What is Mathematical: Definition and 1000 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.

View More On Wikipedia.org
Recent contents Top users
  1. T

    Inequality Mathematical Induction

    Homework Statement Prove the Inequality for the indicated integer values of n. n!>2^n,n\geq4 Homework Equations The Attempt at a Solution For n=4 the formula is true because 4!>2^4 Assume the n=k k!>2^k Now I need to prove the equation for k+1 I can multiply both sides by 2 and have...
  2. T

    Proving the Formula Using Mathematical Induction

    Homework Statement Use mathematical induction to prove the formula for every positive integer n. \sum_{i=1}^{5}i^5=\frac{n^2(n+1)^2(2n^2+2n-1)}{12} Homework Equations The Attempt at a Solution I know this will be true because the RHS is just your standard sums of powers of...
  3. S

    {Theoretical, Mathematical, Particle, Plasma} Physics

    I recently acquired my BSc in physics and am planning to continue my studies on the MSc (and hopefully PhD) level. I have reasonable expectations to get into a very good program dealing on the topics of mathematical and theoretical physics. The main topics covered by the courses in that...
  4. H

    Mathematical Proof of String Theory

    I have only taken a few college physics courses, and I have watched some videos about string theory. I would like to get excited about M-theory, but I have always wondered, why there wouldn't be an infinite number of ways to unite the forces, in which case M-theory is just one possibility...
  5. G

    Did Bell's Theory Contain a Mathematical Flaw?

    I would hope this thread could be limited to Bell-mathematical questions and answers concerning just one given Bell paper -- though it may involve many mathematical questions as we follow it to resolution. This thread concerns mathematical questions in one Bell paper and so it should NOT need...
  6. N

    Mathematical relationship to control camera heading

    Hi everyone, I want to find a mathematical relationship between my head direction and the direction of a camera placed at a distance. The objective is to make the camera heading to same point where my head is looking at. Both my head and the camera are in the horizontal plane and the motion...
  7. J

    Math Proving: Are Computer Derived Theorems Accurate?

    If we feed all the existing mathematical axioms to a powerful computer, it should be able to give us all the proofs and theorems that can be derived using the axioms. Is there anything wrong with this logic?
  8. K

    Is the principle of mathematical induction unable to prove cases of negative numbers?

    Is the principle of mathematical induction unable to prove theorem of negative numbers?
  9. K

    Can the strong form of mathematical induction always be used?

    Can i use the strong form every time? Why do people still use the ordinary form?
  10. C

    How Do Tire Forces Relate to Temperature, Pressure, and Speed?

    I am trying to develop an understanding of the forces within a tire when moving. I have been told to examine the tire as a ring of short compression springs, so I would assume each tire essentially has a spring coefficient and spring rate. I don't imagine the tire manufacturer's give...
  11. C

    Translating english into mathematical equations

    I heard that doing physics is basically translating english into mathematical equations, Could someone elaborate on this, and provide an example? thanks"
  12. S

    Question About a Course on Mathematical Thinking

    I have to take a required course for my major this fall titled Mathematical Thinking. The course description says something along the ways of "students are expected to give presentations, for examples and counterexamples and proof mathematical theories..." that's not verbatim, but it's what I...
  13. G

    Piston diameter size, fricition, mathematical proof

    I have already asked the question of piston diameter size and efficiency in an earlier thread but i'll recap. My question was that is it true that the larger diameter piston have a larger propensity to overcome the mechanical loss of fricition due to the piston rings contact with the mating...
  14. R

    Mathematical Statistics- two sample t-test

    Homework Statement At the 5% level, will the two means be equal ? Homework Equations The Attempt at a Solution I tested the variances and found out that it's very likely that the variances are equal, so this is an assumption we make when we do the second test. Now, I'm trying...
  15. W

    Mathematical treatment of Special Relativity

    Does anyone know of any book that treats special relativity from a mathematical standpoint? I want to learn SR before starting to read Schutz/ Hartle/ Carrol.
  16. J

    Mathematical Biology: Exploring Biology & Physics Combined

    I have just started back to school studying math and physics prior to chasing an advanced degree when I retire in two years (yeah, but that's a whole nother topic). My original BS was in Biology, and I would like to do research that includes that science. So, as I was looking over a local...
  17. K

    Guidance in Mathematical Physics

    Hey all, in need of some advice. I'm currently at UWO going into second year. I'm enrolled in an Astrophysics and Pure Mathematics program, and need to choose which courses I need to take, specifically math courses, for my future. I have a 4.0 and am strong in math. I'm wondering which...
  18. Z

    How can we prove that p(n) is true for all integers n with 1 <= n < v.size()?

    I cannot get my head around this at all. Suppose that v is of type Vector of Integers and we know the following:- 1. v is sorted. 2. No two items in v are the same. 3. v.at(1) is 12. For an integer n, where 1 <= n <= v.size(), let p(n) be the following proposition: p(n): v.at(n) => 11...
  19. E

    Mathematical induction and arithmetic progression

    Homework Statement All the terms of the arithmetic progression u1,u2,u3...,un are positive. Use mathematical induction to prove that, for n>= 2, n is an element of all positive integers, [ 1/ (u1u2) ] + [ 1/ (u2u3) ] + [ 1/ (u3u4) ] + ... + [ 1/ (un-1un) ] = ( n - 1 ) / ( u1un)...
  20. R

    What is this mathematical trickery? (removing a square root?)

