Algebraic Operations on Real Scalar Variables

Algebra is a branch of mathematics that was invented by Greek mathematicians and developed by Hindu, Arab, and European mathematicians. It was apparently the first branch of symbolic mathematics. Its great utility comes from the fact that letters are used to represent constants and variables and that operations are indicated by symbols such as, x, /, and so on. Operations can be carried out symbolically instead of numerically so that formulas and equations can be modified and simplified before numerical calculations are carried out. This ability allows calculations to be carried out that arithmetic cannot handle. 

The numbers and variables on which we operate in this section are called real numbers and real variables. They do not include imaginary numbers such as the square root of — 1, which we discuss later. They are also called scalars, to distinguish them from vectors, which have direction as well as magnitude. Real scalar numbers have magnitude, a specification of the size of the number, and sign, which can be positive or negative. 

EXAMPLE 2.1 Write the following expression in a simpler form  

SOLUTION We multiply out the factors in the numerator and combine terms, factor the denominator, and cancel a common factor  

The van der Waals equation of state provides a more nearly exact description of real gases than does the ideal gas equation. It is 

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In this chapter, we discuss symbolic mathematical operations, including algebraic operations on real scalar variables, algebraic operations on real vector variables, and algebraic operations on complex scalar variables. We introduce the concept of a mathematical function and discuss trigonometric functions, logarithms and the exponential function. 

Section 2.1 Algebraic Operations on Real Scalar Variables... 

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