Modeling of Combustion Systems A Practical Approach 6 pdf

609 Appendix F Numbers in Binary, Octal, and Hexadecimal Representations For readers unfamiliar with binary and related bases, we digress here to consider three important numerical systems besides the decimal (base 10) system; these are binary (base 2), octal (base 8), and hexadecimal (base 16). Obviously, the decimal system uses ten number symbols (0, 1, 2, 3, 4, 5, 6, 7, 8, 9) that multiply 10 raised to some exponent. The system is positional, with columns to the left of the decimal point indicating increasing powers of 10, and those to the right, decreasing powers. For example, 234.5 may be written as (2)(10 2 ) + (3)(10 1 ) + (4)(10 0 ) + (5)(10 –1 ) = 200 + 30 + 4 + 5/10 = 234.5. Table F.1 illustrates the procedure. By analogy, octal (base 8) uses an analogous scheme: eight number symbols (0, 1, 2, 3, 4, 5, 6, 7) and a base of 8. To see what the octal equivalent of 234.5 is, we refer to Table F.2 and find that 352.4 8 = 234.5. The subscript after the number indicates the base. Obviously, if there is no subscript we are referring to base 10. TABLE F.1 Positional Number Representation in Base 10 (e.g., 234.5) Exponent … 3 2 1 0 . –1 –2 –3 … Exponential notation … 10 3 10 2 10 1 10 0 .10 –1 10 –2 10 –3 … Decimal notation … 1000 100 10 1 . 1/10 1/100 1/1000 … Decimal multipliers 2 3 4 . 5 Decimal sum 200 + 30 + 4 . + 5/10 = 234.5 TABLE F.2 Positional Number Representation in Base 8 (e.g., 352.4 8 = 234.5) Exponent … 3 2 1 0 . –1 –2 –3 … Exponential notation … 8 3 8 2 8 1 8 0 .8 –1 8 –2 8 –3 … Decimal notation … 512 64 8 1 . 1/8 1/64 1/512 … Octal multipliers 3 5 2 . 4 Decimal sum 192 + 40 + 2 . + 4/8 = 234.5 © 2006 by Taylor & Francis Group, LLC 610 Modeling of Combustion Systems: A Practical Approach Base 2 is ideal for constructing factorial designs because the system com- prises only two states for any factor: high and low. In base 2, the only numbers we may use are 0 or 1. (We can use – and + in lieu of numeric symbols, but the point is that we only have two symbols at our disposal.) As an example of binary math, the decimal number 14.75 is equivalent to 1110.11 2 . Table F.3 shows why. To find the octal equivalent for 14.75, we could go through the same routine as before. However, we can take a shortcut whenever two bases are related by the formula base B = base (A) n where A, B, and n are integers. In such a case, we may group the base A symbols in groups of n and convert each group directly to its base B equivalent. In the present case, base 8 = base 2 3 . Therefore, we can group the base 2 symbols in groups of three (starting from the octal point and moving in each direction) and then convert each to its octal representation. For example, 1110.11 2 = 1,110.110 2 = 16.6 8 , because 1 2 = 1 8 , 110 2 = 6 8 . Conversion to base 4 follows the same pattern: base 4 = 2 2 , so grouping 1110.11 2 in groups of two gives 11,10.11 2 = 32.3 4 , since 11 2 = 3 4 and 10 = 2 4 . For bases less than 10, we use a subset of the base 10 numerical symbols. For bases greater than 10, we use letters as additional numerical symbols. Thus, for base 16 we augment the symbols 0 through 9 with A = 10, B = 11, C = 12, D = 13, E = 14, and F = 15. So a number like 234.5 equals EA.8 16 using the process already shown. But conversion of EA.8 16 to related bases is much simpler: EA.8 16 = 352.4 8 = 32,22.2 4 = 11,0010,1011.1 2 . To find this, we first con- vert to base 2 using Table F.3. From it we note that E = 1110, A = 1010, and 8 = 1000. Putting these in series and eliminating leading and trailing zeros after the binary point, we have 1110,1010.1, and this is indeed the base 2 representation EA.8 16 or 234.5, base 10. Delimiting the numerals in groups of threes (always beginning from the binary point and proceeding in either direction) rather than fours gives 11,101,010.100. Converting these to their octal equivalents gives 352.4 8 . Delimiting the binary number in groups of twos gives 11,10,10,10.10. Con- 4 the procedure easier. TABLE F.3 Positional Number Representation in Base 2 (e.g., 1110.11 2 = 14.75) Exponent … 3210. –1 –2 –3… Exponential notation … 2 3 2 2 2 1 2 0 .2 –1 2 –2 2 –3 … Decimal notation … 8421. 1/21/41/8… Binary multipliers 1110. 