Concepts of Algebra

  • The Algebra of Real Numbers

    Algebra begins with a systematic study of the operations and rules of arithmetic. The operations of addition, subtraction, multiplication and division serves as a basis for all arithmetic calculations. In order to achieve generality, letters of the alphabets are used in algebra to represent numbers. A letter such as or can stand for a particular […]

  • Basic Algebraic Properties of Real Numbers

    The numbers used to measure real-world quantities such as length, area, volume, speed, electrical charges, probability of rain, room temperature, gross national products, growth rates, and so forth, are called real numbers. They include such number as , , , , , , , and . The basic algebraic properties of the real numbers can […]

  • Sets of Real Numbers

    Grouping or classifying is a familiar technique in the natural sciences for dealing with the immense diversity of things in the real world. For instance, in biology, plants and animals are divided into various phyla, and then into classes, orders, families, genera, and species. In much the same way, real numbers can be grouped or […]

  • Introduction to Scientific Notation

    In applied mathematics, very large and very small numbers are written in compact form by using integer powers of . For instance, the speed of light in vacuum, meters per second (approximately), can be written more compactly as meter per second. More generally, a real number is said to be expressed in scientific notation if […]

  • Approximation of Numbers

    Numbers produced by a calculator are often inexact, because the calculator can work only with a finite number of decimal places. For instance, a -digit calculator gives and , both of which are approximations of the true values. Don’t be too quick to pick up your calculator –answers such as , , and are often […]

  • Concept of Rounding Off

    Some scientific calculators can be set to round off all displayed numbers to a particular number of decimal places or significant digits. However, it’s easy enough to round off numbers without a calculator: Simply drop all unwanted digits to the right of the digits that are to be retained and increase the last retained digit […]

  • Mathematical Models and Idea of Direct and Inverse Variation

    In the later years of his life, the Italian scientist Galileo Galilei (1564 – 1642), wrote about his experiments with motion in a treatise called Dialogues Concerning Two New Sciences. Here he described his wonderful discovery that distances covered in consecutive equal time intervals by balls rolling down inclined planes are proportional to the successive […]

  • Concept of Joint and Combined Variation

    Often the value of a variable quantity depends on the values of several other quantities for instance the amount of simple interest on an investment depends on the interest rate, the amount invested, and the period of time involved. For compound interest the amount of interest depends on an additional variable how often the compounding […]

  • Introduction to Algebraic Expression and Polynomials

    An algebraic Expressions is an expression formed from any combination of numbers and variables by using the operations of addition, subtraction, multiplication, division, exponentiation (raising to powers), or extraction of roots. For instance,,, , , , and are algebraic expressions. By an algebraic expression in certain variables, we mean an expression that contains only those […]

  • Addition and Subtraction of Polynomials

    To find the sum of two or more polynomials, we use the associative and commutative properties of addition to group like terms together, and the we combine the like terms by using the distributive property. Example: Find the following sum: Solution:                                    Example: Find the following difference: Solution:                                        […]

  • Multiplication of Polynomials

    To multiply two or more monomials, we use the commutative and associative properties of multiplication along with the following properties of exponents. Properties of Exponents: Let and denotes the real numbers. Then, if and are positive integers, (1)    (2)    (3) We verify (1) as follows: Properties (1), (2) and (3) useful for simplifying […]

  • Factoring Polynomials

    When two or more algebraic expressions are multiplied, each expression is called a factor of the product. For instance, in the product , the factors are the , , and . Often we are given a product in its expended form and we need to find the original factors. The process of finding these factors […]