To read NUMBERS. To the simple value of each figure join the name of its place, beginning at the left hand and reading toward the right. 2. Beside the simple value of the figures, as above noted, they have, each, a local value, according to the following law : Viz. In a combination of figures, reckoning from right to left, the figure in the first place represents its primitive simple value; that in the second place, ten times its simple value; that in the third place, a hundred times its simple value; and so on; the value of the figure in each succeeding place being ten times the value of it in that immediately preceding it. 3. The names of the places are denominated according to their order. The first is called the place of units; the second, tens; the third, hundreds; the fourth, thousands; the fifth, ten thousands; the sixth, hundred thousands; the seventh, millions; and so on. Thus in the number 3456789; 9 in the first place signifies only nine; 8 in the second place signifies eight tens, or eighty; 7 in the third place is seven hundred; 6 in the fourth place is six thousand; 5 in the fifth place is fifty thousand; 4 in the sixth place is four hundred thousand; and 3 in the seventh place is three millions; and the whole number is read thus, three millions, four hundred and fifty six thousand, seven hundred and eighty nine. 4. A cypher, though it signifies nothing of itself, yet it occupies a place, and, when set on the right hand of other figures, increases their value in the same ten-fold proportion; thus, 5 signifies only five, but 50 is five tens or fifty, and 500 is five hundred, &c. 1 13 To write NUMbers. RULE. Write down the figures in the same order their values are expressed in, beginning at the left hand, and writing toward the right; remembering to supply those places of the natural order with cyphers, which are omitted in the question. EXAMPLES. 5. For the more easily reading of large numbers, they are divided into periods, and half periods, each half period consisting of three figures; the name of the first period being units; of the second, millions; of the third, billions; of the fourth, trillions, &c. Also the first part of any period is so many units of it, and the latter part, so many thousands. The following Table contains a summary of the whole do&rine. Periods. Quadril. Trill. Billions. Millions. Units. Half Per. Figures. 1=1 2=11 th. un. th. un. th. un. th. un. cxt cxu www www www A Synopsis of the Roman NOTATION. As often as any character is repeated, so many 3=111 times is its value repeated. 4=IIII or IV A less character before a greater dimio- 6=VI A less character after a greater increases its 8=VIII value. 12 EXAMPLES. Write down in figures the following numbers: Eighty one. Two hundred and eleven. One thousand and thirty nine. A million and a half. A hundred and four-score and five thousand. Eleven thousand million, eleven hundred thousand and eleven. Thirteen billion, six hundred thousand million, four thousand and one. SIMPLE ADDITION. Simple Addition teacheth to collect several numbers of the same denomination into one total. RULE.* 1. Place the numbers under each other, so that units may stand under units, tens under tens, &c. and draw a line under them. * This rule, as well as the method of proof, is founded on the known axiom, "the whole is equal to the sum of all its parts." All that requires explaining is the method of placing the num bers, 2. Add up the figures in the row of units, and find how many tens are contained in their sum. 3. Set down the remainder, and carry as many units to the next row, as there are tens; with which proceed as before; and so on till the whole is finished. Method bers, and carrying for the tens; both which are evident from the nature of notation for any other disposition of the numbers would entirely alter their value; and carrying one for every ten, from an inferior line to a superior, is evidently right, since an unit in the latter case is of the fame value as ten in the former. Beside the method here given, there is another very ingenious one of proving addition by casting out the nines, thus: RULE 1. Add the figures in the uppermost line together, and find how many nines are contained in their sum. 2. Reject the nines, and set down the remainder directly even with the figures in the line. ་ 3. Do the same with each of the given numbers, and set alf these excesses of nine together in a row, and find their sum; then if the excess of nines in this sum, found as before, is equal to the excess of nines in the total sum, the question is right. This method depends upon a property of the number 9, which belongs to no other digit whatever, except 3; viz. that any number, divided by 9, will leave the same remainder as the sum of its figures or digits divided by 9; which may be thus demonstrated. DEMON. Method of PROOF. 1. Draw a line below the uppermost number, and suppose it cut off. 2. Add all the rest together, and set their sum under the number to be proved. 3. Add DEMON. Let there be any number, as 3467; this separated into its several parts becomes 3000+400+60+7; but 3000=3 In like manner 400 4X X 1000 3 X 999+1=3×999+3. 99+4, and 60=6×9+6. Therefore 3467=3×999+3+4× 99+4+6×9+6+7=3×999+4×99+6×9+3+4+6+7. And 3467 — 3 × 999+4×99+6×9, 3+4+6+7. 9 9 + 3+4+6+7 But 3X 9 999+4×99+6x9 is evidently divisible by 9; therefore 3467 divided by 9 will leave the same remainder as 3+4+6+7 divided by 9; and the same will hold for any other number whatever. Q. E. D. The same may be demonstrated universally thus: 1 DEMON. Let N≈ any number whatever, a, b, c, &c. the digits of which it is composed, and n as many cyphers as a, the highest digit, is places from unity. Then Na with n, o's+b with n-1, o's+c with n-2, o's, &c. by the nature of notation; =aXn-1, 9's+a+bxn−2, 9's+b+c×n—3, 9's+c, &c. =aXn-1,9's+b×n—2,9's+cxn-3, 9's, &c. +a+b+c, &c. but aXn-1, 9's+b×n—2, 9's +c ×n—3, 9's, &c. is plainly divisible by 9; therefore N divided by 9 will leave the same remainder, as a+b+c, &c. divided by 9. Q. E. D. In the very same manner, this property may be shown to belong to the number three; but the preference is usually given to the number 9, on account of its being more convenient in practice. Now from the demonstration here given, the reason of the rule itself is evident; for the excess of nines in two or more numbers being taken separately, and the excess of nines taken also out of the |