What is Algebra?

Algebra is a branch of mathematics that is concerned with arithmetic operations using symbols. Algebra helps in the representation of problems or situations as mathematical expressions. It involves variables like x, y, z, and mathematical operations like addition, subtraction, multiplication, and division to form a meaningful mathematical expression.

Reference: https://www.splashlearn.com/math-vocabulary/algebra/algebra

Algebra deals with symbols and these symbols are related to each other with the help of operators. Letters are used to represent unknown values or values that can change. Mathematicians use algebra to solve problems when they don’t know the exact number(s). This branch of mathematics can be used in business when predicting sales, growth, and profit.

The below algebraic expressions are made up of variables, operators, and constants. Here the number 8 is a constant, x and y are variables, and the arithmetic operation of addition is performed.

Reference: https://www.blendspace.com/lessons/gknkwrDvEGrVYw/gr-8-t3-ch1-l4-distributive-property

When it comes to algebra, the complexity of concepts has no limits. The level of education also makes room for adding new concepts and ideas.

Some facts about Algebra

  • Algebra is derived from the Arabic word Al-jabr, which means the reunion of broken parts.
  • Until the 16th century, there were only two subfields of mathematics: geometry and arithmetic.
  • Fields, rings, and groups of abstractions are all studied as part of algebraic studies.
  • Engineering, science, mathematics, economics, and medicine all use elementary algebra.
  • Professional mathematicians are interested in studying abstract algebra as the subject is part of advanced mathematics.
  • Mathematicians who study algebra are known as Algebraists.

Use of Algebra

It is common for us to use algebra in formulas when we are uncertain of one or more numbers or when one or more numbers may change.

Mathematical formulas

For example, the area of a rectangle is calculated by multiplying the base by the height (b is the base, and h is the height). If the base is 5 cm and the height is 8 cm, then the area is 40 cm².

Another example would be a rectangle with a base of 7 cm and a height of 8 cm. The area for this would be 56 cm².

Considering that the value for the base and height can be changed, we can represent the formula using algebra.

What role does algebra play in business?

The use of algebra can help predict sales and how much a customer will spend. Using algebra, you can estimate the lifetime value of a customer, which computes their annual spending percentage.

It is calculated as y = 12 + 8x

In this case, we can use algebra to determine the spend, where x (the number of months the customer has shopped with a business) may change. The percentage of each year’s sales a customer will spend is represented by y.

Let us assume that a customer spends $50 in their first 2 months.

We have to include the first month as x = 0 and the second month as x = 1.

Since we now know that x = 1, we can substitute it into y = 12 + 8x.

The result is y = 12 + (8 × 1)

The value of y = 20%

If 20% = $50 then in the course of the full year, 100% will equal $250 (5 ×$50). This means we can predict a customer will spend $250 a year.

An overview of the branches of Algebra

Algebra includes everything from solving elementary equations to studying abstract concepts. We often see certain values that keep changing in our real-life problems. There is, however, a constant need to represent these changing values. By utilizing numerous algebraic expressions, many of the complex tasks involved in algebra can be simplified. The various branches of algebra, according to how they are used and the complexity of their expressions can be classified:

  • Pre-algebra
  • Elementary Algebra
  • Abstract Algebra
  • Universal Algebra

In many disciplines, including economics and medicine, a basic understanding of algebra is a prerequisite for other studies. Mathematicians study abstract algebra, a major field of advanced mathematics. A fundamental difference between algebra and arithmetic is its use of abstractions, such as using letters to stand for numbers that are either unknown or allowed to take on many values. Let us take a closer look at the various branches

Pre-algebra

By presenting the unknown values as variables, it is possible to construct mathematical expressions. Pre-algebra involves formulating a mathematical expression for a given problem statement.  It helps in the transformation of real-life problems into mathematical expressions.

Elementary Algebra

Mathematical expressions are solved with the help of elementary algebra. Simple variables like x and y are represented as equations in this branch. The equations are classified according to the degree of the variable, such as linear equations, quadratic equations, and polynomials. A linear equation is represented by the form ax + b = c, ax + by + c = 0, and ax + by + cz + d = 0. On the basis of the degree of the variables, elementary algebra branches out into polynomials and quadratic equations. Generally, quadratic equations are represented by ax2 + bx + c = 0, whereas polynomial equations are expressed by axn + bxn-1 + cxn-2 + …..k = 0.

Number operations and simple rules are involved, such as:

  •  Addition
  •  Subtraction
  •  Multiplication
  •  Division
  •  Algebraic Expressions
  •  Variables
  •  Functions
  •  Polynomials
  •  Equation solving techniques

Abstract Algebra

As opposed to simple mathematical number systems, abstract algebra uses abstract concepts like groups, rings, and vectors. By writing the addition and multiplication properties together, we can find rings, which are a simple level of abstraction. Ring theory and group theory are two of the most important abstract algebra concepts. The mathematical concept of abstract algebra finds numerous applications in computer science, physics, and astronomy, and it embraces vector spaces in its representation of quantities.

