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. F

    MHB Prove that the algebra generated is dense

    Hello I have problems with this exercise Prove that the algebra generated by the set $S = \{ 1,x^2 \}$ is dense in $C [0, 1]$. It is $S$ dense in $C [-1; 1]$ I am thinking to apply Stone-weierstrass theorem but I don't know how to use it properly. Thanks
  2. J

    Foundations Klein's Encyclopedia: Is an English Translation Possible?

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  3. F

    Linear Algebra What are good books for a third course in Linear Algebra?

    What are the suitable books in linear algebra for third course for self-study after reading Linear Algebra done right by Axler and Algebra by Artin?
  4. S

    Linear Algebra uniqueness of solution

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  5. F

    Linear Algebra What are good second course books in linear algebra for self-study?

    What are best second course(undergraduate) books in linear algebra for self-study?I have already read Introduction to Linear Algebra by Lang.
  6. D

    I Commutative algebra and differential geometry

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  7. fresh_42

    A Infinite-Dimensional Lie Algebra

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  8. PeroK

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  9. C

    Proving Poincare Algebra Using Differential Expression of Generator

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  10. appletree23

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  11. Mr Davis 97

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  12. DartomicTech

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  13. A

    Linear algebra projections commutativity

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  14. SebastianRM

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  15. P

    I don't understand the algebra in this answer

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  16. Eclair_de_XII

    Am I using quotient spaces correctly in this linear algebra proof?

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  17. J

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  18. Santiago24

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  19. sahilmm15

    B A problem involving direction cosines (Vector Algebra)

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  20. F

    Deriving Casimir operator from the Lie Algebra of the Lorentz Group

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  21. LCSphysicist

    I Poincaré algebra and quotient group

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  22. V

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  23. L

    In algebra what does x represent?

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  24. M

    Help me with this Algebra problem please (quotient of complex numbers)

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  25. K

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  26. A

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  27. M

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  28. J

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  31. M

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  32. B

    A Geometry of matrix Dirac algebra

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  33. E

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  34. K

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  35. K

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  36. L

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  37. T

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  38. MexChemE

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  39. U

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  40. Demandish

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  41. anemone

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  43. S

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  44. pellis

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  45. Math Amateur

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  46. Math Amateur

    I Smallest Sigma Algebra .... Axler, Example 2.28 ....

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  47. T

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  48. hackedagainanda

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  49. X

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  50. hackedagainanda

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