Mathematical Definition and 1000 Threads

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. Q

    New mathematical operation labelled #

    This is neither homework nor independent study, it was a brain teaser( or so you could call it) I found. A new mathematical operation has been implemented , call it # It is used within our knowledge of numbers( it operates with all sets of numbers) and works together with our known operations (...
  2. D

    Mathematical studies before graduate school

    Dear fellows, my undergrad was not in maths, but I have quite a chance of getting into a master's program in pure maths, later this year. Long story, not particularly interesting. My question is due to the fact that I am quite prepared to the examination (basic real analysis and linear...
  3. R

    How Do You Prove Matrix Powers Using Mathematical Induction?

    Hey guys I am in precalculus right now and we just started picking up mathematical induction. Our teacher assigned us a problem that I am stumped over and I tried looking all over for a clear explanation online but I can't find anything remotely helpful. The question is: Use mathematical...
  4. B

    Are Real Coefficient Polynomials Always Rooted in Complex Conjugate Pairs?

    Homework Statement show that the roots of a polynomial with real coefficients are real of form complex conjugate pairs .The inverse is not true ,in general. Homework Equations The Attempt at a Solution
  5. M

    Principles of Mathematical Analysis by Rudin

    I am curious as to where this book falls in the hierarchy of mathematical education. Could it be used effectively before a calculus course? Is calculus necessary before analysis?
  6. T

    Compressor maps for mathematical model of a gas turbine

    Hello, I`m working on a mathematical model of a gas turbine and I need good compressor maps, especially function of efficiency vs. pressure ratio, inlet temperature, mass flow etc I`ve been looking for this for few days, but can`t find anything interesting. Do you know where I can...
  7. R

    Mathematical Analysis - Continuity

    Homework Statement Homework Equations The Attempt at a Solution Basically the first part of the question asks to prove the binomial theorem through induction which I've done. I'm basically lost as to how to even attempt these questions, I'm not asking for answers as I need to know this...
  8. R

    Proving a[n]≤2^n using Mathematical Induction

    1. Define a sequence of numbers in the following way: a[k]=a[k-1]+a[k-2]+a[k-3] for k≥3, s.t. a[0]=1, a[1]=2, a[2]=3, a[3]=6... *The numbers in brackets will be subscripts for the whole problem Prove that a[n]≤2^n using complete mathematical induction. This is what I have so far. We'll use...
  9. K

    Interesting practical mathematical problem

    Homework Statement Hello, I now have a real practical mathematical problem to ask I have 14 files with different sizes to be burnt on DVD discs. Each DVD disc is 4.7GB. How to arrange the files so as to minimize the use of discs? The file sizes of the 14 files are 1) 2.25GB...
  10. S

    Help understanding the mathematical expression of product of Inertia

    Homework Statement Hello! I'm stuck and confused. I've been browsing the web, and reading books but can't seem to get it right. I've been looking for the mathematical expression for product of Inertia. I've managed to understand the 50% of it. But I'm stuck on the first part...
  11. S

    Part 2 of mathematical modeling question

    Okay so after tearing my face off trying to find a 2-cycle, I've managed to make significant process. I divided an enormous polynomial with another polynomial and got a god awful mess which incidentally was correct. Now I'm having trouble solving this equation with x in terms of c...
  12. M

    Advice on a good Mathematical Methods for Physicists Book?

    Advice on a good "Mathematical Methods for Physicists" Book? Hope this is the right area to be posting it in. I've checked out Tai Chow's book with the same title from my school library. There was a pretty brutal Amazon review on it, but overall I find it helpful. Does anyone have...
  13. M

    Mathematical Analysis and Sequences

    Homework Statement The problem is: Show that an \rightarrow \infty iff for all \Delta > 0, \existsN such that n \geq N \Rightarrow an \rightarrow \infty Homework Equations Not sure if there are any The Attempt at a Solution I can't really think of anything to do here because I have...
  14. O

    Dimensional Analysis and the mathematical steps throughout a process.

    One would assume that: $$t \propto h^\alpha m^\beta g^\gamma$$ Where t = time taken for object to fall, h = height dropped from, m = mass, g = acceleration due to gravity. By doing some dimensional analysis one can find that: $$t \propto h^\frac{1}{2} g^\frac{-1}{2}$$ and that t is...
  15. S

    Book Suggestion on Mathematical Methods

    Greetings, I am an enthusiastic undergraduate university student in physics. In order to increase my mathematical knowledge and skills I would like to study a decent mathematical methods book. I came across two books whose names are: Mathematical Methods for Physics and Engineering by...
  16. M

    Where can I go with a degree in mathematical physics?

