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.

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  1. 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...
  2. 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...
  3. 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...
  4. Femme_physics

    What is the Correct Translation for Regression Rule in Mathematics?

    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"...
  5. 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!
  6. 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...
  7. J

    Why do some propositions defy logical proof?

    I recently read about the unexpected hanging problem and I was so surprised that logic actually failed in determining the solution!:( Is this just an isolated exception, or are there more paradoxes like this? And more importantly, why does logic fail? Isn't there any way around this? I just...
  8. V

    Mathematical induction proof concerning polynomials in Z2.

    Homework Statement For each n\,\in\,\mathbb{N}, let p_n(x)\,\in\,\mathbb{Z}_2[x] be the polynomial 1\,+\,x\,+\,\cdots\,x^{n\,-\,1}\,+\,x^n Use mathematical induction to prove that p_n(x)\,\cdot\,p_n(x)\,=\,1\,+\,x^2\,+\,\cdots\,+x^{2n\,-\,2}\,+\,x^{2n}Homework Equations Induction steps...
  9. Y

    Mathematical Induction with an Inequality

    Homework Statement Prove that (n + 1)n - 1 < nn for n ∈ Z+. [Hint: Induction is suggested. Write out the induction statement explicitly. Make one side of the inequality look like your induction hypothesis.] Homework Equations The Attempt at a Solution ^ That's what I have so far. I'm good...
  10. MathWarrior

    What course did you or do you typically learn about mathematical

    What course did you or do you typically learn about mathematical splines?
  11. A

    Mathematica Which Math Program is Best for Differential Equations & Vector Analysis?

    Hi, I am new to this forum, so forgive me if this is a really obvious question. I am a high school student interested in doing a project involving differential equations, and vector analysis (basically calc 1 - III, linear algebra, and a bit of differential equations). I will use this program...
  12. N

    Cant understand the mathematical transition

    http://www.youtube.com/watch?v=JUFZbGv6wdM&feature=relmfu in here in the end i got an arch that startes at A and end in B he wrote this integral \intop_{a}^{d}\frac{{dl}}{r^{2}} but point a and be are not values that i can put in if my function depends on "r" then i need to change...
  13. nukeman

    Does Time have a mathematical equation or Law?

    Does Time have a mathematical equation or Law? Thanks!
  14. G

    Proof by Mathematical Induction

    Homework Statement Prove: 1+2(2+3+4+···+n)+(n+1)=(n+1)2-1 for all n within NHomework Equations PMI The Attempt at a Solution I tried applying PMI starting with the basis step, allowing n=1, assuming the equation would be true. However 1+2(1)+(1+1)=?=(1+1)2-1 yields 5=?=3 which is false. The...
  15. Aaronvan

    Mathematical Skills for Understanding General and Special Relativity

    I'm curious as to the highest level of mathematical skill required for adequate understanding of Einstein's theories. For example, how deep into PDE's do the field equations require?
  16. L

    Bridge to mathematical physics book?

    Background Next year I am going to be a final year student - Msci Physics. I am fairly mathematically competent even though I don't "take" any theoretical/math options (my confidence in this area is another question). In first year we had a standard introduction to all the topics we were...
  17. P

    Solving Mathematical Problems: Find the Answers Here

    I have some problems. Number 1) Two people are playing, they take a three digit number, at this number, we add its inverse, and wins the number before arrives a sum > 1900, what sum or sums are these? What number is that? Number 2) A person receives 1 euro if He arrive on...
  18. micromass

    Do You Have What it Takes to Be a True Mathematician?

    So, do you think you are a mathematician? Prove it and calculate your MPQ (Mathematical Purity Quotient) Here are 125 questions to determine how much of a mathematician you really are. Answer each question with YES/NO and add up the number of times you answered YES. Then divide this score by...
  19. M

    Is g(f) One-to-One if f is Not One-to-One? Prove It.

    [b]1. Assume that f : A -> B and g : B -> C and that f is not one to one. Prove that g(f) is not one to one. [b]2. one-to-one: x1 != x2 and f(x1) != f(x2) not one-to-one: f(x1)=f(x2) and x1 != x2 [b]3. The start of the proof should go as follows: Assume f: A->B, g: B -> C, f is not...
  20. P

    Math Burnout: Managing Academic Pressure & Career Ambitions

    So, right now I'm in the middle of my third year of being a math/financial economics major, and academically I have been strong but now am really taking a hard hit. The main thing that I feel is that the deeper into math I've gone, the less interesting its become. Reading math on a lower-level...
  21. E

    Proof By Mathematical Induction

    Homework Statement Use mathematical induction to prove the following statement. n \geq 2,~ \sum_{k=1}^{n} \frac{1}{k^2}~<~1-\frac{1}{n} Homework Equations The Attempt at a Solution This is the third problem I've done of this type, so I am by no means an expert. So this attempt may be...
  22. F

    Disability solely in mathematical comprehension

    I have a severe learning disability solely in mathematical comprehension, possibly dyscalculia. I was given a lot of special assistance in high school to help me graduate. I was able to graduate with a diploma with only basic arithmetic. Most of the algebra, geometry, trigonometry and so forth...
  23. T

    What Are the Best Mathematical Physics Books for Self-Study?

