nearly 24-we write 23 (22+1) thousandths. The required decimal is thus found to be £723. EXAMPLE II.-Express 98. 10 d. as a decimal of a pound. In 9 shillings there are 4 florins and I shilling; for the 4 florins we write 4 tenths of a pound, and for the I shilling 5 hundredths of a pound: 98.45. In the remainder of the amount (10ąd.) there are 43 farthings, for which-the number being nearly 48-we write 45 (43+2) thousandths. The required decimal is thus found to be (45+045) £495. 87. Rule for the Conversion of Shillings and Pence into a Decimal of a Pound: Set down as many tenths as there are florins (or two-shilling pieces); and if there be a shilling remaining, write 5 in the second decimal place. Reduce the remainder of the amount to farthings, for which write the same number of thousandths-adding 1 for every 24. EXAMPLE III.-Express £723 in shillings and pence. For the 7 tenths (£7) we take 7 florins or 14 shillings. There then remain 23 thousandths, for which-the number being nearly 25-we take 22 (23-1) farthings, or 5d. The required amount is thus found to be 14s. 5d. EXAMPLE IV.-Express £495 in shillings and pence. For the 4 tenths (4) we take 4 florins or 8 shillings, and for 5 of the 9 hundredths an additional shilling. There then remain (9—5—) 4 hundredths and 5 thousandths, or 45 thousandths, for which—the number being nearly 50-we take 43 (45-2) farthings, or 10d. The required amount is thus found to be (8s.+1s.+10 d.) 98. 10 d. 88. Rule for the Conversion of a Decimal of a Pound into Shillings and Pence: Set down twice as many shillings as there are tenths, and an additional shilling if the number of hundredths be not less than 5. Express the remainder of the decimal in thousandths, for which write the same number of farthings-rejecting 1 for every 25. MEASURES AND MULTIPLES: PRIME AND COMPOSITE NUMBERS: &c. 89. Two numbers are said to be related as MEASURE and MULTIPLE when the smaller is contained an exact number of times in the larger-in other words, when the division of the larger by the smaller leaves no remainder. Of two numbers so related, the smaller is called the "measure," and the larger the "multiple.” Thus, we say that 7 is a measure of 21, or that 21 is a multiple of 7: because 7 is contained an exact number of times in 21. The numbers 8 and 20, however, are not related as measure and multiple—the division of 20 by 8 leaving a remainder. Every number* has at least two measures-itself and unity. 90. A number whose only measures are itself and unity is called a PRIME number. The following are "prime" numbers: 1, 2, 3, 5, 7, II, 13, 17, 19, 23, 29, 31, &c. 91. A number which is not "prime"-in other words, a number which has one or more measures besides itself and unity-is called a COMPOSITE number. The following are "composite" numbers: 4, 6, 8, 9, 10, 12, 14, 15, 16, 18, 20, 21, &c. 92. Every number is either ODD or EVEN. The "even" numbers are those of which 2 is a measuresuch as 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, &c. The "odd" numbers are those of which 2 is not a measure-such as 1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23, &c. Of the odd numbers, some-such as 1, 3, 5, 7, 11, 13—are prime; whilst others such as 9, 15, 21, 25, 27, 33-are composite. Of the even numbers, all are composite except 2, which, having no measure except itself and unity, is a prime number. * That is, every number greater than unity. 93. When considered with reference to two or more of its multiples, a number is called a COMMON measure of those multiples. Thus, we say that 7 is a 66 common measure of 21 and 28; that 10 is a 66 common measure of 20, 30, and 40; &c. 94. By the GREATEST common measure of two or more numbers is meant the largest (or "greatest") measure which those numbers have in common. 24 12346 36 I Taking 24 and 36, for example, we see that the measures of 24 are 1, 2, 3, 4, 6, 8, 12, and 24; that the measures of 36 are 1, 2, 3, 4, 6, 9, 12, 18, and 36; that the numbers which measure 24 but do not measure 36 are 8 and 24; that the numbers which measure 36 but do not measure 24 are 9, 18, and 36; that, consequently, (8, 24, 9, 18, and 36 being rejected,) the common measures of 24 and 36 are 1, 2, 3, 4, 6, and 12; and that the GREATEST common measure is 12. Measures of 24. 