# mathematics Definition and Topics - 170 Discussions

Mathematics (from Greek: μάθημα, máthēma, 'knowledge, study, learning') includes the study of such topics as quantity (number theory), structure (algebra), space (geometry), and change (analysis). It has no generally accepted definition.Mathematicians seek and use patterns to formulate new conjectures; they resolve the truth or falsity of such by mathematical proof. When mathematical structures are good models of real phenomena, mathematical reasoning can be used to provide insight or predictions about nature. Through the use of abstraction and logic, mathematics developed from counting, calculation, measurement, and the systematic study of the shapes and motions of physical objects. Practical mathematics has been a human activity from as far back as written records exist. The research required to solve mathematical problems can take years or even centuries of sustained inquiry.
Rigorous arguments first appeared in Greek mathematics, most notably in Euclid's Elements. Since the pioneering work of Giuseppe Peano (1858–1932), David Hilbert (1862–1943), and others on axiomatic systems in the late 19th century, it has become customary to view mathematical research as establishing truth by rigorous deduction from appropriately chosen axioms and definitions. Mathematics developed at a relatively slow pace until the Renaissance, when mathematical innovations interacting with new scientific discoveries led to a rapid increase in the rate of mathematical discovery that has continued to the present day.Mathematics is essential in many fields, including natural science, engineering, medicine, finance, and the social sciences. Applied mathematics has led to entirely new mathematical disciplines, such as statistics and game theory. Mathematicians engage in pure mathematics (mathematics for its own sake) without having any application in mind, but practical applications for what began as pure mathematics are often discovered later.

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1. ### I Quantum phase estimation - Question regarding rewriting the state

In Nielsen and Chuang p.223 we have the following situation: $$\frac{1}{2^t} \sum\limits_{k,l=0}^{2^t-1} e^{\frac{-2\pi i k l}{2^t}} e^{2 \pi i \varphi k} |l\rangle$$ Which results from applying the inverse quantum Fourier transform to state ##\frac{1}{2^{t/2}} \sum\limits_{k=0}^{2^t-1}...
2. ### Easy geometry problem, please give me a hint...

I have calculated the height of the segment using the Pythagorean Theorem and that's currently where I am right now. I don't seem to find any equations that can help me. Though I might be not trying hard enough or using the wrong words because I'm not really fluent in mathematical terms as you...
3. ### How to sell math/physics textbooks in Europe?

Hello everyone, I don't know if that's the correct place to ask this (didn't want to pollute the science and math textbooks recommendations), sorry if it is not. I have undergraduate physics and mathematics textbooks that I am trying to sell. I live in Belgium and I bought textbooks...
4. ### I Show that an expression approaches an integer

I came across a rather strange thing in an introductory class I still don't understand. There was a statement that $$lim_n (2+ \sqrt(2))^n$$ is an integer. I recalled that I never understood this and just recently tried to take the limit but just get that the expression diverge? Which I think...
5. ### Mechanics Circular Motion Question

Part (i) *Pink represents mark scheme method Part 2 However I am still confused on the diagram.
6. ### Studying Should I read Introduction to Calculus and Analysis by Courant?

Should i read Introduction to Calculus and Analysis by Courant? I have calculus background I want to study multivariable and single variable in rigorous way There is a lot of books in this subject like Spivak, apstoal,Courant,.... . I am not sure what to choose
7. ### A Help required to sum an infinite series in a given equation

Hi, I have a particular equation in a paper, wherein the author specifies an infinite series. The author has apparently found the sum of the series and calculated the equation. Can anyone please help me in understanding how to sum such a series. I have attached part of the paper with the...
8. ### I Quantum Causal Modelling: Mathematics

I have been trying to read through the following paper: https://iopscience.iop.org/article/10.1088/1367-2630/18/6/063032/pdf, but am stuck at the parts from page 6 onwards. What is the "approach/formalism" used in the following sections of the paper, and are there any gentle introductions...
9. ### Find the atom using a quantum oracle

My attempted solutions was, for example let's say we have 4 atoms, and if i ask the oracle about any two atoms that are connected by edge, i can narrow done some possibilities to two atoms. I'm still not sure where i am going with my solution, but if any of you can think this through and come up...
10. ### I'm still trying to solve this derivative :(

I have attached a word document demonstrating the working out cos i was too lazy to learn how Latex primer works and writing it like I did above would've been too hard too read. I tried to make it as understandable as possible, presenting fractions as ' a ' instead of ' a / b ' . ------ b
11. ### Studying Is this math book enough for self-studying undergrad physics?

I'm talking about "Mathematical Tools for Physics by James Nearing". https://www.amazon.com/dp/048648212X/?tag=pfamazon01-20 it seems it cover lots of math... but it feels like it's just a reference book.. will it be enough? or should i get more specialized books to learn each subject thoroughly?
12. ### I Can the volume of a control volume change with time?

