THE EXTRACTION OF THE SQUARE ROOT. RULE. *1. Distinguish the given number into periods of two figures each, by putting a point over the place of units, another over the place of hundreds, and so on, which points shew the number of figures the root will consist of. 2. Find the greatest square number in the first, or left hand period, place the root of it at the right hand of the given number, (after the manner of a quotient in division) for the first figure of the root, and the square number, under the period, and subtract it therefrom, and to the remainder bring down the next period for a dividend. 3. Place the double of the root, already found, on the left hand of the dividend for a divisor. 4. Seek how often the divisor is contained in the dividend, (except the right hand figure) and place the answer in the root for the second figure of it, and likewise on the right hand of the divisor: Multiply the divisor with the figure last annexed by the figure last placed in the root, and subtract the product from the dividend : To the remainder join the next period for a new dividend. 5. Double the figures already found in the root, for a new divisor, (or, bring down your last divisor for a new one, doubling the right hand figure of it) and from these, find the next figure in the root as last directed, and continue the operation, in the same manner, till you have brought down all the periods. Note 1. If when the given power is pointed off as the power requires, the left hand figure should be deficient, it must neverthe less stand as the first period. Note 2. If there be decimals in the given number, it must be pointed both ways from the place of units: If, when there are * In order to shew the reason of the rule, it will be proper to premise the following Lemma. The product of any two numbers can have, at most, but so many places of figures as are in both the factors, and at least but one less. Demonstration. Take two numbers consisting of any number of places; but let them be the least possible of those placss, viz. Unity with cyphers, as 100 and 10: Then their product will be 1 with so many cyphers annexed as are in both the numbers, viz. 1000; but 1000 has one place less than 100 and 10 together have: And since 100 and 10 were taken the least possible, the product of any other two numbers, of the same number of places, will be greater than 1000; consequently, the product of any two numbers can have, at least, but one place less than both the factors. Again, take two numbers, of any number of places, which shall be the greatest possible of those places, as 99 and 9. Now, 99 x 9 is less than 99 X 10; but 99 × 10 (=990) contains only so many places of figures as are in 99 and 9; herefore, 99 X 9, or the product of any other two numbers, consisting of the same number of places, cannot have more places of figures, than are in both its factors. Corollary 1. A square number cannot have more places of figures than double the places of the root, and at least but one less. Corollary 2. A cube number cannot have more places of figures than triple the places of the root, and at least but two less. integers, the first period in the decimals be deficient, it may be completed by annexing so many cyphers as the power requires: And the root must be made to consist of so many whole numbers and decimals as there are periods belonging to each; and when the periods belonging to the given number are exhausted, the operation may be continued at pleasure by annexing cyphers. The Rule for the extraction of the square root, may be illustrated by a tending to the process by which any number is raised to the square. The several products of the multiplication are to be kept separate, as in the proof of the rule for Simple Multiplication. Let 37 be the number to be raised to the square. 37X37 136937 X37 3d. What is the square root of 10342656 ? 5th. What is the square root of 234-09 ? 4th. What is the square root of 964-5192360241? Ans. 31·05671. Ans. 3216. Ans. 15.3. Ans. 00563. Ans. 213. 6th. What is the square root of ⚫0000316969? 7th What is the square root of '045369? RULES For the Square Root of Vulgar Fractions and Mixed Numbers. After reducing the fraction to its lowest terms, for this and all other roots; then, 1st. Extract the root of the numerator for a new numerator, and the root of the denominator for a new denominator, which is the best method, provided the denominator be a complete power. it be not, 2d. Multiply the numerator and denominator together; and the root of this product being made the numerator to the denominator of the given factor, or made the denominator to the numerator of it, will form the fractional part required.