Mathemathics Definition and 80 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. T

    Linear Algebra or Calculus?

    Summary: Hello! I'm an high school student and i want to study more math but I'm not sure where to start. Should i first study linear algebra or calculus? Hello! I'm an high school student and i want to study more math but I'm not sure where to start. Should i first study linear algebra or...
  2. Magnetons

    Find z^n+ 1/ z^n

    Firstly I converted the given equation to a quadratic equation which is ##z^2- (\sqrt3)z+1=0## I got two solutions: 1st sol ##z=\frac {(\sqrt3 + i)} {2}## 2nd sol ## z=\frac {(\sqrt3 -1)} {2}## Then I found modulus and argument for both solution . Modulus=1 Arguments are ##\frac...
  3. F

    I Demonstration of inequality between 2 variance expressions

    Just to remind, ##C_\ell## is the variance of random variables ##a_{\ell m}## following a Gaussian PDF (in spherical harmonics of Legendre) : ##C_{\ell}=\left\langle a_{l m}^{2}\right\rangle=\frac{1}{2 \ell+1} \sum_{m=-\ell}^{\ell} a_{\ell m}^{2}=\operatorname{Var}\left(a_{l m}\right)## 1)...
  4. P

    I Notation of the approximation in quantum phase estimation algorithm

    I'am interested in the notation of the approximation in quantum phase estimation algorithm. In the literature there are different definitions, which I divide into two cases here. Both different in their definition of the ##\delta##. In both cases I start with a quote of the source and show an...
  5. rudransh verma

    Intro Math Maths needed to advance my physics understanding

    I am a 10(high school)+ 2 years student. I have a basic knowledge of calculus, trigonometry, algebra, arithmetic. I need to advance as an undergraduate to start my physics journey. I want a mathematics book for physics which if it will cover some basics in early chapters will be of much help...
  6. E

    Programs I chose math instead of physics. Was it a mistake?

    Long story short, my career goals are to work as a researcher/research scientist in machine learning or to develop models/algorithms for forecasting such as in financial markets or otherwise. Fundamentally, I find think this is quite similar to physics as a science, but that is for another...
  7. C

    Does the stereotypical nerd or geek really exists?

    Do the stereotypical nerds and geeks really exist? I am talking about socially awkward timid loners who are also highly intelligent in Science, Mathematics, Technology, and Engineering. What happens to them later in real life? Do they really end up as unemployed or menial workers because of...
  8. J

    Interaction energy of two interpenetrating spheres of uniform charge density

    I am trying to calculate the interaction energy of two interpenetrating spheres of uniform charge density. Here is my work: First I want to calculate the electric potential of one sphere as following; $$\Phi(\mathbf{r})=\frac{1}{4 \pi \epsilon_{0}} \int...
  9. Which rating is better, mathematically speaking? | Probabilities of probabilities, part 1

    Which rating is better, mathematically speaking? | Probabilities of probabilities, part 1

    Part 1 of 3 videos on probability by 3blue1brown
  10. Lapse

    Comp Sci How do I determine the linear/non-linearity of this problem?

    A system is linear if it satisfies the properties of superposition and homogeneity. Superposition: adding the inputs of two systems results in the addition of the two outputs. Ex) x1(t) + x2(t) = y1(t) + y2(t) Homogeneity: multiplying the input by some scalar value is equal to the output...
  11. W

    I How to visualize the maths behind a formula?

    I just finished my freshman year in computer engineering at my college, and after some physics and materials science classes I noticed a pattern. Everybody—at least in my class, I don't about the rest of the world—is just memorizing formulas, myself included. I'm really trying to not just...
  12. A

    Is it possible to have physics without mathematics?

    or in general, is it possible to create science without mathematics? thanks
  13. EchoRush

    I Questions about derivatives

    As I have said before, I am in calculus class for the first time. I am doing really well in the class, however because of how my mind works, I’m always asking questions to know more, even when it’s too advanced for me. I just like to ponder and think about “what if” I know it’s probably not good...
  14. Morbidly_Green

    Fourier sine series of cos 2x

    I am attempting to find the sine representation of cos 2x by letting $$f(x) = \cos2x, x>0$$ and $$-\cos2x, x<0$$ Which is an odd function. Hence using $$b_n = \dfrac{2}{l} \int^\pi _0 f(x) \sin(\dfrac{n\pi x}{l})dx$$ I obtain $$b_n = \dfrac{2n}{\pi} \left( \dfrac{(-1)^n - 1}{4-n^2} \right)$$...
  15. H

    I Block Diagram manipulation

    Hello I hope someone can help me, as i am kinda stuck for the moment. As you can see, the assignment states that I need to find the poles from the closed loop transfer function. I plan on doing so, by using block diagram reduction method. This is as fare as I've come, and can't come...
  16. M

    Calculus with two variables.

