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and those units which do require us to think of other things of which they are parts, are called fractions.

What is the difference between units that are whole numbers, and units that are fractions ?

Reduce two yards to quarters, and which kind of reduction is it?

Reduce twenty-four inches to feet, and which kind of reduction is it?

Reduce three feet to inches, and which kind of reduc. tion is it?

Which is of highest value, a half or a quarter ?

Reduce eight quarters to halves, and which kind of reduction is it?

Reduce two halves to quarters, and which kind of re. duction is it?

Reduce sixteen quarters to halves, and which kind of reduction is it ?

Reduce two fifths to tenths ; six tenths to fifths ; eight tenths to fifths ; twelve tenths to fifths; three fifths to tenths ; six fifths to tenths.

Reduce one seventh to fourteenths; four fourteenths to sevenths ; four sevenths to fourteenths ; eight fourteenths to sevenths.

Reduce two sixths to twelfths; four twelfths to sixths; eight twelfths to sixths; five sixths to twelfths.

SUMMARY OF DEFINITIONS.

A unit is any whole thing of a kind.
A fraction is a part of a thing.
Addition is uniting several numbers in one.

Subtraction is taking one number from another, to find the remainder.

The largest number is the minuend, the smallest number is the subtrahend.

Multiplication is repeating one number as often as there are units in another number.

The multiplicand is the number to be repeated; the multiplier is the number which shows how often the multiplicand is to be repeated; the factors are both the multi

plier and multiplicand; and the product is the number ob. tained by multiplying,

Division is finding how often one number is contained in another number, and thus finding what part of one num. ber, is another number.

The dividend is the number to be divided. The divisor is the number by which you divide. The quotient is the answer obtained by dividing.

Reduction is changing units of one order, to units of another order.

Reduction ascending, is changing units of a lower, to a higher order.

Reduction descending is changing units of a higher, to a lower order. NOTE TO TEACHERS.—

A review of this First Part, will be found more useful than an increased number of ex. amples.

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ARITHMETIC.

SECOND PART.

NUMERATION. Numeration is the art of expressing numbers by words, or by figures.

Figures are sometimes called numbers, because they are used to represent numbers. Thus the figure 4, is often called the number four, because it is used to represent that number.

There are thirty-five words, that are commonly used in numeration ; viz. i one, two, three, four, five, six, seven, eight, nine, ten, eleven, twelve, thirteen, fourteen, fifteen, sixteen, seventeen, eighteen, nineteen, twenty, thirty, forty, fifty, sixty, seventy, eighty, ninety, hundred, thousand, million, billion, trillion, quadrillion, quintillion, sextillion.

Those words ending in teen, are the words three, four, &c. with teen, which signifies and ten, added to them.

What is the meaning of fourteen? Ans. Four and ten. What is the meaning of thirteen of nineteen? of seventeen?

Those ending in ty, are the words two, three, four, &c. with ty, which means tens, added to them.

What is the meaning of sixty ? of seventy ? of eighty? of twenty ? of thirty?

The words of spoken numeration would be more uni. form, if eleven and twelve, had been called oneteen and two. teen.

The Latin and Greek numerals are so often used in the various sciences, that it is important for pupils to learn

What is numeration? Why are figures sometimes called numbers ? How many words are used in numeration, and what are they? What does teen signify? What is the meaning of fourteen? seventeen? nine teen? What does ty signify? What is the meaning of sixty ? seventy? eighty? twenty ? thirty ?

Latin Names.

Greek Names.

their names. They are therefore put down with the fig. ures, and the English names. The figures are called Arabic, because first introduced into Europe from Arabia.

ENGLISH, LATIN, AND GREEK NUMERALS. Arabic Figures. English names. 1 One.

Unus.

Eis. 2 Two.

Duo.

Duo. 3 Three. Tres.

Treis. 4 Four.

Quatuor. Tessares. 5 Five.

Quinque. Pente. 6 Six.

Sex.

Hex. . ng Seven.

Septem.

Hepta. 8 Eight.

Octo.

Okto. 9 Nine.

Novem. Ennea.
10 Ten.

Decem. Deka.
11 Eleven. Undecim. Endeka,
12 Twelve. Duodecim. Dodeka.
13 Thirteen. Tredecim. Dekatreis.
14 Fourteen. Quatuordecim. Dekatessares.
15 Fifteen. Quindecim. Dekapente.
16 Sixteen. Sexdecim. Dekaex.
17 Seventeen. Septendecim. Dekaepta.
18 Eighteen. Octodecim. Dekaocto.
19 Nineteen. Novemdecim. Dekaennea.
20 Twenty. Viginti. Eikosi.
30 Thirty. Triginta. Triakonta.
40 Forty.

Quadraginta. Tesserakonta.
50 Fifty.

Quinquaginti. Pentakonta. 60 Sixty. Sexaginta. Hexakonta. 70 Seventy. Septuaginta. Hebdomekonta. 80 Eighty. Octoginta. Ogdoekonta. 90 Ninety. Nonaginta. Ennenekonta. 100 Hundred. Centum. Hekaton. 1000 Thousand. Mille.

Chilio. 1000000 Million.

Billion, Trillion, Quadrillion, Quintillion, Sextillion, &c. are made by adding ciphers to 1.

If any higher number than sextillion is to be expressed,

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Why are the figures used called Arabic ?

the names are made by the Latin numbers, with illion add. ed to them; as septillion, octillion, &c.

A unit has been defined as a single thing of any kind.”

But a unit of one kind, may be made up of several units of another kind. Thus the unit one dollar is made

up

of ten units, of the kind, or order called dimes; and one dime is made up of ten units of the order called cents. Order means the same as kind.

A unit which is of the most value, is called a unit of a higher order.

Which unit is of the highest order, a dollar or a cent?

How many units of the order of dimes, are there in one unit of the order of dollars?

How many units of the order of mills, make one unit of the order of cents ?

How many units of the order of cents make one unit of the order of dimès ?

Every figure expresses a certain number; but the num. ber it expresses, depends upon the order in which it is placed.

If the figure (2) stands alone, it expresses two units, and is said to be in the first or unit order.

But if it has a figure at the right of it, thus (20) it ex. presses two tens, or twenty, and is in the second order, or the order of tens.

The cipher is put at the right, to make the 2 stand in the order of tens, and to show that there are no units of the unit order. If some figure was not placed there, the 2 would be in the unit order.

If the figure 2 has two figures at the right of it, thus (200) it represents two hundreds, and stands in the third order, or the order of hundreds.

From this it appears, that in numeration, the number expressed by any figure, depends upon the order in which it stands.

How are higher numbers than sextillion expressed? What is a unit? What distinction is made in regard to units on page 38 ? What is the meaning of the word order ? What is meant by a unit of a higher order ? What does every figure represent? What does the number which any figure represents depend upon ? If a figure stands alone, in what order is

it? If it has one figure at the right of it, in what order is it? If it has two figures at the right of it, in what order is it? In this number, (234) in what order is the 2 ? the 3 ? the 4?

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