    Greetings everyone, I'd like to continue my physics and mathematics education but I'll have to do that on my own since I'm heading into engineering. I'm looking for a book that would cover a lot of the mathematics that are needed at the undergrad physics level. I'll have Calc 1,2,3, Linear...
  24. pairofstrings

    What is the use of mathematical induction

    In the context of Set theory and Relations why do we use mathematical induction. Is there any deep relation between all these concepts or mathematical induction is only a separate concepts introduced in the textbooks after Sets and Relation ; Functions ; and then Mathematical induction.
  25. C

    What is the mathematical basis of quantum mechanics?

    Someone posted this on another forum, and, not knowing enough about it to supply a satisfying answer, I figured I'd ask you guys.
  26. K

    Principles of Mathematical Analysis help

    I recently picked up Walter Rudin's undergraduate analysis textbook. I've read through the first chapter and understood the material without much trouble. I am, however, having trouble with a few of problems at the end of the chapter. My general frustration stems from not knowing what I should...
  27. P

    Is geometry the heart of mathematical insight?

    Richard Feynman once remarked during a lecture that the invention of analytical methods such as calculus allowed people to "be more stupid", in reference to solving problems. I think he was on to something with that remark. Often when faced with a problem which requires me to set up some...
  28. T

    Mathematical Induction am I on the right track?

    Homework Statement The sequence a0 -> n is defined by ai = b+i*c. Prove by induction on n that the sum of the terms in the sequence is (n+1)(a0 + an)/2.Homework Equations The Attempt at a Solution I defined predicate P(n) as (n+1)(a0+an)/2. My goal is P(n+1), which is (n+2)(a0+an+1)/2. I...
  29. T

    Are There Any Recommended Books for Undergraduate Math Competition Preparation?

    Hi. I am from Greece and I am an undergraduate student at the University of Patra and I study pure Mathematics. I am interested in mathematical competitions a lot, but I have never gone before in a mathematical competition. I want to study so I can be able to take part in competitions for...
  30. F

    Choosing boundary conditions for mathematical model

    Greetings, everyone! The problem below is actually a task on Numerical Methods. But I have difficulties making a mathematical model. Homework Statement Let us have a longitudinally homogeneous system of a pipe of radius R and a propeller of nearly the same radius inside it (we shall...
  31. B

    Mathematical models, the 'real world' and quantum mechanics

    A mathematical model may show that there are two possible states for something in the real world, let's say a coin, that may be either heads or tails; make a computer program to simulate coin flipping and you soon discover that the odds of heads or tails is 50/50. In the real world however...
  32. D

    Mathematical prerequisites for Quantum Mechanics

    So, I am about to read Landau's and Lifschitz's textbook on Quantum Mechanics. What kind of mathematics I should be already familiar with in order to completely understand the above mentioned material? Thanks for all the advice.
  33. D

    Mathematical prerequisites for Classical Mechanics

    So, I am about to read Landau's and Lifschitz's textbook on Classical Mechanics. What kind of mathematics I should be already familiar with in order to completely understand the above mentioned material? Would real-variable calculus and linear algebra be sufficient for the task? Thanks for all...
  34. V

    How to Describe the Magnetic Field of a Folded Metal Band with Current?

    Homework Statement hello everyone there, i really hope that someone can help me with my homework,because i don't havee a clue where to start. so the instructions are like this: wide metal band alternately folded on three sides of the square. Through it, along the edges,is electrical...
  35. F

    What is the most challenging subject in math?

    Everyone seems to have their preferences, so I wonder what people prefer the least, or what they find to be the most difficult subject in math, sunject you could never get your head around etc etc? (I know it's different for everybody, that's why I'm asking)
  36. D

    What Are the Best Physics Courses to Apply Advanced Mathematics?