* Or, Now, it is evident that 9, in the place of hundredths, is the greatest square in this product; put its root, 3, in the quotient, and 900 is taken from the product. The next products are 21+21=2×3×7, for a dividend. Double the root already found, and it is 2×3, for a divisor, which gives 7 for the quotient, which annexed to the divisor, and the whole then multiplied by it, gives 2×3×7(=42) +7x7 (49) which placed in their proper places, completely exhausts the remainder of the square. The same may be shown in any other case, and the rule becomes obvious. Perhaps the following may be considered more simple and plain. Let 37,30+7, be multiplied, as in the demonstration of simple multiplication, and the products kept separate. The reason of which is, that the value of a fraction is not altered by multiplying both its parts by the same quantity. Thus√ VIX2, and √2%√2=2 evidently. 3d. Reduce the vulgar fraction to a decimal, and extract its root. 4th. Mixed numbers may either be reduced to improper fractions, and extracted by the first or second rule, or the vulgar fraction may be reduced to a decimal, then joined to the integer, and the root of the whole extracted. 31)81 Therefore, the root of the given fraction. 81 By Rule 2. 16×1681=26896, and √26896=164. Then, 1681 16 M=M=409756+ Ans. 1681)16(0095181439+. And ✓.0095181439=·09756+. Ans. 61. Note. In extracting the square or cube root of any surd number, there is always a remainder or fraction left, when the root is found. To find the value of which, the common method is, to annex pairs of cyphers to the resolvend, for the square, and ternaries of cyphers to that of the cube, which makes it tedious to discover the value of the remainder, especially in the cube, whereas this trouble might be saved if the true denominator could be discovered. As in division the divisor is always the denominator to its own fraction, so likewise it is in the square and cube, each of their divisors being the denominators to their own particular fractions or numerators. In the square the quotient is always doubled for a new divisor; therefore, when the work is completed, the root doubled is the true divisor or denominator to its own fraction; as, if the root be 12, the denominator will be 24, to be placed under the remainder, which vulgar fraction, or its equivalent decimal, must be annexed to the quotient or root, to complete it.* If to the remainder, either of the square or cube, cyphers be annexed, and divided by their respective denominators, the quotient will produce the decimals belonging to the root. Although these denominators give a small matter too much in the square root, and too little in the cube, yet they will be sufficient in common use, and are much more expeditious than the operation with cyphers. A a APPLICATION AND USE OF THE SQUARE ROOT. PROB. I. To find a mean proportional between two numbers. RULE. Multiply the given numbers together, and extract the square root of the product; which root will be the mean proportional sought. Note. When the first is to the second as the second is to the third, the second is called a mean proportional between the other two. Thus, 4 is a mean proportional between 2 and 8, for 2: 4 :: 4X4 4 :: 2 -=8, or 4 is as much greater than 2, as 8 is greater than 4. By Theorem I. of Geometrical Proportion 2×8=4x4=4'. To find a mean proportional between 2 and 8, take the square root of their product. The same must be true in every case, and is the rule. EXAMPLE. What is the mean proportional between 24 and 96 ? 96x24 43. Answer. PROB. II. To find the side of a square equal in area to any given superficies whatever. KULE. Find the area, and the square root is the side of the square sought.* EXAMPLES. 1st. If the area of a circle be 184-125, What is the side of a square equal in area thereto ? 184 125-13-569+ Answer. 2d. If the area of a triangle be 160, what is the side of a square equal in area thereto ? √160=12.649+ Answer. PROB. III. A certain general has an army of 5625 men: pray How many must he place in rank and file, to form them into a square? 5625 75 Answer.1 PROB. IV. Let 10952 men be so formed, as that the number in rank may be double the file. 74 in file, and 148 in rank. PROB V. If it be required to place 2016 men so as that there may be 56 in rank and 36 in file, and to stand 4 feet distance in rank and as much in file, How much ground do they stand on? To answer this, or any of the kind, use the following proportion: As unity: the distance :: so is the number in rank less by one to a fourth number; next, do the same by the file, and mul A square is a figure of four equal sides, each pair meeting perpendicularly, or, a figure whose length and breadth are equal. As the area, or number of square feet, inches, &c. in a square, is equal to the product of two sides which are equal, the second power is called the square. Hence the rule of Prob. II. is evident. + If you would have the number of men be double, triple, or quadruple, &c. as many in rank as in file, extract the square root of 4, 3, 4, &c. of the given number of men, and that will be the number of men in file, which double, triple. quadruple, &c. and the product will be the number in rank. |