Algebra Definition and 999 Threads

Algebra (from Arabic: الجبر‎, romanized: al-jabr, lit. 'reunion of broken parts, bonesetting') is one of the broad areas of mathematics, together with number theory, geometry and analysis. In its most general form, algebra is the study of mathematical symbols and the rules for manipulating these symbols; it is a unifying thread of almost all of mathematics. It includes everything from elementary equation solving to the study of abstractions such as groups, rings, and fields. The more basic parts of algebra are called elementary algebra; the more abstract parts are called abstract algebra or modern algebra. Elementary algebra is generally considered to be essential for any study of mathematics, science, or engineering, as well as such applications as medicine and economics. Abstract algebra is a major area in advanced mathematics, studied primarily by professional mathematicians.
Elementary algebra differs from arithmetic in the use of abstractions, such as using letters to stand for numbers that are either unknown or allowed to take on many values. For example, in



x
+
2
=
5


{\displaystyle x+2=5}
the letter



x


{\displaystyle x}
is an unknown, but applying additive inverses can reveal its value:



x
=
3


{\displaystyle x=3}
. Algebra gives methods for writing formulas and solving equations that are much clearer and easier than the older method of writing everything out in words.
The word algebra is also used in certain specialized ways. A special kind of mathematical object in abstract algebra is called an "algebra", and the word is used, for example, in the phrases linear algebra and algebraic topology.
A mathematician who does research in algebra is called an algebraist.

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  1. V

    MHB Solving Algebra equation 3x=15

    Can anyone please help me with this equation 3x =15 My workings ;3x=15 Any help is appreciated, thanks = 5x, as 15 divide by 3x=5x
  2. C

    Schools Where should I take my calculus and linear algebra online?

    Has anyone taken these two courses online in a self-paced course for credit? If so, where and how was it in terms of quality? How about price? Opinions/thoughts are much appreciated. I'm working and the closest community college is a commute away, so that's out. I'm finding $1100-3000~ for...
  3. U

    Motorboat Ques: Gelfand Algebra 131 | Hints to Help

    I am struggling on how to approach this problem. Hints would help greatly. This is also from Gelfand's Algebra problem 131
  4. Eclair_de_XII

    If you were a linear algebra teacher, would you dock points for this?

    Let's say you were proctoring some test that required proofs of Jordan canonical forms and rational canonical forms. Would you dock points from a lazy student abbreviating the former as "J-canonical forms" and the latter as "##\mathbb{Q}##-canonical forms" in their proofs?
  5. F

    B Why is the distributive law correct in algebra, like in arithmetic?

    When they give reason for multiplication the negative numbers leading to positive number, they base on distribute law.But why the distribute law in algebra is correct like in arithmetic?(e.g why -5(8-6)=-5.8+-5.-6?).In abstract algebra they use distribute law as axiom.But in elementary algebra...
  6. G

    MHB Solve linear algebra equation 2x – 1 = 9 – 3x

    Hi, Could someone help me with this question: 2x – 1 = 9 – 3x Thank you in advance!
  7. W

    Equivalent representations for Dirac algebra

    One thing I was thinking about doing was to consider these representations as a basis for the gamma matrices vector space, then try to determine what the change of basis from one to the other would be. However I'm unsure if it's correct to treat the representations as a basis, or whether this is...
  8. RikaWolf

    I Linear Algebra - Inner Product problem

    I need help to know if I'm on the right track: Prove/Disprove the following: Let u ∈ V . If (u, v) = 0 for every v ∈ V such that v ≠ u, then u = 0. (V is a vector-space) I think I need to disprove by using v = 0, however I'm not sure.
  9. R

    How Can I Verify the Solution to the Equation 2y + 30 = 8y?

    8y-2y=6y 6 y+30 6y/6=30/6 Y=5 Now how can I verify this equation?
  10. R

    Is 4m + 2k Equal to 6mk in Algebraic Simplification?

