EVOLUTION. Art. 171. EVOLUTION is the reverse of involution. It explains the method of resolving a number into equal factors, which factors are called roots. When a number is resolved into two equal factors, each of these factors is called the square root of the number. When a number is resolved into three equal factors, each of these factors is called the cube root of the number. Any required power of a number may be found, but there are numbers whose exact root cannot be obtained; yet by the use of decimals, an approximate root may be obtained sufficiently exact for all practical purposes. A number whose exact root cannot be found is called a surd number, and its approximate root is called a surd root. The square root of a number is indicated by this character ✔placed before the number; the other roots are indicated by the same character with the index of the root placed over it, or by the fractional index placed at the right. The square root of 64 is expressed thus, and the cube root of 64, thus, $64, or 64*. 64, or 643; When the power is expressed by several numbers, with the signor between them, a line, called a vinculum, is drawn from the top of the sign over all the numbers; thus the_square root of 25+11 is 6; the cube root of 64-2788 is 5. EXTRACTION OF THE SQUARE ROOT. Art. 172. THE square root of a number is one of its two equal factors. The extraction of the square root of a number is the method of finding one of its two equal factors. The second powers or squares of the first twelve integral numbers are exhibited in the following table. 1st powers or roots. 2d powers or square numbers. 1? 9? | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | 4 | 9 | 16 | 25 | 36 | 49 | 64 | 81 | 100 | 121 | 144 1. What is the square root of 3. What is the square root of 2. What is the square root of 4? 4. What is the square root of 16? The second power of each of the numbers 1, 2, and 3, is expressed by a single figure. The second power of each of the numbers 4, 5, 6, 7, 8, and 9, is expressed by two figures. The second power of each of the numbers 10, 11, and 12, is expressed by three figures. Hence, the second power of any given number can never contain more than twice its number of figures, and never but one less than twice its number. Therefore, we can determine the number of figures that the square root of any number will contain, by dividing the given number into periods of two figures each, by points, counting from the right; the period on the left may contain one or two figures. The root will contain as many figures as there are periods. We will now find the second power of 48, and then extract its root. 48 X 48= 2304. The number 48 is composed of 4 tens or 40, plus 8 units. We will find the second power of 40+8, the several products will exhibit its component parts. 40+8. 40+8. 320+64. 1600+320 1600+640+64. By examining the several component parts of the second power of 40+8, we perceive that the first part, 1600, is the square of the tens, or the product of the tens multiplied by the tens; hence, each of its equal factors must be the tens, or 40, and 1600 ÷ 40=4 tens or 40, which is the first part of the root. We see that the second part, 640, is twice the product of the tens multiplied by the units; hence, its greater factor must be twice the tens, or 80, and 640808, the less factor or the units. We see, also, that the third part, 64, is the square of the units, or the product of the units multiplied by the units; hence, each of its equal factors must be the units, or 8; and 6488, the units. Moreover, we see that 80+8, or 88, must be the greater factor of 64064, or 704, and 704888, the less factor, or the units, which is the second part of the root. Therefore, the whole root of 1600+640+64, is 40+8, or 48. We may consider every number greater than 9, whose right hand figure is not a cipher, to be composed of tens and units, because all the figures which compose any number, except its right hand figure, express tens; hence, we infer that the second power of every number composed of tens and units must contain the square of the tens, plus twice the product of the tens multiplied by the units, plus the square of the units. We will now reverse the process of finding the second power of 40+8, and we shall obtain one of its two equal factors or root. Divisor 40)1600 + 640 + 64( 40+8=48, the root. 1600 Divisor completed 40+8)640+64 640+64 As 1600 is the square of the tens, we find by trial that one of its two equal factors is 40, which we write at its left for a divisor; we also place it at the right for the first part of the root. We then multiply the divisor by the first part of the root, and subtract the product from 1600+640+64; the remainder is 640+64, which is a new dividend. As 640 contains twice the product of the tens multiplied by the units, twice 40, or 80, must be the greater factor of 640, which we write at its left for a trial divisor; this trial divisor is contained 8 times in 640, which is its less factor, and it is also one of the