English For Math

Suryakancana University, Third Lecture

Algebra and its characteristics

• Algebra (from Arabic and Farsi "al-jabr" meaning "reunion of brokenparts") is one of the broad parts of mathematics, together with numbertheory, geometry and analysis. In its most general form, algebra is thestudy of mathematical symbols and the rules for manipulating thesesymbols.

• The word algebra comes from the Arabic language ( الجبر al-jabr"restoration") from the title of the book Ilm al-jabr wa'l-muḳābala by al-Khwarizmi. The word entered the English language during Late MiddleEnglish from either Spanish, Italian, or Medieval Latin. Algebra originallyreferred to a surgical procedure, and still is used in that sense in Spanish,while the mathematical meaning was a later development.

• it includes everything from elementary equation solving to the study ofabstractions such as groups, rings, and fields. The more basic parts ofalgebra are called elementary algebra, the more abstract parts are calledabstract algebra or modern algebra.

History

The roots of algebra can be traced to the ancient Babylonians, whodeveloped an advanced arithmetical system with which they were ableto do calculations in an algorithmic fashion. The Babylonians developedformulas to calculate solutions for problems typically solved today byusing linear equations, quadratic equations, and indeterminate linearequations.

• This is a page from Al-Khwārizmī's al-Kitāb al-muḫtaṣar fī ḥisāb al-ğabr wa-l-muqābala

• Another Persian mathematician OmarKhayyam is credited with identifyingthe foundations of algebraic geometryand found the general geometricsolution of the cubic equation. Yetanother Persian mathematician,Sharaf al-Dīn al-Tūsī, found algebraicand numerical solutions to variouscases of cubic equations.

Areas of Mathematics with The Word Algebra in Their Name

Some areas of mathematics that fall under the classification abstract algebra have the wordalgebra in their name; linear algebra is one example. Others do not: group theory, ringtheory, and field theory are examples. In this section, we list some areas of mathematicswith the word "algebra" in the name.• Elementary algebra, the part of algebra that is usually taught in elementary courses of

mathematics.• Abstract algebra, in which algebraic structures such as groups, rings and fields are

axiomatically defined and investigated.• Linear algebra, in which the specific properties of linear equations, vector spaces and

matrices are studied.• Commutative algebra, the study of commutative rings.• Computer algebra, the implementation of algebraic methods as algorithms and computer

programs.• Algebraic number theory, in which the properties of numbers are studied from an

algebraic point of view.• Algebraic geometry, a branch of geometry, in its primitive form specifying curves and

surfaces as solutions of polynomial equations.• Algebraic combinatorics, in which algebraic methods are used to study combinatorial

questions.

Basic Concepts of Algebra

• In algebra, letters are used as variables. A variable can assumevalues of numbers. Numbers are called constants.

• Math Note: In some cases, a letter may represent a specificconstant the Greek letter pi (𝜋) represents a constant.

• In mathematics, an expression (or mathematical expression) is afinite combination of symbols that is well-formed according to rulesthat depend on the context.

• An algebraic expression consists of variables, constants, operationsigns, and grouping symbols. In the algebraic expression 3𝑥, the ‘‘3’’is a constant and the ‘‘𝑥’’ is a variable. When no sign is writtenbetween a number and a variable or between two or morevariables, it means multiplication. Hence the expression ‘‘3𝑥 ’’means ‘‘3 times 𝑥 or to multiply 3 by the value of 𝑥. The expression𝑎𝑏𝑐 means a times b times c or 𝑎 × 𝑏 × 𝑐.

Basic Concepts of Algebra

• The number before the variable is called the numericalcoefficient. In the algebraic expression ‘‘3𝑥’’, the 3 is thenumerical coefficient. When the numerical coefficient is 1, it isusually not written and vice versa. Hence, 𝑥𝑦 means 1𝑥𝑦.Likewise, 1𝑥𝑦 is usually written as 𝑥𝑦. Also −𝑥𝑦 means −1𝑥𝑦.

• An algebraic expression consists of one or more terms. A termis a number or variable, or a product or a quotient of numbersand variables. Terms are connected by + or − signs. Forexample, the expression 3𝑥 + 2𝑦 − 6 has 3 terms. Theexpression 8𝑝 + 2𝑞 has two terms, and the expression 6𝑥2𝑦consists of one term.

Elementary algebra

Elementary algebra is the most basic form of algebra. It is taught to students whoare presumed to have no knowledge of mathematics beyond the basic principlesof arithmetic. In arithmetic, only numbers and their arithmetical operations (suchas +, −, ×, ÷) occur. In algebra, numbers are often represented by symbols calledvariables (such as a, n, x, y or z). This is useful because:• It allows the general formulation of arithmetical laws (such as 𝑎 + 𝑏 = 𝑏

+ 𝑎 for all a and b), and thus is the first step to a systematic exploration of theproperties of the real number system.

• It allows the reference to "unknown" numbers, the formulation of equationsand the study of how to solve these. (For instance, "Find a number x such that3𝑥 + 1 = 10" or going a bit further "Find a number x such that 𝑎𝑥 + 𝑏 = 𝑐".This step leads to the conclusion that it is not the nature of the specificnumbers that allows us to solve it, but that of the operations involved.)

• It allows the formulation of functional relationships. (For instance, "If you sell xtickets, then your profit will be 3𝑥 − 10 dollars, or 𝑓(𝑥) = 3𝑥 − 10, where f isthe function, and x is the number to which the function is applied".)

Elementary algebra

Algebraic expression notation:1 – power (exponent)2 – coefficient3 – term4 – operator5 – constant termx y c – variables/constants

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