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.

View More On Wikipedia.org
Recent contents Top users
  1. J

    Calculating the sum of a series of cubes

    Mentor note: Moved from technical math section, so missing the homework template filling in the boxes is easy, 1,3,6,10,15 second is just squares of that, 1,9,36,100,225 But how I anyone supposed to find and expression for this? This is from a textbook on elementary algebra, the specific...
  2. bremenfallturm

    Expression, modular arithmetic

    This is basic modular arithmetic but I just can't get it to work no mather how many different methods I try. I probably have failed to understand some basics of modular algebra... Help is appreciated! Correct is supposed to be ##16##
  3. Rick16

    I Trying to learn tensor algebra

    This is exercise 1.8.3 from Foster & Nightingale: Show that if ##\sigma_{ab} = \sigma_{ba}## and ##\tau^{ab} =-\tau^{ba}## for all ##a##, ##b##, then ##\sigma_{ab}\tau^{ab}=0##. I began writing down ##\sigma_{ab}\tau^{ab}=\sigma_{ba}(-\tau^{ba})=-\sigma_{ba}\tau^{ba}##. Here I got stuck and...
  4. billtodd

    A Scalars, Vectors, Matrices,Tensors, Holors....

    Is there something beyond Holors? :cool:
  5. P

    Equality sign and equivalence relations

    I solved this exercise a long time ago, and I am now reviewing my solution. I think it is correct, but more importantly, when I look at this exercise now, I have a hard time understanding it and my solution. Hence I've got some questions. If this is correct, why does "if ##a=b##, then ##b=a##"...
  6. M

    How to prove this without using Cardano's formula?

    Proof: Let ## x=\sqrt[3]{18+\sqrt{325}}+\sqrt[3]{18-\sqrt{325}} ##. Then ## x^3=(\sqrt[3]{18+\sqrt{325}}+\sqrt[3]{18-\sqrt{325}})^3 ##. Note that ## (a+b)^3=a^3+3ab(a+b)+b^3 ## where ## a=\sqrt[3]{18+\sqrt{325}} ## and ## b=\sqrt[3]{18-\sqrt{325}} ##. This gives ##...
  7. L

    Free Abstract Algebra curriculum in Urdu and Hindi

    I am trying to create a complete college curriculum on abstract algebra in Urdu as there are very few advanced resources for students who want to learn mathematics but are not proficient in English. In South Asia and the Middle East, only privileged people are proficient in English and...
  8. S

    The value of (b - c) / (c - a)

    $$(b-a)^2-4(b-c)(c-a)=0$$ $$b^2-2ab+a^2=4(bc-ab-c^2+ac)$$ $$b^2-2ab+a^2+4ab=4bc-4c^2+4ac$$ $$(b+a)^2-4ac=4c(b-c)$$ $$b-c=\frac{(b+a)^2-4ac}{4c}$$ I don't know how to continue and not even sure what I did is useful. Thanks
  9. N

    B What unique contributions to math did linear algebra make?

    I've been struggling to understand what was the key insight or insights that linear algebra brought to math, or what problems it allowed the solving of that couldn't be solved before. To make a comparison with calculus, I understand that calculus' two key insights were finding a method to...
  10. paulb203

    Is this a simultaneous equation question?

    hb=54 2h+2b=33 h=54/b therefore, 2(54/b)+b=33 108/b + b = 33 I’ve got a feeling I’ve gone down a blind alley here. Any hints?
  11. felizgu

    Hello From NYC!

    My name is Feliz. I am doing a self-study of college algebra. I am not a classroom student. My student days ended in December 1993. I've always enjoyed mathematics and decided to conduct a review of material learned long ago.
  12. G

    I Do hyperbolic harmonics exist?

    With the algebra so(3) are associated the spherical harmonics. I would assume that comparably with the algebra so(2,1) are associated functions that can be addressed as hyperbolic harmonics. But I nowhere found any reference to them. Do they exist and if so, where can they be found? Thank you...
  13. brotherbobby

    Factorise a six-termed quadratic in ##a## and ##x##

    Statement : I copy and paste the problem as it appeared in the text. Attempt : I confess I couldn't go much far at all. Here's my attempt below in ##\text{Autodesk Sketchbook}^{\circledR}##. The underlined , wavy underlined and box brackets below are my attempts to see what terms can be...
  14. chwala