    So I just got accepted into the University of Waterloo's mathematical physics program and I don't know anything about the program. What kind of career could I expect to get if I graduate from the program? If I take this program could I still get a major in accounting, actuary sciences, etc...
  17. A

    Mathematical Definition of Energy

    What is the general definition of energy? I already know that it means ability to perform work and that Work = ∫Force d(displacement) = Δ Kinetic Energy = -Δ Potential Energy ( in a conservative field "a closed path integral of the force = 0"), Σ Kinetic-Potential = constant, ∫Kinetic-Potential...
  18. O

    Gershgorin Circle Theorem, mathematical derivation of eigenvalue estimates

    I intend to use the Gershgorin Circle Theorem for estimating the eigenvalues of a real symmetric (n x n) matrix. Unfortunately, I'm a bit confused with the examples one might find on the internet; What would be the mathematical formula for deriving estimates on eigenvalues? I understand that...
  19. S

    Need Help with a mathematical modeling question

    This question deals with a 1-parameter equation: fc(x)= (4/x)+(x/2)-c for 0≤c≤ 2.3 a.) Use algebra to find the positive fixed point p(c) of fc(x) (in terms of c) and identify its exact interval of existence a.) Ok so for the fixed point I got p(c)= -c+√(c2+8) I'm not sure if this is...
  20. H

    Mathematical induction problem solving question help

    hi this question i just cannot do. i have no idea where to start: a student is trying to recall the formula for the sum of cubes of consecutive numbers. she thinks it may be (n^3(n+1)^3) /8 or (n^2(n+1)^2)/4 . show by counter example one is incorrect and the other is correct by induction...
  21. C

    Where Can I Find Resources for Self-Studying Trigonometry and Pre-Calculus?

    Hello, this is my first post on this forum. I am a high school junior, and I am attempting to build onto my math foundation. Currently, I am finishing Algebra II. Next year, I will not be taking Trigonometry, as my teacher did not recommend me. We were supposed to have straight A's...
  22. S

    Which Universities Use Apostol's Mathematical Analysis Text?

    Which universities or instructors---anywhere in the world and particularly in the USA, Canada, UK, Australia, New Zealand, or India---use or have used Tom M. Apostol's landmark textbook on Mathematical Analysis? Are there any video lectures from a course based on the text available on the...
  23. N

    Difference Between Theoretical and Mathematical Physics

    Hey everyone If you want the concise version, skip over the bit in italics. First I'll quickly introduce myself. I am an undergraduate student, studying physics. I have been reading these forums for some time now (mostly just re-reading the stickies to be honest). In Australia, the first...
  24. M

    Physics and mathematical terms dealing with 1_

    Hey everyone, This is a really strange question and I wasn't sure where to post it, but I need those who are well versed in physics and math to help with this. I recently started a new band and we came up with the symbol "1_" as our logo. We had a name, One Unknown, but we found out that it...
  25. K

    Are Calculus and Linear Algebra prerequisites for Intro to Mathematical Proof?

    Hi, all Going to take the Intro to mathematical proof class this coming quarter. It says the prerequisite is Cal III and linear algebra. I have taken those prereq years ago and forgot most of the materials. I don't know if the Intro to mathematical proof class has a lot of materials from...
  26. J

    Will a GUT or TOE be mathematical

    Most if not all laws of physics are not mathematical, rather statements that define the way mathematical models must follow. They have been laid down by intuition and observation. So how can something like string theory based on maths be expected to achieve a GUT or TOE if it does not know...
  27. I

    Proving inequality with mathematical induction

    I am having trouble proving these. I cannot figure out how to get to the conclusion. Here is my attempt. The stuff in red is just side work and is not part of the proof. I always get stuck on these types of problems, can someone offer some tips on how to approach these kind of problems in...
  28. L

    Advanced Mathematical Methods Class - Question about Utility

    My school is offering a advanced mathematical methods classes next semester, taught by a supposedly world-renowned professor. Everyone to whom I've spoken, students and faculty alike, says he's an amazing teacher who gives you a unique education. However, here is the course description: My...
  29. L

    Understanding Mathematical Induction: Clarifying Starting Points and Examples

    Hi, Some times the starting point in MA confuses me, for example (\sum_{i=1}^ni)^2=\sum_{i=1}^ni^3 have we start with 2 or it is enough to show it when n=1 Thanks in advance.
  30. C

    Mathematical Statistics Xi~N(6,25) P[1.860 < 3(Xbar - 6]/S

    Homework Statement Let X1, X2, ..., X9 be a random sample from a normal distribution, Xi~N(6,25), and denote by Xbar and S^2 the sample mean and sample variance. Use the standard statistical table for normal distribution. Homework Equations E[S^2] = δ^2 = 25 S^2 = (ƩXi^2 -...
  31. C

    I believe that I have discovered a mathematical formula

    I believe that I have discovered a mathematical formula related to trigonometry. It is used for finding the ratio of areas of triangles (not just similar triangles, but all kinds). I asked my Mathematics professor if any such formula for finding ratios exist. He said no. I also counter-checked...
  32. T

    Is this statement true about mathematical propositions?

    Homework Statement "The propositions of mathematics work like this: we believe them first, and only later do we try to prove them."Homework Equations N/A The Attempt at a Solution I read it in some philosophy book but I'm not sure if it's true because I thought the propositions were not...
  33. T

    Some mathematical insight please.

