What is Mathematics: Definition and 980 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. docnet

    B What is a closed form solution?

    Hi friends, I was wondering if you could give the definition of 'closed form', with examples of closed form solutions and open? form solutions. Foe example, is this a closed form solution? $$\sum_{k=1}^\infty \frac{1}{2^k}$$ Or this? $$\sum_{k=1}^5 \frac{1}{2^k}$$ Thanks.
  2. M

    Partial Differentiation of this Equation in x and y

    Hi; please see below I am trying to understand how to get to the 2 final functions. They should be the same but 6 for the first one and 2 for the second? (I hope my writing is more clear than previously) There is an additional question below. thanks martyn I can't find a standard derivative...
  3. gentzen

    A Concrete examples of randomness in math vs. probability theory

    A recent question about interpretations of probability nicely clarified the role of the Kolmogorov axioms: [... some excursions into QM, negative probabilities, and quasiprobability distributions ...] Conclusion: the Kolmogorov axioms formalize the concept of probability. They achieve this by...
  4. chwala

    I Determining the dimension of a given PDE

    Now in my understanding from text ...just to clarify with you guys; the pde is of dimension 2 as ##t## and ##x## are the indepedent variables or it may also be considered to be of dimension 1, that is if there is a clear distinction between time and space variables. Your insight on this is...
  5. neilparker62

    Insights The Lambert W Function in Finance

    [url="https://www.physicsforums.com/insights/the-lambert-w-function-in-finance/"]Continue reading...
  6. jedishrfu

    B Maze Proof and Statistical Mechanics

    https://www.quantamagazine.org/maze-proof-establishes-a-backbone-for-statistical-mechanics-20240207/
  7. chwala

    Find the time taken by the truck to move ##1## metre.

    Unless i am missing something; there is an error with the textbook It ought to be , ## t= \sqrt {\dfrac{1}{0.04}}= \sqrt 25 = 5## seconds.
  8. paulb203

    Work out an estimate for the total number of ponies in the forest

    My question; Which branch of maths is this? Also, can you give me a clue as to where to start regards solving this. Just a hint please, not a full explanation. I'm struggling to even guess at this one. I did think, '60 ponies, 5 of which are tagged, so, 5/60 tagged, which is 1/12 1/12 of the 60...
  9. A

    Number of ways of arranging 7 characters in 7 spaces

    The rightmost position has 3 possibilities: ##x,y,z## The remaining two letters are to be arranged in 6 spaces: ##\frac{6!}{4!}## Now the 3 can be placed in ##\frac{4!}{3!}## Total no of ways =$$3×\frac{6!}{3!}=12×30$$ $$OR$$ Since ##x,y,z## are three different boxes/variables, we can use the...
  10. C

    A SDE valuation equation (stochastic calculus)

    I read from a text: "suppose a stock with price ##S## and variance ##v## satisfies the SDE $$dS_t = u_tS_tdt+\sqrt{v_t}S_tdZ_1$$$$dv_t = \alpha dt+\eta\beta\sqrt{v_t}dZ_2$$ with $$\langle dZ_1 dZ_2\rangle = \rho dt$$ where ##\mu_t## is the drift of stock price returns, ##\eta## the volatility of...
  11. A

    Given value of vectors a,b, b.c and a+(b×c), Find (c.a)

    I thought this was too easy $$a+(b\times c)=0\implies a=-(b\times c)=(c\times b)$$ Then $$3(c.a)=3(c.(c\times b))=0$$ Since cross product of vectors is perpendicular to both vectors and dot product of perpendicular vectors is zero. Now here's the problem, correct answer given is 10. But how do...
  12. M

    DeepMind AlphaGeometry: Is AGI just around the corner?