    Homework Statement So I was doing the following relativity problem: An atomic clock moves at 1 000 km/hr for 1 hr as measured by an identical clock on the Earth. At the end of the 1 hr interval, how many nanoseconds slow will the moving clock be compare with the Earth clock? The problem...
  21. I

    What Math Should I Know for Classical and Quantum Mechanics?

    I'm interested in taking Classical and Quantum Mechanics courses next year at my university; however, I won't be able to take the prerequisite mathematics courses due to scheduling conflicts. I have taken differential calculus, integral calculus, and vector calculus and I will be taking an...
  22. M

    Help with simple proof by mathematical induction

    Homework Statement prove: 0^2 + 1^2 + 2^2 + ... + n^2 = n(n+1)(2n+1)/6 Homework Equations The Attempt at a Solution I'm confused on how to prove this by induction. I'm not exactly sure what the goal of the rearrangement is after substituting (n+1). Any help is much...
  23. A

    Solving for Car Depreciation: Linear, Quadratic, and Exponential Models

    Homework Statement A car depreciates over time. x(years) y ($) 1 14000 2 9100 3 6200 4 4000 5 3000 i) determine the linear model (1 mark) ii) determine the quadratic model (1 mark) iii) determine the...
  24. A

    Mathematical Induction Proof

    Homework Statement Prove that for all the natural numbers n that 2^n > n Homework Equations The Attempt at a Solution Base Case is easy Then the inductive step you have 2^k > k as the inductive hypothesis show that p[k+1] holds 2^(k+1) > k+1 on the left side 2^(k+1) = 2^k * 2...
  25. D

    Mathematical notation to multiply matrix elements

    Hi all, I need to figure out the mathematical notation to multiply matricies A and B = C so that... If matrix A is... a1 a2 a3 a4 a5 and matrix B is... b1 b2 b3 b4 b5 so that matric C is... a1*b1 a2*b2 a3*b3 a4*b4 a5*b5 Would this involve using a the transverse of a...
  26. S

    Mathematical programming (simplex method)

    Homework Statement This question is about simplex method A full time staff works seven hrs which includes an hour break in the middle (he is paid for his break). A part time staff works four hrs. The no of staff needed is shown below Time No of staff needed 9-10am 6...
  27. J

    Mathematical Modeling

    Homework Statement When certain viral particles enter the bdoy, the replicate to 160% every four hours and the immune system eliminates these particular viral particles at the rate of 50000 viral particles per hour. Find an equation modeling this viral growth. Homework Equations N/A...
  28. B

    How do you generate a mathematical model?

    For example, suppose you have a random set of points (x, y). How do you generate a mathematical model/function for these points? Is it possible to have a set of information that cannot be described by a mathematical model? How accurate are mathematical models? What determines how accurate...
  29. U

    Vehicle Mathematical modeling and dynamics

    i am working on the vehicle full suspension model. I am in modeling phase , i am electrical student and i am facing some difficulty in the dynamic results . When i am giving road disturbance to the vehicle the pessanger seat is getting more bounce and oscillations as compared to body to which...
  30. thrill3rnit3

    Mathematical Physics: What Do Mathematicians Do?

    Is there a separate area of study/specialization dedicated to mathematical physics? If so, what do mathematical physicists usually do?
  31. W

    Arithmetic in Mathematical Fields (in particular GF(2^8) )

    Homework Statement * note that I meant fields in the abstract mathematical sense, notphysical (i.e. electric) fields! * Finding the determinant of a matrix in GF(28) I want to know if it is using mod 256 or mod 255 in the field of GF(28) Homework Equations Just math The...
  32. M

    Mathematical Induction on rationals

    Hi I'm a high school student. I gave a proof for the following theorem, but I was told by some professors that this is trivial and using natural induction twice for the rationals will do the same thing. What do you think? Is it just redundant? Theorem: Let P(r) be a statement about r...
  33. A

    Quick one about first steps in mathematical modelling of nuclear stuff

    Hi, I am looking to get some good information sources on modelling nuclear power in power plants and related fields. Anyone got any good starting points ? P.S. I am a biomedical engineer looking into different fields.
  34. G

    Discrete Math Problem : Mathematical Induction

    Homework Statement Prove that H1 +H2 +...+Hn = (n +1)(Hn-n)? Homework Equations Hn denotes the nth harmonic number. The nth harmonic number is the sum of 1+1/2+...1/n, which is n / n +1. I'm not really sure if Hn = (1/ n) . Prove by Mathematical Induction Hn denotes the...
  35. Fredrik

    Mathematical statements with logical symbols

    I'd like to get better at writing mathematical statements using logical symbols. Let's start out with something simple. "For every A with property x, there's a B with property y". How should I write that? I don't even know if there's a symbol for "with property". How about this? \forall...
  36. T

    Beginner's mathematical proof / composition of relations

    Homework Statement Suppose r and s are two positive real numbers. Let Dr and Ds be defined as in part 3 of Example 4.3.1. What is D_r \circ D_s? Justify your answer with a proof. (Hint: In your proof, you may find it helpful to use the triangle inequality.) Homework Equations Example 4.3.1...
  37. T

    What are the possible values of a in the inequality a < 1/a < b < 1/b?