    3m+m=4m 4m +2k =6mk Is this correct, help please if not correct Thanks
  11. F

    Master Algebra Revision with Exam Tips & Methods | Boost Your Summer Exam Prep

    have a Summer Exam , at the begging of next week on all topics that we have done , so I decided to do Algebra , but I forgot have to do it , because it was a while ago, can anyone show me the method again please to memorize. Thanks in advance . Workings : add 4x+5=9 3x+6=9 9/3 =3/3 3x=3
  12. C

    MHB Either or statement from Abstract Algebra Book

    Dear Everyone, What are the strategies from proving a either-or statements? Is there a way for me to write an either-or statement into a standard if-then statements? For example, this exercise is from Dummit and Foote Abstract Algebra 2nd, "Let $x$ be a nilpotent element of the commutative...
  13. M

    Linear Algebra: LU Decomposition

    Here is the initial matrix M: M = \begin{bmatrix} 3 & 1 & 6 \\ -6 & 0 & -16 \\ 0 & 8 & -17 \end{bmatrix} I have used the shortcut method outlined in this youtube video: LU Decomposition Shortcut Method. Here are the row reductions that I went through in order to get my U matrix: 1. R_3 -...
  14. R

    Calculating the half maximum point of a function

    This is the form of the function above: I started by equating (1) to 1/2: $$T(\varphi)=\frac{r^{2}+\tau^{2}-2\tau\cos\varphi}{1+\tau^{2}r^{2}-2\tau r\cos\varphi} = \frac{1}{2},$$ which can be rearranged to: $$2r^{2}+2\tau^{2}-1-\tau^{2}r^{2}=2\tau\left[2-r\right]\cos\varphi$$ using...
  15. J

    Courses Which Abstract Algebra Sequence is Best for Aspiring High Energy Physicists?

    My university offers two different two-semester sequences for learning abstract algebra, and I can't decide which one would be better for me, a physics major. Here are the two sequences and their course descriptions, copied and pasted from the university website: Algebra 1: Theory of groups...
  16. F

    Computer Algebra System: Physics Self-Study Suggestions

    I am stuying physics by myself and I would like to get a Computer Algebra System. Does anybody has any suggestions?
  17. M

    MHB Solve Algebra Word Problem: Overpayment Amount of \$85.00

    March bill was \$400.00 for 25% to be paid of that. There is a recurring overpayment amount of \$85.00 so \$85.00 was to be withdrawn from the 25% of \$400.00 for a total of \$15.00. \$400.00 x .25 = \$100.00 \$100.00 - \$85.00 = \$15.00 However, rather than doing that; the following was...
  18. M

    MHB Two-element Boolean algebra: How are the equalities derived?

    Hey! :o We have the following equalities: \begin{align*}\left (\overline{y}\land z\lor x\land \left (\overline{z}\lor y\right )\right )\land \left (\overline{x}\lor \overline{y}\right )& \overset{(1)}{=}\left (\overline{y}\land z\lor x\land \overline{z}\lor x\land y\right )\land \left...
  19. koiuuuuuuuuuuu

    B Is 3x=15 Really That Hard to Solve?

    Can you help me with this question. 3x=15 Do you have to divide both sides by 3 which, is x=5 Am I right. If not explain please, Thank-you
  20. G

    A Is the Exponential Map Always Surjective from Lie Algebras to Lie Groups?

    Is it correct saying that the Exponential limit is an exact solution for passing from a Lie Algebra to a Lie group because a differential manifold with Lie group structure is such that for any point of the transformation the tangent space is by definition the Lie algebra: is that the underlying...
  21. Zbaldridge

    Schools College Algebra And Trig at the same time?

    I am currently a community college student majoring in Computer Science, and I was placed into Calculus 1. I had to withdraw from this course for two semesters now. My professor advised me that I have "Algebra Issues". With my weaker foundation of Algebra, I plan to take the College Algebra...
  22. S

    MHB Solving Cubic Root Algebra: Don't Know the Steps

    I don't understand this. a is not suppose to be -1; this is the only rule in the equation The answer is the second picture, I just don't know the steps that lead to that answer.
  23. chwala

    Is There a Universal Standard for Order of Operations in Mathematics?

    attempt 1. 6 /2×3 = 3×3=3×3=9 attempt 2 6/2(3)= 6 divide 6 = 1. sorry i cannot see the divide symbol.
  24. caters

    Are my calculations of the pregnancy ratio of the population correct?