    Hello, I am new to the forum and I have made an account to ask this one simple question. Am I good at math or not? I have been deciphering whether I am or not for some time now. It's like a pendulum, I can pathetically fail a math test but answer immense logic problems easily. I taught myself...
  34. Z

    Writing Mathematical Formula in MS Word

    Homework Statement I'm writing up a mathematics-based assignment and I want to type some formula into a MS Word document. I'd appreciate if anyone could let me know how to go about this. Thank you. Homework Equations The Attempt at a Solution
  35. S

    Please point me a way to a mathematical project

    Please point me a way to a mathematical project! Hello everybody, As a part of my Maths program I need to write an essay of about 25-30 pages. Game theory is the main topics and I now need a direction to analyse in game theory using mathematical method such as linear algebra. As I don't have...
  36. F

    Level V universes in Tegmark's mathematical universe hypothesis.

    Hello, I have been reading quite a bit about Tegmark's mathematical universe hypothesis. I realize that it is overly speculative in nature, or at least perceived that way. I am unsure if my question is better suited for the philosophy forum as it seems to me inherently unfalsifiable. Despite...
  37. J

    Creating mathematical model from table of permutations?

    I am trying to create a mathematical model from a table of possible permutations. The table essentially consists of a list of various combinations of variables (there are 7 of them) and then an education guesstimate of how long that combination would take. Each variable is restricted to a...
  38. W

    Mathematical Induction question

    A question I'm working on and my math book doesn't clarify the answer well enough for me to follow. I'm having some issues at getting the math symbols to work correctly so bare with me!Prove by mathematical induction that if A1, A2, ..., An and B are any n + 1 sets, then: Base step = n = 1 so...
  39. M

    Crossover between mathematical analysis and other subjects?

    Hi, I'm wondering how far one can go in learning graduate level real analysis or functional analysis with only background in undergraduate analysis 1 and 2 and none in algebra or topology? That is, is there any requirement or benefit to taking courses in algebra and topology to understand...
  40. O

    Math Seminar Ideas: Advanced Topics Beyond Fin Math, Stochastic

    class. I did a lecture last semester on financial mathematics and stochastic processes. Know of any cool topics that I could lecture on?
  41. P

    How Can I Improve My Mathematical Maturity?

    Hello, I am having some doubts/problems regarding my abilities in math. I am a math major, currently at a junior standing. I have taken calc I & II, linear algebra, probability, history of mathematics and an introduction to proofs class thus far, and I am taking calc III, Differential...
  42. D

    Looking for Books on Mathematical Neurology?

    Does anyone have any suggestions for a good book or two on mathematical neurology? Thanks.
  43. Sigurdsson

    Generalized functions (distributions) problem - Mathematical physics

    Homework Statement Find a distribution g_n which satisfies g'_n(x) = \delta(x - n) - \delta(x + n) and use it to prove \lim_{n \to \infty} \frac{\sin{nx}}{\pi x} = \delta(x) Homework Equations Nothing relevant comes up at the moment. The Attempt at a Solution Well the first...
  44. I_am_learning

    Interesting Practical Mathematical Problem

    I was making a PCB drilling machine at home with my brother, and came across this mathematical? problem. I needed to fix the drill machine. But I needed to determine where should it be fixed so that I could drill largest possible PCB boards. In other words, find the value of x so that you...
  45. D

    Beat frequency- universal mathematical proof?

    It is easily derived from trigonometric identities that cos(2*pi*f1*t)+cos(2*pi*f2*t)=2cos(2*pi*(f2+f1)*t)*cos(2*pi*(abs(f2-f1)*t)) which proves that superposition of two cosine wave will generate a beat frequency, but what about about a universal proof that applies to any kind of...
  46. J

    Writing mathematical proofs has greatly improved my life

    After 1-2 years of writing formal math proofs in undergraduate school, I now speak and write much more eloquently than I used to. Now, before uttering or writing a statement, I take a quick pause to ask myself whether it's logically valid; it's unambiguous; it's relevant and sequentially...
  47. H

    Interchanging mathematical operations proof

    I'm working on the following two proofs: 1.) (x+y)2n+1 = x2n+1 + y2n+1 if and only if x=0, y=0 or y=-x and 2.) (x+y)2n = x2n + y2n if and only if x=0 or y=0 I've tried using induction and get stuck at a certain point. I've also tried playing around with...
  48. A

    Mathematical Logic, Interpretation, Satisfiable, Consequence relation

    Theorem : let A be a set of formulas, a be a formula For all A and all a, Every interpretation which is a model of A is also a model of a iff not (Sat A) U {~a} Proof Every interpretation which is a model of A is also a model of a iff(1) there is no interpretation which is a model of A but...
  49. F

    Simplifying Mathematical Induction for Sequences Using Algebra

    I have an idea of what to do and I have reached the stage but when i am at final stage- I am struggling to simplify as i simply don't understand (1/2)+(2/2^2) +(3/2^3)+...+ (n/2^n) = 2 - (n+2/2^n) so I have done p(k) and p(k+1) this gives me p(k) (1/2)+(2/2^2) +(3/2^3)+...+ (k/2^k)...
  50. K

    Developing Mathematical maturity

    I have a question about development of mathematical matury. First off, I'm a high school senior with almost all my college general education credits due to dual enrollment in Florida. I have been doing independed study since 10th grade and have self-taught everything (computation) from...
Back
Top