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B, we have a XB
m measures y. As m measures x Xy, the prime factors of m all occur amongst those of x Xy—that is, amongst the prime factors of x and those of y, taken collectively. But, m being prime to x, the prime factors of m do not, any of them, occur amongst those of x; therefore the prime factors of m must all occur amongst those of y—in other words, m must measure y.
A (II.) Let
and let be in its simplest form: to prove
b that a measures A, and that b measures B. Multiplying by
A; so that b measures ax B. But b is
b prime to a, the fraction being in its simplest form ; conse
7 quently (I.) b measures B. Next, by inverting the given frac
b B tions, or dividing unity by each, we have and (multiply
A bXA ing by A) -B; so that a measures 6X A. But a is prime to b; therefore (I.) a measures A.
Let us now take 11 as an illustration of a fraction in its simplest form, and in whose denominator neither 2 nor 5 occurs as a factor.
For a reason already explained, this fraction will become a circulating decimal- either“ pure” or 6 mixed” -when the numerator is divided by the denominator. vertible into a pure circulating decimal, is ($ 153) equivalent to a fraction having some number of nines for denominator; whilst (154), if convertible into a mixed circulating decimal, Hi is equivalent to a fraction having for denominator some number of nines followed by some number of ciphers—in other words, the product of some number of nines by a power of
In the latter case, 21 would (II.) measure the product of some number of nines by a power of 10—the fraction ni being in its simplest form. But (I.) 21 could not measure such a product without measuring the nines: because, as neither 2 nor 5 occurs amongst its prime factors, 21 is prime to 10, and to every power of 10. So that li is equivalent to a fraction having some number of nines for denominator; and as (§ 153) every such fraction is the equivalent of a pure circulating decimal having the figure or figures of the numerator for period, it follows that is convertible into a pure-not a mixed-circulating decimal.
13 II X2
2 x 5
13 X 5
Х II IO
-10=24+10=2.36_10= = '236
2 X II 5 X2
5 x 5 _7 X 5 X 5 175 =-X
-100=581 : 100=58-3-100=.583. 3X2 X 2 3 2 X 2 3 2 X 2 X 5 X 5 3 IOO 3 7 I
2 X 2 –7 X 2 X2
-X :-100=9}: 100=9:3-100='093. 3 X5 X5 3. 5x535 X 5 X 2 X2 3
Х +100=113-100=116--100='116. 3 X 2 X 2 X 5 3 2 X 2 x 5 3 2 X 2 X 5 X 5 3 100 3
=2}+100=23-100='023. 3X 2 X 2 * 5 * 5 3 2 x 2 x 5 x 5 3 100
3 7 X 2 X 2 X 2
5 x 5 x 5 3x 5 x 5 x 5 x 7 2 X 2 X 2 7 2 X 2 X 2 X 5 X 5 X 5 7
* 1000=534 = 1000= 7
3 3x 3x
2 X 2 X 2 3X2 X2x2
• 1000=39+1000=3:428571; 7 * 5 * 5 * 5 7 5x5x5 75 X 5 X 5 X 2 X 2 X 2 7
5x 5 3x
3 X 5 X 5
15 7 x 2 X 2 X 2 X5X5 72 x 2 X 2 * 5 * 5 72 X 2 X 2 X 5 5*5
7 I 000
= '000857142. 3 3х.
+ 1000 = 428571---1000 = '000428571. 7X2X2X2 X 5 X 5 X 5 7 2X2X2X5X5 X5 7
Let us next take the fraction 1.5, which is in its simplest form, and the prime factors of whose denominator are 2 and
This fraction will become either a pure or a mixed circulating decimal, when the numerator is divided by the denominator. Now, 11, if convertible into a pure circulating decimal, would be equivalent to a fraction having some number of nines for denominator; and this denominator would be measured by 22, and therefore by 2, a factor of 22. But 2 is not a measure of any such denominator (9,or 99, or 999, &c.); and consequently the circulating decimal must be mixed—not pure.
Again: 1 , the prime factors of whose denominator are 3 and 5, will, when the numerator is divided by the denominator, become either a pure or a mixed circulating decimal. Now, 14, if convertible into a pure circulating decimal, would be equivalent to a fraction having some number of nines for denominator, and this denominator would be measured by 15, and therefore by 5, a factor of 15. But 5 is not a measure of 9, or of 99, or of 999, &c.—the division of every such number by 5 leaving 4 for remainder; consequently the circulating deci. mal must, as in the last case, be mixed—not pure.
From an examination of the examples in the two preceding pages,
it will be seen that, in the case of a fraction convertible into a mixed circulating decimal, the number of figures in the non-circulating part of the decimal is always indicated by the number of twos or of fives (see p. 152) found amongst the prime factors of the denominator, when the fraction is in its simplest form. In each case we first obtain (every such frac
13X5, 13X2, &c. producing) a PURE circulating decimal, and this we afterwards convert into a MIXED circulating decimal by removing the decimal point to the left: the number of places the point is removed—and therefore the number of places occupied by the non-circulating part of the decimal — being the same as the number of twos or of fives found amongst the prime factors of the given fraction's denominator. Because the lowest power of 10 which is measured by the product of (I.) a number of twos (II.), a number of fives, or (III.) a number of twos AND fives, must contain as many ciphers as there are twos or fives: 10=2X5; 100 = 2 X 2 X5X 5; 1,000=2X2X2X5X5X5; &c
SIMPLE PROPORTION, OR THE
RULE OF THREE." 155. When we speak of a number as “large” or “small,” we necessarily compare it—although unconsciously, sometimes—to another number of the same kind.
A shilling, for example, is large in amount compared to a penny, but small compared to a sovereign ; a rood of land is large compared to a perch, but small compared to an acre quart is large compared to a pint, but small compared to a gallon ; and so on.
In like manner, 10 pounds of tea are large in quantity compared to 2 pounds, but small compared to 8o pounds; a piece of cloth 12 yards long is large compared to a piece of the same cloth 4 yards long, but small compared to a piece 30 yards long; a flock of 20 birds is large compared to a flock of 6, but small compared to a flock of 100; &c.
156. The relation-as to largeness or smallnessof one number to another of the same kind, is called the RATIO of the one number to the other.
Thus, the relation --as to largeness—of 10 pounds of tea to 2 pounds, is called the “ratio” of 10 pounds to 2 pounds ; the relation—as to smallness—of 12 yards of cloth to 30 yards, the “ ratio” of 12 yards to 30 yards; and so on.
157. Of two numbers so compared—and which are spoken of as the terms of the ratio—the first is called the ANTECEDENT; the second, the CONSEQUENT.
In the case of the ratio of 10 pounds to 2 pounds, the “ antecedent” is 10 pounds, and the " consequent 2 pounds ; in the case of the ratio of 12 yards to 30 yards, the antecedent” is 12 yards, and the “consequent” 30 yards; &c.
158. The terms of a ratio are written one after the other,—the consequent after the antecedent,-and separated by two dots, placed one below the other.
Thus, the ratio of
10 lbs. :
12 yds. : 30 yds.