    DeepMind's new geometry problem-solver AlphaGeometry can solve Euclidean plane geometry questions from the International Mathematics Olympiad (IMO) almost as well as a human gold medallist. https://deepmind.google/discover/blog/alphageometry-an-olympiad-level-ai-system-for-geometry/...
  13. chwala

    Find an irrational number that satisfies the given inequality

    I just came across this question and the ms indicates, Would ##31.5## be correct? ...i think it is rational as it can be expressed as ##31.5 = \dfrac{63}{2}##.
  14. A

    A,B,C are 3 points with position vectors a,b,c Find length of median

    Heres how I proceeded, Equation of line ##AC## in vector form: $$\vec r=a+t(c-a)$$$$\vec r=(1i+4j+3k)+t(2i-6j+2k)$$ Since ##B## doesn't lie on ##AC## ##b\neq (1+2t)i +(4-6t)j+(3+2t)k## The following equation is derived: $$2\hat i+\alpha \hat j+4\hat k\neq (1+2t)\hat i +(4-6t)\hat j+(3+2t)\hat k...
  15. caffeinemachine

    I New Video Course on General Topology on YouTube

    Hello Everyone! I created a YouTube channel (here's the link) a few years ago in which I post detailed lectures in mathematics. I just started a series on General Topology. Following is a snapshot from a video. I mean to deliver a comprehensive course with a lot of pictures and intuition and...
  16. docnet

    B Confusion about infinity

    We have come to accept that Infinity times two is infinity. In the sense of 'size' we use to think about everyday numbers, the rules of arithmetic with infinities seem like nonsense. For example, consider the computable number $$0.100100100100100....$$ In the decimal expansion, there are...
  17. J

    A Klein's Protocols

    https://www.uni-math.gwdg.de/aufzeichnungen/klein-scans/klein/ I hope you enjoy folks. It would be nice to see an English translation somewhere. Same for his encyclopedia.
  18. Greg Bernhardt

    Mathematics Award

    Please select up to 3 members who were most impactful in the Mathematics forums in 2023. This is a popular vote. Polls were created by weighing activity and measure of helpfulness. Everyone nominated should feel honored. Many more could be added to this poll, we can never realistically add...
  19. S

    B More examples of equations that unexpectedly model nature in similar ways

    Been dipping my toes into maths by examining how equations work on the most basic level, and I love encountering equations that turn out to model similar aspects in nature, for example the inverse square law is apparent in equations for gravity and for electromagnetism. In the thumbnail of...
  20. M

    Number Theory What books or materials are needed to understand primitive roots?

    do you know any books, videos, or notes that can help me to understand these topics: 1-primitive roots for primes 2-the existence of primitive roots I am using right now elementary number theory and its application by Kenneth H. Rosen to understand these topics, do you know another source that...
  21. G

    I Integrating 1/x with units (logarithm)

    Hi. What exactly is happening mathematically when you integrate ##\frac{1}{x}## $$\int_a ^b \frac{1}{x} dx=\ln{b}-\ln{a}=\ln{\frac{b}{a}}$$ if there's units? Sure, they cancel if you write the result as ##\ln{\frac{b}{a}}##, but the intermediate step is not well-defined, so why should log rules...
  22. LCSphysicist

    I Page 183-184 of Howard George's group book

    I would appreciate if someone could help me to understand what is happening in section 12.3 from the Howard George's book. First of all, the propose of the section is to show how $SU(3)$ decomposes into $SU(2) \times U(1)$. But i can't understand what is happening. First of all, i can't get the...
  23. A

    Job Skills What Can I Do Science/Math Related While Working on Undergrad Degree?

    I only have my AA so far (working towards physics degree), but I would like to start changing directions career-wise. I'm completely burnt out on my current field (went through several careers over the years, but lately, it's been sales, and I HATE sales, just good at it). I would love to do...
  24. I

    Prove ##a \cdot 1 = a = 1 \cdot a## for ##a \in \mathbb{N}##

    I have to prove ##a \cdot 1 = a = 1 \cdot a## for ##a \in \mathbb{N}##. The book I am using ("The real numbers and real analysis" by Ethan Bloch) defines Peano postulates little differently. Following is a set of Peano postulates I am using. (Axiom 1.2.1 in Bloch's book) There exists a set...
  25. A

    Good series of books to brush up on math, physics, and computer programming?