    I have almost finished my bachelor's degree in mathematics, with a minor in physics. A while ago I decided not to continue with physics on my master's degree, but to focus on pure mathematics while learning a bit of computer science as well. However, after planning out the two semesters of...
  37. K

    Mathematical Induction Problem

    Homework Statement Prove that \frac{1}{\sqrt{1}}+\frac{1}{\sqrt{2}}+...+\frac{1}{\sqrt{n}}\geq\sqrt{n} for all n \in N Homework Equations The Attempt at a Solution p(1): \frac{1}{\sqrt{1}} = \frac{1}{1} = 1 = \sqrt{1} \geq \sqrt{1} Let \frac{1}{\sqrt{1}} + \frac{1}{\sqrt{2}} +...+...
  38. D

    Is ecology inherently mathematical?

    Hello I'm interested in pursuing ecological studies. I'm trying to find out as much as I can about the nature of ecological science. I have seen some books in my school library about mathematical ecology and my ecology professor has said that ecology is pretty quantitative. I have hopes that...
  39. F

    Which is the best Mathematical Physics book

    Prefer me a book that covers entire Mathematics which is required for Physics Note: please consider me a new(3 years no touch of maths due to health issue) and the book you prefer me must have clear understanding of mathematical concepts subjects as follows #Algebra #Trigonometry...
  40. N

    How Enzymes/Proteins Work at Atomic/Molecular Level

    So I've learned in biology about the amino acids forming chains/trees, and having different interacting "R groups" Can somebody link to me an animation or careful description of how a specific enzyme/protein moves its parts step by step on an atomic/molecular level? I cannot find one. I want...
  41. D

    Chennai Mathematical Institute(India) entrance exam

    What is the difficulty level of the CMI entrance exam?Is it tougher than IIT JEE?I am in std. 11 and I want to take up BSc Physics at UG level.Can one get through the exam by having a grip over the basic concepts of physics or does it require rigorous preparation like the IIT JEE(joining...
  42. 8

    Proving the Dogs Meet at a Single Point: A Mathematical Analysis

    Homework Statement There are N dogs on the plane. Each dog chases the tail of the next dog with a speed of 1 (and the last dog is chasing the first). I want to prove that the dogs will eventually meet at a single point. 2. The attempt at a solution I defined a function that is sum of...
  43. C

    Mathematical model of holographic universe ?

    A light wave can be modeled by a complex number U which represents the electric or magnetic field of the light wave. The amplitude and phase of the light are represented by the absolute value and angle of the complex number. The object and reference waves at any point in the holographic system...
  44. W

    Four-Momenta derivation: is this mathematical step even valid?

    Homework Statement Hey all, this is my first post to the forum, but I've been reading for years. Anyways, I was reviewing a solution to a Compton Scattering derivation we learned in class, and am still trying to make sense out of it. Homework Equations Given, we have the initial and...
  45. M

    What is mathematical analysis and/or discrete mathematics used for?

    I am starting a maths major and I will going to go into pure maths. I am going to specialize in either analysis or discrete maths. I understand that mathematical analysis has a very strong connection to calculus and that discrete mathematics is used mainly in the cryptography and security...
  46. F

    Linear Algebra + Mathematical Proof - Summer

    I was planning to take differential equations with Linear Algebra, but there is a conflict so I can't take both. Finally there is a course called Introduction to Linear Programming In my Physics/Math Calendar, it doesn't say I have to take it, so I am wondering what is it really...
  47. A

    Proving x_n+1 > x_n with Mathematical Induction | Real Number Sequence

    Homework Statement Consider the sequence of real numbers x1, x2, x3,... defi ned by the relations x1 = 1 and xn+1 =\sqrt{1 + 2xn} 1. Use mathematical induction to show that xn+1 > xn for all n \geq 1. The Attempt at a Solution I'm a bit thrown off by this question because it...
  48. N

    Using mathematical methods to trade Forex

    Hello, so I've been in the forex market (currency trading and exchanges) for quite some time, I've done reasonably well, however I was curious as to how perhaps calculus and other mathematical methods could benefit me. Basically, I am a day trader...I wanted to know if there is a way to predict...
  49. A

    Interested in Mathematical Modelling? How Can You Specialize in Energy Systems?

    Hi I will like some advice. I intend to know more about mathematical modelling. Can anyone recommend any useful books, websites of software? How does one develop an interest in it? I will like to specialise in the modelling of energy systems and not sure how to do it. Please advise
  50. A

    Mathematical Concepts Vs Problems

    This has been bothering me for a long time. I hope someone in this forum will help me clear my confusions. So the way I approach mathematics is by working carefully over the concepts. I reread the concepts till they become crystal clear.( This has been painful for some of my classes like real...
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