    Solve the given word problem: Selecting 2 numbers from a watch face

    I honestly do not understand this question, my thoughts; ignoring the diagram and using algebra i can see that the step size [1,5] → [2,6] can be found by adding 1 (common difference) to each number meaning that the answer is A... ...the other options B,C,D and E can not be related by a...
  15. RChristenk

    Solve ##\sqrt{\dfrac{x}{y}}+\sqrt{\dfrac{y}{x}}=4\dfrac{1}{4} \cdots##

    ##\Rightarrow \begin{cases} (\sqrt{\dfrac{x}{y}}+\sqrt{\dfrac{y}{x}})^2=(4\dfrac{1}{4})^2\\ (\dfrac{x}{\sqrt{y}}+\dfrac{y}{\sqrt{x}})^2=(16\dfrac{1}{4})^2 \end{cases}## ##\Leftrightarrow \begin{cases} \dfrac{x}{y}+\dfrac{y}{x}+2=\dfrac{289}{16}\\ \dfrac{x^2}{y}+\dfrac{y^2}{x}...
  16. MatinSAR

    Why Is My Argument Calculation for Complex Number Incorrect?

    Question 1: Find the modulus and argument of ##z=-\sin \frac {\pi}{8}-i\cos \frac {\pi}{8}##. The modulus is obviously 1. I can't prove that the argument is ##\frac {-5\pi} {8}##. I think ##\frac {-5\pi} {8}## is not correct ... What I've done: $$\tan \theta=\cot \frac {\pi}{8}$$$$\tan...
  17. mathgenie

    I Taking the derivative of a function

    I would like to take the derivative of the following function with respect to Gt: $$\mathrm{G}_{t+1}=\mathrm{g}_{0}\mathrm{e}^{-qHt}$$ I think that the answer is either -1 or ##\mathrm{e}^{-qHt}-1## If you could show the calculations that would be a great help. Thanks very much.
  18. A

    I Practice With Proofs? (Algebra, Trig, and Calc)

    I'm trying to brush up on my algebra, trig, and calculus, and one thing I know I was always weak on before was proofs. I was never sure what equations would suffice as "proof," and which equations did not. Maybe this is an inane question, and maybe there is a really simple answer to this. I...
  19. jeff einstein

    B What went wrong with (-x)^2=x^2?

    I have a very basic confusion that supports some basic elements of algebra. Being a high school student my teacher couldn't answer this, hope someone could help here. We know this equation is true: (-x)^2=x^2 but once we square root both sides it becomes this: -x=x we can see this equation was...
  20. brotherbobby

    Factor an equation second order in ##x,y## with other variables

    Statement of the problem : Let me copy and paste to the right the problem as it appears in the text. Attempt : I couldn't go far into the solution. Below is my hopeless attempt. Request : Any hints would be welcome. [There are no solutions provided, but the answers are at the back of the...
  21. M

    Solving Number Triangle Puzzle with Trial and Error

    First I tried to solve this with algebra, but there are not enough equations: a+ b + c + d + e + f + g + h = 36 S = 12 + (d +f + a)/3 ........... ( d +f + a has to be a multiple of 3) a + b + c = e + f a + h + g = d + e So I had to resort to the trial and error to find the solution...
  22. F

    Insights Why Division by Zero is a Bad Idea

    Continue reading...
  23. E

    Why is the square root of x^2 = |x|?

    If I reason this as follows, I run into problems. Please help me understand what is wrong with reasoning like this. a) I start with the left hand side of the equation and let that x be -2. b) I square it. This gives me 4. So I now have the square root of 4. c) The square root of 4 is +/- 2. The...
  24. M

    I Algebra Homomorphisms as Subsets of the Cartesian Product

    Let ## \varphi \subseteq A \times B; \psi \subseteq B \times C ##. Then ## \varphi \circ \psi = \left \{ (a, c)| \exists b: (a,b) \in \varphi, (b,c) \in \psi \right \} \subseteq A \times C##. Task: Let ##\varphi## and ##\psi## are subalgebras of algebras ##A \times B## and ##B \times C##...
  25. brotherbobby

    To prove the "##m^{\text{th}}## Powers Theorem"

    Statement : Let me copy and paste the statement as it appears in the text on the right. Attempt : I could attempt nothing to prove the identity. The best I could do was to verify it for a given value of the ##a's, m, n##. I am not even sure what this identity is called but I will take the...
  26. T

    B Question on basic linear algebra (new to the subject)

