232. Fractions in the Binary System a. The system of expressing fractions with binary numbers is similar to the decimal num3 bering methods. For example, the common fraction may be expressed in binary numbers as 11 101 Also, binary fractions may be expressed as decimal fractions when the powers of 2 are used with negative exponents. The binary fraction 0.011 is equivalent to the decimal fraction 0.375 and may be written as: b. The following table lists some of the fractional values and their equivalents in both systems: c. Using values from the table, the decimal fraction 0.375 is equal to .25 + .125 and hence has the binary equivalency of .01 + .001 = .011. 233. Conversion of Decimal Fractions to Binary Numbers Usually, the decimal fractions are converted to binary fractions by performing a series of multiplications by 2. This method is directly opposite to the method explained in paragraph 224. As a rule, decimal fractions cannot be converted to exact binary equivalents. The extent of error must be tolerable for a given application and the number of bits used must be reasonable. a. To convert 0.375 to a binary number, proceed as follows: (1) Multiply the decimal 0.375 by 2 to obtain a new integer (whole number) and a new decimal, 0.75. Since in 0.75 the integer to the left of the decimal point is 0, place a 0 in the binary equivalent as .0. Decimal X 2 0.375 X 2 New integer and decimal 0.75 Partial binary equivalent .0 (2) Multiply the decimal 0.75 by 2 to obtain a new integer and decimal. Since the integer to the left of the decimal point is a 1, place a 1 in the binary equivalent as .01. Decimal X 2 0.75 X 2 New integer and decimal Partial binary equivalent · .01 2 New integer and decimal 'Partial binary equiva .011 the operation ends when the decimal part has been expanded to 0.00. raction, 0.375, is equivalent to the binary fraction, .011. In this instance, d decimal fractions have exactly the same value. ampie illustrates the conversion of 0.3465 to its binary equivalent. Note equivalents are added at the end of the operation to obtain the comp the binary and decimal fractions differ in value and the amount of error ma otraction: 0.3465 0.34375 0.00275 (fraction of error) y Numbers ary number is a combination of whole numbers and binary fractions. Exam) nental operations (addition, subtraction, multiplication, and division) for mi or binary fractional numbers alone are in accordance with the principles alre chapter. APPENDIX I BASIC SLIDE RULE OPERATIONS 1. General This appendix describes the basic slide rule and covers the operations of multiplication, division, squaring, and square root. 2. Description of Slide Rule a. Slide rules are made in several different sizes and styles, and in an assortment of scales. However, they all contain the same basic scales and use them in the same manner. b. The most common type of slide rule is about 10 inches long and generally has scales on both sides. The most frequently used scales, and the ones covered here, are the A, B, C, and D. Figure 95 is a simplified drawing of a slide rule of this type, showing these scales and the other essential parts of the rule. Note that these scales have indexes (the number 1) on both ends. Also note that the A and B scales have an additional index in the center that divides these scales into two equal parts. The left-hand part of the scales is called Aleft or B-left, and the right-hand part, A-right or B-right. 3. Basic Principles of Operation The slide rule is based on the principle of the logarithm; that is, the segments on the rule represent exponents, or logarithms, but are indicated by the antilogs, or numbers corresponding to those logarithms. Consequently, when the slide rule is used so that two INDEX line segments are added, the logarithms of the numbers shown are actually being added, and the sum of the two line segments is represented by the antilog of the sum of the logarithms. Since the sum of the logarithms of two numbers is equal to the logarithm of the product of the two numbers (par. 121), adding two line segments on a slide rule will give the product of the two numbers represented by the line segments. This is the technique used in multiplication with a slide rule (par. 6 of this app). In the division process, the reverse procedure is used; that is the two line segments are used so that one is subtracted from the other. 4. Accuracy The accuracy of a slide rule depends on the length of the rule and on the portion of the rule being used. With the 10-inch rule shown in figure 95, numbers can be approximated to four significant figures on the left-hand end of the C or D scales, but only to three significant figures on the right-hand end of these scales. Despite this fact, the results obtained with the slide rule are sufficiently accurate for many practical purposes; in any case, the results serve as a rapid and efficient check of more complex computations. 5. Reading the Scales a. Since the scales on a slide rule do not have uniform increments along their lengths, INDICATOR be careful when approximating numbers at different points on the scales. For example, the space between the larger numbers 1 and 2 on the D scale (fig. 96) is divided into 10 subgroups (identified by the small numbers 1 through 10) of 10 increments each; thus there are 100 increments between 1 and 2 on the D scale, and each increment is equal to one onehundredth of the difference. Between 4 and 5 on the D scale, however, there are only 20 increments, and each increment therefore, is equal to five one-hundredths of the difference. Consequently, the number 105 would be located 5 increments above 1, whereas 405 would be 1 increment above 4 on the scale. Figure 96 shows the location of these and other numbers on the D scale. b. To locate a number on a scale, first determine its general location between two of the numbers on that scale; then determine the value of each increment between the numbers. Finally, determine its exact location based on the value of the increments. c. In reading the scale, as in logarithms, the decimal point is neglected until after the absolute value of the result is obtained; therefore, in figure 96, the number 1245 could actually represent 1.245, 12.45, 124.5, .001245, etc. The use of scientific notation (par. 106) will greatly simplify the handling of very large or very small numbers. 6. Multiplication a. Normally, the process of multiplication is performed by using the C and D scales. The A and B scales may also be used, but they are not as accurate because the increments are smaller. To multiply two numbers, proceed as follows: (1) Locate one number on the D scale. Slide the indicator until the hairline is over the number to mark its location. (2) Place one of the indexes of the C scale above the number on the D scale. Use the hairline of the indicator to aline the index and the number. (3) Locate the second number on the C scale. If the number is located on the portion of the C scale beyond the end of the D scale, reposition the slide so that the other index on the C scale is above the number on the D scale. (4) Slide the indicator so that the hairline is over the number on the C scale. The product of the two numbers is read under the hairline on the D scale. b. The two examples below illustrate the method of multiplication described above. They also point out the use of the two indexes on the C scale. Example 1: Multiply 2 by 3. product of 2 times 3 or 6 is read Example 2: Multiply 6 by 3. Step 2. Step 3. 7. Division Locate the number 6 on the D scale and slide the indicator so that the hairline is over it. Place the right-hand index of the C scale above the number 6 on the D scale. Use the hairline on the indicator for alinement. (The right-hand index is used because the number 3 on the C scale would be beyond the end of the D scale if the lefthand index were used.) Locate the number 3 on the C scale and slide the indicator so that the hairline is over it. The product of 6 times 3 or 18 is read under the hairline on the D scale. Figure 98 shows a slide rule arranged for this product. a. The process of division, like multiplication, generally is performed by using the C and D scales. To divide one number by another number, proceed as follows: (1) Locate the dividend (number to be |