Can an expanding balloon be considered a control volume?
13. ### Insights Lie Algebras: A Walkthrough The Representations - Comments

Greg Bernhardt submitted a new blog post Lie Algebras: A Walkthrough The Representations Continue reading the Original Blog Post.
14. ### Insights Lie Algebras: A Walkthrough The Structures - Comments

Greg Bernhardt submitted a new blog post Lie Algebras: A Walkthrough The Structures Continue reading the Original Blog Post.
15. ### Insights Lie Algebras: A Walkthrough the Basics - Comments

Greg Bernhardt submitted a new blog post Lie Algebras: A Walkthrough the Basics Continue reading the Original Blog Post.
16. ### Math Job opportunities USA that combine data science with physics

Hello, I am a Belgian master in physics and astronomy with a couple of years of working experience under his belt, who recently decided to obtain a second master in statistics. Normally I should graduate in 2020. My goal is to become a data scientist, but preferably within a scientific context...
17. ### A Order of poles of z∕Sin(πz²)

When The denominator is checked, the poles seem to be at Sin(πz²)=0, Which means πz²=nπ ⇒z=√n for (n=0,±1,±2...) but in the solution of this problem, it says that, for n=0 it would be simple pole since in the Laurent expansion of (z∕Sin(πz²)) about z=0 contains the highest negative power to be...
18. ### I Compatibility of Physics and Abstract Algebra

One of the last classes I'm taking before finishing my degrees as an undergraduate is abstract algebra. My professor uses the textbook 'Contemporary Abstract Algebra' by Joseph Gallian. The book isn't written terribly nor is the teacher a poor one, but I just find this subject so...
19. ### Programs Doing a joint honours in maths and physics or just physics?

Hi, I'm currently a freshman and I need to choose my program next year! My goal is to pursue physics and go to graduate school for theoretical physics (experimental is still an option). I firstly planned to do a joint honours in mathematics and physics which is offered by the Math department in...
20. ### B Spivak's Calculus as a Prerequisite for General Topology

High school student here... Recently, I've found an interest in topology and am trying to figure out the correct path for self-studying. I am familiar with set theory and some concepts of abstract algebra but have not really studied any form of analysis, which from what I've read is a...
21. ### I Mathematical series in physics - Why and when do we need them?

Hi, Before I post my question, let me admit that my foundation on mathematics is poor. I am trying to work on it, specifically on the application part. When I came through the following image, I was stuck to understand why I will need one like Taylor's series in a simple case like "F+ΔF = F...
22. ### Schools Getting a mathematics mentor in high school?

High school student here... Over the past year or so, I've developed an intense interest in mathematics. I work problems every day and I am constantly working to grow my knowledge. As a junior, I am currently in AP calculus and will most likely be eligible for Calc 3 my senior year. I'm...

hi everyone, i'm electrical engineer student and i like a lot arnold's book of ordinary differential equations (3rd), but i have a gap about how defines action group for a group and from an element of the group.For example Artin's algebra book get another definition also Vinberg's algebra book...
24. ### Physics How to do independent research in Physics?

Hey folks! I have a degree in Electrical,Electronics and Communication Engineering and I work in Computer networking/Cybsersecurity field. Lately, I met one guy and we both decided to do some indepdendent research in electromagnetism as we both have fare amount of love for Physics( I'm in...
25. M

### Programs Best complementary degree for computer science?

What would be the best degree to have along side a degree in computer science? I am thinking it is either physics or mathematics. My interests involve, robotics, artificial general intelligence, 3D graphics, computer vision, neural nets, file compression and aerospace engineering(It’s an...
26. ### I Mathematical Structure and Mathematical Space Hierarchies

Hi. I am trying to understand the images that I have posted below. Each layer of Mathematical Structure Hierarchy in the image in this post and Mathematical Space Hierarchy in the image in this post are: statements. 1. What do these statements of each layer of these hierarchies in the...
27. ### Need help with an angular momentum problem...

Homework Statement I've spent at least 1.5 hours on this problem trying to figure out what i did wrong and I can't find anything. With an exam in two days plus another chapter to go through. Regardless, here are the problem(6) and answer, as well as my work. Hope you can read it, and the...
28. ### Sequencing game

<Moderator's note: LaTeX code edited.> 1. Homework Statement Hi, I have a question. I have a sequencing problem ##(N, \sigma_{0}##, ##\{p_{j}\}_{j\in J}, \{c_{i}\}_{i\in N})## with ##N = \{1,2,3\}## the set of players, and ##J = \{A,B,C,D,E,F\}## the set of jobs. The processing times of the...
29. ### Applied Numerical Analysis book question

So in my future numerical analysis class the recommended book i liked the most was Numerical mathematics by Quarteroni : https://books.google.pt/books/about/Numerical_Mathematics.html?id=m-bHBAAAQBAJ&printsec=frontcover&source=kp_read_button&redir_esc=y#v=onepage&q&f=false Since i like this...
30. ### Plasma Which order should I read these in?

I'm a computer scientist with aspirations to apply myself to fundamental research, data analysis, or computation, in physics; And these are the books I've purchased so far but haven't read. Which order do you think I should read these? Or more generally, how should I schedule the reading of...