    It would be nice if someone could find the history of why we use the letters i and j or m and n for the basics when working with Matrices ( A = [aij]mxn ). I tried looking up the information and I was not successful. I understand what they represent in the context of the matter, but not why they...
  27. RChristenk

    Show that the ratio ##x+y:x-y## is increased by subtracting ##y##

    ##x+y:x-y=\dfrac{x+y}{x-y} \tag1## Subtract ##y## from each term: ##x:x-2y=\dfrac{x}{x-2y} \tag2## Assume ##k=\dfrac{x}{y} \Rightarrow x=ky## ##(1)= \dfrac{ky+y}{ky-y}, (2)= \dfrac{ky}{ky-2y}## Subtract ##(1)## from ##(2)## since we are told by the problem statement ##(2)## is bigger...
  28. question_asker

    I Tracing parabolic motion with only current velocity and position?

    Is it possible to trace the trajectory of an object using only its velocity and position, both of which are given as components. My method of doing so involves using the time until max height is reached, and using that time value to calculate the max height itself (h,k), then plugging in the...
  29. azizlwl

    I'm getting the wrong answer for the Indefinite Integral of: (x^2+2x)/(x+1)^2

    ((x+1)^2 -1)/(x+1)^2 dx 1-1/(x+1)^2 dx Let u=x+1 1-1/u^2 du u+1/u +c (u^2+1)/u +c Not as answer given in the book.
  30. F

    Insights Introduction to the World of Algebras

    Continue reading...
  31. chwala

    Solve the given problem that involves binomial theorem

    part (a) ##(4+3x)^{1.5} = 2^3+ 9x+ \left[\dfrac {1}{2} ⋅ \dfrac {3}{2} ⋅\dfrac {1}{2}⋅\dfrac {1}{2}⋅9x^2\right]+ ...## ##(4+3x)^{1.5}=8+9x+\dfrac {27}{16} x^2+...##part (b) ##x≠-\dfrac {4}{3}##part (c) ##(8+9x+\dfrac {27}{16} x^2+...)(1+ax)^2 = \dfrac{107}{16} x^2## ... ##8a^2+18a+\dfrac...
  32. M

    How Did the Author Derive the Perfect Square from the Algebraic Equation?

    Hello, While following problem solution found this $$ 4a^4 + 8 a^3 + 8 a^2 + 4a + 1 = ( 2a (a+1) + 1 )^2 $$ Trying to figure out how did author do it but failed. Anyone?
  33. Charles Link

    So many people are unable to do simple algebra

    I find it interesting that so many know how to use all kinds of apps on their cell phones, but so few are able to do simple algebra any more. If you ask around, engineers not included, I think you would find very few people e.g. to be able to find the axis of symmetry of the parabola ##...
  34. A

    A Extending reals with logarithm of zero

    What do you guys have to say about this Mathoverflow post? Do you have any interesting ideas about this? https://mathoverflow.net/questions/432396/extending-reals-with-logarithm-of-zero-properties-and-reference-request
  35. S

    I A wonderful flow chart for taxonomy of matrices

    https://upload.wikimedia.org/wikipedia/commons/d/d1/Taxonomy_of_Complex_Matrices.svg
  36. H

    A About universal enveloping algebra

    Please, I have a question about universal enveloping algebra: Let ##U=U(\mathfrak{g})## be the quotient of the free associative algebra ##\mathcal{F}## with generators ##\left\{a_i: i \in I\right\}## by the ideal ##\mathcal{I}## generated by all elements of the form ##a_i a_j-a_j a_i-\sum_{k \in...
  37. H

    A How do we prove that a nonzero nilpotent Lie algebra has a nontrivial center?

    Please, in the book of Introduction to Lie Algebras and Representation Theory J. E. Humphreys p.12, I have a question: Proposition. (3.2). Let ##L## be a Lie algebra. (c) If ##L## is nilpotent and nonzero, then ##Z(L) \neq 0##. how we prove this, Thanks in advance,
  38. H

    A Questions about solvable Lie algebras

    Please, in the book of Introduction to Lie Algebras and Representation Theory J. E. Humphreys p.11, I have a question: Proposition. Let ##L## be a Lie algebra. (a) If ##L## is solvable, then so are all subalgebras and homomorphic images of ##L##. (b) If ##I## is a solvable ideal of ##L## such...
  39. H

    A What is the definition of quotient Lie algebra?