Universal Algebra

All other mathematical forms are universal algebra apart from trigonometry, calculus, coordinate geometry, which involve algebraic expressions. Universal algebra concentrates on mathematical expressions and does not study models of algebra. In a sense, all the other branches of algebra can be seen as subsets of universal algebra. Almost any real-life problem can be classified into one of the branches of mathematics and can be solved using abstract algebra.

Topics of Algebra

To facilitate a thorough understanding of algebra, the subject is divided into various topics. Among the topics, we will discuss in this section are algebraic expressions and equations, sequences and series, exponents, logarithms, and sets.

Algebraic Expressions

Equations in algebra are derived by an integer constant, a variable, and basic arithmetic operations, which include addition(+), subtraction(-), multiplication(×), and division(/).

Here is an example of an algebraic expression: 7y + 3. This example has fixed numbers 7 and 3, and a variable.

Furthermore, we can either use simple variables like x, y, z, or we can use complex variables like x2, x3, xn, xy, x2y etc.

Algebraic expressions are also called polynomial expressions.

Variables (also called indeterminates), coefficients, and exponents of variables are the components of polynomials.

An example would be 5x3 + 4x2 + 7x + 2 = 0.

The image given above is an example of algebraic expression. Here x is the constant whereas a, b and c are the integer constants.

A mathematical equation consists of two algebraic expressions that have equal values, linked by ‘equal to’ symbols. Below are a few examples of different types of equations with variable degrees, where we apply the algebraic concept:

  • Linear Equations: These equations describe a relationship between variables such as x, y, and z and are expressed as exponents in one degree. To solve linear equations, we need to use algebra, starting with addition and subtraction of algebraic expressions.
  • Quadratic Equations: A quadratic equation can be expressed in the standard form as ax2 + bx + c = 0, In this form, the first term, a, b, and c are constants and x is the variable. This type of equation has at most two solutions when x is the value that satisfies the equation.
  • Cubic Equations: In algebra, these are equations whose variables have power 3. The cubic equation in generalized form is written as ax3+ bx2 + cx + d = 0. Numerous applications of cubic equations can be found in calculus and three-dimensional geometry.

Sequence and Series

A set of numbers that have a relationship across the number system is called a sequence. The numbers have a common mathematical relationship, and the terms of a sequence make up a series. In mathematics, there are two broad number sequences and series, namely arithmetic progression and geometric progression. Among these series, some are finite and some are infinite. The two series can be represented as follows:

  • Arithmetic Progression: An arithmetic progression (AP) involves terms whose difference between successive terms is always a constant. An arithmetic progression series can be represented as a, a+d, a + 2d, a + 3d, a + 4d, a + 5d…
  • Geometric Progression:  Any progression where the relationship between adjacent terms remains the same is a geometric progression. A geometric sequence is generally represented as a, ar, ar2, ar3, ar4, ar5, …..
Reference:https://isequalto.com/iet-app/ask-the-world/PPmy8850-What-is-AP-and-GP-series?

Exponents

Exponents are mathematical operations written as an. In this expression, an involves two numbers, a and n. Here ‘a’ is the base and the exponent or power is ‘n’. To simplify algebraic expressions, exponents are used. The various topics of exponents involve squares, cubes, square root, and cube root. The names are derived from the powers of these exponents. An exponent is represented by the formula an = a x a x a x … n times.

Reference: http://mathsmd.com/124/algebra/exponents-and-power-examples-rules/

Logarithms

The concept of logarithms was discovered by John Napier in 1614. An algebraic logarithm is the inverse of an exponent. Logarithms are useful for simplifying large algebraic expressions. The exponential form is represented as bx = a and can be transformed and represented in logarithmic form as logba = x. Logarithms are now considered an integral part of modern mathematics.

Reference: https://d138zd1ktt9iqe.cloudfront.net/media/seo_landing_files/log-formula-1613465313.png

Sets

Sets represent algebraic variables as well-defined collections of distinct objects. Using sets, we describe and represent a collection of related and relevant objects in a group.

Example: Set A = {1, 3, 5, 7}……….(A set of odd numbers), Set B = {a, e, i, o, u}……(A set of vowels).

Reference: https://calcworkshop.com/wp-content/uploads/unequal-sets-example.png

Algebraic Formulas

An algebraic identity is an equation that is true regardless of the values associated with its variables. It means that when all the variables have the same value, the left and right sides of the equation are identical. Using these formulae, we can solve algebraic expressions quickly by using squares and cubes. Below are a few commonly used algebraic formulas.

