(i. e., have no common divisor), a2 and b2 are also prime to each other;* there fore a2 is an irreducible fraction, and can not be equal to a whole number. 85. The difference between the squares of two consecutive whole numbers is greater in proportion as the numbers themselves are greater; the expression for this difference can easily be found. Let a and a+1 be two consecutive whole numbers; that is to say, the difference of the squares of two consecutive whole numbers is equal to twice the less of the two numbers plus unity. Thus, the difference between the squares of 348 and 347 is equal to 2x3471, or 695. *This depends upon the principle that, if any prime number, P, will divide the product of two numbers, it must divide one of them, which may be demonstrated as follows: Let A and B be the two numbers, and let it be supposed that P will not divide A, we are to prove that it must divide B. Dividing A by P, and denoting the quotient by Q and the remainder by P', we have AB P'B A=PQ+P'.. multiplying by B, AB=PQB+P'B .. dividing by P, P ·=QB+· P Since by hypothesis AB is divisible by P, P'B must be, else we should have a whole number, equal to a whole number plus a fraction, which is impossible. Proceed now with P and P' after the method for finding a common divisor, and let P", P"", &c., be the suc cessive remainders, which can none of them be zero, because P is by hypothesis a prime number (i. e., a number divisible only by itself and unity): these remainders must go on diminishing till the last becomes unity, and we shall have the series of equalities, P=P'Q'+P", P=P"Q"+P"", &c.; or, multiplying by B and dividing by P, P'Q'B PB PB P"Q"B, P""B The first of these equalities shows that if P'B is divisible by P, P/B must also be divisi ble; and if both these are divisible, the second equality shows that PB is divisible by P, and so on. But the remainders, P", P"", &c., diminish till the last becomes unity, and we shall thus have, finally, 1XB, or B divisible by P. Q. E. D. Now, since a2 is the product of a and a, any prime number which divides a must divide a, or which divides b3 must divide b, so that any prime number which divides both a2 and must divide a and b. Every number is either prime or composed of prime numbers as factors, and if this number will divide the two terms of a fraction, its prime factors will successively divide them. This follows from (10, I., 2). As an addition to this note may be demonstrated the following theorem: A literal quantity can not be decomposed into prime factors in different ways. Let ABCD... be a product of prime factors, and suppose that it could be equal to another product, abcd..., the factors a, b, c, d... being also prime. The factor a, dividing abcd, must divide the equal ABCD...; but if the prime quantity a is different from each of the quantities A, B, C, D, &c., it can not divide any of them. Not dividing either A or B according to the above theorem, it can not divide the product AB. Not dividing either AB or C, it will not divide the product ABC, and so on. The factor a must, therefore, necessarily be equal to one of the factors A, B, C, &c. Suppose a=A. Dividing the two products by A, the remaining products, BCD... and bed..., are still equal, and applying to them the preceding reasoning, we conclude that bought to be equal to one of the factors of the product, BCD..., and so on. The two products, ABCD... and abcd..., must, therefore, be composed of the same prime factors. Q. E. D The square of a number will always consist of twice as many digits, or one less than twice as many, as the number itself. Thus, the square of 10 is 100, and the square of any number less than 10 must be less than 100, or contain not more than two figures. The square of 100 is 10000, and the square of all numbers between 10 and 100 must be between 100 and 10000; i. e., consist of 3 or 4 figures. In the same way it may be shown that the square of a number containing three figures must be one containing five or six figures, and so on; i. e., the square of a number consists of twice as many digits as the number itself, or one less than twice as many. Let us now proceed to investigate a process for the extraction of the square root of any number, beginning with whole numbers. EXTRACTION OF THE SQUARE ROOT OF WHOLE NUMBERS. 86. If the number proposed consist of one or two figures only, its root may be found immediately by inspecting the squares of the nine first numbers in (Art. 83). Thus, the square root of 25 is 5, the square root of 42 is 6 plus a fraction, or 6 is the approximate square root of 42, and is within one unit of the true value; for 42 lies between 36, which is the square of 6, and 49, which is the square of 7. Let us consider, then, a number composed of more than two figures, 6084 for example. Since this number consist of four figures, its root must necessarily consist of two figures, that is to say, of tens and units. Designating the tens in the root sought by a, and the units by b, we have 6084=(a+b)2=a2+2ab+b2, 60'841 118'4 Q. 78 which shows that the square of a number consisting of tens and units is composed of the square of the tens, plus twice the product of the tens by the units, plus the square of the units. This being premised, since the square of a certain number of tens must be a certain number of