Algebra

Algebra - A Clear Presentation This is about the fundamental ideas of Algebra, and understanding why and how Algebra works. The present text is unusually accessible to readers who want to acquire algebraic skills. The clear presentation allows the reader to focus on the crucial facts of Algebra. The text is not cluttered with unnecessary details. That is why 500 plus pages are not necessary. We do not use the devastating phrase "it is obvious", because nothing is obvious to a person learning any subject. The ideas of digit position and digit position weight are introduced to show how integers greater than 9 are created. In this way understanding replaces rote learning. And, the real number system is reviewed. Fractions appeared when division created remainders. Fractions are numbers. The text shows how fractions are manipulated by the four operations addition, multiplication, subtraction, and division. Decimal integer and fractional parts are created when q divides p in the fraction p/q. The ideas of digit position and digit position weight are extended past the decimal point to show how the fractional part is valued. The text shows how to manipulate decimals. A focus on general methods for solving algebraic equations allows one to know how to solve any problem. The numerous special methods are distractions that have limited value. Sometimes an equation is not in the desired form. Algebraic operations are used modify the form of the equation by making the same changes to both sides of =, which does not upset the equality. A polynomial in one variable x is defined and its essential properties are presented. The text shows how to manipulate polynomials. The Remainder Theorem is explained. The theorem simplifies finding factors of polynomials. Newton's method for finding polynomial zeros is explained. Cramer's Rule is the straightforward way to find solutions by determinants of algebraic equations. How to find solutions of linear equations by addition, subtraction or substitution is also explained. The formula solving quadratic equations is derived and explained. An exponent n is a symbol written above, and on the right of, another symbol known as the base x as in x to the n. The text shows how arithmetic operations manipulate exponents. The Binomial Theorem shows how to expand (a+x) to the n when a and n are any numbers, positive, negative, integral or fractional. The Exponential and Logarithmic Functions are explained. The text shows how to manipulate them. Many problems are simplified when a rational function, the ratio of two polynomials, is decomposed into a sum of partial fractions with denominators of lower degree. Partial fractions have many applications such as simplifying many algebraic problems as well as the important inverse Laplace Transform process. Matrix algebra allows one to write and process equations efficiently. Furthermore, in many problems, the matrix format makes the next step easier to perceive. The concept of Mathematical Induction is explained and applied to problems. The presentations are eminently clear, because they are based on the policies assume nothing and nothing is obvious. The present text's contents are topics one actually uses when solving algebraic problems.