    TL;DR Summary: Got my AA with a focus in physics nearly a decade ago. Looking to go back and finish what I started, but need to brush up and looking for good resources to do so? Looking for resources that are thorough that can help me brush up on calculus 1-3, physics 1-2, and possibly...
  26. George Jones

    Mathematicians Have Modest Goals

    From the Introduction to https://arxiv.org/abs/2106.11285 "Since the dawn of time, human beings have asked some fundamental questions: who are we? why are we here? is there life after death? Unable to answer any of these, in this paper we will consider cohomology classes on a compact projective...
  27. M

    I Eccentric anomaly of ellipse-circle intersections

    I want to calculate eccentric anomaly of all points of ellipse-circle intersection. Ellipse is not rotated and its center is in origin. Circle can be translated to (Cx, Cy) coordinates. I am using python for calculations. Only solution I found, is this...
  28. H

    A gardener collected 17 apples...

    In fact, it WAS a homework couple of years ago, and I've solved it, kind of (below). I still would like to find a cleaner solution. Here is what I did. Let's say, the apples are labeled, and their weights are ##x_1, x_2, ...##. He takes out the apple #1 and finds that, e.g., ##x_2+x_5+x_9+... =...
  29. S

    Why some students are good at math and others lag behind

    "Former math teacher explains why some students are good at math and others lag behind" The title of a news article shown on todays Yahoo site, https://www.yahoo.com/news/former-math-teacher-explains-why-122744193.html Looking in the section called " Why are some students ‘good’ at math and...
  30. S

    Use of calculus in a launch of a rocket - Relevant Assumptions

    I know that we have to assume certain things for the math to be achievable (at my level). for instance, I assumed that the rocket goes in a straight line instead of orbiting around the earth at an angle. but I can't develop any further assumptions as the task is so generalised and open-ended...
  31. H

    What is a point of using complex numbers here?

    Firstly, the exercise itself is not difficult: On one hand, $$|(a + ib)(c + id)|^2 = |a + ib|^2|c + id|^2 = (a^2 + b^2) (c^2 + d^2) = MN.$$ On the other hand, ##(a + ib)(c + id) = p+ iq## for some integers p and q, and so $$|(a + ib)(c + id)|^2 = |p + iq|^2 = p^2 + q^2.$$ Thus, ##MN = p^2 +...
  32. O

    Calculus, it's personal for me

    How did you find PF?: I was randomly searching the net for info on calculus books for self study, found a math reddit that brought me here. I'm 65. Not working since 2000. In HS, 9th Algebra 1 = A, 10th Geometry/Trig =A, 11th, Algebra 2= D, (long story)....so no Calculus in Sr yr. Through...
  33. Ali sulieman

    Engineering Looking for resources to develop my skills in undergraduate Physics and Mathematics

    I want YouTube channels that systematically raise my mathematics and physics level
  34. N

    B 1 divided by infinity equals zero (always?)

    I understand the mathematics that 1 divided by infinity is virtually zero and so equals zero. I look on the internet and that is the answer that I get. Is this a simplification for early mathematics learning and, if I continue, will I find a more complex answer? The reason that I ask is that I...
  35. fresh_42

    Insights Epsilontic – Limits and Continuity

    Epsilontic – Limits and Continuity I remember that I had some difficulties moving from school mathematics to university mathematics. From what I read on PF through the years, I think I’m not the only one who struggled at that point. We mainly learned algorithms at school, i.e. how things are...
  36. chwala

    Find the roots of the given cubic equation

    Going through this, am still checking but will post all the same; which method did they apply to find the roots of the attachment below. My thinking; Let ##p+qi## be the cube root of ##x^3-6x+2=0## then, ##\sqrt{x(x^2-6)}=i\sqrt{2}## ##(p^2-q^2+2pqi)(p+qi)= x^3-6x+2## We know that...
  37. mcastillo356

    B Cosecant function: notation for the domain and range -- Am I right?