hundreds, or have two ciphers on the right, it follows that the squares of the tens contained in the root must be found in the part 60 (or 60 hundreds), to the left of the last two figures of 6084 (which written at full length is 6000+80+4), the 84 forming no part of the square of the tens; we, therefore, separate the last two figures from the others by a point. The part 60 is comprised between the two perfect squares 49, and 64, the roots of which are 7 and 8; hence 7 is the figure which expresses the number of tens in the root sought; for 6000 is evidently comprised between 4900 and 6400, which are the squares of 70 and 80, and the root of 6084 must, therefore, be comprised between 70 and 80; hence, the root sought is composed of 7 tens and a certain number of units less than ten. The figure 7 being thus found, we place it on the right of the given number, in the place of tens, separated by a vertical line as in division; we then subtract 49, which is the square of 7, from 60, which leaves as remainder 11 (which is 11 hundreds), after which we write the remaining figures, 84. Having taken away the square of the tens, the remainder, 1184, contains, as we have seen above, twice the product of the tens multiplied by the units plus the square of the units. But the product of the tens multiplied by the units must be tens, or have one cipher on the right, and, therefore, the last figure 4 can not form any part of the product of the tens by the units; we, therefore, separate it from the others by a point. If we double the tens, which gives 14, and divide the 118 tens by 14, the quotient 8 is the figure of units in the root sought, or a figure greater than the one required. It may manifestly be greater than the figure sought, for 118 may contain, in addition to twice the product of the tens by the units, other tens arising from the square of the units, which may exceed the denomination units. In order to determine whether 8 expresses the real number of units in the root, it is sufficient to place it on the right of 14, and then multiply the number 148, thus obtained, by 8. In this manner we form, 1o, the square of the units; 2°, twice the product of the units by the tens. This operation being effected, the product is 1184; subtracting this product, the remainder is 0, which shows that 6084 is a perfect square, and 78 the root sought. It will be seen, in reviewing the above process, that we have successively subtracted from 6084, the square of 7 tens or 70, plus twice the product of 70 by 8, plus the square of 8, that is, the three parts which enter into the composition of the square of 70+8, or 78; and since the result of this subtraction is 0, it follows that 6084 is the square of 78. The quotient obtained from dividing by double the tens is a trial figure; it will never be too small, but may be too great, and on trial may require to be diminished by one or two units. Take as a second example the number 841. This number being comprised between 100 and 10000, its root must consist of two figures, that is to say, of tens and units. We can prove, as in the last example, that the root 8'41 29 4 49 44'1 441 0. of the greatest square contained in 8, or in that portion of the number to the left of the last two figures, expresses the number of tens in the root required. But the greatest square contained in 8 is 4, whose root is 2, which is, therefore, the figure of the tens. Squaring 2, and subtracting the result from 8, the remainder is 4; bringing down the figures of the second period 41, and annexing them on the right of 4, the result is 441, a number which contains twice the product of the tens by the units, plus the square of the units. We may farther prove, as in the last case, that if we point off the last figure 1, and divide the preceding figures 44 by twice the tens, or 4, the quotient will be either the figure which expresses the number of units in the root, or a figure greater than the one sought. In this case the quotient is 11, but it is manifest that we can not have a number greater than 9 for the units, for otherwise we must suppose that the figure already found for the tens is incorrect. Let us try 9; place 9 to the right of 4, and then multiply this number 49 by 9; the product is 441, which, when subtracted from the result of the first operation, leaves a remainder 0, proving that 29 is the root required. Let us take, as a third example, a number which is not a perfect square, such as 1287. Applying to this number the process described in the preceding example, we find that the root is 35, with a remainder 62. This shows that 1287 is not a perfect square, but that it is comprised between the square of 35 and that of 36. Thus, when the number is not a perfect square, the above 12'87 35 65 38'7 62 process enables us at least to determine the root of the greatest square contained in the number, or the integral part of the root of the number. 87. Let us pass on to consider the extraction of the square root of a number composed of more than four figures. Let 56821444 be the number. Since the number is greater than 10000, its root must be greater than 100; that is to say, it must consist of more than two figures.* But, whatever the number may be, we may always consider it as composed of units and of tens, the tens being expressed by one or more figures. (Thus, any number such as 37142 may be resolved into 37140+2, or 3714 tens, plus two units.) 