PROOF OF THE RULES FOR THE MULTIPLICATION AND DIVISION OF DECIMALS.

93. To prove the Rule for the Multiplication of Decimals.

Let P′ and Q' be numbers, which contain (p) and (9) decimal places respectively, and let P, Q, denote the numbers when the decimal points are removed.

P

10P

Then we may represent P' and Q by the fractions and spectively, (Art. Arith. 83. Art. Alg. 17).

Q

re

109

"Hence the Multiplication of Decimals is performed as in whole numbers, and the product contains as many decimal places as the multiplier and multiplicand together contain.”

94. To prove the Rule for the Division of Decimals.

or after the division is effected as integers, the quotient will contain (p-q) decimal places.

which is a whole number, if P be divisible by Q without a remainder.

or we must affix (q-p) cyphers to the dividend, and the result up to this

point of the division will be a whole number.

95. To convert a vulgar fraction into its equivalent decimal.

which is a decimal containing n decimal places after the division of P× 10"÷Q has been effected: hence the following Rule: "Divide the numerator by the denominator affixing to the numerator as many cyphers as necessary, and then mark off from the quotient as many decimal places as equal in number the cyphers affixed to the numerator.”

96. To find under what circumstances vulgar fractions are convertible into finite or recurring decimals.

Now it is clear that the division of P× 10" by Q will terminate or not, as the division of 10" by Q will terminate or not. Since 10 is only divisible by 2 and 5, it follows that all powers of 10 are only divisible by 2 and 5, or by their powers; hence the division of 10" by Q can never terminate, unless Q be made up entirely of the factors 2 and 5, or one of them, or (which is the same thing) of powers of 2 and 5, or one of them. P

Therefore will be convertible into a finite decimal, if it be of the form Q

P

2P51'

but into a recurring decimal in all other cases.

97. When a fraction is converted into a finite decimal, to find the number of decimal places.

If it be convertible into a finite decimal, then, by the last Article, it

which is a decimal containing q decimal places.

98. If a fraction be converted into a recurring decimal, the remainders must recur in a certain order.

In the division of P× 10" by Q, each remainder must be less than Q, therefore there can, at the most, only be Q-1 different remainders. Hence if no remainder become zero, a remainder must recur within Q-1 operations at the farthest; the figures will then recur, and the result of course will be a recurring decimal.

=

Ex. XLIV.

Miscellaneous Questions and Examples.
A.

1. Shew that (a−b) (c+d)=ac-bc + ad- bd, a being > b.

that if m and n be positive integers, and n > m, that a”÷a” =

2. Simplify as much as possible each of the following expressions:

(3) (a−b) (b−c) (c− a) + a2 (b−c) + b2 (c− a) + c2 (a − b).

3. Investigate a Rule for finding the square root of a compound Algebraical quantity, and find the value of

4. Define the terms "Measure,"

"Greatest Common Measure."

State and prove the Rule for finding the Greatest Common Measure of

two compound Algebraical expressions.

Ex. Find the G. C. M. of the quantities

x2+4 (x3—30)—x(18x+104) and x2-10(x3+12) +x (24x+36).

5. Prove that

and that

(ab) (bc)+(a-d) (c-d) b-d
(a−b) (a−d) + (b−c) (c−d) a-c
(x-a)(x-b) (x−b) (x−c) (x−c) (x-a)
(c-a) (c-b) (a−b) (a−c) (b−c)(b−a)

+

+

=1.

6. State and prove the Rules for the multiplication and division of decimals. Divide 688896 by 89'7, and 688896 by '0897.

B.

1. Reduce to its most simple form the difference between the

sions,

expres

x3-y3+6{x2y-(xy2+3xy)}-14 {8 (x2-y3)-12(2x-3y+1)},
y3-x3-3{2xy (y-x)+6xy}-56 {2y2-2x2+9y-6x-3},

and find the value of 2a-la-[b (p+q)—a (1+b)]}; when a=b=p=q=8.

2. What is meant by the symbol am when m is a positive integer? Prove that ama"=am+", when m and n are positive integers: supposing that when m and n are fractions the same rule holds; what does a mean?

Multiply (a ̃3)*+ {(a3b)*;* by (a3) ̃a—{(a*b)*;*;

and divide 1+ 2x by 1–3x, to four terms, stating the correct remainder.

+

c2

(a-b) (a-c)(x+a) * (b− a) (b−c) (x + b) * (c− a) (c—b) (x+c) *

(1)

4. Investigate the condition under which a vulgar fraction is expressible in the form of a finite decimal. If this condition be not fulfilled.