# 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. ### 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...
2. ### 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...
3. ### 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}...

39. ### 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...
40. ### 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?
41. ### 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...
42. ### 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...
43. ### 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...
44. ### 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...
45. ### 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...
46. ### 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...
47. ### Linear algebra problem with a probable typo

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48. ### 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...
49. ### 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...
50. ### 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}##...