    Quotient rule: z= f/g ------ z'= (f'g - g'f)/g^2 starting with finding the derivative in respect to x, i treated y^2 as constant 'a': z'= [(a*2x*cos a*x^2)(sin a*x^2) - (- a*2x*sin ax^2)]/cos(a*x^2)^2= [(a*2x*cos a*x^2)(sin a*x^2)+(a*2x*sin ax^2)]/cos(a*x^2)^2 For the derivative in respect to y...
  17. S

    Regression for a first order system

    Homework Statement I am carrying out a regression for diameter of a part Homework Equations Diameter = -0.0531052 + 0.0443237 * exp (-0.0103633 * 'Time elapsed') if diameter is -0.052 then can some one please calculate the value for time elapsed would you please explain the steps The...
  18. Celso

    Simple harmonic motion interpretation problem

    I'm in trouble trying to understand the expression ##t= \frac{1}{\omega} cos^{-1}(x/A)## that comes from ##x = Acos(\omega t)##, in which ##A## is the amplitude, ##t## is time and ##x## is displacement. When ##x = 0##, ##t = \frac{\pi}{2\omega} ##, shouldn't it be 0 since there was no movement?
  19. Boltzman Oscillation

    Square of the sum of two orthonormal functions?

    Homework Statement Given: Ψ and Φ are orthonormal find (Ψ + Φ)^2 Homework Equations None The Attempt at a Solution Since they are orthonormal functions then can i do this? (Ψ + Φ) = (Ψ + Φ)(Ψ* + Φ*)?
  20. P

    Silicon not heating with induction heater?

    I’m using an inductive heater to try to melt some 99.85% polycrystalline silicon, as would occur in the Czochralski process, but the silicon workpiece is not melting let alone even getting hot. The silicon workpiece is about the size of a playing dice. It becomes conductive at 200 F and becomes...
  21. A

    I Discrete to continuum Gaussian function

    I have a question regarding a paragraph in "Radiation detection and measurement" by Knoll. In the chapter about the discrete Gaussian it states that "Because the mean value of the distribution ##\bar{x}## is large , values of ##P(x)## for adjacent values of x are not greatly different from each...
  22. S

    I Could this be simplified?

    I am sorry but i am not a mathematician Could this be simplified ?
  23. S

    French student look after pre-calculus books

    Hi there , well my questions is in the title but i have to say some things : I have multiples mathematiques and physics courses calc-based and so i want to review the basic . I already search around this forum and others and i find some books who looks popular here there are : - Pre-Calculus...
  24. C

    B Does anybody think that there is an underlying reality?

    It seems like we keep chasing "reality", and by "reality" I think Physicists would mean the apparent rules of quantum physics which we hope would (if applied) lead to all the apparent known rules of macro-physics. However ... It seems like we have had to create a few things to do that: 1. Ideas...
  25. Robert Wilson

    Simple Way to Learn Algebra

    Is there any simplest way to learn algebra?
  26. Subrahmanyan

    To find the nature of roots of a quintic equation....

    The asks for us to find the nature of roots of the following equation ,i.e,rational or irrational nature of the roots: the Equation is :^5+x=5 I have been able to...
  27. IonizingJai

    Implicit differentiation problem

    Homework Statement If ##x\sqrt{1+y} + y\sqrt{1+x } = 0##, then prove that ##\frac {dy} {dx} = \frac {-1}{(x-1)^2}##. 2.Relevant Equations: $$ \frac {dy} {dx} = - \frac {\left (\frac {\partial f}{\partial x} \right)} {\left( \frac {\partial f} {\partial y} \right)}.$$ 3...
  28. R

    Schools Germany - Masters EE/mathematics/physics online

    Hello Everyone, So I have been looking for some time now to find an online program in EE/mathematics/physics. There are a number of programs I have found (purdue, johns hopkins, etc) but the prices are absolutly staggering. I have paid my dues to the student loan machine with my BS degree in...
  29. pairofstrings

    I System to represent objects in Mathematics

    Hi. Usually, Computer Programmers use Flow Charts, Algorithms, or UML diagrams to build a great software or system. In the same manner, in Mathematics, what do Mathematicians use to build a great system that they want to build. Category Theory is at the highest level of abstraction; then...
  30. M

    Intro Physics I’d like to learn physics

    Hello. I’d like to self-learn physics. basically starting from a zero knowledge background to keep things simple. I have asked before and some individuals said that it is using your time to learn it once you know calculus? Is this true or should I start learning the basics now so that when I...
  31. Jeviah

    Answer to cos^2x^2

    Homework Statement hello, I'm currently studying simpsons rule (unrelated) however the method requires the answer to cos^2(1^2) the answer given by my tutor is 0.2919, I have been unable to get this answer after inputting cos in various ways I always get 0.9997, which is right and if 0.2919 is...
  32. M

    Programs Physics or mathematics degree?

