# What is Mathemathics: Definition and 104 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. ### Courses Choosing courses for mathematics minor to support my physics major

I'm interested in having a career in physics (I'm leaning more towards theoretical work instead of experimental aspect of the subject). I'm currently doing my undergraduate studies and decided to pursue Mathematics as a minor subject. These are the courses that I'm planning to take...
2. ### Integrate [cosec(30°+x)-cosec(60°+x)] dx in terms of tan x

I proceeded as follows $$\int\frac{2(\sqrt3-1)(cosx-sinx)}{2(\sqrt3+2sin2x)}dx$$ $$\int\frac{(cos(\pi/6)-sin(\pi/6))(cosx-sinx)}{(sin(\pi/3)+sin2x)}dx$$ $$\frac{1}{2}\int\frac{cos(\pi/6-x)-sin(\pi/6+x)}{sin(\pi/6+x)cos(\pi/6-x)}dx$$ $$\frac{1}{2}\int cosec(\pi/6+x)-sec(\pi/6-x)dx$$ Which leads...
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### Complex contour integral proof

I’ve attached my attempt. I’ve tried to use triangle inequality formula to attempt, but it seems I got the value which is larger than 1. Which step am I wrong? Also, it seems I cannot neglect the minus sign in front of e^(N+1/2)*2pi. How can I deal with that?
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### Put the eigenvalue function in self-adjoint form

Here’s my work: The integrating factor I find is (x^(2)-1)^1/2. The self adjoint form I find is -d/dx (((1-x^(2))^(3/2))*dy/dx))=k(x^(2)-1)^(1/2). Am I right?
6. ### I'm studying to be a year 7-9 (age 12-15) science and math teacher.

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7. ### Foundations High school and undergraduate math textbooks for refreshing my skills

Hi everyone, Im new to the forum and physics in general. I recently started to dig a bit into popular science on quantum mechanics, cosmology, general relativity etc. and found it super interesting ( as everyone on this forum I assume ). I think the time has come for me to dig a bit deeper into...
8. ### Foundations Books and websites about manipulatives in mathematics?

Do you know about books and websites about manipulatives in mathematics teaching for secondary schools? I think manipulatives are incredibly useful in teaching Mathematics. Thank you for your support.

10. ### Problems in manipulating these 4 radical equations

In order to write next step in all four equations above l used the definition of radicals. ##\sqrt a=b## means ##b^2=a##. Squaring both sides also works. I don’t know if it’s right. I mean I read that ##(\sqrt a)^2=a##. But I don’t know if we can apply this on expressions. Main problem is if we...
11. ### What are the two different senses of since and how are they used?

Summary: Hello! I'm an high school student and i want to study more math but I'm not sure where to start. Should i first study linear algebra or calculus? Hello! I'm an high school student and i want to study more math but I'm not sure where to start. Should i first study linear algebra or...
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### I’m Cosby, a Math student from Ghana 🇬🇭

I’m Cosby from Ghana.A student of the University of Education-Winneba in Ghana.. I’m in my first year and I’m trying to get help yo develop my mathematical skills in order to become a good mathematician.And that is why I joined this family.
13. ### Find z^n+ 1/z^n: Why Consider Only One Argument?

Firstly I converted the given equation to a quadratic equation which is ##z^2- (\sqrt3)z+1=0## I got two solutions: 1st sol ##z=\frac {(\sqrt3 + i)} {2}## 2nd sol ## z=\frac {(\sqrt3 -1)} {2}## Then I found modulus and argument for both solution . Modulus=1 Arguments are ##\frac...
14. ### I Demonstration of inequality between 2 variance expressions

Just to remind, ##C_\ell## is the variance of random variables ##a_{\ell m}## following a Gaussian PDF (in spherical harmonics of Legendre) : ##C_{\ell}=\left\langle a_{l m}^{2}\right\rangle=\frac{1}{2 \ell+1} \sum_{m=-\ell}^{\ell} a_{\ell m}^{2}=\operatorname{Var}\left(a_{l m}\right)## 1)...
15. ### I Notation of the approximation in quantum phase estimation algorithm

I'am interested in the notation of the approximation in quantum phase estimation algorithm. In the literature there are different definitions, which I divide into two cases here. Both different in their definition of the ##\delta##. In both cases I start with a quote of the source and show an...
16. ### Intro Math Maths needed to advance my physics understanding

I am a 10(high school)+ 2 years student. I have a basic knowledge of calculus, trigonometry, algebra, arithmetic. I need to advance as an undergraduate to start my physics journey. I want a mathematics book for physics which if it will cover some basics in early chapters will be of much help...
17. ### Programs I chose math instead of physics. Was it a mistake?

Long story short, my career goals are to work as a researcher/research scientist in machine learning or to develop models/algorithms for forecasting such as in financial markets or otherwise. Fundamentally, I find think this is quite similar to physics as a science, but that is for another...
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### Does the stereotypical nerd or geek really exists?

Do the stereotypical nerds and geeks really exist? I am talking about socially awkward timid loners who are also highly intelligent in Science, Mathematics, Technology, and Engineering. What happens to them later in real life? Do they really end up as unemployed or menial workers because of...
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### What Can You Expect to Learn from Studying Math and Physics in College?

hi all! I study both math and physics, studying for a bachelor degree at college. I hope it will be useful and fruitful for everyone here!
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### What are the key topics in Advanced Calculus and Algebraic Geometry?

Hello, I am a very experienced Mathematician with a BSc Honours degree in Mathematics and one year MSc studies in Operational Research in Sussex and London Universities respectively. I am interested in Advanced Calculus, Algebras, Positivity in Algebraic Geometry, The standard Model, and many...

45. ### Schools Germany - Masters EE/mathematics/physics online

Hello Everyone, So I have been looking for some time now to find an online program in EE/mathematics/physics. There are a number of programs I have found (purdue, johns hopkins, etc) but the prices are absolutly staggering. I have paid my dues to the student loan machine with my BS degree in...
46. ### I System to represent objects in Mathematics

Hi. Usually, Computer Programmers use Flow Charts, Algorithms, or UML diagrams to build a great software or system. In the same manner, in Mathematics, what do Mathematicians use to build a great system that they want to build. Category Theory is at the highest level of abstraction; then...
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### Intro Physics Is Learning Calculus First Necessary for Self-Studying Physics?

Hello. I’d like to self-learn physics. basically starting from a zero knowledge background to keep things simple. I have asked before and some individuals said that it is using your time to learn it once you know calculus? Is this true or should I start learning the basics now so that when I...
48. ### Solving Simpson's Rule: Finding cos^2(1^2)

Homework Statement hello, I'm currently studying simpsons rule (unrelated) however the method requires the answer to cos^2(1^2) the answer given by my tutor is 0.2919, I have been unable to get this answer after inputting cos in various ways I always get 0.9997, which is right and if 0.2919 is...
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### Programs Physics or mathematics degree?

Hi, currently I am having a dilemma trying to decide whether I want to do a 4 year Msci in physics or mathematics. I am about to finish a degree in computer science I had started when I was 13 or 14 with the open university. Although, I still want to go to a real university for mathematics or...
50. ### Linear algebra, field morphisms and linear independence

Homework Statement Let f1,f2, ..., fn : K -> L be field morphisms. We know that fi != fj when i != j, for any i and j = {1,...,n}. Prove that f1,f2, ..., fn are linear independent / K. Homework Equations f1, ..., fn are field morphisms => Ker (fi) = 0 (injective) The Attempt at a Solution I...