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. Sahan Naveen

    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. A

    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...
  3. G

    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?
  4. G

    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?
  5. macforlife

    Foundations High school and undergraduate math textbooks for refreshing my skills

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  6. robertphy

    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.
  7. M

    Problems in manipulating these 4 radical equations

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  8. T

    What are the two different senses of since and how are they used?

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  9. C

    I’m Cosby, a Math student from Ghana 🇬🇭

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  10. Magnetons

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  11. F

    I Demonstration of inequality between 2 variance expressions

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  12. P

    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...
  13. rudransh verma

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  14. E

    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...
  15. C

    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...
  16. S

    What Can You Expect to Learn from Studying Math and Physics in College?

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  17. J

    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...
  18. J

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  19. Which rating is better, mathematically speaking? | Probabilities of probabilities, part 1

    Which rating is better, mathematically speaking? | Probabilities of probabilities, part 1

    Part 1 of 3 videos on probability by 3blue1brown
  20. Lapse

    Comp Sci How do I determine the linear/non-linearity of this problem?

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  21. W

    I How to visualize the maths behind a formula?

    I just finished my freshman year in computer engineering at my college, and after some physics and materials science classes I noticed a pattern. Everybody—at least in my class, I don't about the rest of the world—is just memorizing formulas, myself included. I'm really trying to not just...
  22. A

    Is it possible to have physics without mathematics?

    or in general, is it possible to create science without mathematics? thanks
  23. EchoRush

    I What can we learn from higher derivatives in calculus?

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  24. M

    Can Math Doubts Help You Solve Your Math Problems?

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  25. Morbidly_Green

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  26. H

    I Pole Identification Using Block Diagram Reduction

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  27. M

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  28. S

    Regression for a first order system

    Homework Statement I am carrying out a regression for diameter of a part Homework Equations Diameter = -0.0531052 + 0.0443237 * exp (-0.0103633 * 'Time elapsed') if diameter is -0.052 then can some one please calculate the value for time elapsed would you please explain the steps The...
  29. Celso

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  30. Boltzman Oscillation

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    Homework Statement Given: Ψ and Φ are orthonormal find (Ψ + Φ)^2 Homework Equations None The Attempt at a Solution Since they are orthonormal functions then can i do this? (Ψ + Φ) = (Ψ + Φ)(Ψ* + Φ*)?
  31. P

    Silicon not heating with induction heater?

    I’m using an inductive heater to try to melt some 99.85% polycrystalline silicon, as would occur in the Czochralski process, but the silicon workpiece is not melting let alone even getting hot. The silicon workpiece is about the size of a playing dice. It becomes conductive at 200 F and becomes...
  32. StenliSuryadinata

    Who Is Stenli Suryadinata and What Does He Want to Share?

    Hello every one, My name is Stenli Suryadinata from Indonesia. I found that this site was a great place to learn and share. I wish to learn and share interesting things with all of you Thank you
  33. A

    I Discrete to continuum Gaussian function

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  34. S

    I Is There a Simpler Way to Understand Mathematics?

    I am sorry but i am not a mathematician Could this be simplified ?
  35. S

    French student look after pre-calculus books

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  36. C

    B Does anybody think that there is an underlying reality?

    It seems like we keep chasing "reality", and by "reality" I think Physicists would mean the apparent rules of quantum physics which we hope would (if applied) lead to all the apparent known rules of macro-physics. However ... It seems like we have had to create a few things to do that: 1. Ideas...
  37. Robert Wilson

    Easy Algebra Learning: Simplify the Process

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  38. Subrahmanyan

    To find the nature of roots of a quintic equation....

    The asks for us to find the nature of roots of the following equation ,i.e,rational or irrational nature of the roots: the Equation is : http://www5a.wolframalpha.com/Calculate/MSP/MSP7106108efdacf0gb84ie000039f29d33b6hie7ic?MSPStoreType=image/gif&s=56&w=63.&h=18.x^5+x=5 I have been able to...
  39. A

    Deleting Old Accounts on Scientific Forums

    Hello! New here. Sort of. I have a major in math. I'd like to specialize in data science for my master's degree. This is my 2nd account here. I don't know how to delete the other one. Can anyone tell? Thanks.
  40. IonizingJai

    Implicit differentiation problem

    Homework Statement If ##x\sqrt{1+y} + y\sqrt{1+x } = 0##, then prove that ##\frac {dy} {dx} = \frac {-1}{(x-1)^2}##. 2.Relevant Equations: $$ \frac {dy} {dx} = - \frac {\left (\frac {\partial f}{\partial x} \right)} {\left( \frac {\partial f} {\partial y} \right)}.$$ 3...
  41. R

    Schools Germany - Masters EE/mathematics/physics online

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  42. pairofstrings

    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...
  43. M

    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...
  44. Jeviah

    Solving Simpson's Rule: Finding cos^2(1^2)

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  45. M

    Programs Physics or mathematics degree?

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  46. M

    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...