    Hi, currently I am having a dilemma trying to decide whether I want to do a 4 year Msci in physics or mathematics. I am about to finish a degree in computer science I had started when I was 13 or 14 with the open university. Although, I still want to go to a real university for mathematics or...
  33. M

    Linear algebra, field morphisms and linear independence

    Homework Statement Let f1,f2, ..., fn : K -> L be field morphisms. We know that fi != fj when i != j, for any i and j = {1,...,n}. Prove that f1,f2, ..., fn are linear independent / K. Homework Equations f1, ..., fn are field morphisms => Ker (fi) = 0 (injective) The Attempt at a Solution I...
  34. H

    Find the limit as h --> 0 for this trigonometery equation

    Homework Statement ##\lim_{h \to 0} \frac{f(x - 2h) - f(x + h)}{g(x + 3h) - g(x-h)}## While f(x) = cos x g(x) = sin x Homework Equations The Attempt at a Solution Using L Hopital i couldn't make it more simple. I tried to divide it by cos and sin Can you give me clue?
  35. KFSKSS

    B Need some help with understanding linear approximations

    Hello. My problem is that I began with Linear Approximation and I'm terribly stuck. I have problems understanding its very concept and with calculations. (It may sound stupid but I'm autodidact and I'm studying mathematics in english (not my mother tongue) and sometimes it gets hard). It would...
  36. pairofstrings

    B Constants in Mathematics

    Hi. π is a constant. The purpose of π is to use it to find circumference of a circle or area of a circle. Correct? Can you please elaborate? Can you point out any other two constants and its purpose? Thanks.
  37. S

    B Why does the multiverse theory have only 4 levels?

    Why does max tegmark's and brian greene's levels of multiverses consider as the highest level the mathematical multiverse hypothesis if other authors like David Lewis consider that there could be also universes non described by maths? Why does it stop at level iv (mathematical universes...
  38. F

    I Please suggest whether I should use delta or dx method.

    I previously made a derivation of Neumann potential. It can be found in the pdf file below. I originally made it in the ##dx## method. It involved equations like ##dm=I dS##. My maths teacher told that such an expression has no meaning, at least in elementary calculus. However I argued that my...
  39. C

    Mathematica Is this a bug in Mathematica 11 ?

    I calculated a simple complex function integration using Mathematica 11. As the screenshot shows, the only difference between the two inputs is that I set a = 1 in the second one. However, the two outputs differ by a factor of 2. What's wrong? Is this a bug?
  40. R

    Broken Stick Math Discussion

    Here's a math question I've been thinking about lately. We have a stick of length one which is broken in one spot (with that spot chosen randomly and uniformly). Of the two broken pieces we take the one on the right and break it into two pieces in the same manner as before. With our three...
  41. S

    What is 'phase space in chaos theory and nonlinear dynamics?

    The term 'phase space' is often used in the study of nonlinear dynamics.What is it.
  42. S

    I What is the difference between phase space and state-space?

    In state space, the coordinates are the state variables of a system.So,each point in state space represents a specific value of state variables.Thus,state space representation represents the changes in a dynamical system. The state variables are the minimum number of variables which uniquely...
  43. S

    Why is chaos not random?

    Chaos is deterministic behavior.Why is chaos deterministic.Why chaos is not random. Chaos is sensitive dependence on initial conditions,a slight change in initial condition can give rise to totally different trajectories.
  44. ohwilleke

    I Do these equations have a non-trivial solution?

    I was reading the following article that tries to use some equations originally proposed by Pauli in 1951 to reason from one of two reasonably plausible axions that there are tight constraints on the fundamental particle content and mass spectrum of the Standard Model together with BSM...
  45. AD MCFC

    Engineering BS Physics to Materials Science Graduate Advice/Thoughts

    Hello All, I am currently a junior in physics and have set sights on Materials Science (Electronic Materials/Nanotechnology) for grad school. My school does not offer MS.Eng as a degree and so the only Materials Science course I can enroll in as the general "Introduction to Materials Science...
  46. pairofstrings

    B A single or multiple equations of 'y'?

    Hello. I have an object that I want to talk about. The object above does not seem to have numbers in a definite pattern, so, I think that this object will not have a single equation that describes the object, but instead there will be multiple number of equations of 'y' in 'x'. Correct?
  47. daniele1234

    Number Theory Undergraduate Number Theory Book Recommendations

    Hi , everyone! This is my first post/thread/anything on this forum so first I apologise if I slip up or make any mistakes. Anyway, my question is about recommendations for textbooks for Undergraduate Number Theory. So far, I have studied Calculus 1-3 (including things like line integrals...
  48. H

    How to find the angles of a triangle in a semicircle?

    Homework Statement Homework Equations d(y)/d(x) --> max area area of triangle = 1/2 . base . height The Attempt at a Solution for number (2) [/B] x^2 + y^2 = r^2 --> circle equation base = 2R, height = y Area = 1/2 . 2R . y area = 1/2 . 4. √ (r^2 - x^2) area now is half of max = 2...
  49. pairofstrings

    Job Skills Artificial Intelligence

    Hello. I am looking for guidance. I have been working in software industry for about eight years now. I see that the trend is moving towards making a machine intelligent. I want to be part of this trend and move forward. My questions is: What skills do I need to become part of this era of...