Using Exponential Scientific Notation

Numbers such as these are very awkward to work with. For example, neither of the numbers just written could be entered directly on a calculator. Operations involving very large or very small numbers can be simplified by using exponential (scientific) notation. To express a number in exponential notation, write it in the form... 

It is often difficult to express very large numbers to the proper number of significant figures using conventional notation. The solution to this problem lies in the use of scientific notation, also referred to as exponential notation, which involves the representation of a number as a power of ten. 

Chapter 1 describes the use of scientific notation (exponential notation) to represent such numbers more conveniently. The rules for scientific notation, as summarized there, are as follows ... 

The advantage of the SI system is that it is a measuring system based on a decimal system. With calculations written in groups of ten, results can be easily recorded as something called scientific notation. There are written prefixes that indicate exponential values as well. Some of these are listed in Table 2.2 which lists terms used in scientific notation. 

The SI system is based on seven fundamental units, or base units, each identified with a physical quantity (Table 1.1). All other units are derived units, combinations of the seven base units. For example, the derived unit for speed, meters per second (m/s), is the base unit for length (m) divided by the base unit for time (s). (Derived units that are a ratio of base units can be used as conversion factors.) For quantities much smaller or larger than the base unit, we use decimal prefixes and exponential (scientific) notation (Table 1.2). (If you need a review of exponential notation, see Appendix A.) Because the prefixes are based on powers of 10, SI units are easier to use in calculations than English units. 

Chemists frequently work with measurements that are very large or very small. A mole, for example, contains 602,213,670,000,000,000,000,000 particles, and some analytical techniques can detect as little as 0.000000000000001 g of a compound. For simplicity, we express these measurements using scientific notation thus, a mole contains 6.0221367 X 10 particles, and the stated mass is 1 X 10 g. Sometimes it is preferable to express measurements without the exponential term, replacing it with a prefix. A mass of 1 X 10 g is the same as 1 femtogram. Table 2.3 lists other common prefixes. 

In general, any ambiguity concerning the number of significant figures in a measurement can be resolved by using exponential notation (often referred to as scientific notation ), discussed in Appendix 3. 

Discover howto deal with, organize, and use all the numbers that play a huge role in chemistry. In particular, find out about exponential and scientific notation as well as precision and accuracy. 

Addition or subtraction gets easier when you express your numbers as coefficients of identical powers of 10. To wrestle your numbers into this form, you may need to use coefficients less than 1 or greater than 10. So scientific notation is a bit too strict for addition and subtraction, but exponential notation still serves you well. 

Be a good chemist. Report your measurements in scientific notation to avoid such annoying ambiguities. (See the earlier section Using Exponential and Scientific Notation to Report Measurements for details on scientific notation.)... 

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. 

To use exponential notation to work with very large and very small numbers To use the basic elements of the metric system—a system of units and prefixes designed to make scientific calculations as easy as possible... 

Exponential notation enables easy reporting of extremely large and extremely small numbers. A number in scientific notation consists of a coefficient times 10 to an integral power, where the coefficient is equal to or greater than 1 but less than 10. Learn how to convert numbers from exponential notation to ordinary decimal values, and vice versa, and also how to use exponential numbers in calculations. Also learn to use effectively an electronic calculator with exponential capability (see Appendix 1). (Section 2.2)... 

Scientists report numbers from literally astronomical to almost infinitesimal. In order to do so conveniently, we use scientific notation, also known as standard exponential notation. Scientific notation is a form of a number with a decimal coefficient times a power of 10. The following number is in scientific notation, with its parts identified ... 

Scientists must use extremely small and extremely large numbers to describe the objects in Figure 1. The mass of the proton at the center of a hydrogen atom is 0.000000000000000000000000001673 kg. HIV, the virus that causes AIDS, is about 0.00000011 m. The temperature at the center of the Sun reaches 15,000,000 K. Such small and large numbers are difficult to read and hard to work with in calculations. Scientists have adopted a method of writing exponential numbers called scientific notation. It is easier than writing numerous zeros when numbers are very large or very small. It is also easier to compare the relative size of numbers when they are written in scientific notation. 

It is inconvenient to be limited to decimal representations of numbers. In chemistry, very large and very small numbers are commonly used. The number of atoms in about 12 grams (g) of carbon is represented by 6 followed by 23 zeros. Atoms typically have dimensions of parts of nanometers, i.e., 10 decimal places. A far more practical method of representation is called scientific or aqponential notation. A number expressed in scientific notation is a number between 1 and 10 which is then multiplied by 10 raised to a whole number power. The number between 1 and 10 is called the coefficient, and the factor of 10 raised to a whole number is called the exponential factor. 

Scientific calculators are generally able to convert numbers to exponential notation using one or two keystrokes frequently SCI for scientific notation will convert a number into exponential notation. Consult your instruction manual to see how this operation is accomplished on your calculator. 

The prefixes in Table 2.4 are represented by the powers of 10 used in scientific, or exponential, notation for writing large and small numbers. For example, 10 = 10 X 10 X 10 = 1000. Appendix B reviews this notation. 

The multiplication and division of numbers written in scientific notation can be done qnite simply by using some characteristics of exponentials. Consider the following multipUcation ... 

Science has constantly pushed the boundaries of the very large and the very small. We can, for example, now measure time periods as short as 0.000000000000001 seconds and distances as great as 14,000,000,000 light-years. Because the many zeros in these numbers are cumbersome to write, scientists use scientific notation to write them more compactly In scientific notation, 0.000000000000001 is 1 X 10 and 14,000,000,000 is 1.4 X 10. A number written in scientific notation consists of a decimal part, a number that is usually between 1 and 10, and an exponential part, 10 raised to an exponent, n. 

On all scientific calculators it is possible to enter numbers in exponential notation. The method used depends on the brand of calculator. Most often, it involves using a key labeled [exp), [ ee 1, or [eex]. Check your instruction manual for the procedure to be followed. To make sure you understand it, try entering the following numbers ... 

Many of the numbers you will deal with will either be very large (e.g., Avogadro s number— 6.02 X 1023) or very small (e.g., Planck s constant—6.63 X 10 34 J s). Rather than write these numbers with all of the zeros, it is much easier to use scientific (or exponential) notation ... 

In using such large and small numbers, it is inconvenient to write down all the zeroes. In scientific (exponential) notation, we place one nonzero digit to the left of the decimal. 

In chemistry we frequently use very large or very small numbers. Such numbers are conveniently expressed in scientific, or exponential, notation. 

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