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II. If the work is correct, the last excess will equal the excess in the minuend.
Subtract and prove the following: 2. 4736_2431.
5. 233461–87563. 3. 57973—44567.
6. 4465612345612. 4. 98793—47867.
7. 876543—625781. PROOF OF MULTIPLICATION. 984. The Proof of Multiplication by casting out 9's is based upon the following principle:
Prin. The excess of 9's in the product of two numbers equals the excess of 9's in the product of the excesses of those numbers.
Each number is a multiple of 9, plus its excess, hence the product will be a multiple of 9, plus the product of the excesses, and the excess in this product of excesses will therefore evidently be the excess in the product of the two numbers.
1. Multiply 346 by 68.
SOLUTION.—The excess in the multipli- 346 excess 4 cand is 4, in the multiplier 5, and in the
5 product of these excesses 2.
2768 20 excess 2. in the product is also 2; hence the work is correct.
23528 excess 2. Rule.-I. Find the excess of 9's in the multiplier and multiplicand, the excess in the product of these excesses, and also the excess in the product of the numbers.
II. If the work is correct, the last two excesses will be equal.
Multiply and prove the following: 2. 6563 x 736.
5. 68735 x 5642. 3. 4918 X 875.
6. 79636 X 4876. 4. 15978 x 6353.
7. 387981 x 3578.
PROOF OF DIVISION. 985. The Proof of Division by casting out 9's is based upon the following principle:
Prin. The excess of 9's in the dividend equals the product of the excesses in the divisor and quotient, plus the excess in the remainder.
For D=dxq+r. Now the excess of 9's in the product dx q equals the excess in the product of the excesses of these terms (Prin. Art. 984); and the excess in this product plus the excess in r must equal the excess in the dividend. (Prin. Art. 982.)
1. Divide 2443 by 56 and prove the result. SOLUTION.—The excess in
OPERATION. the divisor is 2; in the quotient
56)2443(43 excess in d, 2 7; in qxd it is the excess in 2
224 excess in
7 X7 or 14, which is 5; in r, 8; in qxd+r it is the excess in
203 excess in qxd, 5 5+8, or 13, which is 4; and in
excess in 8 the dividend it is 4; hence the
35 excess in qxd+1, 4 work is correct.
excess in D, 4 Rule.-I. Find the excess of 9's in the divisor and quotient, the excess in the product of these excesses, the excess in the remainder, then the excess in the sum of the last two excesses, and then the excess in the dividend.
II. If the work is correct, the last two excesses will be equal.
Divide and prove the following: 2. 6734;371.
5. 793742 : 4242. 3. 59453;276.
6. 8746391;3792. 4. 679432 ;-4833.
SCALES OF NOTATION. 986. The Scale of a system of notation is the law of relation between its successive orders of units.
987. The Radix of the scale is the number which expresses the relation of the successive orders.
Any number might have been taken as the basis of the scale of Notation. The use of ten, the basis of the decimal scale, originated from the counting of the fingers of the two hands, which was the primitive method of calculation.
988. A scale whose radix is two is called Binary; three, Ternary; four, Qualernary; five, Quinary; six, Senary; seven, Septenary; eight, Octary; nine, Nonary; ten, Denary; twelve, Duodenary or Duodecimal, etc.
989. In expressing a number in any one of these scales, there must be as many significant characters as there are units in the basis of the scale, less 1. Thus in the decimal
scale there are 9, in the octary 7, in the quinary 4, etc. In each the zero, 0, is used to fill vacant places.
990. In expressing numbers in scales higher than the decimal, it is necessary to introduce some new characters; thus o may stand for ten, and 11 for eleven.
In order to use any scale of notation with facility, the names of numbers should also be based on the same scale. Thus, in the quinary scale we should count one, two, three, four, five, one and five, two and five, etc., to two fives, and then two fives and one, two fives and two, etc.
Not having these names, we may read by powers of the radix. Thus, 4234 in the quinary scale may be read, four 5's cubed, two 5's squared, three 5's and 4 units. The scale in which a number is expressed may be indicated by writing the radix as a subscript.
991. To pass from any scale to the decimal scale.
1. 24325 is a number in the quinary scale; express the same number in the decimal scale.
