Since x equals the (+ 1)th convergent of (2), by the recurrence formula, Clearing (3), xqr + gr-1 Irx2 - (Pr - Ir-1)x —Pr−1 = 0, a quadratic in x whose positive real root (§ 304, I) is x = Pr - qr−1 + √(Pr−¶r−1)2 + 4 pr-19r. 2qr Since a recurs, a1 must be equal to or greater than 1. .. Pr>r>gr-1, Pr - qr-1 is a positive integer, and x is real. To prove that the radical in (4) is a quadratic surd, write √(Pr − ¶r−1)2 + 4 Pr−19r=√(Pr+ Įr−1)2 + 4(Pr−19r — Pr¶r−1) Prin. 1, formula (1), = = √(Pr+ 9r−1)2 ± 4. (4) Since no positive integral square increased or diminished by 4 is a perfect square, the radical is a quadratic surd. Hence, x is a quadratic surd. Therefore, if the value of x is substituted for x in (1) and the result is simplified by repeated rationalization of denominators and reduction to higher terms, y will finally be reduced to the form of a quadratic surd. EXAMPLES Reduce each of the following to a periodic continued fraction and find a near approximation (§ 525) to its value : THEORY OF NUMBERS SCALES OF NOTATION 528. The theory of numbers treats of the properties of positive integers. In this chapter the word number will be used in the sense of positive integer, and the terms integer, even number, odd number, power, root, etc., will be employed in the same sense as in arithmetic, unless the contrary is stated. 529. The Arabic system of notation is called the decimal system, because ten units of any order make one unit of the next higher order. 10 is called the radix of the scale of notation. Any number in the decimal system may be expressed in a scale of descending powers of 10, the exponent of the highest power being one less than the number of digits of the number. Thus, 256473200000 + 50000 + 6000 + 400 + 70 + 3 = 2(10)6 + 5(10)1 + 6(10)3 + 4(10)2 + 7(10) + 3. 530. Any number greater than 1 may be used as the radix of a scale of notation. The meaning of the expression 256473 in the decimal scale has just been shown. But in the scale whose radix is 8, 256473 = 2(8)5 + 5(8)1 + 6(8)3 + 4(8)2 + 7(8) + 3. In the binary scale the radix is 2; in the ternary, 3; in the quaternary, 4; in the quinary, 5; in the senary, 6; in the septenary, 7; in the octary, 8; in the nonary, 9; in the denary, or decimal, 10; in the undenary, 11; in the duodenary, or duodecimal, 12; and so on. The general symbol for any radix is r. 531. PRINCIPLE. Any integer N may be expressed in the scale whose radix is r in one and only one way. The above principle may be established as follows: Divide N by r, and let the quotient be N1 and the remainder s. Divide N1 by r, and let the quotient be N2 and the remainder q. (1) Continuing the process, a quotient less than r is finally obtained. Suppose this occurs after r- 1 divisions, and denote the last quotient and remainder by a and b, respectively. Then, Nam-1+ brn−2+ ... + gr+s. The coefficients a, b, ..., q, s are the digits of the number, and since each is a remainder of some division by r, each is less than r. Also, since there can be but one remainder less than r for each division, there is but one way of expressing N in the scale of r. COROLLARIES.-1. The exponent of the highest power of the radix is one less than the number of digits. 2. In any scale the number of different characters, including 0, that are required to express all numbers is equal to the radix. For example, in the duodecimal scale the ten Arabic figures and two other characters, t for ten and e for eleven, are required; in the quinary scale only the figures 0, 1, 2, 3, 4 are required. 532. To change from the decimal to another scale. EXAMPLES 1. Change 3010 from the decimal to the senary scale. PROCESS 63010 6/501 6 83 6 13 2 ... ... 3 5 EXPLANATION. - Since 6 units of any order make one unit of the next higher order, 3010 units of the first order make 501 units of the second order and 4 units of the first order; 501 units of the second order make 83 units of the third order and 3 units of the second order; and so on. Hence, 3010 when expressed in the senary scale is the sum of 2 units of the fifth order, 1 unit of the fourth order, 5 units of the third order, 3 units of the second order, and 4 units of the first order. For convenience the radix of the scale is indicated by a subscript figure, which may be omitted in the decimal scale. Thus, 3010: = 215346. 2. Express in the octary scale 50, 128, and 5283. 3. Express in the quinary scale 12, 342, and 6627. 4. Express in the duodecimal scale 15, 100, and 6053. 533. To change from any scale to the decimal scale. EXAMPLES 1. Change 21534, to the decimal scale. PROCESS 21534 6 13 6 83 6 501 6 3010 EXPLANATION. -Since each unit of any order is equal to 6 units of the next lower order, 2 units of the fifth order are equal to 12 units of the fourth order, and adding 1, the number of units of the fourth order in 215346, the whole number of units of the fourth order is 13. Continuing in a similar way to reduce the units of each order to units of the next lower order, the whole number of units of the lowest order is found to be 3010, which is expressed in the decimal scale. Change the following to the decimal scale: 2. 427. 3. 6654. 4. 4t611. 5. 6e412. 534. Arithmetical processes in any scale. 6. 1111001, The processes are performed in the same manner as in the deci、 mal scale. The student must simply bear in mind each time the number of units of each order required to make one of the next higher order. When the process is complicated or when different scales are involved, all the numbers may first be reduced to the decimal scale. EXAMPLES Perform the operations indicated: 1. 3812 + 4512 + €612. 3. 4241, 3323,. 5. 1304,25,. 2. 101, 110, +101101. 4. 4e582312 x 1512 6. √1321. 7. Multiply 2112 by 13, and express the product in the decimal scale. 8. Show that in any scale the expression 121 represents a per fect square. 9. In what scale is 5 times 6 expressed by 36? 10. In what scale is of the number 100 equal to the number 30? 11. Which of the weights of 1, 2, 4, 8, ... pounds must be taken to aggregate 75 pounds, if not more than one of each is used? PRINCIPLES OF DIVISIBILITY 535. In this subject 'divisible' is used for 'exactly divisible.' Also, since there is no remainder when zero is divided by any number, zero is regarded as exactly divisible by any number. ... 536. If the number N = arn¬1 + bpn-2 + 1 + pr2 + qr + s is divided by r, the remainder is s, the last digit; if N is divided by , the remainder is the number that is expressed by the last two digits; if N is divided by 3, the remainder is the number that is expressed by the last three digits; and so on. Likewise, if N is divided by a factor of r, the remainder is the same as that obtained by dividing the last digit by that factor; if N is divided by a factor of 2, the remainder is the same as that obtained by dividing qr+s, the number expressed by the last two digits, by that factor; and so on. Hence, PRINCIPLE 1.- (a) If a number in the scale of r is divided by , the remainder is the number that is expressed by the last m digits of the given number. (b) If the number is divided by a factor of rm, the remainder is the same as that obtained by dividing by that factor the number that is expressed by the last m digits. COROLLARIES. 1. In the decimal system a number ending in m ciphers is divisible by 10m. 2. In the decimal system (a) Every number ending in 0 or 5 is divisible by 5. (b) Every number ending in 00, 25, 50, or 75 is divisible by 25. (c) Every number ending in three ciphers or in three digits that express a multiple of 125 is divisible by 125. And so on. 3. In the decimal system (a) Every number ending in 0, 2, 4, 6, or 8 is divisible by 2. |