PRINCIPLES. 1. The square of a number of two figures equals the TENS2+2 times TENSX UNITS+ UNITS 2. 2. The square of a number of three figures equals HUNDREDS 2+2 times HUNDREDS TENS+TENS2+2(HUNDREDS+ 794. These principles may also be expressed in symbols. Let u represent units figure, t tens, h hundreds, and T thousands, and two letters written together denote multiplication; then we have (t+u)2 = t2+2tu+u2. (T+h+t+u)2 = T2+2Th+h2+2(T+h)t+t2+ 2(T+h+t)u+u2. CUBING NUMBERS. 795. There are Two Methods of cubing numbers, called the Analytical or Algebraic, and the Synthetic or Geometrical methods. 796. The object of these methods is to find the law of forming the cube, and thus prepare for corresponding methods of explaining evolution. equals 2x40x52; 5 times 402 equals 402×5; 40 times 52 equals 40x52; 40 times 2×40×5 equals 2×402×5; 40 times 402 equals 403. Taking the sum of these products, we have 53; next, 40×52 plus 2×40×52 equals 3x40x52; next, 2x402×5 plus 402×5 equals 3×402×5; and next we have 403; hence 453=403+3×402×5+3x 40×52+5. Therefore the cube of 45 equals the cube of the tens, plus 3 times the square of the tens into the units, plus 3 times the tens into the square of the units, plus the cube of the units. 2. Find the cube of 45 by means of the cubical blocks. OPERATION. 40364000 402x5x3=24000 40x52x3= 3000 53= 125 Hence 453 = = 91125 GEOMETRICAL SOLUTION.-Let A, Fig. 1, represent a cube whose sides are 40 units, its contents will be 40364000. To increase its dimensions by 5 units we must add, 1st, the three rectangular slabs, B, C, D, Fig. 2; 2d, the three corner pieces, E, F, G, Fig. 3; 3d, the little cube H, Fig. 4. The three slabs, B, C, D, are 40 units long and wide and 5 units thick; hence their contents are 402x5x3=24000; the contents of the corner pieces, E, F, G, Fig. 3, whose length is 40 and breadth and thickness 5, equal 40x52x3=3000; and the contents of the little cube H, Fig. 4, equal 53125; hence the contents of the cube represented by Fig. 4 are 64000+24000+3000+125=91125. NOTE.-When there are three figures in the number, complete the second cube as above, and then make additions and complete the third in the same manner; or let the first cube represent the cube already found, and then proceed as at first. EXAMPLES FOR PRACTICE. Cube the following numbers by both methods: 7. 123. Ans. 1860867. 12. 5678. Ans. 183056925752. 797. The following principles are important, and should be committed to memory. PRINCIPLES. 1. The cube of a number consisting of two figures equals TENS3+3 times TENS2 X UNITS +3 times TENS UNITS2+ UNITS3. 2. The cube of a number consisting of three figures equals HUNDREDS 3+3 times HUNDREDS2 × TENS +3 times HUNDREDS XTENS2+TENS3+3 times (HUNDREDS+TENS)2 X UNITS +3 times (HUNDREDS+TENS) × UNITS2+UNITS3. 798. These principles may also be expressed in symbols as follows: (t+u)3= t3+3t2u+3tu2+u3 3(h+t)u2+u3. EVOLUTION. 799. Evolution is the process of finding a root of a number. 800. A Root of a number is one of its equal factors. Roots are of different degrees; as, second, third, etc. 801. The Square Root, or second root, of a number is one of its two equal factors. Thus, 8 is the square root of 64, since 8×8=64. 802. The Cube Root, or third root, of a number is one of its three equal factors. Thus, 4 is the cube root of 64, since 4x4x4=64. 803. The Fourth Root is one of the four equal factors; the fifth root is one of the five equal factors, etc. 804. The Symbol of Evolution is; thus, 2/64 or ✓64, denotes the square root of 64; 3/64 denotes the cube root of 64. 805. The Index of the root is a small figure placed in the angle of the symbol. The index indicates the degree of the root. Roots are also indicated by the denominator of a fractional exponent; thus 9 denotes ✔9; 27 denotes 27, etc. 806. The following principles of involution are given to enable us to determine the number of figures in the root. PRINCIPLES. 1. The square of a number contains twice as many figures as the number itself, or twice as many, less one. 12= 1 92= = 81 102 992 = 100 = 9801 DEM.-The square of 1 is 1, and the square of 9 is 81, hence the square of a number consisting of one figure is a number consisting of one or two figures. The square of 10, the smallest number of two figures, is 100, the square of 99, the largest number of two figures, is 9801, hence the square of a number consisting of two figures is a number consisting of three or four figures, that is, twice two, or twice two, less one, etc. The same may be shown for the square of a number consisting of any number of figures. 2. The cube of a number contains three times as many figures as the number itself, or three times as many, less one or two. 13 = 1 93729 1031000 993970299 DEM.-The cube of 1 is 1, and the cube of 9 is 729, hence the cube of any number consisting of one figure is a number consisting of one, two, or three figures. The cube of 10 is 1000, a number of four figures, the cube of 99 is 970299, a number of six figures, hence the cube of a number consisting of two figures contains four, five, or six figures, that is, three times two, or three times two, less one or two. The same may be shown for the cube of a number consisting of any number of figures. EVOLUTION BY FACTORING. 807. When the number is a perfect power and the factors are easily found, the root of a number can be readily obtained by the following Rule.-Resolve the number into its prime factors, and for the square root form a product by taking ONE of every Two equal factors; for the cube root, ONE of every THREE equal factors; etc. 1. Find the square root of 2025. SOLUTION.-We first resolve the number into its prime factors. Since the square root of a number is one of its two equal factors, we take one of every two equal factors, and have 3×3×5, which equals 45. Hence the square root of 2025 is 45. NOTE. It will be well for the pupil to mark the factor taken with a star, as in the margin. OPERATION. 3)2025 3)225 *3)75 *5 808. There are Two Methods of explaining the general process of extracting the square root, called the Analytic or Algebraic Method, and the Geometrical Method. 809. The Analytic Method of square root is so called because it analyzes the number into its elements and derives the process of evolution from the law of involution. 810. The Geometrical Method is so called because it makes use of a geometrical figure to explain the process of extracting the root. 1. Extract the square root of 2025. OPERATION. t2+2tu+u2 402 = =2025(40 =1600 5 ANALYTIC SOL. - Since the square of a number contains twice as many figures as the number itself, or twice as many less one, the square root of 2025 will consist of two figures, and hence consist of tens and units, and 2025 consists of (2t+u)u=(80+5)×5= 425 tens2+2x tens units +units". 2tu+u2 = 425 45 The greatest number of tens whose square is contained in 2025 is 4 tens; squaring the tens and subtracting, we have 425, which equals 2× tens x units+units". Now since 2x tens units is much greater than units2, 425 must consist principally of twice the tens into the units; hence if we divide by 2xtens we can ascertain the units. Twice the tens equal 40×2=80; dividing, we find the units to be 5; now finding 2x tens X units+units", or, what is the same, 2x tens+units, both multiplied by units, which equals (80+5) × 5 = 425, and subtracting, nothing remains. Hence the square root of 2025 is 4 tens and 5 units, or 45. GEOMETRICAL SOL.-Let Fig. 1 represent a square which contains 2025 square units, then our object is to find the number of linear units in the edge. Since the square of a number consists of twice as many places as the number itself, or twice as many less the square root of 2025 will consist of two places, and hence will consist of tens and units. one, OPERATION. 2025(40 402 =1600 5 40×2=80 425 45 (80+5)×5 = = 425 |