INTRODUCTION TO ARITHMETIC. PART XI. CONTAINING, EXTRACTION OF THE SQUARE ROOT, OF THE CUBE ROOT, AND SINGLE AND DOUBLE POSITION. و EXTRACTION OF THE SQUARE ROOT. А square is the product of a number multiplied by itself; as, 4x4=16; here 16 is the square of 4: This is called Involution ; and when this sign is inserted 7 it signifies, the square of any number is required, as, 74, is as much as to say, “ multiply 4 by itself,” or, “ the square of 4." The Square Root is a number extracted, which being multiplied by itself will produce the given number it was extracted from; as, the square root of 16=1; and this root 4 multiplied by itself, will produce its resolvend or dividend 16. Extraction of Roots is called Evolution; and when the square root of any number is required, it may be signified by this mark placed before the number thus, V16; that is to say, “ the square root of 16.” A TABLE OF ROOTS AND SQUARES. 5 7 8 9 Squares, 1 9 16 25 36 49 36 49 64 81 The dividend is commonly called the Resolvend. The quotient is the Root. That root which can be found to exactness, is a rational root. That root which cannot be found to exactness, is a surd. 6 When odd places are given in the decimals, make them even by annexing ciphers. When a root proves to be a surd, annex ciphers by pairs, till the fraction is reduced to a sufficient degree of minute ness. When preparing the dividend for extraction, begin with the integers at units and count two figures to the left, there make a distinction; then count two more, and continue this operation till the whole numbers are marked into pairs: Begin again with the decimals, and proceed to the right marking them also into pairs. EXAMPLES. LESSON 1. 3 6,27,50.25, d root 4 225 25025 Point off so many decimals in the quotient or root, as there are pairs in the fractions. In this case there is one pair in the decimals, therefore we point off one figure in the root for decimals. OPERATION WITH WORDS. First seek the root of 6; it must be 2; place 2 under in the quotient; multiply it by itself; set its square 4 under 6; subtract, the remainder is 2. Second : Bring down 27 from under b to the right of 2, makes 227 for a new dividend: Multiply the quotient by 2 makes 4 ; place this 4 for a divisor as at 4 under e. Seek how many times 4 in the dividend, excepting always the right hand figure; say 5 times; place 5 on the right side of 4, for a new divisor, also place 5 in the quotient as under o; then multiply as in Long Division, and place the product 225 under 227; subtract, the remainder is 2. Third : Bring down the pair 50 from under c to the right side of 2, makes 250; double the quotient and place the product as at 50 under f, for a divisor; seek how many times 50 in 250, rejecting the right hand figure in the dividend; set no times or cipher in the quotient on the right side of 5 under o, and place a cipher also in the divisor at the right hand of 50 under f, which cipher will make the divisor 500. Fourth: Bring down the pair 25 from under d, to the right side of 250, makes 25025 for a new dividend ; seek how many times 500 in 25025, rejecting the right hand 5 in the dividend; say 5 times; set 5 on the right hand of the divisor 500 under f, makes 5005 for a new divisor, and place 5 under t, multiply the divisor by the quotient figure, as in Long Division, and place the product under 25025. The root or answer is 250.5 PROOF OF LESSON 1. RULE. Multiply the root by itself, the product will be the same as the resolvend or dividend. LESSON 2. 250.5 What is the square root of 121377.071681? 12525 2,13,77.07,16,81(146.209 125250 1 5010 Ans. 146.209 Proof, 62750.25 24)113 96 1716 i. 2922)6107 Continued. 72250.5 2922)6107 5844 2631681 1 OPERATION WITH WORDS: First: Seek the root up to the first comma; the root of 2 is 1: place 1 in the quotient and I under the first figure 2; subtract, the remainder is one. Second: Bring down the pair 13 to the right side of 1, makes 113 for a new dividend ; double the quotient figure 1 and place the product for a divisor as at 2 under seek how many times 2 in the new dividend 113, rejecting the right hand figure 3; say 4 times; set 4 in the quotient; set 4 also on the right of 2 under g, makes 24 for a new divisor; multiply this divisor by the last figure 4 in the quotient; set the product 96 under 113; subtract, the remainder is 17. Third : Bring down the pair 77 to the right side of 17, makes 1777 for a new dividend ; double the 14 in the quotient makes 28; place 28 for a divisor as under h. seek how many times 28 in 1777, rejecting the right hand 7; say 6 times; set 6 in the quotient at the right side of 14; and place 6 also on the right of the divisor 28, which makes 286 for a new divisor; multiply as in Long Division, and place the product under 1777; subtract; the remainder is 61. Fourth; Bring down the pair 07 to the right side of 61, makes 6107 for a new dividend; double the three figures in the quotient; set their product for a divisor as at 292 under i ; seek how many times 292 in 6107; say 2 times; place 2 in the quotient on the right of 6; place 2 also on the right of 292, makes 2922 for a new divisor; multiply as in Long Division, and place the product under 6107 ; subtract, the remainder is 263. |