    Please, in the definition of quotient Lie algebra If ##I## is an ideal of ##\mathfrak{g}##, then the vector space ##\mathfrak{g} / I## with the bracket defined by: $$[x+I, y+I]=[x, y]+I, for all x, y \in \mathfrak{g}$$, is a Lie algebra called the quotient Lie algebra of ##\mathfrak{g}## by...
  40. H

    I About derivations of lie algebra

    Please, I am looking for a simple example of derivation on ##sl_2##, if possible, I try to use identity map, but not work with me, A derivation of the Lie algebra ##\mathfrak{g}## is a linear map ##\delta: \mathfrak{g} \rightarrow \mathfrak{g}## such that ##\delta([x, y])=[\delta(x), y]+[x...
  41. BvU

    How Did 1886 Dutch Students Solve This Complex Math Problem?

    Book answer is ##\qquad a≥0\qquad x={ 7\over 9}\;a\ \ \lor\ \ x={ 13\over 14}\;a\ \ ##but I fail to see how to get there ! Stunned by an 1886 dutch high school exam exercise. Hats off for the 17 year olds that did it ! ##\ ##
  42. H

    A Understanding the Second Direction in Semi Simple Lie Algebra

    Please, I need some clarifications about second direction, in the file attached, $$ \text { Then ad } x \text { ad } y \text { maps } L \rightarrow L \rightarrow I \text {, and }(\text { ad } x \text { ad } y)^2 \text { maps } L \text { into }[I I]=0 \text {. } $$Thank you in advance,
  43. K

    A The Exceptional Jordan Algebra in physics

    I found 3 papers on The Exceptional Jordan Algebra in physics arXiv:2305.00668 (hep-ph) [Submitted on 1 May 2023] CKM matrix parameters from an algebra Aditya Ankur Patel, Tejinder P. Singh Download PDF We report a theoretical derivation of the Cabibbo-Kobayashi-Maskawa (CKM) matrix...
  44. F

    Intro to Linear Algebra - Nullspace of Rank 1 Matrix

    The published solutions indicate that the nullspace is a plane in R^n. Why isn't the nullspace an n-1 dimensional space within R^n? For example, if I understand things correctly, the 1x2 matrix [1 2] would have a nullspace represented by any linear combination of the vector (-2,1), which...
  45. graviton_10

    I Showing that operators follow SU(2) algebra

    For two quantum oscillators, I have raising and lowering operators and , and the number operator . I need to check if operators below follow commutation relations. Now as far as I know, SU(2) algebra commutation relation is [T_1, T_2] = i ε^ijk T_3. So, should I just get T_1 and T_2 in...
  46. H

    About semidirect product of Lie algebra

    Homework Statement: About semidirect product of Lie algebra Relevant Equations: ##\mathfrak{s l}_2=## ##\mathbb{K} F \oplus \mathbb{K} H \oplus \mathbb{K} E## Hi, Please, I have a question about the module of special lie algebra: Let ##\mathbb{K}## be a field. Let the Lie algebra...
  47. H

    How to compute the Casimir element of Lie algebra sl(2)?

    Homework Statement: please, could you help me to know hoe I compute the Casimir element of lie algebra sl(2), I know the basis and their relations, but i could not find the book explain in details how we get the Casimir element.. I think it is related to killing form, but also I could not find...
  48. M

    Algebra Algebra 2 textbook recommendations please

    I am currently learning some maths from “Precalculus by James Stewart”. I was wondering if that’s ok? Is it ok to just dive straight into it or go back and brush up my algebra 2 ? I was wondering what are some good textbooks on algebra 2 by the way? Thank you. (This is all for the love of physics).
  49. bella987

    Deriving the commutation relations of the Lie algebra of Lorentz group

    This is the defining generator of the Lorentz group which is then divided into subgroups for rotations and boosts And I then want to find the commutation relation [J_m, J_n] (and [J_m, K_n] ). I'm following this derivation, but am having a hard time to understand all the steps: especially...
  50. V9999

    I A doubt about the multiplicity of polynomials in two variables

    Let ##P(x,y)## be a multivariable polynomial equation given by $$P(x,y)=52+50x^{2}-20x(1+12y)+8y(31+61y)+(1+2y)(-120+124+488y)=0,$$ which is zero at ##q=\left(-1, -\frac{1}{2}\right)##. That is to say, $$ P(q)=P\left(-1, -\frac{1}{2}\right)=0.$$ My doubts relie on the multiplicity of this point...
Back
Top