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THE

PUPILS's GUIDE, &c.

ARITHMETIC is the art of reckoning by numbers, and is founded on the five following rules, viz. Numeration, Addition, Subtraction, Multfplication, and Divifion. By the right applica tion of thefe, all other rules of Arithmetic are wrought.

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NUMERATION.

NUM

UMERATION teaches to express any number, compofed of these ten characters; 1, 2, 3, 4, 5, 6, 7, 8, 9, o. The firft nine of these, or rather all of them, are called figures or digits-o is called nought, or cypher: and when alone, is of no fignification; but when annexed to the right hand of other figures, it makes the number ten times as much as it was before; thus, 7, by annexing a cypher, (70) becomes feventy.

When a number of figures are fet together, the first, at the right hand, reprefents its own.

value; that in the fecond place, ten times its own value; that in the third place, a hundred times its own value, &c. according to the following table.

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In the foregoing table, I have divided the figures by commas into periods of three figures each. The firft period. (beginning at the right hand,) is a period of units; the fecond is a period of thousands; the third is a period of millions.

The first figure at the right hand of the first period is one; the two firft, taken together,

are twenty one; and the whole period is three hundred and twenty one,

The first figure of the fecond period is four thousand; the two firft, taken together, are fifty four thoufand; and the whole period is fix hundred and fifty four thousand.

The first figure of the third period is feven millions; the two firft are eighty feven mil lions; and the whole period is nine hundred and eighty seven millions; which, joined to the other two periods, is nine hundred and eighty feven millions, fix hundred and fifty four thousand, three hundred and twenty one, 987,654,321.

In like manner, you may enumerate figures which stand in a different order from thofe in the table. In feparating them into periods, you must begin at the right hand, and proceed towards the left; but in expreffing their value, you must begin at the left hand, and read towards the right.

A TABLE to enumerate a greater number of

figures.

w Hund. of thous. of bill.
~ Tens of thous. of bill.
Thous of billions.
Hundreds of billions.
** Tens of billions.

3 2 1,9 8 7,6 5 4,3 2 1,9 8 7 6 5 4

Hunds. of th. of mill.
Tens of thous, of mill.
of millions..
Billions.

Thous.

w Hundreds of millions.
Tens of millions.

Hundreds of thous.
Tens of thousands.
Thoufands.
• Hundreds.

- Millions.

ir Tens.
→ Units,

You may proceed in the fame manner from billions to trillions, quatrillions, quintillions, fextillions, feptillions, octillions, Ronillions, decillions, &c.

Billions are millions of millions; Trillions are millions of millions of millions; Quatrillions are the 4th degree of millions; Quintillions are the 5th degree of millions; Sextillions are the 6th degree of millions, &c.

APPLICATION.

Write down in figures the following numbers. 1. Two hundred and four.*

2. Four thousand, three hundred.

3. Eleven thousand and forty four.

4. One hundred and fifty feven thousand, one hundred and five.

s. Nine million, fix thousand, one hundred and fifty two.

6. Fourteen million and a half.

Write down in words the following numbers.

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Thofe places of the natural order, which are omitted in the question, must be supplied with exphers.

SIMPLE ADDITION.

Simple Addition teaches to colle& two or more numbers of the fame denomination into one Jum.

RULE.

1. Place the numbers under each other, fo that units may ftand under units, tens under tens, hundreds under hundreds, &c. and draw line underneath.

2. Add up every figure in the column of units, and feek how many tens are contained in their fum; fet down what is over exactly under the column added, and carry one for every ten to the next column, with which proceed as before, and fo on, to the laft, at which fet down the whole amount.

PROOF.

1. Cut off the upper line, and add all the reft together, fetting their fum under the num ber to be proved.

2. Add this amount to the top line, and if the work be right, their fum will be the fame as that found by the firft addition.

Or, Begin at the top and reckon the figures downwards, and if the work be right, this sum will be equal to the firft.

* The method of carrying one for every ten from an inferior to a Juperior column, is evident ly right, because one unit in the latter cafe is equal to the value of ten units in the former.

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