The document discusses the topics of algebra and its history. It defines algebra as the study of mathematical symbols and the rules for manipulating symbols. Algebra originated from Arabic and Persian mathematicians in the 9th century who developed methods for solving equations. The basics of algebra taught in elementary courses involve using variables, expressions, and terms. More advanced areas of algebra study abstract algebraic structures and their properties.

1. English For Math
Suryakancana University, Third Lecture

2. Algebra and its characteristics
• Algebra (from Arabic and Farsi "al-jabr" meaning "reunion of broken
parts") is one of the broad parts 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.
• 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 Middle
English from either Spanish, Italian, or Medieval Latin. Algebra originally
referred 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 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.

3. History
The roots of algebra can be traced to the ancient Babylonians, who
developed an advanced arithmetical system with which they were able
to do calculations in an algorithmic fashion. The Babylonians developed
formulas to calculate solutions for problems typically solved today by
using linear equations, quadratic equations, and indeterminate linear
equations.
• This is a page from Al-Khwārizmī's al-
Kitāb al-muḫtaṣar fī ḥisāb al-ğabr wa-
l-muqābala
• Another Persian mathematician Omar
Khayyam is credited with identifying
the foundations of algebraic geometry
and found the general geometric
solution of the cubic equation. Yet
another Persian mathematician,
Sharaf al-Dīn al-Tūsī, found algebraic
and numerical solutions to various
cases of cubic equations.

4. Areas of Mathematics with The Word
Algebra in Their Name
Some areas of mathematics that fall under the classification abstract algebra have the word
algebra in their name; linear algebra is one example. Others do not: group theory, ring
theory, and field theory are examples. In this section, we list some areas of mathematics
with 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.

5. Basic Concepts of Algebra
• In algebra, letters are used as variables. A variable can assume
values of numbers. Numbers are called constants.
• Math Note: In some cases, a letter may represent a specific
constant the Greek letter pi (𝜋) represents a constant.
• In mathematics, an expression (or mathematical expression) is a
finite combination of symbols that is well-formed according to rules
that depend on the context.
• An algebraic expression consists of variables, constants, operation
signs, and grouping symbols. In the algebraic expression 3𝑥, the ‘‘3’’
is a constant and the ‘‘𝑥’’ is a variable. When no sign is written
between a number and a variable or between two or more
variables, it means multiplication. Hence the expression ‘‘3𝑥’’
means ‘‘3 times 𝑥 or to multiply 3 by the value of 𝑥. The expression
𝑎𝑏𝑐 means a times b times c or 𝑎 × 𝑏 × 𝑐.

6. Basic Concepts of Algebra
• The number before the variable is called the numerical
coefficient. In the algebraic expression ‘‘3𝑥’’, the 3 is the
numerical coefficient. When the numerical coefficient is 1, it is
usually 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 term
is a number or variable, or a product or a quotient of numbers
and variables. Terms are connected by + or − signs. For
example, the expression 3𝑥 + 2𝑦 − 6 has 3 terms. The
expression 8𝑝 + 2𝑞 has two terms, and the expression 6𝑥2
𝑦
consists of one term.

7. Elementary algebra
Elementary algebra is the most basic form of algebra. It is taught to students who
are presumed to have no knowledge of mathematics beyond the basic principles
of arithmetic. In arithmetic, only numbers and their arithmetical operations (such
as +, −, ×, ÷) occur. In algebra, numbers are often represented by symbols called
variables (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 the
properties of the real number system.
• It allows the reference to "unknown" numbers, the formulation of equations
and the study of how to solve these. (For instance, "Find a number x such that
3𝑥 + 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 specific
numbers that allows us to solve it, but that of the operations involved.)
• It allows the formulation of functional relationships. (For instance, "If you sell x
tickets, then your profit will be 3𝑥 − 10 dollars, or 𝑓(𝑥) = 3𝑥 − 10, where f is
the function, and x is the number to which the function is applied".)

8. Elementary algebra
Algebraic expression notation:
1 – power (exponent)
2 – coefficient
3 – term
4 – operator
5 – constant term
x y c – variables/constants

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