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(2) The E-6B aerial dead reckoning computer material is included because the computer is a device which applies the principles of mathematics learned. .
(3) Sufficient material has been included to enable the trainee to solve the simpler problems of navigation and aerial bombardment numerically by vector-diagram construction, or by slide rule and computer. The solution of the oblique spherical triangle is given for the purpose of solving the astronomical triangle which is the basis of celestial navigation.
d. In mathematics, as in learning to fly, no amount of reading can replace actual practice. For this reason, many exercises have been included in each paragraph, and at the end of each section a collection of miscellaneous exercises has been added based on the material in that section. It is not contemplated that every student will do all of the exercises. However, an ample number of exercises has been inserted to provide an opportunity for those trainees who may feel the need for extra practice. The answers to the even numbered exercises are given to enable each student to check his own work if he wishes. Illustrative examples are profuse and should help to clarify difficult points which may arise.
e. Undoubtedly, some of the topics will seem very simple to many of the trainees. It must be remembered, however, that the mathematical proficiency demanded of a pilot not only involves an understanding of the various operations, but also the ability to perform these operations accurately and quickly, and often under trying circumstances. Therefore the time spent in practicing such a simple operation as addition, for example, will not necessarily be so much time wasted, no matter how clearly the process is understood.
2. Materials. In addition to pencil and paper, the student will need a ruler, a protractor, graph paper, a compass, and an E-6B computer.
13 Positive and negative numbers ----Addition of positive and negative numbers -Subtraction of positive and negative numbers Multiplication and division of positive and negative Square root ----------Miscellaneous exercises------
3. Purpose and scope.—The purpose of this section is to provide a review of the four fundamental operations of arithmetic: addition, subtraction, multiplication, and division. Upon these fundamental operations all other mathematical calculations are based. One or more of them must be used in solving any problem.
4. Addition.-a. Addition is the operation of finding the sum of two or more numbers. To add several numbers, place the numbers in a vertical column so that the decimal points are all in a vertical line. (When no decimal point is indicated, it is assumed to be on the right.) Then add the figures in the right-hand column and place the sum under this column. If there is more than one figure in this sum, write down only the right-hand figure and carry the others to the next column to the left.
Example: Find the sum of 30.53, 6.475, 0.00035, and 3476.
Figure 1. . b. Units.—Almost all the numbers which arise in practical arithmetic have to do with definite quantities such as 78 feet, 239 miles, 25 degrees, 160 miles per hour, 210 pounds, and so on. In these ex
amples, the words in italics, which state what the quantities are in each case, are called the units. When adding several quantities together, it is clear that the units must all be the same. For example, “the sum of 78 feet and 160 miles per hour” is a completely meaningless statement.
(1) Units are so important and occur so often that standard abbreviations have been adopted for them. A list of the correct abbreviations and the relations which exist between some of the units are given in the appendix. (2) Example: Find the sum of 78 feet and 3 miles.
Solution: In this case, since 1 mile is the same as 5,280 feet (see app. I), then 3 miles are the same as 15,840 feet. Therefore, 78 feet and 15,840 feet may be added together to give 15,918 feet. But the student is cautioned that unless there is a relation between the various units so that all the quantities may be expressed in terms of the same units, the addition cannot be performed.
C. Symbols.-In arithmetic and in other branches of mathematics, much space and effort are saved by using symbols. Thus, in order to write "find the sum of 70.765 and 301.4,” the plus sign (+) is used and this phrase can be written simply as “70.765+301.4=?.” When more than two numbers are to be added, the plus sign is repeated, for example: 70.765+301.4+765.84=1,138.005.
d. Exercises. (1) 30.53 in. +6.475 in.= ? (2) 648.03 cm.+37.895 cm.+219.921 cm.+.08376 cm.=905.92976 cm.
Answer. (3) 100.001+9.098+5678.91= ? (4) 897.1 +0.989+900.76 +91901.359=93700.208
Answer. (5) 9876 ft.+101.109 ft. +77.007 ft. +92.928 ft.+94.987 ft. +60.768
ft.= ? (6) 19.767+43.542+76.305+58.143+13.25=211.007. Answer. (7) 11.1111 miles +66.667 miles +1.222 miles +125.125 miles +
375.375 miles=? (8) 78.908+202.202+62.501 +0.003594+75=418.614594. Answer. (9) 7.8098+20.202+6.2501+000.3594+7.5=?. (10) 78.808 yd.+98.15 yd. +760 yd. +88199.76 yd.=89136.718 yd.
Answer. 5. Subtraction.—a. Subtraction is the operation of finding the difference between two numbers. In order to subtract one number from another, write the smaller number below the larger so that the decimal points are in a vertical column. Beginning with the right column, subtract the figures in the smaller number from the corresponding figures in the larger number above them.
Example: Subtract 765.3 from 986.7.
Answer. b. If, however, the figure in the number being subtracted is larger than the figure directly above it, it is necessary to borrow one unit from the next figure to the left. (1) Example: Subtract 765.3 from 843.1. Solution:
,77,8 - Answer
FIGURE 2. (2) It is better to learn to do the "carrying over” mentally so that the preceding solution looks like this:
Answer. c. When a column has only one figure in it, zeros must be supplied in the blank spaces.
Example: Subtract 765.328 from 843.1.
8431 - EXTRA
: FIGURE 3. d. A problem in subtraction may be checked by adding the answer to the number directly above it. The sum should always be the number in the top row. (1) Example: Check the answer to the example in c(1) above. Solution:
§ 84 31oo.
(2) Units. As in addition, care must be exercised to be sure that the units of the two quantities in a subtraction are the same.
e. Exercises.-In each of the following exercises subtract the smaller number from the larger. The symbol used to indicate subtraction is the minus sign (-). Therefore, an expression such as "1,205.789— 980.833=?” means “What is the remainder when 980.833 is subtracted from 1,205.789?” The number following the minus sign is always subtracted from the number preceding the sign.
(1) 1,205.789 in. —980.833 in.=? (2) 19.52—.78=18.74
Answer. (3) 760,591—674,892=? (4) 73.44 cu. in.-8.7375 cu. in.=64.7025 cu. in.
Answer. (5) 89.73–10.0045=? (6) 941.7-87.372=854.328
Answer. (7) 1,004.78 miles-1,004.164 miles=? (8) 100,433 sq. ft. -99,857 sq. ft.=576 sq. ft.
Answer. (9) 1,000,000.3—998,757.4=? (10) 3,756.04—2,489.7=1,266.34
Answer. 6. Multiplication.—a. Multiplication is a short method of adding a number to itself as many times as may be indicated. The numbers multiplied together are called the factors and the result of the multiplication is called the product. To multiply two numbers together, or in other words, to find the product of two factors, first write the factors one below the other (see example below). It is usually easier to operate with the smaller number of figures in the bottom row. Multiply the factor in the top row by the right-hand figure of the factor in the bottom row, and write this partial product directly under the second factor. If there is more than one figure in the product the same “carrying over” procedure is followed as in addition. Then multiply the factor in the top row by the second figure from the right in the second factor, and write this second partial product so that its right-hand figure is directly under the figure that was used to find it. These partial products are then added together to yield the required product.
Example: Multiply 1,653 by 247.