Evolution. ART. 429.—Evolution is the process of finding any required root of a number. ART. 430.—A Root of a number is one of its equal factors. ART. 431.-The Square Root of a number is one of its two equal factors. Thus, 6 is the square root of 36, because 6 is one of its two equal factors. 4 is the cube root of 64, because 4 is one of its three equal factors. ART. 432.-The Sign generally used to denote that some root of a number is to be extracted is; it is called the Radical Sign, and means that the square root is to be extracted. ART. 433.-The Index of the root is the small figure placed in front and above the radical sign. It indicates what root is to be extracted. Thus, 64 means that the third or cube root of 64 is to be extracted. ART. 434.-Evolution is the reverse of involution. Let us now learn the relation of a number to its square root. 252 = (20+ 5)2 20 + 5 20+ 5 202 +20 × 5 The square of 25 is 625. In squaring 25, we first square the tens, which gives 400. We next multiply the units (5) by the tens (2), which gives 100. We next multiply the tens (2) by the units (5), 202+2(20×5)+52=625 which gives 100. We next square the units, which gives 25. The sum 20 × 5 +52 of these several products is 625, which is the square of 25. By analyzing the foregoing we find that 625, when compared with its square root, contains the square of the tens, plus twice the product of the tens by the units, plus the square of the units. 9281 9929801 From the squares in the margin we may infer that the square of any number con1252 = 15625 tains twice as many figures as the number itself, or twice as many less one. Hence, If a number be separated into periods of two figures each, beginning at the right, there will be as many periods as there are figures in its square root. The left hand period may contain only one figure. WRITTEN EXERCISES. 1. Required the square root of 1,024. Process. 62)124 124 Analysis.-The square root of 1,024 consists of two figures, since the number contains two periods. The greatest square contained in 1,000 is 3 tens, which being squared and subtracted, leaves 124. This remainder must be equal to twice the tens by the units, plus the square of the units, since we have subtracted the square of the tens. Twice the (3) tens is 6 tens, which we place on the left as the first figure of the trial divisor. 6 tens is contained in 12 tens, 2 units times which we write for the second figure of the root. For convenience we write it also on the left as the second figure of the trial divisor. Now when 62 is multiplied by 2, we have multiplied twice the tens in the root by the units, and we have also squared the units, and no remainder is left. For convenience, we omit, in extracting the root, the ciphers in the several partial products, and treat each figure of the root as a simple digit. 2. Extract the square root of 55,225: also of 31,449,664. ART. 435.-Rule for the extraction of the square root.— Beginning at the right hand, separate the number into periods of two figures each. Find the greatest square contained in the left hand period, and place its root at the right for the first figure of the root. Subtract the square of this quotient from the left hand period, and to the remainder annex the second period for a dividend. Double the root already found, and place it on the left of the dividend as a trial divisor. Find how many times it is contained in the dividend (excluding the right hand figure of the dividend). The quotient will be the second figure of the root. Annex the second figure of the root to the trial divisor for the true divisor and multiply by the last quotient figure; subtract the product from the dividend, and annex the third period to the remainder for the next dividend. Double the root already found for a second trial divisor, find the third figure of the root as before, and so continue until all the periods have been used. NOTES (a). If the product of a divisor by the last quotient is greater than the dividend, the figure in the root must be diminished by a unit. (b). If a cipher appears in the root, a cipher must be annexed to the trial divisor, the next period brought down, and the process continued as before. (c). If the number is not a perfect square, the exact root cannot be found. It may be approximated, however, by annexing periods of ciphers. (d). In separating into periods a decimal or a mixed decimal number, we must begin with the units. If the decimal places are uneven, a cipher should be annexed. (e). To find the square root of a common fraction, find the square root of each term. When these are not perfect squares, the exact square root cannot be found. In such cases the fraction may be reduced to a decimal, and its square root approximated. ART. 436. To find the side of a square whose area is given. Since the area of a square is the product of two equal factors expressing the length of its sides, it follows that the square root of the number expressing its area represents each of the sides. 1. What is the side of a square field which contains 9,859,600 square feet? 2. A blackboard contains 10,800 square inches, and its length is 3 times its breadth: what are its dimensions? 3. A general seeking to mass his army in a square, found that by placing 236 men on a side, he lacked 696 men to form the square: how many men were in his army? 4. How much will it cost at $.75 a rod to inclose 10 acres of land in the form of a square ? 5. A certain square room contains 1,296 square feet, and another 729 square feet: how much longer is the side of the first than the side of the second? 6. A.man having a garden 420 yards square, extended it so that it was four times as large: how many yards square was it then? 7. The owner of a tract of land containing 1,200 acres divided it into four equal square farms: what was the length of one of their sides? 8. Illustrate the application of square root by an original problem. ART. 437-To find any side of a right-angled triangle when the other two sides are given. Principles.—1. The square de scribed upon the hypotenuse of a right-angled triangle equals the sum of the squares on the other two sides. 2. The square on a side adjacent to the right angle equals the square on the hypotenuse less the square on the other side. 3 1. The base and perpendicular of a right-angled triangle are 15 inches and 20 inches respectively what is the hypotenuse? |