Algebra is a field of mathematics that uses representative symbols with no set values, usually in the letters x, y, z, p, or q, called variables. From elementary to college, algebra is part of the topics in the mathematics curriculum. The standardized test, which is the GRE Subject Test in Mathematics, includes algebra concepts. It shows that graduate school evaluates aspiring students’ basic knowledge in this field. An algebraic equation represents a scale where what is done on one side of the scale is likewise done on the other side, and the numbers operate as constants. Numerous mathematical representations are included in algebra. Familiarity with concepts, rules, algebraic expressions, and algebraic operations is necessary to ace the GRE Math Exam and other standardized tests.
In a real-life setting, we often observe in our surroundings and day-to-day social interactions that number values constantly change. To constantly keep track of these changing values, a way to represent them has been formulated, bringing forth the existence of Algebra. It is a branch of mathematics that involves representative symbols with no fixed values, usually in the letters x, y, z, p, or q, called variables. These symbols are substituted to find their respective values through various arithmetic operations such as addition, subtraction, multiplication, and division. (1)
According to Jacques Sesiano in "An Introduction to the History of Algebra" (AMS, 2009), this problem is based on a Babylonian clay tablet circa 1800 B.C. (VAT 8389, Museum of the Ancient Near East). Since its roots in ancient Mesopotamia, algebra has been central to many advances in science, technology, and civilization as a whole. The language of algebra has varied significantly across the history of all civilizations to inherit it (including our own).
Algebraic thinking underwent a substantial reform following the advancement by scholars of Islam's Golden Age. Until this point, the civilizations that inherited Babylonian mathematics practiced algebra in progressively elaborate "procedural methods." Sesiano further explains: A student needed to memorize a small number of [mathematical] identities, and the art of solving these problems then consisted of transforming each problem into a standard form and calculating the solution. (As an aside, scholars from ancient Greece and India did practice symbolic language to learn about number theory.)
An Indian mathematician and astronomer, Aryabhata (A.D. 476–550), wrote one of the earliest-known books on math and astronomy, called the "Aryabhatiya" by modern scholars. (Aryabhata did not title his work himself.) The work is "a small astronomical treatise written in 118 verses giving a summary of Hindu mathematics up to that time," according to the University of St. Andrews, Scotland. (1)
Abstract algebra is one of the divisions in algebra that discovers the truths relating to algebraic systems independent of the specific nature of some operations. These operations, in specific cases, have certain properties. Thus, we can conclude some consequences of such properties. As a result, this branch of mathematics is known as abstract algebra.
Abstract algebra deals with algebraic structures like fields, groups, modules, rings, lattices, vector spaces, etc. (2)
The concepts of abstract algebra are below:
- Sets: Sets is defined as the collection of objects determined by some specific property for a set. For example, a set of all the 2×2 matrices, the set of two-dimensional vectors present in the plane and different forms of finite groups.
- Binary Operations: When the concept of addition is conceptualized, it gives rise to binary operations. Without a set, the concept of all the binary operations will be meaningless without a set.
- Identity Element: The numbers 0 and 1 are conceptualized to give the idea of an identity element for a specific operation. Here, 0 is called the identity element for the addition operation, whereas 1 is called the identity element for the multiplication operation.
- Inverse Elements: The idea of inverse elements comes up with a negative number. For addition, we write "a" as the inverse of "a," and for multiplication, we write "a-1′′ as the inverse form of "a."
- Associativity: When integers are added, there is a property known as associativity in which the grouping of the numbers added does not affect the sum. Consider the example, (3 + 2) + 4 = 3 + (2 + 4)
The following are the various branches of algebra in which it can be classified:
- Pre-algebra. Involved in this are the basics of algebra, which is to practically formulate a mathematical expression of a given problem with the unknown values represented through variables.
- Elementary Algebra. This branch is concerned with the solutions of the algebraic expression in order to come up with an answer. The equations represent simple variables such as x and y and are broken down into various degrees, namely linear equations in the form of ax + b = c, ax + by + c = 0, quadratic equations with ax2 + bx + c = 0, and polynomial equations expressed as axn + bxn-1+ cxn-2+....k = 0.
- Abstract Algebra. Unlike the other branches of algebra that deal with number systems and algebraic expressions, abstract algebra leans more towards learning about concepts that involve groups, rings, and vectors and how these concepts can be applied and used to represent quantities in other hard-science fields like computer sciences, physics, and astronomy, among many others, that are observable in a real-life setting.
- Universal Algebra. Trigonometry, calculus, and coordinate geometry, among many other mathematical forms, are all classified under the umbrella term "universal algebra." This is because the concentration of this branch of algebra is the study of mathematical expressions, excluding the concept of studying algebra models.
Each of the aforementioned branches of algebra exists to serve their respective purposes, with varying degrees of difficulty depending on the use: representing variables through equations, applying algebraic concepts to understand other disciplines, or scrutinizing how mathematical expressions are formulated. (2)
Algebra consists nine basic rules and properties:
- Commutative Property of Addition: a + b = b + a
- Commutative Property of Multiplication: a × b = b × a
- Associative Property of Addition: a + (b + c) = (a + b) + c
- Associative Property of Multiplication: a × (b × c) = (a × b) × c
- Distributive Property: a × (b + c) = a × b + b × c or a × (b - c) = a × b - a × c
- Reciprocal: Reciprocal of a = 1/a
- Additive Identity: a + 0 = 0 + a = a
- Multiplicative Identity: a × 1 = 1 × a = a
- Additive Inverse: a + (-a) = 0
These rules and properties serve as a blueprint for the equations that test takers are expected to solve. Familiarizing the steps involved in the operations is a crucial skill towards mastering algebra. (2)
The algebraic operations involved in the solving process are the following:
- Addition: the value of two or more variables is being combined. The plus (+) sign separates the addends in the equation.
