Introduction to Algebra (2021)

This post is about an Introduction to Algebra.

The basics of algebra include numbers, variables, constants, expressions, equations, linear equations, and quadratic equations. Further, it involves the basic arithmetic operations of addition, subtraction, multiplication, and division within the algebraic expressions.

Introduction to Algebra

We work with numbers in arithmetic, while in algebra we use numbers as well as Alphabets such as $A, B, C, a, b$, and $c$ for any numerical values we choose. We can say that algebra is an extension of arithmetic. For example, the arithmetic sum of two numbers $5+3=8$ means that the sum of numbers 5 and 3 is 8. In algebra, two numbers can be summed by the expression $x+y=z$ which is the general form that can be used to add any two numbers. For example, if $x=5$ and $y=3$ then $x+y$ will be equivalent to the left-hand side ($5+3$) and the summation of these numbers will be equivalent to the right-hand side $z$ which is 8.

In algebra all arithmetic operators such as $+, -, \times, =$ and $\div$, etc., can be used used.

For example, $x-y=z$ means that the difference between two numbers is equal to the number represented by the letter z. In algebra, many other notations used are the same as in arithmetic. For example,

$c=a\times b$ means that the product of two numbers represented by $a$ and $b$ is equal to the number $c$.

$x \times x \times x$ can be written as $a^4$.

From the above discussion, note that letters of the alphabet represent variables, and arithmetic operators (+, -, etc) represent the mathematical operations on a variable. The combination of numbers and letters of the alphabet is called an algebraic expression. For example, $8x + 7y$, $x+y$, and $7x^2+2xy-5y^2$ etc. are examples of expression.

Some important points to remember:

• Algebra is like a toolbox for solving mathematics problems with unknowns. Instead of using specific numbers, we use letters like $x$, $y$, and $z$ to represent unknown values. These letters are called variables.
• A variable is a quantity (usually denoted by letters of the alphabet) in algebraic expressions and equations, that changes from place to place, person, to person, and/or time to time. The variable can have any one of a range of possible values.
• A factor that multiplies with a variable. For example, in $2x^3+3x=0$, $x$ is a variable, 2 is the coefficient of $x^3$, and 3 is the coefficient of $x$.

We learning algebra the following concepts are very important to understand the concepts used in algebra:

• Variables are the building blocks, representing unknown numbers.
• Expressions are combinations of variables, numbers, and mathematical operations (such as +, -, *, /) that do not necessarily have an equal sign (=).
• Equations are statements (or expressions) with an equal sign that shows two expressions are equivalent. The equations are solved to find the value of the variable.
• Inequalities are statements (or expressions) having “greater than” (>), “less than” (<), or “not equal to” (≠) symbols for making comparisons between expressions.

Real Life Applications of Algebra

• Finances and Budgeting: Algebra helps you create formulas to track income, expenses, savings goals, and loan payments. You can set up equations to see how much extra money you’d have if you cut certain expenses or how long it’ll take to save for a down payment on a house.
• Mixing and Ratios: Whether you’re baking a cake or mixing paint colors, algebra helps you determine the correct ratios of ingredients to achieve the desired outcome. You can set up proportions to find out how much water to add to a paint concentrate or how much flour you need to double a recipe.
• Motion and Physics: From calculating travel time based on speed and distance to understanding the trajectory of a thrown ball, algebra forms the foundation for many physics concepts. You can use formulas to figure out how long it’ll take to drive somewhere at a certain speed or the angle needed to throw a basketball into the hoop.
• DIY Projects and Home Improvement: From measuring lumber for a bookshelf to calculating the amount of paint needed for a room, algebra helps with planning and executing home improvement tasks. You can use formulas to find the area of a wall to determine how much paint to buy or calculate the volume of wood needed for a project.
• Scientific Research and Data Analysis: Algebra is the backbone of many scientific formulas and equations used in research. It helps analyze data, identify trends, and make predictions.