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. B

    How can I effectively learn mathematical modeling for social science research?

    Hi All, Just wanted to know, what is a good way to learn mathematical modeling? I'm a social scientist working with simulation and data, and would like approach the issue more formally. Before trying modeling textbooks, what should I know beyond calculus? Any suggestions (books, tutorials...
  2. A

    Time dependent PDEs - mathematical modelling - diffusion equation

    Homework Statement Porous membranes are used to separate mixtures in industry, because smaller compounds permeate through them more easily than larger ones. KnoGas Pty Ltd are trialling an experimental separation process, using a membrane to sep- arate compounds A and B: compound A...
  3. H

    Mathematical approximations in Physics for angles <20 degrees

    Homework Statement 62. In physics, it is important to use mathematical approximations. (a) Demonstrate that for small angles (<20°) tan a ~ sin a ~ a = (Pia'/180 degrees) where a is in radians and a' is in degrees. (b) Use a calculator to find the largest angle for which tan a may...
  4. C

    Mathematical probability prerequisites

    For anyone who has taken mathematical probability, what prerequisite(s) did you have? At my school, Calc IV is the only prerequisite. I ask because we are notorious about not having prerequisites lined up properly so I don't want to waste my time with this class if I don't have the proper...
  5. P

    Mathematical Statistics good for Computer Scientists?

    I am finished with the first bachelor year at Computer Science and the next courses are: Assembly, Linear Algebra, Statistics, Java. Many people from my class said that they will not take Statistics as it's not that helpful to them and I was thinking of how accurate that is. Is Statistics...
  6. S

    Where can I find fellow math enthusiasts for online correspondence?

    How can I find people who are interested in correspondence on mathematics and are also around my age on the net? By correspondence I mean emails. I can't see many people interested in mathematics on popular social networks, or if there are, it's not possible to find them through searching...
  7. R

    Linear Algebra Vs Mathematical Modeling -Importance in relation to physics

    Hey as part of my Physics undergrad in second year I have to take a module in either Mathematical Modelling or Linear Algebra (both course descriptions below) In first year I preferred Linear Algebra ( a very basic intro course) but apparently in second year its just all proof and no...
  8. J

    An interesting mathematical property

    3^3 = [3^3 - 3^2] +[3^2 - 3^1] + [3^1 - 3^0] + 3^0 Practical Demonstration: 27 = [27-9] + [9-3] + [3-1] + 1 27 = 18 +6+2+1 27 = 27 Is this property discussed in theory of games?
  9. A

    Understanding Peano's Axioms: The Role of Mathematical Induction

    I'm having a bit of trouble understanding why the principle of induction is included as one of Peano's axioms. It seems like it should not be independent of the others. Obviously it can be stated as: If a predicate P is true only of natural numbers, P(0) is true, and also P(n)\rightarrow...
  10. D

    Scientific Thinking Graphing -Determining the Mathematical Expression

    "Scientific Thinking Graphing" -Determining the Mathematical Expression Homework Statement Ok, so I recently got an assignment for "Scientific Thinking Graphing Practice". Basically, I need to take a data table that will produce a curve shaped graph and turn it into a straight lined graph...
  11. liometopum

    What is the Name of this Mathematical Series?

    Is there a name for this series: 1/2 + 2/3 + 3/4 + 4/5 + 5/6 + 6/7 + 7/8 + 8/9 +...+ n/(n+1) Thanks.
  12. A

    A simple question from Rudin's principles of mathematical analysis

    It's NOT a homework. I'm self-studying Rudin's Analysis and I came to this part that I can't follow the argument after a certain point: statement: if x,y \in R, and x<y, then there exists a p \in Q such that x<p<y. Proof: since x<y, we have y-x>0, and the Archemedean property furnishes a...
  13. 2

    Mathematical induction problem.