1 1 Decimal sum 8 + 4 + 2 + 0 . + 1/2 + 1/4 = 14.75 © 2006 by Taylor & Francis Group, LLC verting these to their base 4 equivalent yields 32,22.2 . Table F.4 may make Numbers in Binary, Octal, and Hexadecimal Representations 611 Example 3.1 Conversion of Numbers among Bases Problem statement: Convert 100 to its hexadecimal equivalent. Then convert the hexadecimal number into the bases 2, 4, and 8. Which conversions are easier? Why? Solution: We use the following methodology: Step 1: Ten is not an integer power of 2; therefore, we find the largest power of 16 that is less than 100: 16 0 = 1, 16 1 = 16, 16 2 = 256. Step 2: Divide 100 by the largest integer power of 16 and retain the integer. This will be the first hexadecimal digit: 100/16 = 6.25 (use 6). Step 3: Subtract 60 16 from 100 by converting to base 10: 6·16 = 96. Subtract this from 100, leaving 4. Step 4: Repeat the process until closure: 100 – 96 = 4. Therefore, 100 = 64 16 , and we are done. Now that we have found the hexadecimal equivalent, we may convert this directly to binary integers one integer at a time: 64 16 = 110,0100. Regrouping in threes gives 1,100,100, which converts to 144 8 . Regrouping in twos gives 1,10,01,00, which yields 12,10 4 . TABLE F.4 Base Equivalents Base 10 16 8 4 2 000 0 0 111 1 1 222 2 10 333 3 11 44410 100 55511 101 66612 110 77713 111 8 8 10 20 1000 9 9 11 21 1001 10 A 12 22 1010 11 B 13 23 1011 12 C 14 30 1100 13 D 15 31 1101 14 E 16 32 1110 15 F 17 33 1111 16 10 20 100 10000 © 2006 by Taylor & Francis Group, LLC 612 Modeling of Combustion Systems: A Practical Approach Clearly, the latter method is easier, but it is only possible if an integer power relates the bases. Conversely, because 16, 8, 4, and 2 relate by integer powers, one can convert among them with relative ease. An Apologetic for Octal In the usual case, humans possess 10 digits, 5 on each hand. Our counting system seems to derive from this. The linguistic term betrays an association — we use the term digits to refer to either fingers or numerals. Were our counting system to have ignored thumbs, it may have grown to become octal rather than decimal. This would have some advantages. First, it is much easier to convert octal to binary or hexadecimal — the arithmetic of com- puters. Second, 24 hours (30 8 ) would equal three revolutions on a clock (numbered 0, 1, 2, 3, 4, 5, 6, 7), dividing the work–rest–play cycle evenly. And instead of counting only to 10 with our fingers and thumbs, we would be able to count to 24 (30 8 ) using our fingers to represent units and thumbs to represent octals. The ancient tally system of counting is somewhat closer to this procedure in the sense that it uses four vertical strokes to represent numbers up to 4 and one cross-stroke for the number 5. This is actually a base 5 system, and it unmistakably resembles the four fingers and one thumb of the human hand. The leap to base 10 (two hands) is not so hard to imagine. Considering their love for the number 7, one would have thought that the ancient Hebrews and their predecessors would have preferred octal to dec- imal, since it gives 7 as the largest numeral.* The relationship between a 24- hour day and a 360° rotation of the Earth gives 15° per hour; octal would have allowed a slightly more precise division into 384° (600 8 °), each hour representing 16° of Earth’s rotation (20 8 °), making angles easier to repeatedly bisect. So, the Greeks with their love for geometry and integers should have loved such a system. An octal world is a world where children never labor to convert fractional measures; they become trivial: 1/2 = 0.4 8 , 1/4 = 0.2 8 , 1/8 = 0.1 8 , 1/16 = 0.04 8 , 1/32 = 0.02 8 , 1/64 = 0.01 8 , etc. Even music with its octave scales and whole, half, quarter, eighth, and sixteenth notes would be better served by an octal system. Alas, for all these advantages and ancient possi- bilities, we live in a decimal world, not an octal one. But is it too much to hope for change? The metric system has unified and harmonized scores of other units and divisions. The French still count by 20s (e.g., 90 is spoken quatre vingt dix, literally “four 20s, 10”). Perhaps an octal future is not so far- fetched. The author eagerly awaits this brave new world (along with the return of the slide rule, which, by the way, would be easier to construct). * This is attested to in various Hebrew literature. For example, see various books of the Bible, such as Genesis 2:1, 7:2, Exodus 20:10, and Revelation 1:12–20, 5:5, 8:6, 10:4, 15:1, 16:1. © 2006 by Taylor & Francis Group, LLC . & Francis Group, LLC 61 0 Modeling of Combustion Systems: A Practical Approach Base 2 is ideal for constructing factorial designs because the system com- prises only two states for any factor:. 1101 14 E 16 32 1110 15 F 17 33 1111 16 10 20 100 10000 © 20 06 by Taylor & Francis Group, LLC 61 2 Modeling of Combustion Systems: A Practical Approach Clearly, the latter method is easier, but. take a shortcut whenever two bases are related by the formula base B = base (A) n where A, B, and n are integers. In such a case, we may group the base A symbols in groups of n and convert each group