56'82'14'44 7538 49 145 78'2 0. Now the square of the root sought, that is, the proposed number, contains the square of the tens, plus twice the product of the tens by the units, plus the square of the units. But the square of the tens must give at least hundreds; hence the last two figures, 44, can form no part of it, and it is in the portion of the number to the left hand that we must look for that square. But this portion containing more than two figures, its root will consist of units and tens; it will, therefore, be necessary to commence the process for finding the root of this portion by cutting off its two right-hand figures, 14, and the square of the tens of the tens is to be sought in the figures now remaining at the left, 5682. This number being the square of two figures, we again separate 82, and seek for the square of the tens of the tens of the tens in the two remaining figures, 56. The given number is thus separated into periods of two figures each, beginning on the right. We then go on to extract the root of the number 5682, as in the previous examples; this will give the tens of the root of the number 568214. We then double these tens for a divisor, and take the remainder after the last operation, with 14 annexed for a dividend; we divide this dividend, after cutting off the right-hand figure, and the quotient will be the units of the root of 568214. All the figures now found of the root will constitute the tens of the root of the given number, and we find the units by the rule previously given. The detail of the whole operation is as follows: Extracting the root of 56, we find 7 for the root of 49, the greatest square contained in 56; we place 7 on the right of the proposed number, and squaring it, subtract 49 from 56, which gives a remainder 7, to which we annex the following period, 82. Separating the last figure to the right of 782, and then dividing 78 by 14, which is twice the root already found, we have 5 for a quotient, which we annex to 14; we then multiply the whole number 145 by 5, and subtract the product 725 from 782. We next bring down the period 14, annex it to the second remainder 57, and point off the last figure of this number 5714. Dividing 571 by 150, which is twice the root already found, the quotient is 3, which we place to the right of 150, and multiplying the whole number 1503 by 3, we subtract the product 4509 from 5714. Finally, we bring down the last period 44, annex it to the third remainder 1205, and point off the last figure of this number 120544. Dividing 12054 by * We have seen in the last article that it will consist of four figures, half as many as the given number. Had the given number contained but seven figures, the root would still be composed of four. 1506, which is twice the root already found, the quotient is 8, which we place on the right of 1506, and multiplying the whole number 15068 by 8, we subtract the product 120544 from the last result 120544. The remainder is 0; hence 7538 is the root sought. From what has been said above, it is easy to deduce the rule, ordinarily given in Arithmetic, for the extraction of the square root of a number consisting of any number of figures, and which it is unnecessary here to repeat. EXTRACTION OF THE SQUARE ROOT BY APPROXIMATION. 88. When a whole number is not the square of another whole number, we have seen (Art. 84) that its root can not be expressed by a whole number and an exact fraction; but although it is impossible to determine the precise value of the fraction which completes the root sought, we can approximate it as nearly as we please. Suppose that a is a whole number which is not a perfect square, and that 1 we are required to extract the root to within that is, to determine a number n' 1 which shall differ from the true root of a, by a quantity less than the fraction n To effect this, let us observe that the quantity a may be put under the form ; if we designate the integral, or whole number, portion of the root of an2 ana n2 an n2 consequently, the root of a is com by r, this number an2 will be comprised between r2 and (r+1)2; hence, T r+1 that is, between - and Thus, n 1 of the true value. n it appears that represents the square root of a within From this we derive the following RULE. To extract the square root of a whole number to within a given fraction, multiply the given number by the square of the denominator of the given fraction; extract the integral part of the square root of the product, and divide this integral part by the given denominator. Let it be required, for example, to find the square root of 59 within of the true value. Multiply 59 by the square of 12, that is, 144, the product is 8496; the integral part of the root of 8496 is 92. Hence or 7 is the approximate root of 59, the result differing from the true value by a quantity less than 12 So, also, VII 3 true to, 89. The method of approximation in decimals, which is the process most frequently employed, is an immediate consequence of the preceding rule. ... In order to obtain the square root of a whole number within, TOO, TOOO · of the true value, we must, according to the above rule, multiply the proposed number by (10), (100), (1000)3, or, which comes to the same thing, ..... |