12340∞ & # 12 Common measures 12 of 24 and 36. +8 Measures of 36. GREATEST common 95. Two numbers which have no common measure —that is, no common measure greater than unity— are said to be prime to one another, or relatively prime. Such numbers, however, are not necessarily prime in themselves, or absolutely prime. Thus, 15 and 22-having no common measure (except unity)-are prime to one another, or "relatively" prime; but neither 15 nor 22 is a prime number, 15 being measured by 3 and 5, whilst 22 is measured by 2 and 11. 96. To find the Greatest Common Measure of two numbers: Divide the larger number by the smaller, and if there be a remainder, divide the smaller number by it; if there be a second remainder, divide the first remainder by it; if there be a third remainder, divide the second remainder by it; and so on. Continue the process until nothing remains, and the last divisor-that which leaves no remainder—will be the greatest common measure.* EXAMPLE I.—Find the greatest common measure of 92 and 438. Dividing 438 by 92, we obtain 70 for remainder. Dividing 92 by 70, we obtain 22 for second remainder. Dividing 70 by 22, we obtain 4 for third remainder. Dividing 22 by 4, we obtain 2 for fourth remainder. Lastly, dividing 4 by 2, we find that there is no remainder. So that, according to the Rule, 2 is the greatest common measure required. 92)438(4 368 70)92(I 22)70(3 4)22(5 20 2)4(2 4 EXAMPLE II. Find the greatest common measure of 46 and 175. * The following rendering of the Rule occurs in some of the older treatises on Arithmetic, and deserves to be recorded: "The greater by the less divide; "By what remains-till nought remain: EXAMPLE III. Find the greatest common measure of 37 and 185. 37)185(5 As the division of 185 by 37 leaves no remainder, 37 is the greatest common measure required. For, measuring both itself and 185, 37 is a common measure; and it is obviously impossible for two numbers, one of them being 37, to have a greater common measure than 37. Reason of the Rule.-In examining the reason of the foregoing Rule, we must bear the following two facts in mind: (a) A number which measures another will measure any multiple of that other. (b) A number which measures two others will measure both the sum and the difference of those others. : These facts are rendered intelligible by a little reflection :(a) Taking 21, for instance, which contains an exact number of sevens, we see that the double, or the treble, or the quadruple, &c. of 21 must also contain an exact number of sevens. Again if a debt, consisting of a certain number of shillings, could be paid in instalments of (say) 5 shillings each, it is obvious that a debt twice, or 3 times, or 4 times, &c. as large could also be paid in 5-shilling instalments. m) a (q mq General Demonstration.-Let m measure a: to prove that m measures ax, a multiple of a. As m measures a, the division of a by m will give (say) the quotient q, and leave no remainder. We therefore have a=mq; (multiplying by x) ax=mqx; and (dividing by m) ax÷m= (mqx÷m=) qx. It thus appears that m measures ax-the division of ax by m leaving no remainder. O a=mq ax=mqx ax÷m (mqx÷m=)qx (b) As 33 and 12 contain, each, an exact number of threes, both 3312 and 33-12 must also contain, each, an exact number of threes. Thus, there being 11 threes in 33, and 4 threes in 12, there must be (11-+4=) 15 threes in 33+12, and (11—4=) 7 threes in 33-12. Again: if there were two papers of pins, one paper containing a larger number than the other, but each containing an exact number of rows of (say) 12 pins each, it is evident that the two papers, taken together, would contain an exact number of such rows. It is also evident that, if a row were torn off the smaller paper, and a row off the larger; then, another row off the smaller, and another off the larger; then, a third row off the smaller, and a third off the larger; |