Mathematics scientific notation

The abbreviation log stands for logarithm. In mathematics, a logarithm is the power (also called an exponent) to which a number (called the base) has to be raised to get a particular number. In other words, it is the number of times the base (this is the mathematical base, not a chemical base) must be multiplied times itself to get a particular number. For example, if the base number is 10 and 1,000 is the number trying to be reached, the logarithm is 3 because 10 x 10 x 10 equals 1,000. Another way to look at this is to put the number 1,000 into scientific notation ... 

In science, we often encounter very large and very small numbers. Written in standard decimal notation, these numbers can be cumbersome. There are, for example, about 33,460,000,000,000,000,000,000 water molecules in a thimbleful of water, each having a mass of about 0.00000000000000000000002991 gram. To represent such numbers, scientists often use a mathematical shorthand called scientific notation. Written in this notation, the number of molecules in a thimbleful of water is 3.346 X 1022, and the mass of a single molecule is 2.991 X 10 23 gram. 

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). 

To use the scientific notation of numbers in mathematical operations, we must remember the laws of exponents. 

In practice, we perform mathematical operations on numbers in scientific notation according to the simple rules that follow. It is not necessary to know these rules when calculations are done with a calculator (you need only know how to enter numbers), but many calculations are so simple that no calculator is needed, and you should be able to handle these operations when your calculator is broken down or not available. The simple rules are the following. 

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. 

A good portion of the AP Chemistry Test deals with calculations, either with or without the aid of a calculator. For all of these problems, there are two different components—the chemistry component and the math component. Most of this book is devoted to a review of the chemistry component of the problems, but this chapter is designed to review a few important mathematical skills that you will need to know as you work through the problems. Three skills that are critical to success on the AP Chemistry Test use significant figures, scientific notation, and dimensional analysis. 

To simplify reporting large numbers and doing calculations, you can use scientific notation. As you learned in an earlier mathematics course, scientific notation works by using powers of 10 as multipliers. 

When a number is expressed as a logarithm, this refers to the power n that the base number a must be raised to give that number, e.g. logjoflOOO) = 3, since 10 = 1000. Any base could be used, but the two most common are 10, when the power is referred to as logio or simply log, and the constant e (2.718282), used for mathematical convenience in certain situations, when the power is referred to as loge or In. Where a coefficient would be used in scientific notation, then the log is not a whole number. 

A Scientific Notation and Experimental Error A.2 B SI Units, Unit Conversions, Physics for General Chemistry A.9 C Mathematics for General Chemistry A.21 D Standard Chemical Thermodynamic Properties A.37 E Standard Reaction Potentials at 25°C A.45 F Physical Properties of the Elements A.47 G Solutions to the Odd-Numbered Problems A.57... 

Consider the number four hundred and fifty trillion 450,000,000,000,000. Would you find it easy to work with this number in mathematical equations Let s say you needed to divide four hundred and fifty trillion by nine thousand. Sounds messy This is where scientific notation becomes helpful. Instead of writing 450,000,000,000,000, you can write an equivalent value of 4.5 times the appropriate power of 10. Let s look at the powers of ten ... 

Finally, the nvalue column of the vla4.property table can be populated when possible. This column stores the numerical value of the property. Since not all values are numerical, this column may have null entries. The purpose is to enable efficient use of numerical data when appropriate, for example, to select by value, sort, apply mathematical functions, etc. The following SQL will update the nvalue column when possible with a numeric value. The tilde operator in the where selects text values that match the regular expression. The expression shown here allows integers, decimal values, and scientific notation using E or e for the exponent, for example 6.023E23. 

One last thing If you are given a number between 1 and 10 and need to write it in scientific notation, the power on 10 would be zero. In scientific notation, the number 8 would be written 8 x 10°. In mathematics, 10° equals 1. 

The Math Handbook helps you review and sharpen your math skills so you get the most out of understanding how to solve math problems involving chemistry. Reviewing the rules for mathematical operations such as scientific notation, fractions, and logarithms can also help you boost your test scores. 

Science often involves very large and very small numbers. Such numbers may be cumbersome to write down, and an abbreviated notation (known as standard or scientific notation) is often used. This relies upon the following mathematical symbolism ... 

Skill 18.1 Apply appropriate mathematical skills (e.g., algebraic operations, graphing, statistics, scientific notation) and technology to collect, analyze, and report data and to solve problems in chemistry. 

One reason chemistry can be challenging for beginning students is that they often do not possess the required mathematical skills. Thus we have paid careful attention to such fundamental mathematical skills as using scientific notation, rounding off to the correct number of significant figures, and rearranging equations to solve for a particular quantity. And we have meticulously followed the rules we have set down, so as not to confuse students. 

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. 

NEW BASIC MATH SKILLS APPENDIX To aid the flow of introductory chemistry material in Chapter I, a review of topics in basic mathematics skills, including scientific notation and use of significant figures, with numerous examples, now appears in Appendix A. Related exercises remain in the Measurements and Calculations section at the end of Chapter 1. 

FOR PRACTICE Mathematical Operations with Scientific Notation... 

In chemistry, chemical structures have to be represented in machine-readable form by scientific, artificial languages (see Figure 2-2). Four basic approaches are introduced in the following sections trivial nomenclature systematic nomenclature chemical notation and mathematical notation of chemical structures. 

More sophisticated number communication systems (as opposed to numbers themselves) are a cultural invention, such that in Western society, we think of numbers quite differently, and have developed a complex system for talking and describing them. Most scientifically and mathematically minded people believe that numbers and mathematics exist beyond our experience and that advancing mathematics is a process of discovery rather than invention. In other words, if we met an alien civilisation, we would find that they had calculus too, and it would work in exactly Ihe same way as ours. What is a cultural invention is our system of niunerical and mathematical notation. This is to some extent arbitrary while we can all easily imderstand the mathematics of classical Greece, this is usually only after their notational systems have been translated into Ihe modem system Pythagoras certainly never wrote + = z. Today, we have a single commonly... 

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. 

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. 

---