algebra

Algebra (from Arabic: الجبر‎ al-jabr, meaning "reunion of broken parts") is one of the broad areas of mathematics, together with number theory [数论], geometry [几何] and analysis [分析]. In its most general form, algebra is the study of mathematical symbols and the rules for manipulating these symbols. 1859年我国数学家李善兰 coined the word "代数"，代数者，代替数也。

It includes everything from elementary equation solving to the study of abstractions such as groups, rings, and fields. 伽罗瓦与阿贝尔并称为现代群(group)论的创始人。zdic.net: 伽qié jiā gā. 某输入法只认jialuowa，不对。百度翻译可以听发音。

In elementary [初等] algebra, the quadratic formula [求根公式] is a formula that provides the solution(s) [解] to a quadratic equation [一元二次方程]. In mathematics, quadratic means "involving the second and no higher power [幂mì] of an unknown quantity or variable [变量]". But the prefix quad- usually describes something that has to do with four, such as quad-core processor and quadrilateral [四边形]. Why does "quadratic" describe second power when "quad" means "four"?

Because the second power is a square [平方], and a square [正方形] has four sides. In fact the word square itself comes from quad-, too. 正方形的面积是边长的平方。几何的历史比代数悠久。

The more abstract [抽象的] 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 [算术] in the use of abstractions, such as using letters to stand for numbers that are either unknown or allowed to take on many values. A mathematician who does research in algebra is called an algebraist.

Linear algebra is the branch of mathematics concerning linear equations and their representations in vector spaces [向量空间] and through matrices [矩阵]. In three-dimensional Euclidean [欧几里得] space, these three planes [平面] represent solutions [解] of linear equations [线性方程式], and their intersection [交集] represents the set of common solutions: in this case, a unique point. The blue line is the common solution to two of these equations.

Linear algebra is central to almost all areas of mathematics. Linear algebra is also used in most sciences and fields of engineering, because it allows modeling many natural phenomena [自然现象], and computing efficiently with such models. For nonlinear systems, which cannot be modeled with linear algebra, it is often used for dealing with first-order approximations [一阶近似], using the fact that the differential of a multivariate function at a point is the linear map that best approximates the function near that point.

Jane overheard [不经意地听到] that her friend-of-a-friend Marcus got a deep learning [深度学习] research job and is getting paid over $400,000 a year in cash and stock [股票加现金40万美元/年, 2019年数据]. Jane did some research and found a book "Deep Learning" . The introduction read much like the technical blog posts [博客帖子] she was used to, and got her even more excited about learning the topic. But as she kept reading, the content got harder. The first chapter covered the required math concepts and introduced a lot of terminology [术语] and notation [符号] that Jane had never seen. She skimmed [浮光掠影] it and tried to get on to the meat of the book, but it got harder and harder. Jane decided she needed to pause her study of AI until she learned some math. The math chapter of Deep Learning listed a reference [参考文献] on linear algebra for students who had never seen the topic before.

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