    Homework Statement I originally made this thread for something else, but I have another problem that I need help with. Suppose a and b are nonzero real numbers. Prove that if a < 1/a < b < 1/b then a < -1. Homework Equations A hint was given for the problem: Assume a < 1/a < b <...
  38. A

    Mathematica Proving 1/2 Series Converges Using Mathematical Induction

    (a) Give a recursive definition of the set P of all non negative integers, (b) formulate the applicable induction principle and (c) then apply the induction principle to prove that 1/2^0+1/2^1+1/2^2...+1/2^i = 2-1/2^n for n>=0 I have solved parts a and b and stuck on c (a) P is the...
  39. F

    Terence tao's list of mathematical embarrassments

    http://rjlipton.wordpress.com/2009/12/26/mathematical-embarrassments/: i didn't think the invariant subspace problem would be an embarrassment. i guess it's not all that hard for finite dimensions. only one person mentioned goldbach's conjecture, & nobody said anything about perfect numbers...
  40. P

    Mathematical Methods for Physicists - Arfken

    I don't know where to post this message, excuse me. I am taking the course "Matmematical Methods for Physicists." And my textbook is by Arfken. I wonder anyone here knows online sources that I might benefit for studying. I am in lack for good sources please help.
  41. C

    What is the mathematical genealogy of Felix Klein?

    For another BRS thread, I looked up some information at http://www.genealogy.ams.org/ and was completely sidetracked by my attempt to look up the genealogies of my undergraduate mentors :smile: Seriously, physics needs a tool like this, because it really does give amazing insight into the...
  42. K

    How important is probability and mathematical statistics?

    I'm a sophomore in math.I intend to learn mathematical and physics something like differential geometry,topology and QFT in my next years and do some research in some relative topics.I know I need to learn many courses and I feel I don't have enough time.This semester we open a course called...
  43. M

    Mathematical notation question

    Hi, I need to translate some conditions/statements into mathematical notation but my mind is drawing blank right now. There are a set of processes p={1,2,...n} and a class of resources r={a,b,...x}. 1 set of resources must be assigned to each existing process (e.g. resource a is...
  44. S

    Understanding the cdf of a Sum of Independent Random Variables

    I am have trouble with a question, and how exactly each given part of this question relates to getting the answer. [b]1. The problemb] Assume that N,X1, X2...are independent. Let P(N = k) = qk-1p , k\geq1, p+q = 1 and let X1,X2,X3,...be iid with a common pdf f(x)...
  45. M

    Proving 2^n>=(n+1)^2 by mathematical induction

    Hello, In this problem I am trying to Determine the exact set of natural numbers n for which the inequality 2^n>=(n+1)^2 holds. (equation (1)) I have already dealt with the base case where n=6, (since the inequality does not hold for n<6), and so (1) holds for n=k, and I need to show that it...
  46. A

    Prove Inequality by Mathematical Induction

    Homework Statement \forall n \in Z^+, \sum_{i=1}^n \frac{\sqrt{i+1}}{2i} > \frac{\sqrt{n}}{2} Homework Equations I have to prove the above via mathematical induction. The Attempt at a Solution I did the base case, n = 1 and found it true for the base case. Then I assumed that...
  47. L

    Mathematical Probability Question

    I spent quite a lot of time trying to figure out this question that I read in a book. Through trial and error, I got a rough idea of what the solution should be like but can't figure out how I would derive it mathematically. Can anyone give me some directions? :) QUESTION: n people stand...
  48. W

    Mathematical treatment of analog electronics

    Hello. I was wondering if anyone here could recommend a book on analog electronics that doesn't shy away from the math, but isn't too advanced for my level (electronics-wise, not math-wise). For reference, I've gone through Floyd's Electronics Fundamentals (although I don't necessarily...
  49. C

    Using the method of mathematical induction

    -------------------------------------------------------------------------------- 1. Homework Statement Let H be a ten- element set of potive integers ranged from 1 to 99. Prove that H has two disjoint subsets A and B so that the sum of the elements of A is equal to the sum of the elements of B.
  50. H

    Peskin Eq 11.72, mathematical identity

    In Eq 11.72 in the QFT text by Peskin, the following equality is stated: i\int\frac{d^{d}k_{E}}{(2\pi)^{d}}\log(k_{E}^{2}+m^{2})=-i\frac{\partial}{\partial\alpha}\int\frac{d^{d}k_{E}}{(2\pi)^{d}}\frac{1}{(k_{E}^{2}+m^{2})^{\alpha}}|_{\alpha=0} This suggests that...
Back
Top