two equal factors of 64. We write this factor 8 for the second part of the root, and also at the right of the trial divisor; the sum is the divisor completed. We then multiply this completed divisor by 8, the product is 640+64, which we write under the dividend; this product being equal to the dividend, the operation is completed, and the root is 40 +8=48. We will now find one of the two equal factors, or the square root, of 2304, by a shorter process. Divisor 4)2304 ( 48, the square root. 16 Trial divisor 80)704, new dividend. Divisor completed 88)704 We first divide 2304 into periods of two figures each, by placing a point over the units, and another over the hundreds; the number of periods always indicating the number of figures that there will be in the root. The first period on the left is 23, (hundred) which must contain the square of the tens, because tens multiplied by tens produce hundreds. The greatest square number contained in 23 (hundred) is 16, (hundred) and each of its two equal factors is 4, (tens) one of which we write at the left for a divisor, and the other on the right, for the first figure of the root. We then multiply the divisor by the figure placed in the root, and subtract the product from the first period, the remainder is 7, (hundred) we then annex the next period of figures to this remainder, and we have 704 for a new dividend. This dividend contains twice the product of the tens multiplied by the units, plus the square of the units. Twice 4, (tens) the root already obtained, is 8, (tens) which is the greater factor of twice the product of the tens multiplied by the units; we then write the 8 (tens) with a cipher annexed at the left of the new dividend, for a trial divisor; this trial divisor is contained 8 times in the dividend; then 8 must be the less factor of twice the product of the tens multiplied by the units, and it must also be one of the two equal factors of the square of the units; we then write the 8 at the right of the 4 (tens) in the root, and also under the trial divisor; the sum of the trial divisor and this less or equal factor is the divisor completed; we then multiply this completed divisor, 88, by the figure last placed in the root, the product is 704, which we write under the dividend; this product being equal to the dividend, the operation is completed, and we have found one of the two equal factors or square root of 2304 to be 48. From the preceding illustrations, we obtain the following rule for extracting the square root of numbers. RULE. Divide the given number into periods of two figures each, by placing a point over the units, and another over every second figure at the left; also, over every second figure at the right, when the number contains decimals. Find, by trial, the greatest square number in the first period on the left, and write one of its two equal factors at the left for a divisor, and the other on the right for the first figure of the root; then multiply this divisor by the figure in the root, subtract the product from the first period, and annex the next period of figures to the remainder for a new dividend. Write twice the number of the root already found with a cipher annexed at its left for a trial divisor; then find the number of times this trial divisor is contained in the new dividend, and write a figure expressing the number for the second figure of the root, and write it also under the new trial divisor; add this figure to the trial divisor, their sum is the divisor completed. Multiply this completed divisor by the figure last placed in the root, subtract the product from the dividend, and annex the next period of figures to the remainder for a new dividend. Write twice the number of the root already found, with a cipher annexed at the left of the new dividend, for a new trial divisor; then repeat each of the successive steps of the previous operation, and thus continue until the required root is obtained. If there should be a remainder after the root of all the periods has been extracted, the operation may be continued by annexing periods of ciphers. 729 ? 4096. 15625. 46656? 117649. Ans. 343. 262144. 531441? 1048576. 9765625. 1. What is the square root of 2. Find the square root of 3. Extract the square root of 4. What is the square root of 5. Find the square root of 6. Extract the square root of 7. What is the square root of 8. Find the square root of 9. Extract the square root of 10. What is the square root of 60466176? 11. Find the square root of 282475249. 12. Extract the square root of 3486784401. 13. What is the square root of 17.3056? 14. Extract the square root of 373. 15. What is the square root of 3.15? Ans. 59049. Ans. 19.3132079+ Ans. 1.7748239 16. What is the square root of 8.93? Ans. 2.9883105 Art. 173. To EXTRACT THE SQUARE ROOT OF A VULGAR FRACTION. Reduce the fraction to its lowest terms, then extract the square root of the numerator and denominator, if each of them is a square number; if not, reduce the vulgar fraction to a decimal and then extract its root. |