FUNDAMENTAL CONCEPT OF ALGEBRA -
Algebra is a branch of mathematics that deals with symbols and the rules for manipulating those symbols to solve equations and study relationships between variables. Its fundamental concepts include:
This is the first time you are introduced by the term algebra. Algebra is the branch of mathematics.
What is the missing number?
? - 2 = 4
OK, the answer is 6, right? Because 6 − 2 = 4. Easy!
Well, in Algebra we don't use blank boxes, we use a letter (usually an x or y, but any letter is fine). So we write:
x − 2 = 4
It is really that simple. The letter (in this case an x) just means "we don't know this yet", and is often called the unknown or the variable.
In algebra we have two types of symbols - Constants and Variables (literals), Let us now understand these terms better -
A variable is a symbol or letter, such as "x" or "y," that represents a value. As the name implies, the value of a variable can change. It is a symbol for a number we don't know yet.
Example:- In the expression 2x + 5, x is the variable.
2. Constants:- Constants are specific, unchanging values like numbers (e.g., 1, 2, 3) or mathematical constants (e.g., π or e).
Variables that store values that do not change are called constant. Constant is a quantity which has a fixed value.
Example:- In the expression 2x + 5, 5 is the constant.
3. Expressions:- An expression is a combination of variables, constants, and mathematical operations (addition, subtraction, multiplication, division, exponentiation, etc.). For example, 2x + 3 is an algebraic expression.
4. Equations:- An equation is a statement that two expressions are equal. Equations are used to describe relationships between variables and constants. For example, x + 5 = 10 is an equation, and solving it means finding the value of x that makes both sides equal.
5. Functions:- A function is a rule that assigns a unique output (or value) to each input (or value) based on some mathematical relationship. Functions are often represented by equations, and they are a fundamental concept in algebra. For example, y = 2x is the equation of a linear function.
6. Solving Equations:- Algebraic techniques are used to solve equations, which means finding the values of variables that make the equation true. This involves various operations like simplification, isolating the variable, and applying inverse operations.
7. Graphing:- Graphs are essential tools in algebra. They visually represent relationships between variables and functions. The Cartesian coordinate system is frequently used to plot points and draw graphs.
8. Inequalities:- Inequalities involve expressions or equations where the relationship is not one of equality but rather less than (<), greater than (>), less than or equal to (≤), or greater than or equal to (≥).
9. Polynomials:- Polynomials are algebraic expressions that consist of variables raised to non-negative integer exponents, multiplied by constants. For example, 3x^2 - 2x + 5 is a polynomial.
10. Factoring:- Factoring involves breaking down algebraic expressions, especially polynomials, into their simpler, irreducible forms. It's a crucial technique for solving equations and simplifying expressions.
11. Quadratic Equations:- Quadratic equations are a special type of equation where the highest power of the variable is squared (e.g., ax^2 + bx + c = 0). The quadratic formula is often used to solve them.
12. Systems of Equations:- This involves dealing with multiple equations simultaneously, often with multiple variables. Solving systems of equations helps find solutions that satisfy all the equations.
13. Rational Expressions:- These are expressions involving fractions where polynomials are in the numerator and denominator. Simplifying and manipulating rational expressions is an important algebraic skill.
14. Exponents and Radicals:- Understanding and manipulating expressions involving exponents (e.g., x^2, x^3) and radicals (e.g., √x) are fundamental to algebra.
These fundamental concepts of algebra provide the foundation for solving a wide range of mathematical problems and are essential in various fields such as science, engineering, economics, and computer science. Algebra is a bridge between arithmetic and more advanced mathematics, serving as a powerful tool for problem-solving and modeling real-world situations.