    Monthly Cycle numbers Here is the cycle ratio: $$2_{early}:2_{fertile}:1_{late}$$ And the numbers: $$20,000_{early}:20,000_{fertile}:10,000_{late}$$ Now, let's divide the early into 2 groups, pre-fertile, and safe and assume there is a 50/50 split between those 2 groups. Let's also assume...
  25. A

    Matrix algebra : Find the matrix C such that N(A) = R(C)

  26. J

    Implications of zero in the denominator when solving equations

    My book states the following : ##{\frac{(y_1)-(y_2)}{(x_1)-(x_2)}}*\frac{(3+1)}{(0-0)} = -1## ...(1) ##\implies y_1 - y_2 = 0## (A) Is this a valid deduction? Context : The Problem :If ##A(0,-1)## and ##B(0,3)## are two opposite vertices of a square, then find the other two vertices...
  27. M

    Poincare algebra and its eigenvalues for spinors

    Homework Statement Show that for $$W^\mu = -\frac{1}{2}\varepsilon_{\mu\nu\rho\sigma}M^{\nu\rho}P^{\sigma},$$ where ##M^{\mu\nu}## satisfies the commutation relations of the Lorentz group and ##\Psi## is a bispinor that transforms according to the ##(\frac{1}{2},0)\oplus(0,\frac{1}{2})##...
  28. S

    I Can Fundamental Theorem Algebra be proven by simple DOF?

    A monic polynomial of degree N has N number of coefficients. The product of N number of linear factors has N number of free terms. A complex number has 2 DOF. Therefore, both a monic polynomial and the product of free terms have 2N number of DOF of real values. Thus, it must be possible to...
  29. S

    I On the topological proof of the Fundamental Theorem of Algebra

    Sorry for the misspelling, but this forum doesn't allow enough characters for the title. The title should be: For the topological proof of the Fundamental Theorem of Algebra, what is the deal when the roots are at the same magnitude, either at different complex angles, or repeated roots? I...
  30. Edge5

    I An exponential number algebra problem

    Let a,b,c and n are real numbers.a-b = C I want to get rid of a,b and find the following expression in terms of C and n. How can I do that? (an-bn)= ? (in terms of C and n) Thank you.
  31. A

    B Why is x2 . y2 not equal to (x-y) (x+y)?

    why x2 . y2 = (x-y) (x+y) and xy . y2 cannot be (x-y) (y+y)
  32. H

    MHB Linear Algebra Rank of a Matrix Problem

    Let A be a n x n matrix with complex elements. Prove that the a(k) array, with k ∈ ℕ, where a(k) = rank(A^(k + 1)) - rank(A^k), is monotonically increasing. Thank you! :)
  33. Eclair_de_XII

    Linear Algebra Can anyone recommend an advanced linear algebra book?

    I have already taken two elementary linear algebra courses, and have taken the upper-division linear algebra course offered at my school. However, I feel that I did not learn as much from the latter as I should have. I can owe this to not applying myself as much as I should have, due to other...
  34. F

    B Are Algebra, Geometry, Probability innate or cultural?

    Are the concepts of the rules of Algebra, Geometry and Probability things that all humans have some instinctive grasp at some level, or things that we basically learn from others, therefore cultural? Let me explain. I once saw an experiment with a mommy rat. She had 4 puppies, and someone put a...
  35. josephweeds

    Intro Math Math Books: Algebra, Form & Function, Trigonometry

    Hi everybody. I am looking for some math books (Algebra Form and Function...Trigonometry...) I found a lot but i don't know how to download it :cry:
  36. topsquark

    MHB Graded Algebra: Understanding Color Dirac Spinors in Space-Time

    I just read through a paper on a \mathbb{Z} _ 3 graded Algebra. In this instance we are talking about color Dirac spinors in space-time. It looks like the author is talking about \left ( SU(3) \otimes L^4 \otimes \mathbb{Z}_2 \otimes \mathbb{Z} _2 \right ) \otimes \mathbb{Z} _3. ( SU(3) is...
  37. V

    Final Grades for Linear Algebra Course

    Homework Statement Consider the below vector x, which you can copy and paste directly into Matlab. The vector contains the final grades for each student in a large linear algebra course. x = [59 70 83 89 72 70 54 55 68 61 61 58 75 54 65 55 62 39 43 53 67 100 60 100 61 100 77 60 69 91 82 71 72...
  38. E

    I Deriving reduction formula in Geometric Algebra

    Hi, I am trying to learn Geometric Algebra by going through the book "New Foundations for Classical Mechanics" by David Hestenes. I was reading the part about reduction formula (shown below) but couldn't get the result the shown in the book. Can someone show me how iterating (1.15) gives the...
  39. W

