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ARITHMETIC.

ARITHMETIC is the science of Numbers.

Numbers are applied to express either the repetitions or the parts of an object or a quantity, and according to these applications they are called either whole or fractional.

A single object is sometimes called unity with reference to several individuals of the same sort; as one man with reference to several men.

A single dimension of value, weight, measure, &c. is called an integer, or an integral quantity; as one pound, one yard.

A simple quantity is a dimension of value, &c. expressed in only one denomination; as seven pounds, or five yards.

A compound quantity is a dimension expressed in more denominations than one; as seven pounds, five shillings, and sixpence, when used to express one sum of money.

NUMERATION.

Numeration is the valuation of numbers expressed by figures. There are 9 figures, and to fill up a vacant place, a cipher, also called a nought, is used.

Numbers are divided into classes and periods; there are 6 figures in a class, and 3 in a period.

The figures in the first period are called units, tens, and hundreds; those in the second period are called thousands, ten thousands, and hundred thousands.

The first class is called the class of units, the second that of millions, and the third that of billions; therefore if a number expressed at length has not more than 6 figures, it will not contain millions; and if it has not more than 12 figures, it will not contain billions.

Table of the first three periods.

In this number 9 8 7 6 5 4 3 2 1, the 9 is hundred-millions, the 8 is ten-millions, the 7 is millions, the 6 is hundred-thousands, the 5 is ten-thousands, the 4 is thousands, the 3 is hundreds, the 2 is tens, and the 1 is a unit.-The whole is read,

Nine hundred and eighty seven millions, six hundred and fifty four thousand, three hundred and twenty one.

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Two thousand, six hundred and twenty seven

Twenty five thousand, eight hundred and ninety two

One hundred and seventeen millions, two hundred and sixty five thousand, one hundred and five.

ROMAN NUMERALS.

The following table exhibits this mode of expressing all the units, tens, and hundreds.

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SIMPLE ADDITION.

Simple Addition is finding the amount of two or more quantities of the same denomination, expressed by whole numbers.

Rule. Place the numbers so that the figures of the same class may be one under the other.

Then, beginning with the units, find the amount of each column of figures, put down the unit figure of each total, and carry the tens to the next column, except for the last, of which set down the whole of the amount.

Proof. Separate the numbers into two or more parts, find the amount of each, and add these amounts together.

Or, reverse the addition; that is, if the work is begun with the bottom figure, let the proof begin with the top figure.

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N. B. It was formerly the custom to place a dot against every ten found in the addition of each column; and then rejecting the ten thus formed, to add in only the units; as 9 and 2 are 11, dot and go 1; 1 and 4 are 5 and 4 are 9 and 8 are 17, dot and go 7; then putting down 7, carry 2 to the next row because there are 2 dots.

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SIMPLE SUBTRACTION.

Simple Subtraction is finding the difference between two simple quantities of the same denomination, expressed by whole numbers.

RULE.

Place the less number under the greater, with the figures of the same class under each other, as in Addition.

Then, begin with the lower unit figure, (and so proceed with the rest,) and find the difference between that and the figure above it, when the lower figure is not the greater; but when the lower figure is greater, borrow 1 from the next column, call it 10, add it to the upper figure, subtract the lower figure from the amount, set down the remainder, and carry one to the next lower figure.

Proof. Add the remainder to the lower line.

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Note. In the Examples and Exercises to this and the following Rules on simple quantities, any denomination that is preferred may be given.

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