Some Basic Terms Associated with Algebra
When it comes to studying Algebra, there are several basic mathematical terms that you will go through. Before we move into the detailed study of Algebra, it’s good to familiarize yourself with a few basic Algebraic terms.
An equation can be defined as a statement involving symbols (variables), numbers (constants) and mathematical operators (Addition, Subtraction, Multiplication, Division etc) that asserts the equality of two mathematical expressions. The equality of the two expressions is shown by using a symbol “=” read as “is equal to”. For example: 3X + 7 = 16 is an equation in the variable X.
A variable is a symbol that represents a quantity in an algebraic expression. It is a value that may change with time and scope of the concerned problem. For example: in the equation 3X + 7 = 16, X is the variable. Also in the polynomial X2 + 5XY – 3Y2, both X and Y are variables.
One Variable Equation
An equation that involves only one variable is knows as a One Variable Equation. 3X + 7 = 16 is an example of it.
Two Variable Equation
An equation that involves two variables is knows as a Two Variable Equation. 2X + Y = 10 is a Two Variable Equation of where X and Y are variables. Please note that here both X and Y have a power or exponent of 1. Hence it is an equation with degree 1. The degree is equal to the highest power of the variable(s) invloved. X2 + 5XY – 3Y2 = 25 is an example of a Two Variable Equation of degree 2.
Three Variable Equation
An equation that comprises three variables / symbols is called a Three Variable Equation
x + y − Z = 1 -------------(1)
8x + 3y − 6z = 1 -------------(2)
−4x − y + 3z = 1 -------------(3)
The above three equations form a system of 3 equations in 3 variables X, Y and Z. Each of these equations is a Three Variable Equation of degree 1. Also these equations are called Linear equations in three variables.
A monomial is a product of powers of variables. A monomial in a single variable is of the form xn where X is a variable and n is a positive integer. There can also be monomials in more than one variable. For example xm yn is a monomial in two variables where m,n are any positive integers. Monomials can also be multiplied by nonzero constant values. 24x2 y5 z3 is a monomial in three variables x,y,z with exponents 2,5 and 3 respectively.
A polynomial is formed by a finite set of monomials that relate with each other through the operators of addition and subtraction. The order of the polynomial is defined as the order of the highest degree monomial present in the mathematical statement. 2x3 + 4x2 + 3x – 7 is a polynomial of order 3 in a single variable.
Polynomials also exist in multiple variables. x3 + 4x2y + xy5 + y2 – 2 is a polynomial in variables x and y.
Exponentiation is a mathematical operation written as an where a is the base and n is called power or index or exponent and it is a positive number. We can say that in the process of exponentiation, a number is repeatedly multiplied by itself, and the exponent represents the number of times it is multiplied. In a3, a is multiplied with itself 3 times i.e. a x a x a. a5 translates to a x a x a x a x a (a multiplied with itself 5 times).
Shown below is a graph that shows exponentiation for different values of bases a.
By looking at the graph we conclude that the numbers less than one approach to zero as the exponent value grows. On the other hand, the exponentiation values tend to infinity as exponentiation index grows for numbers greater than 1.