    Hi. I am learning mathematical induction on my own and I came across this problem: Homework Statement Prove: 1*4 + 4*7 + 7*10 + ... + (3n - 2)(3n + 1) = n(n + 1)²2. The attempt at a solution Quick test for n=1: (3 -2)(3 + 1) = 1(1 + 1)² 4 = 4 Alright, so I rewrite this with, on the left...
  14. matqkks

    Motivate Students to Understand Logical Deduction of Proofs

    I have found that for a number of students doing proofs is an unnecessary evil. For some of them it is like a painful visit to the local dentist. I thought I had explained the need for proves but it still puts a number of students off. What motivating reasons can I give students for them to at...
  15. T

    What is the mathematical relationship bewtween pressure and displacement?

    Homework Statement I'm having trouble understanding the relationship between pressure, speed, and displacement to answer the following question correctly: While you are stopped at an intersection turn lane, a large truck zooms by, just a few feet from your car. As the truck passes, your car...
  16. O

    What is the best Unix like OS for mathematical computation?

    I want to basically have a dedicated machine for running scientific computation programs and be able to access it remotely anywhere using a laptop/netbook. That means I don’t care much about GUIs. I’m used to using Unix and am perfectly fine with doing everything through a command window. At the...
  17. G

    Mathematical methods of physics problem

    Homework Statement Here's the problem. Verify the operator identity x - d/dx = -exp (-x^2 / 2) d /dx exp (-x^2 / 2) Can someone please help me? Thanks :) Homework Equations The Attempt at a Solution
  18. K

    Help with Undergraduate Math, Algebra & PDE Implicit Scheming

    Hi a want to code implicite scheme for PDEs u_t =a laplace u + L p_t p_t = laplace p + u pls help I made a classical FEM discretization, but i don't know how to programe it. (u(n+1)ij - un)/t = a ((u(n+1)(i-1j) -2 u(n+1)(ij) +u(n+1)(i+1 j)) /x2 + (u(n+1)(i j-1)-2u(n+1)(ij) +...
  19. E

    Roulette vs Mathematical Betting Strategy. Please Comment and Criticize

    Ok so on the roulette table the odds against winning on red or black are 1.111 to 1. This is about a 47% chance to win. Payout is 1 to 1. Say minimum bet is 1$ and bets are allowed in multiples of 1$. up to some limit. Suppose I want to win 1$ for every spin I play. I bet 1$, and I...
  20. B

    Mathematical Induction: Power Rule for Differentiation

    Homework Statement Prove that \frac{d}{dz}z^n = nz^{n-1}\;\;\; \forall n\in\mathbb{N} using the Product Rule for differentiation and mathematical induction. Homework Equations \frac{d}{dz} f(z) = \lim_{\Delta z\to 0} \frac{f(z+\Delta z) - f(z)}{\Delta z} \frac{d}{dz}[f(z)g(z)] = f\,'(z)g(z) +...
  21. fluidistic

    Boas Mathematical physics book, definition of center of mass

    In Boas' book I can read that the definition of center of mass of a body has coordinates x_{CM}= \int x_{CM}dM= \int x dM. Shouldn't it be this same integral but divided by M?! Also, I didn't find the definition of center of mass for particles or any non continuous bodies. I'd be grateful if...
  22. N

    Mathematical Requirements for Basic Quantum Physics

    I want to be able to follow basic books for quantum physics to get a taste of some of the underlying principles. The problem is: I don't even know half of the symbols. So basically, what mathematical knowledge is needed to be able to follow the formulas and principles in a quite basic quantum...
  23. F

    Mathematical purity vs real world understanding

    This is a very philosophical question, so i understand the answer is going to be very arbitrary and subject to opinion. I am interested in both Pure mathematics and Theoretical Physics. The time is soon approaching, where i must decide upon which path to take. I love proofs, and i love...
  24. K

    What Careers Combine High-Level Mathematics and Programming?