(a + b)2 = a2 + 2ab + b2

(a – b)2 = a2 – 2ab + b2

(a + b)(a – b) = a2 – b2

(x + a)(x + b) = x + (a + b)x + ab

(a + b + c)2 = a2 + b2 + c2 + 2ab + 2bc + 2ca

(a + b)3 = a3 + 3a2b + 3ab2 + b3

(a – b)3 = a3 – 3a2b + 3ab2 – b3

Using the following example, we can see how these formulas are applied to algebra.

Example: Using the (a + b)2 formula in algebra, find the value of (101)2.

Solution:

Given: (101)2 = (100 + 1)2

Using algebraic formula (a + b)2 = a2 + 2ab + b2, we have,

(100 + 1)2 = (100)2 + 2(1)(100) + (1)2

(101)2 = 10201

Algebraic Operations

The basic operations covered in algebra as already mentioned are addition, subtraction, multiplication, and division.

  • Addition: In algebra, the addition operation is performed by separating two or more expressions with a plus sign (+).
  • Subtraction: The subtraction operation involves the use of a minus symbol (-) between two or more expressions in algebraic operations.
  • Multiplication: The multiplication operation in algebra consists of a multiplication sign (x) between two or more expressions.
  • Division: In algebra, the division operation involves two or more expressions separated by a “/”.

Basic Rules and Properties

Following are the basic rules or properties of algebra for variables, algebraic expressions, or real numbers a, b and c:

  • Commutative Property of Addition: a + b = b + a
  • Commutative Property of Multiplication: a × b = b × a
  • Additive Identity: a + 0 = 0 + a = a
  • Additive Inverse: a + (-a) = 0
  • Associative Property of Addition: a + (b + c) = (a + b) + c
  • Associative Property of Multiplication: a × (b × c) = (a × b) × c
  • Multiplicative Identity: a × 1 = 1 × a = a
  • Distributive Property: a × (b + c) = a × b + b × c or a × (b – c) = a × b – a × c
  • Reciprocal: Reciprocal of a = 1/a

A working example

Simplify 3a – 5b + 7a using proper steps.

We are going to do the exact same algebra we have always done. The answer looks like this:

3a – 5b + 7a :  original (given) statement

3a + 7a – 5b :  Commutative Property

(3a + 7a) – 5b :  Associative Property

a(3+7) – 5b :  Distributive Property

a(10) – 5b :  simplification (3 + 7 = 10)

10a – 5b :  Commutative Property

To keep your negatives straight, convert “– 5b” to “+ (–5b). Keep the minus sign in mind.

Solved Examples on Algebra

Example 1: Expand (2x + 3y)2 using the algebraic identities.

Solution:

In this case to solve the equation, we will use an algebraic identity, (a + b)2 = a2 + 2ab + b2

(2x + 3y)2 = (2x)2 + 2(2x)(3y) + (3y)2

= 4x2 + 12xy + 9y2

Therefore the answer is (2x + 3y)2 = 4x2 + 12xy + 9y2

Example 2: The age of a person is double the age of his son. Ten years ago, it was four times the age of his son. Use the concept of algebra and find the present age of the son.

Solution:

Let us assume the son is ‘x’ years old now. Since the person is twice as old as his son, let’s use ‘2x’ years as the age of the person. Ten years ago, the person was (x – 10) years old and the son was (2 x – 10) years old. We also know from the question that the age of the person 10 years ago was four times the age of the person today.

2x – 10 = 4(x – 10)

2x – 10 = 4x – 40

2x – 4x = -40 + 10

-2x = -30

2x = 30

x = 30/2

x = 15

The son’s age is therefore 15 years old.

Conclusion

Even if you think that you don’t need algebra outside of high school, managing budgets, paying bills, and even deciding health care costs and planning for the future will require a fundamental understanding of algebra. In addition to developing critical thinking, including logic, patterns, problem-solving, deductive and inductive reasoning, understanding basic algebra concepts can help individuals better handle problems involving numbers, especially in real-world situations where expenses and profits can be unknown and require employees to use algebraic equations to determine the missing factors. Ultimately, the more math a person knows, the more likely they are to succeed in engineering, actuarial, physics, programming, or any other tech-related field. In fact, most colleges and universities require algebra and other higher math courses for their applicants.

References:

  1. https://www.ipracticemath.com/learn/algebra/algebra_definition
  2. https://orion.math.iastate.edu/dept/links/formulas/form1.pdf
  3. https://brainly.in/question/6157572?tbs_match=3
  4. https://www.purplemath.com/modules/numbprop.htm
  5. https://isequalto.com/iet-app/ask-the-world/PPmy8850-What-is-AP-and-GP-series?
  6. https://www.myinterestingfacts.com/algebra-facts/
  7. https://www.thoughtco.com/what-is-algebra-why-take-algebra-2311937