What is Mathematics: Definition and 997 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. TurtleKrampus

    Why are Physicists so informal with mathematics?

    I realize of course that this will probably not apply to all physicists, but at least every physicist in my university's math department is very unrigorous when it comes to mathematics. This is frustrating because some of the physics material seems genuinely interesting, but the lack of an...
  2. G

    Difference between -3² and (-3)² ?

    What is the difference between these to. -3² and (-3)² ? I know - x - = + I am told -3²= -9 not +9 -3² means -3 x -3 = 9 I am told (-3)² = 9
  3. A

    A Summing over continuum and uncountable numerocities

    Here I want to address of the question if it is possible to make a sum over an uncontable set and discuss integration rules involving uncountably infinite constants. I will provide introduction in very condensed form to get quicker to the essense. Conservative part First of all, let us...
  4. hackedagainanda

    Partial Fraction Decomposition

    ##\frac {1} {x^2 -c^2}## with ##c \neq {0}## So the first thing I do is split the ##x^2 -c^2## into the difference of squares so ##x +c## and ##x - c## I then do ##\frac {A} {x + c}## ##+## ##\frac {B} {x-c}##, and then let ##x=c## to zero out the expression. And that is where I am getting...
  5. fibrebundle

    Looking to meet other people who like math

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  6. C

    Programs What subject is better for an aspiring experimental physicist?

    I'm in my last 2 years of high school, and I have to pick a speciality to study before becoming an undergraduate and studying in college. In the future, I'm hoping to become an experimental physicist. My high school offers 3 specialities that are relevant to physics to pick from, all of them...
  7. Jonathan West

    New book out by Routledge called "Mathematical Conundrums" with many interesting problems

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  8. bhobba

    Insights What Are Numbers? Table of Contents

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  10. yucheng

    Style of teaching/learning mathematics: by proofs of theorems only?

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  11. A

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  12. chwala

    I Order of an element in ##\mathbb{Z}_n##

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  13. A

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  14. C

    How Can a Web-App Make Theoretical Physics and Mathematics More Accessible?

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  15. nicolegeogery

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  16. chwala

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  17. azizlwl

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  18. chwala

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  19. Mark44

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  20. S

    I P-adic numbers and the Ramanujan summation

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  21. E

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  22. fresh_42

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  23. advhaver

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  24. DaveE

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  25. Feynstein100

    B Is everything in math either an axiom or a theorem?

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  26. Vectronix

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  27. G

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  33. D

    Will the Riemann hypothesis be solved by 2100?

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  34. chwala

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  35. Slimy0233

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  36. RChristenk

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  37. Slimy0233

    Studying How to self study Physics and Mathematics

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  38. Charles Link

    So many people are unable to do simple algebra

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  39. Slimy0233

    Overcoming English Difficulty to Learn Physics

    I am an Indian and even though my English knowledge is somewhat better than that of an average Indian, there isn't a day I try to real highly technical books and fail to understand it at the first glance (or don't understand it at all). I understand why they use language which might seem...
  40. Abhikamya M S

    Schools Which colleges in India are best for physics and/or mathematics BSc MSc and PhD?

    In 12th and considering a career in research. Ik most of the collages ( IIT Kanpur, IISc Bangalore, IISERs etc ) but I am not sure of the research possibilities provided so if anyone can help out please do
  41. M

    Does 'rest' have a precise mathematical definition?

    Math qyuestion for AI (Skype) include an expression that x is resting on y (both straight line segments). AI insisted that x coincides with y, while my intent was only placing x on y. Does 'rest' have such a narrow definition?
  42. T

    A Separation of variables is possible only in 11 coordinate systems?

    I vaguely (strong word there because I can no longer remember the source, but the idea sticks in my head for 30 years now) recall reading (somewhere long forgotten) that method of separation of variables is possible in only 11 coordinate systems. I list them below: 1.Cartesian coordinates...
  43. S

    Solving this definite integral using integration by parts

    Using integration by parts: $$I_n=\left. x(1+x^2)^{-n} \right|_0^1+\int_0^{1} 2nx^2(1+x^2)^{-(n+1)}dx$$ $$I_n=2^{-n} + 2n \int_0^{1} x^2(1+x^2)^{-(n+1)}dx$$ Then how to continue? Thanks
  44. T

    I Understanding Vectors: Magnitude & Direction

    Good Morning (And apologies if this is not the right forum -- it is not a homework problem.) On the one hand, a vector is an arrow and a tail: it has magnitude and direction. It is used to describe direction, forces, acceleration, etc. However, there are more mathematical definitions: a...
  45. Euge

    POTW An Integral with Fractional Part

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  46. S

    What type of mathematics did you find the most difficult?

    Mathematics tends to be more progressive than any other field. But I've heard some people say that some math classes that they took were more difficult than the most advanced math classes that they took. For instance, I've heard people say that Calculus II is more difficult than Calculus III...
  47. S

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  48. Theia

    Geometry Looking for a good book about trigonometry

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  49. J

    Other Math Olympiad Advice Please (I will be 13 years old next year in 9th grade)

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  50. C

    Why Can We Take Limits of Both Sides? [Answered]

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