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Appendix A: Powers of Ten

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Appendix A: Powers of Ten Appendix A: Powers of Ten The power-of-ten notation is a wonderfully compact way of expressing not only very large numbers but also exceedingly small ones. At its core is the fact that numbers are normally expressed using the base 10. As an illustrative reminder, consider the number 234. In it, the digit 4 is in the unit’s column, 3 is in the ten’s column, and 2 is in the hundred’s column. That is, 234 means 2 × 100 + 3 × 10 + 4 × 1. Each of the three members of the preceding sum is a number times a power of 10: 100 = 10 × 10 = 102, so that 100 equals 10 to the power 2. Similarly, 10 = 101 (i.e., 10 to the power 1), and by definition of the power 0, 100 = 1. The general case is 10n, where for the present the exponent n (i.e., the power to which 10 is raised) is a positive integer or zero. 10n is a compact way of writing 1 followed by n zeros, for instance 104 = 10,000. Furthermore, any number greater than 1 can be expressed this way. A simple example is 40,000: 40,000 = 4 × 10,000 = 4 × 104. This notation allows you to express exceedingly large numbers very neatly: it is just a question of determining the correct exponent. Another example is the once-famous gigantic number 10100, denoted a google: it equals 1 followed by 100 zeros. The salutary nature of the notation should be evident. The power-of-ten notation can be extended to include numbers less than one, as follows. First recall that a fraction such as 1/m has a specific meaning: it represents one of the portions obtained when 1 is divided into m equal parts (m is assumed to be an integer). When m is a power of ten, the decimal notation comes into play: 1/10 = 0.1, 1/100 = 0.01, 1/1000 = 0.001, etc., where, for reasons of clarity, the decimal point is preceded by a 0. Each of these examples is of the form 1/10n, and the general rule for expressing such a number as a decimal is a decimal point followed by n − 1 zeros followed by 1. That is, 1/10n means 1 in the nth place to the right of the decimal point. Rather than retain the power of ten in the denominator of this fraction, an altered notation has been introduced: one defines 1/10n as 10−n, which 226 Appendix A: Powers of Ten transforms a fraction into a negative exponent power of ten. Hence, 10−n becomes a compact, non-fraction means of writing 0.000 . 001, where the total number of zeros is n − 1. If instead of 1/10n one has m/10n, the rule becomes m/10n = m × 10−n. As an instance of this rule, the very small number five tenths of one bil- lionth, or 0.0000000005, becomes 5 × 10−10. You should not regard the introduction of the 10−n notation as a purposeless piece of pedagogy: it serves a highly needful func- tion, since exceedingly small numbers occur again and again in modern cosmology. Two among many examples are the mass of a hydrogen atom, which is about 1.67 × 10−27 kg, and the possibility of the event known as inflation, during which the very early Uni- verse underwent a period of immense expansion lasting about 10−34 seconds. (Masses are introduced in Chapter 2; inflation is discussed in Chapter 9.) The foregoing description is summarized in Table A.1, which also includes the names of the numbers and a few of the common symbols and prefixes in current use. Mega and kilo are frequently encountered, as in megawatts (MW) for electrical power and kilo- meters (km) for distance. Ditto for centi, whereas nano—for example nanotechnology, referring to microscopic machines of size 10−9 m or so—has only recently become an element of common usage. An example of a prefix not listed in Table A.1, one Table A.1. Powers of Ten* Number Name Power Symbol Prefix 0.000,000,000,000,001 Quadrillionth 10−15 0.000,000,000,001 Trillionth 10−12 0.000,000,001 Billionth 10−9 n nano 0.000,001 Millionth 10−6 0.001 Thousandth 10−3 0.01 Hundredth 10−2 c centi 0.1 Tenth 10−1 1 One 100 10 Ten 101 100 Hundred 102 1,000 Thousand 103 k kilo 1,000,000 Million 106 1,000,000,000 Billion 109 M mega 1,000,000,000,000 Trillion 1012 1,000,000,000,000,000 Quadrillion 1015 *Adapted from Berman and Evans (1977). Appendix A: Powers of Ten 227 hardly known outside the community of nuclear and elementary- particle physicists and engineers, is femto, which means 10−15. It is named after Enrico Fermi, the Nobel