    I just have a few quick questions on mathematical neuroscience. What are the branches of mathematics that are used frequently in neuroscience? And finally, is it common for math majors to enter neuroscience in graduate school? Yes, I'm thinking about it micro, now don't pester me!
  25. H

    Mathematical Modeling of an Elevator System

    Hi, I'm a physics major working an internship for an engineering company at present. I have been trying to model the energy consumption and/or power for an elevator system. The goal is to be able to input different variables (cab weight, capacity, counterweight percentages, motor types...
  26. W

    Appropriate Journal for mathematical perspective

    Hello. I'm a college undergrad who has developed a mathematical perspective technique for drawing that I am convinced is original enough to be meaningful. Without going into too much detail, I think I have the mathematics background to formalize it, and I have the visual arts background to make...
  27. J

    Pictures of abstractions behind mathematical equations?

    I need to understand physics (and other) concepts but I'm having trouble visualizing what the equations mean. One example would be an applied math thing - I want to understand what about electromagnetism exactly the Maxwell equations are describing exactly so I can gain insight into what math...
  28. O

    COMSOL example model results do not match with Mathematical Model

    Hi! I started to look into freq response analysis of a cantilever beam using COMSOL. Luckily COMSOL already has an example model "Model Library>> Structural mechanics module>>Dynamics and Vibrations>>rotating blade". While not changing anything in the model and with Omega=0 and Cloris =0, I get...
  29. N

    Mathematical question i physics

    \vec{E}=f_1(z-ct)\hat{x}+f_2(z-ct)\hat{y}+0\hat{z} f_1(u) and f_2(u) are functions of "u" u=z-ct i have the formula \nabla \times \vec{E}=-\frac{db}{dt } \nabla\times\vec{E}=|\begin{array}{ccc} \hat{x} & \hat{y} & \hat{z}\\ \frac{{d}}{dx} & {\frac{{d}}{dy}} & \frac{{d}}{dz}|=\\...
  30. B

    Does chess ability and mathematical ability go hand-in-hand?

    I do not know if this is the correct forum to post this, but I have always wondered if the ability to play chess and mathematical ability go hand-in-hand with each other. A lot of people around me always tell me that. I like playing chess, I don't know if I'm good, but I like playing it. I also...
  31. B

    What should I consider when applying for a PhD in mathematical physics?

    right now I'm finishing up a master's degree in quantum chemistry, but I much prefer the development of the methods than the actual use of them (i.e. the more theoretical side of things). And I've been keeping on top of my mathematics of all flavor, and would really be interested in doing a phd...
  32. O

    Mathematical Logic by Cori and Lascar: Incomplete proof of Lemma 1.9?

    "Mathematical Logic" by Cori and Lascar: Incomplete proof of Lemma 1.9? I have a question on the book "Mathematical Logic: Propositional calculus, Boolean Algebras, predicate calculus" by Rene Cori and Daniel Lascar. Proof of Lemma 1.9 given on...
  33. O

    Mathematical Logic by Cori and Lascar : Possible typo?

    I have a question on the textbook "Mathematical Logic: Propositional calculus, Boolean Algebras, predicate calculus" by Rene Cori and Daniel Lascar. This is not about an exercise but about the conceptual content of the book. So I did not post this in the "Coursework and Homework questions"...
  34. N

    Mathematical meaning of measurement

    I pasted a paragraph from http://ocw.mit.edu/courses/electrical-engineering-and-computer-science/6-845-quantum-complexity-theory-fall-2010/lecture-notes/MIT6_845F10_lec02.pdf" below. ------------------------------ Consider the following matrix which represents a 45◦ counter clock wise...
  35. Phrak

    Is mass conserved by mathematical identity?

    In electromagnetism J denotes the oriented charge-current density, J=d*F. Conservation of charge immediately follows. J=d2*F=0(identically). All exact forms are closed. We can identify scalar mass as the norm of the one-form, μ=(E/c2,-p/c). *μ is then spatial mass density. Like...
  36. M

    Theorem 2.13 of Principles of Mathematical Analysis

    This question is directed toward people who have a copy of the text--in other words, I'm too lazy to retype the whole theorem and proof (sorry). First, I'm having trouble simply understanding the meaning of certain parts of the theorem itself. I'm taking Bn to include only n-tuples (i.e. no...
  37. A