SOLUTION.—The given number consists of 2 ones, 3 fives, 4 fives squared, and 2 fives cubed. Two fives cubed
53 x2= 250 equal two hundred and fifty; 4 fives squared equal one
52X4=100 hundred; 3 fives equal fifteen; 2 ones equal 2 ones;
5 X3= 15 the sum of all is three hundred and sixty-seven, which
1 X2= 2 expressed in the decimal scale is 367.
367 Change each of the following to the decimal scale: 2. 32045; 60358; 210324 ; 25348. Ans. 429; 3101, etc. 3. 1011012; 785036, ; 3742081 ; 20116038 12.
Ans. 45; 469509; 599200; 6211916.
992. To pass from the decimal scale to any other scale.
1. 45789 is a number in the decimal scale; express the same number in the quinary scale.
OPERATION. SOLUTION.—To express any number in the quinary 5)45789 scale, we ascertain how many fives, how many fives squared, how many fives cubed, etc., the number con
5) 9157+4 tains. Dividing by five, we ascertain the number of 5) 1831+2 fives and units; dividing the number of 5's by 5, we
5) 366+1 ascertain the number of 5's squared; dividing these
In by 5, we ascertain the number of 5's cubed, etc.
5) 73+1 this manner we find 45789 equals 4 ones, 2 fives, 1 five 5) 14+3 squared, 1 five cubed, etc., which written in the quinary
2+4 scale gives 24311245.
2. Express 3478 and 79437 in the octary scale.
Ans. 66263; 233115g. 3. Express 54321 and 33787 in the senary scale.
Ans. 6552536; 4202316. 4. Express 67893 and 59466 in the duodecimal scale.
Ans. 333592; 204116 12
993. To pass from one scale to another, neither being the decimal scale.
1. 3464, is a number in the octary scale; express the same number in the quinary scale.
SOLUTION.-Remembering that the given number
5)3464 is in the scale of eight, and dividing successively
5) 560+4 by 5, we find the number contains 4 ones, 3 fives, 3 fives squared, 4 fives cubed, and 2 fives fourth power, 5) 111+3 which, expressed in the quinary scale, gives the num
5) 16+3 ber 243345.
Ans. NOTE.—In making the division, it must be remembered that the number divided is in the octary scale, and hence any remainder, instead of being so many tens, is so many eights. 2. Reduce 2433, and 10111, to the quaternary scale.
Ans. 320224; 1134. 3. Reduce 157742, and 34581u to the ternary scale.
Ans. 42212111023; 21121112003. 4. Reduce 303214 and 45324g to the nonary scale.
Ans. 1116, ; 86779.
CASE IV. 994. To perform arithmetical operations on numbers in any scale. 1. Add 23673, 50623, 75064g.
Ans. 104535. 2. Subtract 75981112 from $2811612
Ans. 2802712 3. Multiply 540812 by 311712.
Ans. 195376812 4. Divide 195376812 by 311712.
Ans. 549812 5. Extract the square root of 11530112. Ans. 34712 6. Add 23124, 43245, 543416, 373463, 29491112.
Ans. 2435253. NOTE.—For a fuller discussion of this subject, sce Philosophy of Arith
995. Mensuration treats of the measurement of geometrical magnitudes.
996. Geometrical Magnitudes consist of the Line, Surface, Volume, and Angle.
997. A Line is that which has length without breadth or thickness. Lines are either straight or curved.
998. A Straight Line is one that has the same direction at every point.
999. A Curved Line is one that changes its direction at every point. The word line used alone means a straight line.
1000. Parallel Lines are those which have the same direction. Parallel lines, it is thus seen, will never meet.
1001. One line is said to be perpendicular to another when the adjacent angles formed by the two lines are equal.
1002. An Angle is the opening between two lines which diverge from a common point.
1003. A Right Angle is an angle formed by one line perpendicular to another; as, ABC.
1004. An Acute Angle is an angle less than a right angle; An Obtuse Angle is one larger than a right angle; as, DEG.
MENSURATION OF SURFACES. 1005. A Surface is that which has length and breadth without thickness. Surfaces are plane or curved.
1006. A Plane Surface is a surface such that if any two of its points be joined by a straight line, every part of that line will lie in the surface.