16

To find how often the divisor (211) is contained in the numbers of the several steps of the operation, first enquire how many times 2 (the left figure of the divisor) is contained in 6 (the left figure of the dividend); this gives 3 for the first figure in the quotient; next, the 2's in 4 are 2 for the second figure; thirdly, 211 the divisor being greater than 30, a cipher or 0 will be the third figure; fourthly, the 2's in 3 give 1; next, the 2's in 9 give 4; and lastly, the 2's in 10 are 5.

27. But when the dividend is a large number, and the divi-sor consists of several figures, a table may be formed containing the products of the divisor by the several digits, as in the next example:

Ex. 6. Divide 1447859740478 by 1783.

28. Those who are expert in the practice of division, sometimes omit the products, and set down the remainders only.

Thus, (taking the last example.)

1783) 1447959740478 (812035749

2145

3629

6374

10250

13354

8737

16058

remainder.

And the division is sometimes performed without bringing down the figures of the dividend.

Where the remainders stand under the corresponding figures of the dividend, as before.

In these contracted methods, the remainders are obtained by performing the subtraction while you multiply. Thus to find 214 the first remainder: 8 times 3 make 24, and 4 make 28, therefore 4 is the right-hand figure of the remainder; next 8 times 8 make 64, and 2 (the tens carried) make 66, and I make 67, consequently 1 is the next figure; again, 8 times 7 are 56, and 6 (the tens carried) make 62, and 2 make 64, therefore 2 is the other figure of the remainder. And in the same manner the other remainders are found.

29. When the divisor is a number with ciphers on the right, cut them off, and also the like number of figures from the right of the dividend, then divide the remainder of the dividend by that of the divisor in the usual manner, and bring down the figures cut off from the dividend to the right of what remains after this division, if any thing, for the whole remainder; otherwise the figures cut off will be the true remainder.

642 quotient, the remainder being 1.

30. When the divisor is the product of two or more single figures, divide by one of those figures, and the quotient by another, and so on.

Ex. 11. Divide 332250 by 72, or 9 times 8. (See Example 7, in Multiplication.)

9) 332280
8)36920

4615 quotient.

The method of finding the true quotient when there are remainders, belongs to Vulgar Fractions, to which we refer for an example.

Since the product of the divisor and quotient (without the fractional part, should there be any) gives the dividend lessened by the remainder, it is evident that division may be proved by casting out the 9's exactly in the same manner as multiplication.

OF VULGAR FRACTIONS.

31. THE operations by common arithmetic extend to integers only, unity or one being the least number in the computations. When parts or quantities less than 1 are the subject of consideration, it is called Fractional Arithmetic. A fraction there

force is properly an expression for part of an unit or the integer 1. This integer 1 may represent a whole of any kind, and the parts into which it is broken, or supposed to be divided, are fractions of that whole.

Thus if 1 pound is the integer, and we divide it into 20 equal parts, 1 of these parts, or a shilling, will be represented by the fraction (one twen• tieth); and 7 shillings by the fraction (seven twentieths). If a foot in length is the integer, the expression for 1 inch will be 1⁄2 (one twelfth); but if we make a yard the integer, 1 inch will be denoted by (one thirtysixth), because 36 inches make a yard,

Σ

32. A fraction also arises from division in whole numbers when there is a remainder; or when the divisor is greater than the dividend: in the former case it is part of the quotient (sce examples 2, 4, &c. in simple division), and in the latter, the quotient itself,

Thus if 5 be divided by 2 the quotient is 24. And 3 divided by 4 gives for the quotient. Here the fractions are and 3: the former (1) being half, or 1 divided by 2; and the latter (3) three-fourths, or 3 divided by 4, or the 4th of 3.

33. The lower figure of a fraction (denoting the number of parts into which the integer or 1 is supposed to be divided is called the denominator; and the upper figure (which shews the number of those parts expressed by the fraction) the numerator; thus 4 is the denominator, and 3 the numerator of the fraction 4. Also both are generally named the terms of the fraction.

34. Fractions are either proper, improper, simple, or compound.

A proper fraction is when the numerator is less than the denominator, as, or 4, or, &c. and therefore it is always less than 1.

An improper fraction has the numerator equal to, or greater than the denominator, and consequently its value must be equal to, or greater than 1. Thus 4 is an improper fraction, because

it denotes l'or a whole; for four fourths make a whole.

is also an improper fraction, it being the same as 7 quarters or 1 and 4.

A simple fraction is any fraction having only one numerator, and one denominator, as, or T

:

3

A compound fraction is the fraction of a fraction, or several single or simple fractions connected with the word of between them thus of 4, and of of, are compound fractions. Also if 1 pound be the integer, the compound fraction of will denote sixpence, it being the of 1 shilling or of of a 12/24 pound.

A mixt number is composed of an integer and a fraction, as 5, 20, &c.

A whole number may be expressed like a fraction by placing 1 under it as a denominator: thus denotes 12 units,

or 12.

A prime number is that which can only be measured by 1, or unity thus 2, 3, 5, 7, 11, &c. are prime numbers.

A composite number is that which can be measured by some number greater than 1 or it is the product of two or more numbers: thus 4, 6, 8, &c. are composite numbers.