Prize–winning physicist for whom the Fermi National Accelerator Laboratory is also named. A femtometer, initially denoted a Fermi, is 10−15 m, a length approximately equal to the radius of a proton, a particle first discussed in Chapters 3 and 4. Sometimes it is necessary to multiply numbers expressed as powers of ten. The rule for accomplishing this is to multiply the numerical prefactors and add the exponents, as in 10n × 10m = 10n+m. An instance is the following simple product: (2 × 102) × (3 × 104) = (2 × 3) × (102 × 104) = 6 × 106 = 6,000,000. Negative exponents also obey the addition rule: 10n × 10−m = 10n−m. In this case, the power of ten that is the numerically larger number determines whether the result of the multiplication is greater or less than one. Thus, the product of one tenth and one hundred, which equals ten, is (1/10) × 100 = 10−1 × 102 = 10−1+2 = 101 = 10, whereas the product of one hundredth and ten, which equals one tenth, is (1/100) × 10 = 10−2 × 101 = 10−2+1 = 10−1 = 1/10. The addition-of-exponents rule also explains why 100 = 1. To show this, divide any power- of-ten number, for instance 10n, by itself: 10n/10n = 10n × 10−n = 10n−n = 100 = 1, as must be, since any number divided by itself is one. A useful feature of this notation follows from expressing big exponents as a sum of smaller ones, as it often allows a very large number to be stated in ordinary words, for example, in millions or billions. This procedure is activated by searching for ways of writing the exponent as a sum involving the numbers 6 and 9 (or 12, if trillions are desired), since from Table A.1, 106 is a million and 109 is a billion. For instance, the sun’s luminosity, or energy radiated per second (its power), is about 4 × 1026 W, obvi- ously a very BIG number. To get a feeling for how big, let us re- express it in terms of millions and billions by seeking a breakdown of 26 into 6s and 9s. One way to achieve this is via the sum 2 + 6 + 9 + 9, so that 1026 becomes 102+6+9+9 = 102 × 106 × 109 × 109, and therefore 4 × 1026 W is equal to four hundred million billion billion watts!! (That’s the same as four hundred trillion trillion watts. Either way, the sun is a bit stronger than any terrestrial light bulb....) 228 Appendix A: Powers of Ten The preceding breakdown into more familiar elements also works for negative exponents. Consider the “size” of a proton, which is about 10−15 m (that is, a femtometer). This number can be re-expressed in terms of millionths and billionths, as follows: 15 = 6 + 9, from which one gets 10−15 = 10−6−9 = 10−6 × 10−9. Hence, a femtometer is one millionth of a billionth of a meter. It should be clear that the femtometer is a much more appropriate unit for nuclear sizes than the meter. The preceding comment, on the appropriateness of the meter for nuclear sizes, brings us to the question of how many significant figures are appropriate in stating a number, especially a very large or a very small one when expressed as a power of ten. As noted in Chapter 2, the quantity 10n generally carries the most important information in a very large or very small number. A case in point is the speed of light, denoted c, whose value is 299,792,438m/sec. Equivalent statements of this speed are 2.99792438 × 108 m/sec and 2.99792438 × 105 km/sec. There are circumstances where the full nine-digit numbers must be used, but none of them will arise in the context of this book. In general, it is sufficient in the preceding power-of-ten expressions to replace all nine digits simply by 3, so that the speed of light is, to an excel- lent approximation, either 3 × 105 km/sec or 3 × 108 m/sec. In each of these numbers, the only significant figure turns out to be the numeral 3; the significant information resides in the exponent. However, it is unwise to blindly discard all but one of the digits: the factor multiplying 10n is important in some situations. A case in point is the astronomical unit, AU, introduced in Chapter 2. Needful information could be lost if at least its two-digit, approx- imate, numerical value of 9.3 × 107 miles (or 1.5 × 108 km) were not used. Appendix B: Primordial Nucleosynthesis In this appendix, I will examine some of the predictions concern- ing the relative abundance of the very light nuclei that were pro- duced in the 17-minute period beginning at 3 minutes aBB.
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