    Maxwell's equations and exterior algebra

    Maxwell's equations in differential form notation appeared as a motivating example in a mathematical physics book I'm reading. However, being a mathematical physics book it doesn't delve much into the physical aspects of the problem. It deduces the equations by setting dF equal to zero and d(*F)...
  40. Mutatis

    Find the eigenvalues and eigenvectors

    Homework Statement Find the eigenvalues and eigenvectors of the following matrix: $$ A = \begin{bmatrix} 3 & 0 & 0 \\ 0 & 3 & 2 \\ 0 & -1 & 0 \end{bmatrix} $$ Homework Equations Characteristic polynomial: $$ \Delta (t) = t^3 - Tr(A) t^2 + (A_{11}+A_{22} +A_{33})t - det(A) .$$ The Attempt at...
  41. Mutatis

    Find the eigenvalues and eigenvectors

    Homework Statement Find the eigenvalues and eigenvectors fro the matrix: $$ A=\begin{pmatrix} 0 & -i \\ i & 0 \end{pmatrix} $$. Homework Equations Characteristic polynomial: ## \nabla \left( t \right) = t^2 - tr\left( A \right)t + \left| A \right|## . The Attempt at a Solution I've found...
  42. C

    Finding the minimum number of boxes to pack products

    Homework Statement This was a coding challenge, and I've already completed it. But I feel there must be a better solution to my approach at the moment. Especially maybe there's a mathematical approach. You have 3 sizes of boxes, small, medium, large: small: holds 3 items medium: holds 5 items...
  43. W

    Other Using Michael Artin's "Algebra" for Group Theory

    Hi all, I have stumbled upon Artin's book "Algebra" and was wondering if I could use it to do some self-study on Group Theory. Some background: I am a physics undergraduate who has some competence in elementary logic, proofs and linear algebra. It seemed to me that ideas related to Group...
  44. F

    Solving for y: e^(y) = y^(2) - 2

    Homework Statement Hi I am having trouble solving for y for the following equation: e^(y) = y^(2) - 2 Homework Equations ln(x)^(y) = y*ln(x) ln(e^(y)) = y ln(2) = 1 The Attempt at a Solution My attempt: e^(y) = y^(2) -2 ln(e^(y)) = ln(y^(2)) - ln(2) y = 2*ln(y) - 1 y - 2*ln(y) + 1 = 0...
  45. L

    Studying Why Am I Struggling with Algebra 2 in College?

    Hi, I'm currently having trouble getting past Algebra 2 at my college. I got local credit in Algebra 2 in high school. My current college major is Chemistry with an emphasis on materials science/engineering. I obviously need Calculus 3 and differential equations and my ultimate roadblock is...
  46. Destroxia

    I DSP: Recurrence Relations in a Linear Algebra Equation

    Hello, I've been working through some Digital Signal Processing stuff by myself online, and I saw a system that I wanted to write down as a Linear Algebra Equation. It's a simple delay feedback loop, looks like this: The (+) is an adder that adds 2 signals together, so the signal from x[n]...
  47. T

    I Compatibility of Physics and Abstract Algebra

    One of the last classes I'm taking before finishing my degrees as an undergraduate is abstract algebra. My professor uses the textbook 'Contemporary Abstract Algebra' by Joseph Gallian. The book isn't written terribly nor is the teacher a poor one, but I just find this subject so...
  48. E

    I Linear Algebra and Identity Operator Generalized to 3D

    I'm just getting into 3D quantum mechanics in my class, as in the hydrogen atom, particle in a box etc. But we have already been thoroughly acquainted with 1D systems, spin-1/2, dirac notation, etc. I am trying to understand some of the subtleties of moving to 3D. In particular, for any...
  49. M

    Linear Algebra: Question about Inverse of Diagonal Matrices

    Homework Statement Not for homework, but just for understanding. So we know that if a matrix (M) is orthogonal, then its transpose is its inverse. Using that knowledge for a diagonalised matrix (with eigenvalues), its column vectors are all mutually orthogonal and thus you would assume that...
  50. J

    Using binomial coefficients to find sum of roots

    Homework Statement >Find the sum of the roots, real and non-real, of the equation x^{2001}+\left(\frac 12-x\right)^{2001}=0, given that there are no multiple roots. While trying to solve the above problem (AIME 2001, Problem 3), I came across three solutions on...
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