SQUARING NUMBERS. 791. There are Two Methods of squaring numbers, called the Analytic or Algebraic, and the Synthetic or Geo metrical methods. 792. The object of these methods is to find the law of forming the square, and thus prepare for corresponding methods of explaining Evolution. 1. Square 45 analytically and synthetically. five equals 40 plus 5, or 4 tens plus 45 OPERATION. 40+5 45 = 40+5 225 = 40×5+52 180 402+40×5 = = 402+2(40×5)+5a 2025 the square of 45 equals the square of the tens, plus twice the product of the tens by the units, plus the square of the units, which we find to be 2025. SYNTHETIC SOLUTION.-Let the line AB represent a length of 40 units, and BK 5 units. Upon AB construct a square; its area will be 4021600 square units. On the two sides BC and DC, construct rectangles each 40 units long and 5 units wide; the area of each will be 40×5, and the area of both will be 2 (40×5), or 400 square units. Now add the little square on CH; its area will be 52 = 25 square units; and the sum of the different areas, 1600+400+25=2025, is the area of a square whose side is 45. D E A F B K NOTE.-When there are three figures, after completing the second square as above, we must make additions to it as we did to the first square. When there are four figures there are three additions, etc. Square the following numbers by both methods: 793. The following principles derived from the above solutions are important, and should be committed to mem ory. PRINCIPLES. 1. The square of a number of two figures equals the TENS 2+2 times TENS UNITS + UNITS 2. 2. The square of a number of three figures equals HUNDF EDS2+2 times HUNDREDS TENS+TENS2+2(HUNDREDS+ TENS) UNITS+ UNITS2. 794. These principles may also be expressed in symbols. Letu represent units figure, t tens, le hundreds, and T thousands, and two letters written together denote multipli. cation; then we have (t+u)2 = t2+2tu+u2. (h+t+u)2 = h2+2ht+t2+2(h+t)u+u2. (T+h+t+u)2= T2+2 Th+h2+2(T+h)t+t2+ 2(T+h+t)u+u2. CUBING NUMBERS. 795. There are Two Methods of cubing numbers, called tle 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 2×40×52; 5 times 402 equals 402x5; 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 3×40×52; next, 2×402X5 plus 402×5 equals 3×40×5; and next we have 403; hence 453403+3×402x5+3x 40x52+58. 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 45391125 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 HUNDREDS 2 × TENS+3 times HUNDREDS XTENS2+TENS3+3 times (HUNDREDS+TENS) 2 × 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 (h+t+u)3=h3+3h2t+3ht2+t3 +3(h+t)2u+ 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; 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. 12=1 9281 102 100 9929801 1. The square of a number contains twice as many figures as the number itself, or twice as many, less one. 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 103 1000 993-970299 = 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 5/25 *5 |