Measurement exponential notation

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. 

Several aspects of measurement will be considered in this chapter. First, Section 2.1 presents the factor label method, which makes calculations with measured quantities easier. This method will be used in the sections that follow and throughout the book. Next, in Section 2.2, we consider how to calculate with extremely large and extremely small numbers, using exponential notation. Section 2.3 introduces the metric system, a system of weights and measures... 

When measurements expressed in exponential notation are added or snbtracted, both the units and the exponents must be the same. When such measurements are mnltipHed or divided, nnits and exponents can be different. 

The SI uses seven base units, which are listed in Table B.l. All other units can be written as combinations of the base units. In writing the units for a measurement, we abbreviate them (see Table B.l), and we use exponential notation to denote the power to which a unit is raised a minus sign appears in the exponent... 

A problem arises when a number ends with zeros but contains no decimal point. In such cases, it is normally assumed that the zeros are not significant. Exponential notation (Appendix A.l) can be used to indicate whether end zeros are significant. For example, a mass of 10,300 g can be written to show three, four, or five significant figures depending on how the measurement is obtained ... 

In scientific measurement and calculations, we often encounter very large and very small numbers—for example, 0.00000384 and 602,000,000,000,000,000,000,000. These numbers are troublesome to write and awkward to work with, especially in calculations. A convenient method of expressing these large and small numbers in a simplified form is by means of exponents, or powers, of 10. This method of expressing numbers is known as scientific, or exponential, notation. 

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... 

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. 

Usin0 Exponential and Scientific Notation to Report Measurements... 

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.)... 

Fig. 2. (a) The free induction decay, G(t) for 19F in a single crystal of CaFi for B0 along [1,0,0]. The experimental points are given by circles and crosses from the CW and pulse measurements, respectively, and the theoretical curve is that of Eq. (14), corresponding to an exponential decay multiplied by a sine function. Note that F(t) is equivalent to G(t) in the present notation. Reproduced with permission from A. Abragam, The Principles of Nuclear Magnetism, p. 121, Oxford University Press, London, 1961. (b) The lineshape in the frequency domain corresponding to the Fourier transform of the theoretical curve. 

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. 

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. 

In this case there is no exponential growth and one is left with estimating the mass renewal function the notation is the same as for > 1, just keep in mind that now b = 0, so A° (n) = Ma, (n), Ta,f n) = Ma p n) fi/ and P( ) is the Markov transition matrix of J, with invariant measure V, Va = Ca a- We Will uot give the details of the proof, that can be found in [Caravenna et al. (2005)], but we stress that the proof is a matter of deahng with a return distribution that is a random superposition of return laws with 0=1/2 and trivial L( ), so the N dependence in Theorem 3.4(2) does not come as a surprise once we consider the corresponding result (2.15)... 

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