    100% Mathematical Proof vs Vellmen's How to Prove it

    "100% Mathematical Proof" vs Vellmen's "How to Prove it" Hello, I'm looking for books that teach proof methods and techniques. I know Vellemen's book is a popular choice but a few dissenting reviews among unanimous praise, on its amazon page, caught my attention. Like this one: And here are...
  38. B

    Is 11^n - 4^n a Multiple of 7? Proving with Mathematical Induction

    Homework Statement Prove that 11^n - 4^n is a multiple of 7 Homework Equations N/AThe Attempt at a Solution I substituted k+1 in for n and simplified to get 11(11^k)-4(4^k) but after this point I get stuck. Any help would be appreciated.
  39. J

    Mathematical induction w/ Summation question

    Homework Statement Summation of i(i + 1) (with i going from i = 2 to i = n-1) = n(n-1)(n=1) / 3 a. Write P(2). Is P(2) true? b. Write P(k) c. Write P(k+1) d. Prove by mathematical induction that the formula holds true for all integers n \geq 2 Homework Equations...
  40. D

    Mathematical modeling of a feedback control system

    My system is a compound pendulum hinged at some distance from one end and have a propeller mounted dc motor at the other end. I wish to position the pendulum with the help of motor propeller system. I tried but i am unable to get a mathematical model for it. I am getting a non linear model...
  41. A

    Two courses in Mathematical Methods for Physics, or higher mathematics?

    I have a problem. The amount of units I have taken up until now (this coming semester I'll be starting my Junior year) and the amount of time I have left is restricting my options for math courses. I do want to become a physics major and I intend on going to grad school. I have a strong...
  42. Z

    Engineering physics or Mathematical physics

    Hi, So I'm with a real dilemma and I only have a few days to make a choice. My situation is the following: I'm 18 years old and next year I'm going to start university in Canada (Montreal). I have already applied in engineering physics and got accepted. However, the problem is that I'm really...
  43. Q

    The Mathematical Method in Economics

    In the introduction to their book 'The Theory of Games and Economic Behaviour' von Neumann and Morgenstern consider the arguments against the use of mathematics in economics. One of the reasons they believe that previous applications of mathematics to economics had been unsuccessful was that...
  44. G

    Mathematical Operations of General Relativity

    How are the tensors of General Relativity simplified and operated? And can someone give me a mathematical example of General Relativity being done with just some basic values being plugged in for G=kT?
  45. MathWarrior

    Mathematical functions from data sets?

    I feel like I have gone pretty far in math now, but I keep finding myself asking the same question. Say I had a series of data points from like a randomly collected survey or stock stock price graph over time etc. Is there a way to take this seemingly random and scattered data and turn it into...
  46. P

    Mathematical Induction Problem

    I was working on my homework and I did two of the mathematical induction problems before this one and they were super easy but I must be forgetting something because I just can't seem to solve this one. 2+7+12+17...+(5n-3) = (n/2)(5n-1) So I know that step 1 is to prove that it works with...
  47. S

    Can someone explain HUP with some mathematical rigor?

    I was watching one of Susskind's videos and I had some questions: Briefly summarizing from the video: E = hf (Planck's constant times frequency) E = cp (speed of light times momentum) or p = E/c = hf/c And we know that if we have wavelength lambda, then if t = 1/f and vel = dist/time...
  48. Femme_physics

    Trying to translate a mathematical term from Hebrew to English ( regression rule )

    Trying to translate a mathematical term from Hebrew to English ("regression rule") If I translate it word by word in Hebrew, it's the "regression rule" Such as, I am told that "a series is defined for every natural n by the regression rule"...
  49. R

    Mathematical formulation of Kaluza-Klein theory

    Good morning, what is the mathematical formalism used for a rigorous formulation of the Kaluza-Klein theory? Is there any reference? Thanks!
  50. P

    Proving by Mathematical Induction

    so the problem is 3+7+11+15+...+(4n-1) = n(2n+1) so I know that step 1 is to plug in 1 for the right side and check that it equals three... 3=1(2(1)+1) and yes it equals 3 Then I know that you assume that 3+7+11+15+...+(4n-1) = n(2n+1) The next step is where I get confused I know that...
Back
Top