- Subtraction: the value of the minuend is diminished by the value of the subtrahend. The minus (-) sign separates the expressions.
- Multiplication: the value of a particular expression is increased a number of times as indicated by the multiplier. In between two or more expressions is the multiply (x) sign.
- Division: This operation distributes the value of the expressions into equal parts. The sign (/) is placed in between the expressions.
These operations are fundamental to the mathematical solving process. To simplify algebraic expressions, test takers have to use the aforementioned list. (2)
The concept of algebra can be applied to different algebraic equations depending on the degree of the variable. These mathematical statements can be expressed in the following types:
- Linear equations: The relationship between variables is represented with linear equations that utilize basic operations such as addition and subtraction. These are expressed in exponents of one degree.
- Quadratic equations: The standard form of the quadratic equation ax2 + bx + c = 0 consists of two solutions at most. x is the variable while a, b, and c are the constants.
- Cubic equations: This equation is frequently applied in calculus and three-dimensional geometry. It has a cubic variable and can be expressed in the generalized form ax3 + bx2 + cx + d = 0.
To better recall the differences between the equations, remember that the exponents of linear equations are expressed to one degree, the exponents of quadratic equations are powered to the second degree, and the exponents of cubic equations reach the third degree of power. (2)
These four stages were recognized in the development of algebra:
- Geometric Stage. The Babylonians were recognized as the pioneers of algebra. At this stage, geometric concepts were predominant. This knowledge continued with the Greeks and was revived by Omar Khayyam later on.
- Static equation-solving stage. When generalized algorithmic processes for solving algebraic equations were introduced by Al-Khwarizmi, algebra gradually transitioned to the static equation-solving stage wherein numbers were sought to satisfy numeric relationships. This transition dates back to Diophantus and Brahmagupta.
- Dynamic function stage. Gottfried Leibniz motivated the transition to the dynamic function stage, wherein the idea of motion and function emerged in the mathematical discussions. Sharaf al-Din al-Tusi pioneered this knowledge.
- Abstract stage. Algebra becomes more structured and organized in the solving process.
Mathematical knowledge emerged when people drew geometric forms to visualize their calculations. As scholars developed it to become the numerical expression we know today, concepts such as recognizing mathematical relationships and including the idea of motion and function were introduced. Mathematical structures became more important as a product of the developments that occurred in the 19th and 20th centuries. (3)
When a set of numbers shares a mathematical relationship with other numbers, it is called a sequence. When the terms of a sequence are added, the result is called a series. Series can be represented in two ways: Arithmetic Progression and Geometric Progression. An arithmetic progression has a constant difference between two consecutive terms. On the other hand, a geometric progression has a fixed ratio of adjacent terms.
Exponents, which are expressed as an, are used to simplify algebraic expressions. This mathematical operation includes squares, cubes, square roots, and cube roots. When exponents are inverted, the operation is called a logarithm. As an integral part of modern mathematics, logarithms simplify larger algebraic expressions conveniently.
As proven through polynomial division, a complex root can be extracted from every non-constant single-variable polynomial with complex coefficients. It is also known as the d’Alembert-Gauss Theorem.
Algebra can be tricky if a test taker will not take their time to patiently study its processes. Hence, here are some tips that can be used to increase the mastery of this knowledge:
- Do not forget your arithmetic lessons. If you want to become good at solving algebraic expressions, you must understand how it works. Being familiar with the fundamental operations that were taught to you earlier is a helpful skill in achieving your desired level of mastery.
- PEMDAS is the key. The flow of mathematical structures is systematic and organized. It has a pattern that you can follow to solve a problem successfully. PEMDAS is an acronym that describes the order of the operations. It stands for parentheses, exponents, multiplication, division, addition, and subtraction.
- Don’t let negative numbers intimidate you. To overcome your fear of making mistakes in solving problems involving negative numbers, you must first cure your ignorance. To achieve this, keep performing basic operations with two negative numbers or with a positive and a negative number.
- Don’t miss any steps. When solving a math problem, show your solution as much as possible so you can trace back when you make mistakes and make justifications for the result. Strategies that do not precisely follow the innate process are welcome. However, it’s still better to systematically follow the steps involved.
- Be comfortable with letters. Letters represent the unknown. These variables can be substituted with integers when their values are provided. By carefully performing the operations in the solving process, one variable at a time can be found.
- Know your formulas. Formulas serve as a tour guide in your journey toward finding the solution. Make sure you are familiar with the variables and patterns involved in their process and make sure you understand how the steps are performed. As much as they recommend memorizing formulas, we cannot expect everyone to remember them. Therefore, know the formulas by heart and find your way out of the problem.
- Understand what is being asked. When facing a mathematical problem, it is important to know what the variables mean and what should be done to find what the problem is looking for. Reread the statement, identify the problem, and understand how it works so you’ll know the appropriate formula to use.
These strategies are just some of the tips you can follow to increase your skills in mathematical solving. Whichever method you choose to employ is valid as long as it effectively achieves your goal. (4)