{K} \in \{\mathbb{R}, \mathbb{C}\}$). So intuitively I need to leave out on some open sets of $\tau^*$ in order for all functions $x^*$ to be continous. Is this right? And if I want to compare $\tau_{w^*} \supseteq \tau_w$ I can then argue that (since the weak$*$ topology only considers any functions in the embedding $\iota(X) \subseteq X^{**}$) that by considering less functions I can admit more open sets?


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Mathematics – Wolfram Blog News, Views and Insights from Wolfram

  • Hypergeometric Functions: From Euler to Appell and Beyond
    by Tigran Ishkhanyan on January 25, 2024 at 5:35 pm

    Hypergeometric series appeared in the mid-seventeenth century; since then, they have played an important role in the development of mathematical and physical theories. Most of the elementary and special functions are members of the large hypergeometric class. Hypergeometric functions have been a part of Wolfram Language since Version 1.0. The following plot shows the implementation

  • Get Down to Business with Finite Mathematics in Wolfram Language
    by John McNally on December 22, 2023 at 3:41 pm

    “There is every reason to expect that the various social sciences will serve as incentives for the development of great new branches of mathematics and that some day the theoretical social scientist will have to know more mathematics than the physicist needs to know today.” —John G. Kemeny, first author of the original textbook on

  • Don’t Be Discreet and Learn Discrete Mathematics with Wolfram Language
    by Marc Vicuna on November 29, 2023 at 6:00 pm

    “The spread of computers and the internet will put jobs in two categories. People who tell computers what to do, and people who are told by computers what to do.” — Marc Andreessen, inventor of the Netscape browser How is data organized in databases? Why are some computer programs faster than others? How can algorithms

  • Learn Multivariable Calculus through Incredible Visualizations with Wolfram Language
    by Tim McDevitt on November 6, 2023 at 3:57 pm

    Multivariable calculus extends calculus concepts to functions of several variables and is an essential tool for modeling and regression analysis in economics, engineering, data science and other fields. Learning multivariable calculus is also the first step toward advanced calculus and follows single-variable calculus courses. Wolfram Language provides world-class functionality for the computation and visualization of

  • Expand Your Understanding of Statistics with Wolfram Language
    by Jamie Peterson on June 6, 2023 at 4:27 pm

    Statistics is the mathematical discipline dealing with all stages of data analysis, from question design and data collection to analyzing and presenting results. It is an important field for analyzing and understanding data from scientific research and industry. Data-driven decisions are a critical part of modern business, allowing companies to use data and computational analyses

  • Stack the Odds in Your Favor and Master Probability with Wolfram Language
    by Marc Vicuna on March 24, 2023 at 3:46 pm

    “I believe that we do not know anything for certain, but everything probably.” —Christiaan Huygens Have you ever wondered how health insurance premiums are calculated or why healthcare is so expensive? Or what led to the financial crisis of 2008? Or whether nuclear power is safe? The answers to these questions require an understanding of

  • Active Learning with Wolfram|Alpha Notebook Edition
    by Jordan Hasler on January 20, 2023 at 8:16 pm

    As you may know from your own experience (or perhaps from the literature on education), passively receiving information does not lead to new knowledge in the same way that active participation in inquiry leads to new knowledge. Active learning describes instructional methods that engage students in the learning process. Student participation in the classroom typically

  • Wolfram|Alpha Pro Teaches Step-by-Step Arithmetic for All Grade Levels
    by AnneMarie Torresen on August 26, 2022 at 3:12 pm

    In grade school, long arithmetic is considered a foundational math skill. In the past several decades in the United States, long arithmetic has traditionally been introduced between first and fifth grade, and remains crucial for students of all ages. The Common Core State Standards for mathematics indicate that first-grade students should learn how to add

  • Fractional Calculus in Wolfram Language 13.1
    by Tigran Ishkhanyan on August 12, 2022 at 9:10 pm

    What is the half-derivative of x? Fractional calculus studies the extension of derivatives and integrals to such fractional orders, along with methods of solving differential equations involving these fractional-order derivatives and integrals. This branch is becoming more and more popular in fluid dynamics, control theory, signal processing and other areas. Realizing the importance and potential


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