    Hi, PF There are two ways to write domain and range of a function: through set notation, or showing intervals. I've chosen the set notation, and, for ##y=\csc x##, this is the attempt: $$\text{D}:\{x\,|\,x\not\in{n\pi},\,n\not\in{\mathbb{Z}}\}$$ $$\text{R}:\{f(x)\,|\,x=\mathbb{R}\(-1,1)\}$$...
  38. TurtleKrampus

    Why are Physicists so informal with mathematics?

    I realize of course that this will probably not apply to all physicists, but at least every physicist in my university's math department is very unrigorous when it comes to mathematics. This is frustrating because some of the physics material seems genuinely interesting, but the lack of an...
  39. G

    Difference between -3² and (-3)² ?

    What is the difference between these to. -3² and (-3)² ? I know - x - = + I am told -3²= -9 not +9 -3² means -3 x -3 = 9 I am told (-3)² = 9
  40. A

    A Summing over continuum and uncountable numerocities

    Here I want to address of the question if it is possible to make a sum over an uncontable set and discuss integration rules involving uncountably infinite constants. I will provide introduction in very condensed form to get quicker to the essense. Conservative part First of all, let us...
  41. hackedagainanda

    Partial Fraction Decomposition

    ##\frac {1} {x^2 -c^2}## with ##c \neq {0}## So the first thing I do is split the ##x^2 -c^2## into the difference of squares so ##x +c## and ##x - c## I then do ##\frac {A} {x + c}## ##+## ##\frac {B} {x-c}##, and then let ##x=c## to zero out the expression. And that is where I am getting...
  42. C

    Programs What subject is better for an aspiring experimental physicist?

    I'm in my last 2 years of high school, and I have to pick a speciality to study before becoming an undergraduate and studying in college. In the future, I'm hoping to become an experimental physicist. My high school offers 3 specialities that are relevant to physics to pick from, all of them...
  43. Jonathan West

    New book out by Routledge called "Mathematical Conundrums" with many interesting problems

    TL;DR Summary: new book with interesting problems There's a new book out by Routledge called Mathematical Conundrums with many interesting problems in algebra, arithmetic, route-drawing, and logic. Good for schools as algebra is no higher than high school. Challenging though.
  44. bhobba

    Insights What Are Numbers?

    [url="https://www.physicsforums.com/insights/what-are-numbers/"]Continue reading...
  45. revix

    I Where can I find explanations about physics applications in real life?

    I am a curious physics student who wants to learn how to use its knowledge to create things, to understand phenomenons and so on. I am looking for detailed explanations that use physics and maths. (books, websites, videos, etc.)
  46. yucheng

    Style of teaching/learning mathematics: by proofs of theorems only?

    I remember there was a method of learning/teaching mathematics where all they do in class is to force students to prove the theorems themselves. What was this method again? It was named after someone.... @fresh_42 ?
  47. chwala

    I Order of an element in ##\mathbb{Z}_n##

    Doing some self study here; my understanding of order of an element in a group is as follows: Order of ##3## in ##\mathbb{z_4}## can be arrived by having, ##3+3+3+3=12≡0## likewise, the order of ##12## in ##\mathbb{z_{20}}## can be arrived by ##12+12=24 ≡4≠0## ##12+12+12=36≡16≠0##...
  48. A

    Relativity Looking for a good introductory Tensor Analysis Textbook

    Hello all, I've taken math through differential equations and linear algebra, am in my senior year of physics curricula while conducting McNair research regarding General Relativity. I found a NASA document outlining Einstein's field equations, which suggests only preparative familiarity with...
  49. C

    Prospective ArXivLabs project

    In short, I'm interested in working on a web-app to make landmark papers in theoretical physics and mathematics more broadly accessible, especially to undergraduate and graduate students who are looking to catch up to modern topics (without sacrificing rigor or exactness of understanding), and...
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