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COURSE IN MATHEMATICS FOR

ARTILLERY GUNNERS

CHAPTER I

DEFINITIONS AND USE OF MATHEMATICAL SIGNS Mathematics is the science of quantity.

Science is an arrangement of the principles of any subject in regular and proper order.

Quantity is anything that can be increased, diminished, or measured. To measure a quantity is to find how many times it contains some other quantity of the same kind, called the unit of measure. A unit is a single thing of any kind.

In pure mathematics, or mathematics which considers quantity without regard to matter, there are but eight kinds of quantity, and hence only eight kinds of units, viz.: Units of number, of length, of surface, of volume, of weight, of time, of currency, and of angular measure.

Quantity includes number and magnitude. A number is one or more units; a magnitude anything that can be measured. Number answers the question How many? magnitude, How much?

Arithmetic is the science of numbers. In arithmetic numbers are usually expressed by figures; as 1, 2, 3, etc. Numbers are called abstract when the kind of unit is not named; as one, two, three, etc.; denominate, when the unit is named; as one yard, two pounds, three seconds, etc.

A problem is a question regarding quantity proposed for solution.

A solution is a statement of the mathematical work done to obtain the answer to a problem.

A rule is a general direction for solving problems of the same kind.

In stating mathematical work it is often convenient to indicate by characters, called mathematical signs, what is to be done with the quantities considered, or the relations which exist between them.

The signs most used in arithmetic are +, -, X, ÷, and √, which indicate work to be done, and are called signs of operation, and —, (), [], -, :, and ::, which show relation =, between or among quantities, and are called signs of relation.

+ is the sign of addition, and is read "plus." The numbers between which it is placed are to be added. Thus: 4 + 2, is read "4 plus 2," and equals 6.

is the sign of subtraction, and is read "minus." When placed between two numbers, the one on the right is to be taken from that on the left. Thus: 5-3 is read "5 minus 3," and equals 2.

X is the sign of multiplication, and is read "multiplied by," or "times." The numbers between which it is placed are to be multiplied together. Thus 4 X 5 is read "4 multiplied by 5," or "4 times 5," and equals 20.

÷ is the sign of division, and is read "divided by.” When placed between two numbers, the one on the left is to be divided by that on the right. Thus: 6 ÷ 3 is read "6 divided by 3," and equals 2.

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(), and [] are signs of aggregation, or of bringing together. The first is the vinculum; the second, the parentheses; the third, the brackets. They are used for the same purpose to connect several quantities. Numbers placed under the vinculum, or within the parentheses or brackets, are to be considered as one quantity. Thus: 6 − 3 + 2, or 6 – (3+2), or 6−[3+2] means that the sum of 3+2 is to be taken from 6. Brackets are ordinarily used only when the total

relation can not be shown by means of a single vinculum or parentheses.

=

is the sign of equality, and is read "is equal to," or "equals." Quantities between which it is placed are equal. Thus: 3+2×4=(6+4)×2=[(3+2)×(9−1)]÷2=20.

√ is the radical sign;, the sign of ratio; ::, the sign of proportion. Their uses will be explained hereafter.

Arithmetic depends on the general principle that any number may be increased or diminished. The fundamental operations of arithmetic are addition, subtraction, multiplication, and division. Multiplication is simply a short method of adding equal numbers; division, a short method of making several subtractions of the same number. Thus: 5 can be multiplied by 4 by adding four 5's together; and 20 can be divided by 5, or how many times 20 contains 5 can be found by subtracting four 5's from 20.

Numbers are therefore classified as positive numbers, or numbers to be added together; and negative numbers, or numbers to be subtracted from positive numbers, but added to other negative numbers. Positive numbers are preceded by the sign+; negative numbers by the sign —. When a number has no sign before it, it is considered positive.

In every mathematical expression a+ or a- affects the whole result of the work indicated between it and the next followingor, or, between it and the end of the expression. In no case can a X or a ÷ affect any quantity before the preceding, or beyond the following+ or -. Thus: in the expression 5+7×3×6-4X3, the + indicates the addition of 126, not of 7 only; and the indicates the subtraction of 12. The same meaning is better expressed by 5+(7×3×6)−(4×3).

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The signs and simply show what operations are to be performed on the positive or negative numbers which precede them. When they occur in succession they have their effects in the order of their occurrence. Thus: in the expression [30-(6×4)]÷3, the sign X shows that 6 is to be multi

plied by 4, but it does not show what is to be done with the resulting 24; this is shown by the -. 24 is to be taken from 30, and the remainder divided by 3.

PROBLEMS.

1. 4×3+7X2-9×3+6×4-3×3=what? Ans. 14. Solution: 4X3=12; 7X2=14; -9×3=-27; 6×4=24; -3X3=-9. Grouping according to the + and signs, and adding, we have, 12+14+24=50; and -27-9=-36. Therefore, 50-36=14, Ans.

2.

8×2-9÷3+4×5-6÷3-7X5=what?

Ans. -4.

Solution: 16-3+20-2-35. Grouping and adding, we have, 36-40=-4, Ans.

3.

4.

21÷3X7-1X1÷1X4÷2+18÷3×6÷(2×2)+(4−2+6

-7)X4X6÷8=what? Ans. 59.

Solution: Performing operations indicated between + and signs, we have 49-2+9+3, or 61-2-59, Ans.

16X4÷8-7+48÷16-3+24X6÷48-4X9÷÷12-1=

what? Ans. 0.

5. (16÷16X96÷8-7-5+3)× [(24÷4)÷3−1]+(91÷13×7

6.

=

-45-3)X9 what? Ans. 12.

(12+4×9÷3)× [(2+5×2÷1+4×3-10)÷2-1]÷[282X(3X2+8÷4)-2]=what? Ans. 12.

CHAPTER II

COMMON FRACTIONS

A fraction is one or more equal parts of a divided unit.

A whole number is one or more units.

A mixed number consists of a whole number and a fraction. Fractions are divided into two classes-common fractions and decimal fractions; the former are ordinarily called simply fractions; the latter, when expressed as hereafter explained, decimals.

A common fraction is expressed by writing one number above and another below a line. Thus: is a common fraction which is read "three-fourths." The number above the line is called the numerator; that below, the denominator. The numerator and denominator are called the terms of the fraction.

A proper fraction is one whose numerator is less than its denominator; as, 1, 2, .

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An improper fraction is one whose numerator is equal to or greater than its denominator; as, 1, 1.

A whole number may be expressed as an improper fraction by writing 1 for its denominator. Thus 4=1; 5={.

A mixed number may be expressed as an improper fraction by multiplying the whole number by the denominator of the fraction, to the product adding the numerator and writing the sum over the denominator. Thus: 28=4; 32=4.

CANCELLATION.

Cancellation is a process by which operations in fractions may often be shortened. It depends on the principle that if

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