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MATHEMATICAL SIGNS + Plus. The sign of addition : indicates that the quantity is to be added, as Lat. left 27° og'25” N. Long. left 68° 53' 45" W. Dif. + 2 53 45 N. Dif. + 2 17 45 W. Lat. ship 30 03 10 N. Long. ship 71 II 30 W.

– MINUs. The sign of subtraction: indicates that the quantity is to be subtracted, as Lat. left. 38° 16' 15" S. Long. left 35° 22' 15" E. Lat. ship - 32 28 30 S. Long. ship — 33 46 45 E. Dif. 5 47 45 N.

Dif.

I 35 30 W.

X Multiply. The sign of multiplication: indicates that the quantities are to be multiplied, as

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• DIVIDE. The sign of division : indicates that the quantity is to be divided, as 63° 15' 30" • 2, and is usually written thus: 63° 15' 30"

2)63° 15' 30" 2

Ans. 31 37 45

= EQUALITY. The sign of equality: indicates that the quantities between which it is placed are equal, as in latitude 30°, departure 60' = difference of longitude 120'.

DECIMAL POINT. The sign indicating by its position the value of a number, as

151. is one hundred and fifty one.
15.1 is fifteen and one-tenth.
1.51 is one and fifty-one hundredths.
.151 is one hundred and fifty-one thousandths.

DECIMALS

Of all the mathematical signs indicating operation or value, the decimal point is the most important, and is given the least consideration by many who study navigation and nautical astronomy. For that reason it is suggested that the student should not begin advanced work until the use of the decimal point is thoroughly known and practised.

A decimal is a fraction having some power of ten for its denominator which is not used, the value being fixed by the location of the decimal point.

Ciphers to the right of a decimal does not change its value; but between the decimal point and the figures, it does.

ADDITION The addition of decimally expressed quantities is the same as whole numbers, if they are so placed that the decimal points, including that of the sum, are directly under the one uppermost. Examples: Add 45.63

10.II 90.43

.151 1.015

10.3 181.9

1.821 Ans. 318.975

Ans. 22.382

SUBTRACTION The subtraction of one decimally expressed quantity from another is the same as whole numbers if they are so placed that the decimal points, including that of the remainder, are directly under the one uppermost.

In teaching subtraction, it is customary to place the smaller number under the greater. That cannot be adhered to in this work, as they must occupy a proper position. Examples: Subtract — 273.54

8328.469 9321.4306

– 869.783 Ans. 9047.8906

Ans. 7458.686

MULTIPLICATION The multiplication of quantities expressed decimally is the same as whole numbers, except that the product must contain consideration. If the product does not contain a sufficient consideration, and if the product does not contain a sufficient number of figures place the proper number of ciphers to the left. Examples: Multiply 58”.68

3.7 5. X 7.35

x 16.5 29340

1875 17604

22 50 41076

375 Ans. 431.2980

Ans. 6 1.875

DIVISION If the divisor contains decimals, the decimal point in the dividend must be moved to the right as many places as there are decimals in the divisor. Then divide as if a whole number.

Examples:
5 8.6 8)4 3 1.2 9,8(7.35 Ans.

41076

20538
17604
29 340
29340

3.7 5)6 1.87,5 (1 6.5 Ans.

375 2437 2250 1875 1875

THE USE OF COMMON LOGARITHMS. When numbers are to be multiplied or divided, logarithms may be advantageously used, as by doing so those operations are performed by simple addition or subtraction and are essential in the practice of navigation.

The logarithm is composed of two parts: the index and the decimal portion. The index is fixed by the number of figures composing the number for which the logarithm is wanted. The decimal part is found in Table 42.

The index is one less than the number of figures to the left of the decimal point.

4563. index is 3.7
456.3 “ “ 2.
45.63 " ", } .65925
4.563 “ “ o.)

To take a logarithm from Table 42, find the first three figures in the first column and the fourth figure at either the top or bottom and the proper logarithm will be found at the intersection. When using this table, bear in mind that the numbers 1, 2, 3, 4, 5, etc., at the top and bottom of the table are not tenths. They are the fourth figures. The logarithm for a decimal is found in the same manner.

If the number is a decimal without ciphers between the decimal point and the first figure, the index is 9. If there are ciphers between the decimal point and the first figure, subtract i from 9 for each cipher to obtain the index, as

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but the decimal part from Table 42 remains the same.

MULTIPLICATION

To multiply one number by another, select the logarithms from Table 42, and when properly indexed their sum will be the logarithm of the product. This logarithm, when found in Table 42, will be at the intersection of some figures in the first column and under one of the numbers at the top of the page. These four figures pointed off from the right, so there will be one more than the index, will be the product wanted.

ne interse the nun from

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Example: Multiply 92 by 8 and the product by 6.

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