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Multiple of a number. The product obtained by taking the given

number an integral number of times. Multiplicand. The number to be multiplied by another. Multiplication. The operation of finding a number bearing the

same ratio to the multiplicand which the multiplier bears to unity. Multiplier. The number by which the multiplicand is multiplied. Net proceeds. The amount that remains of the money received for

property after paying all expenses incurred in disposing of it. Notation. A system of expressing numbers by symbols. Note. A written agreement to pay a specified sum of money at a

specified time. Number. The answer to the question, How many? Numeration. A system of naming numbers. Obligation. A debt, or liability to pay. Order of units. A name used to designate the number of things

in a group, as tens, hundreds, thousands, etc. Partial payment. Part payment on a note. Partnership. An association of two or more persons to carry on

business. Par value. Face or nominal value. Pendulum. A body suspended by a straight line from a fixed point,

and moving freely about that point as a centre. Percentage. A part of any given number reckoned at some rate

per cent.

Period. A group of three figures.
Policy. A written contract of insurance.
Poll tax. A tax levied by the head or poll.
Power. The product of two or more equal factors.
Premium. The sum paid for insurance computed at some rate per

cent of the amount insured. Also the excess of market value above

par

value. Present worth. The present value of a debt due at some future day. Prime number. A number which has no integral factors except

itself and one. Principal. The sum of money drawing interest. Problem. A question to be solved. Product. The result obtained by multiplying the multiplicand by

the multiplier. Profit. The excess of selling price above cost.

Proof The evidence by which the accuracy of any result is estab

lished. Proper fraction. A fraction, the numerator of which is less than

the denominator. Proportion. A statement that two ratios are equal. Quantity. The answer to the question, How much ? Quotient. The number sought in division. Rate per cent. Rate by the hundred. Ratio. The relative magnitude of two numbers or of two quantities. Reciprocal of a number. One divided by that number. Reduction. The process of changing the unit in which a quantity

is expressed without changing the value of the quantity. Remainder. The number which, added to the subtrahend, gives a

sum equal to the minuend. Root of a number. One of the equal factors of the number. Rule. The statement of a prescribed method. Security. Property used to guarantee the payment of any debt. Share. One of a certain number of equal parts into which the capi

tal of a company is divided. Short division. The method of dividing in which the operations of

multiplying and subtracting are performed mentally. Solid. A magnitude which has length, breadth, and thickness. Solution. The process by which the answer to a question is obtained. Specific gravity of a substance. The ratio of the weight of a

given volume of it to that of an equal volume of water. Square root. One of two equal factors. Stock. Capital invested in business. Subtraction. The process of finding a number which added to one

of two given numbers will produce the other. Sum. The number which results from combining two or more num

bers together. Surd. An indicated root the value of which cannot be exactly ex

pressed in figures. Surface. That which has only length and breadth. Thermometer. An instrument for measuring heat. Unit. A single thing. Also, an arbitrary length, adopted as a stan

dard of measure, in terms of which all measurements are expressed. Verify. To establish, by experiment, the truth of any statement. Volume of a solid. The ratio of a solid to an assumed unit of

measure; usually a cube of the linear unit.

A

GRAMMAR SCHOOL ARITHMETIC.

CHAPTER I.

PRELIMINARY DEFINITIONS.

1. A COLLECTION of several similar objects (as a collection of apples) or the repetition of the same event (as successive peals of thunder) gives the idea of Number.

2. The idea of number presents itself also when we wish to express the values of quantities in terms of some wellknown value.

3. A Unit is a fixed value with which we compare

all quantities of the same kind. Each kind of quantity has its own unit. Thus:

The unit of length is the yard.
The unit of surface is the square yard.
The unit of capacity is the quart.
The unit of weight is the pound.
The unit of money

is the dollar.

4. To measure a quantity is to find the number of times the quantity contains its unit.

5. Number results from measuring a quantity. If the unit is contained in a quantity several times without remainder, the result is an Integral Number. Thus, the integral number three will represent the length of a line, if the line contains the yard, three times exactly.

6. If the quantity to be measured is less than the unit, we divide the unit into equal parts and find how many times one of these parts is contained in the given quantity. Thus, to measure a line less than a yard, we can apply to this line a third part of a yard, and if this third is contained twice in the line exactly, the length of the line is expressed by two-thirds. The expression two-thirds is called a Fraction.

7. If a line contains the yard five times and one-fourth of a yard three times, the length of the line is expressed by five and three-fourths. The expression five and threefourths is called a Mixed Number.

8. Arithmetic comprises all questions that can be proposed upon numbers.

NOTATION AND NUMERATION.

9. The first numbers have special names, as follows:

one, two, three, four, five, six, seven, eight, nine, ten.

10. The first nine of these numbers are called Simple Units, or units of the first order.

11. The group of ten units has received the name of a ten, or a unit of the second order; and we count by tens as by units; thus:

one ten, two tens, three tens ... nine tens, ten tens.

12. The group of ten tens has received the name of a Hundred, or a unit of the third order; and we count by hundreds, as by tens and units; thus:

one hundred, two hundreds ... ten hundreds.

13. A group

of ten hundreds is called a Thousand, or a unit of the fourth order.

14. From ten units of the fourth order is formed a ten thousand, or a unit of the fifth order; and from ten units of the fifth order is formed a hundred thousand, or a unit of the sixth order.

15. Units of the seventh order are called Millions ; of the eighth order, ten millions; of the ninth order, hundred millions. Finally, units of the tenth order are called Billions; units of the thirteenth order, Trillions; and so on.

16. The table of units of different orders is as follows:

first class.

second class.

First order,

simple units,
Second order, tens of units,
Third order,

hundreds of units,
Fourth order, thousands,
Fifth order, tens of thousands,
Sixth order, hundreds of thousands,
Seventh order, millions,
Eighth order, tens of millions,
Ninth order, hundreds of millions,
Tenth order, billions,
Eleventh order, tens of billions,
Twelfth order, hundreds of billions,
Thirteenth order, trillions,

}

third class.

fourth class.

fifth class.

17. The

group

of the first three orders is called the first class of units, and the group of the three following orders, the second class, and so on.

18. The unit of the second class is equal to a thousand units of the first class, and a unit of the third class is equal to a thousand units of the second class, and so on.

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