What is Algebra: Definition and 999 Discussions

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. mathgenie

    A 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.
  2. 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...
  3. 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...
  4. 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...
  5. 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...
  6. fresh_42

    Insights Why Division by Zero is a Bad Idea

    Continue reading...
  7. 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...
  8. 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##...
  9. 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...
  10. 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...
  11. 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...
  12. 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...
  13. 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.
  14. 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...
  15. M

    Stumped by the algebraic steps in this 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?
  16. 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 ##...
  17. 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
  18. S

    I A wonderful flow chart for taxonomy of matrices

    https://upload.wikimedia.org/wikipedia/commons/d/d1/Taxonomy_of_Complex_Matrices.svg
  19. 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...
  20. 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,
  21. 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...
  22. H

    A About 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...
  23. 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...
  24. BvU

    Baffled by old school exam

    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 ! ##\ ##
  25. H

    A Understanding the Second Direction in Semi Simple Lie Algebra: A Guide

    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,
  26. 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...
  27. 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...
  28. 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...
  29. 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...
  30. 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...
  31. 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).
  32. 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...
  33. 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...
  34. bhobba

    Unlock the Power of Calculus: Algebra 1 to Boaz for Students

    Here is an interesting book a student could do after after Algebra 1, or even integrate into an Algebra 1 course: https://www.amazon.com/dp/B077VV95N3/?tag=pfamazon01-20 And a website: https://www.calculussolution.com/ Several topics become easier, such as logarithms, when you know a...
  35. A

    I About writing a unitary matrix in another way

    It is easy to see that a matrix of the given form is actually an unitary matrix i,e, satisfying AA^*=I with determinant 1. But, how to see that an unitary matrix can be represented in the given way?
  36. VX10

    I A question about Young's inequality and complex numbers

    Let ##\Omega## here be ##\Omega=\sqrt{-u}##, in which it is not difficult to realize that ##\Omega ## is real if ##u<0##; imaginary, if ##u>0##. Now, suppose further that ##u=(a-b)^2## with ##a<0## and ##b>0## real numbers. Bearing this in mind, I want to demonstrate that ##\Omega## is real. To...
  37. S

    I Intuition for why linear algebra is needed in quantum physics

    I'm watching a nice video that tries to explain how linear algebra enters the picture in quantum physics. A quick summary: Classical physics requires that physical quantities are single-valued and vary smoothly as they evolve in time. So a natural way to model classical physical quantities is...
  38. James1238765

    I How to do algebra on the Kitaev toric code grid?

    The toric code is a basic computational model as follows: There are 2 operations that can be performed, A and B, on this grid. To compute the value A at each point on the grid, we transform the raw values at each dot (located in between two vertices) according to some predefined operators...
  39. C

    Do We Need Boundaries for Fraction Equations?

    When working with fractions and when we have a fraction or equation with fractions like this one for example ##\frac{x}{x-1}+\frac{x}{x+1}=\frac{9}{4}## do we always need to set boundaries? Like, do we always need to write that x can't be a number that would give the denominator 0? In this...
  40. D

    I The Price of Beer - Linear Algebra Problem

    I came across the following problem somewhere on the web. The original site is long gone. The problem has me stumped. May be sopmeone can provide some insight. (The problem seems too simple to post in the "Linear/Abstract Algebra" forum.) The Cost of Beer It was nearing Easter, and a group...
  41. P

    I Why is the dual of Z^n again Z^n ?

    Hello, how can one proof that the dual of ##\mathbb{Z}^n## is ##\mathbb{Z}^n##? My idea: The definition of a dual lattice says, that it is as set of all lattice vectors ##x \in span(\Lambda)## such that ##\langle x , y \rangle## is an integer. When we now consider ##\mathbb{Z}^n## we see that...
  42. P

    Linear algebra problem with a probable typo

    Well, my guess is that there is something wrong with the factors chosen, because ##\left\Vert \left(0,1,0\right)\right\Vert =1## and \begin{align} \left\Vert F\left(0,1,0\right)\right\Vert &=\left\Vert...
  43. P

    Book recommendations: Abstract Algebra for self-study

    Hello, I am looking for one or more books in combination for self-study of abstract algebra. Desirable would be a good structure of the book with good examples of sentences and definitions. Of course, exercise problems should not be missing. I am now almost tending to buy the Algebra 0 book by...
  44. S

    Is "College Algebra" really just high school "Algebra II"?

    I had learned everything in College Algebra in my Algebra II course in high school, and indeed (at least at my alma mater) in engineering, physics or math, no credit is even given for College Algebra. Perhaps what is going on here is that colleges can't trust that someone who has passed (even...
  45. mcastillo356

    I Understanding Theorem 13 from Calculus 7th ed, R. Adams, C. Essex, 4.10

    The following properties of big-O notation follow from the definition: (i) if ##f(x)=O(u(x))## as ##x\rightarrow{a}##, then ##Cf(x)=O(u(x))## as ##x\rightarrow{a}## for any value of the constant ##C##. (ii) If ##f(x)=O(u(x))## as ##x\rightarrow{a}## and ##g(x)=O(u(x))## as ##x\rightarrow{a}##...
  46. adamaero

    I Algebra equation with variable as exponent

    This equation takes a present value (PV) to find mortgage payments, PMT: Alternatively, switching V for PV and T for PMT: V/T = r(1-r^n)/(1-r) What is an algebraic method to solve for "r"? Can it not be solved for? I realize I can just find out "r" by trial by error in Excel using the PMT...
  47. murshid_islam

    What would be a good book for learning Linear Algebra by myself?

    Summary: What would be a good book for learning Linear Algebra by myself in my situation (which is explained in my post below)? I did an undergraduate Linear Algebra course about 18 years ago. The textbook we used was Howard Anton’s “Elementary Linear Algebra”. The problem is that I never...
  48. BadgerBadger92

    Intro Physics Best Physics, Algebra, and Trigonometry Textbooks (Modern)

    I am looking for good textbooks in physics, algebra, and trigonometry textbooks that are up to date and a good read. I heard that Feynman’s Lectures was really good. Is it still up to date enough? Any opinions?
  49. A

    Book recommendations (abstract algebra and number theory)

    Hi, For an engineer who graduated and finished typical Cal A,B,C + Linear Algebra + ODE, what book do you recommend to start reading to be a transition to advanced pure math subjects like abstract algebra and number theory? I did deep google search & concluded that that book supposed to include...
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