Mathematics exponential notation

Appendix 3 contains a mathematical review touching on just about all the mathematics you need for general chemistry. Exponential notation and logarithms (natural and base 10) are emphasized. 

Exponential notation is an alternative way of expressing numbers in the form fl ( a to the power ), where a is multiplied by itself n times. The number a is called the base and the number n the exponent (or power or index). The exponent need not be a whole number, and it can be negative if the number being expressed is less than 1. See Table 39.2 for other mathematical relationships involving exponents. 

An array is a multidimensional (not linear) data structure. An appointment schedule for the business week, hour by hour and day by day, is an example of an array. A mathematical function can generate data for an array structure. For example, the four-dimensional array shown in Table I (with entries in exponential notation) was obtained by substituting numeric values for x, y, w, and z in the expression ... 

All these forms are mathematically correct as numbers expressed in exponential notation. In scientific notation, the decimal point is placed so that there s one digit other than zero to the left of the decimal point. In the preceding example, the number e]q>ressed in scientific notation is 1.25 x 10 m. Most scientists automatically express numbers in scientific notation. 

In addition to basic arithmetic and algebra, four mathematical operations are used frequently in general chemistry manipulating logarithms, using exponential notation, solving quadratie equations, and graphing data. Each is discussed briefly below. 

Only a few basic mathematical skills are required for the study of general chemistry. But to concentrate your attention on the concepts of chemistry, you will find it necessary to have a firm grasp of these basic mathematical skills. In this appendix, we will review scientific (or exponential) notation, logarithms, simple algebraic operations, the solution of quadratic equations, and the plotting of straight-line graphs. 

At first we will consider = 10 and work only with concentrations that can be expressed with whole-number exponents. In the next section, after you have become familiar with the mathematical procedures, concentrations will be written in the usual exponential notation form, including coefficients. 

Appendix 1 Mathematical Procedures A1 Al.l Exponential Notation A1 A1.2 Logarithms A4 A1.3 Graphing Functions A6 A1.4 Solving Quadratic Equations A7 A1.5 Uncertainties in Measurements AlO Appendix 2 The Quantitative Kinetic Molecular Model A13... 

To understand how this shorthand notation works, consider the large number 50,000,000. Mathematically this number is equal to 5 multiplied by 10 X 10X 10X 10X 10 X 10 X 10 (check this out on your calculator). We can abbreviate this chain of numbers by writing all the 10s in exponential form, which gives us the scientific notation 5 X 107. (Note that 107 is the same as lOx lOx 10x lOx 10 X 10 X 10. Table A. 1 shows the exponential form of some other large and small numbers.) Likewise, the small number 0.0005 is mathematically equal to 5 divided by 10 X 10 x 10 X 10, which is 5/104. Because dividing by a number is exactly equivalent to multiplying by the reciprocal of that number, 5/104 can be written in the form 5 X 10-4, and so in scientific notation 0.0005 becomes 5 X 10-4 (note the negative exponent). 

Notice how numbers that are either very large or very small are indicated in Table 1.4 using an exponential format called scientific notation. For example, the number 55,000 is written in scientific notation as 5.5 X 104, and the number 0.003 20 as 3.20 X 10 3. Review Appendix A if you are uncomfortable with scientific notation or if you need to brush up on how to do mathematical manipulations on numbers with exponents. 

The mathematical constant e = 2.71828, like tt, occurs in many scientific and engineering problems. It is frequently seen in the natural exponential function y = e. The inverse function is called the natural logarithm, x = In y, where In y is simplified notation for log y. 

Since we have no a priori knowledge of the mathematical form of the functions r x) and r ( ), let us make the simple assumptions that all modes of repair are inactivated exponentially with dose, and that the induced components increase as simple saturation functions. On the basis of the notation and assumptions just introduced, we can write... 

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