CHAPTER XIX. SYSTEMS OF NOTATION. 1. A System of Notation is a method of expressing numbers by means of a series of powers of some fixed number called the Radix, or Base of the scale in which the different numbers are expressed. 2. The Radix of any system is the number of units of one order which makes one of the next higher. 4. In writing any number in a uniform scale, as many distinct characters, or symbols, are required as there are units in the radix of the given system. Thus, in the decimal system, 10 characters are required; in the ternary, 3; viz., 1, 2, and 0; in the senary, 6; viz., 1, 2, 3, 4, 5, and 0; and so on. 5. Let be the radix of any system, then any number, N, may be expressed in the form, Divide To express an integral number in a proposed scale : the number by the radix, then the quotient by the radix, and so on; the successive remainders taken in order will be the successive digits beginning from units place. I. Express the common number, 75432, in the senary system. III. 7. .. 75432 in the decimal system=1341120 expressed in the senary system. I. Transform 3256 from a scale whose radix is 7, to a scale whose radix is 12. III. ... 12)166+4 0+8 3256 in the septenary system-814 in the duodenary system. Explanation.-In the senary system, 7 units of one order make one of the next higher. Hence, 3 units of the fourth order = 7X3, or 21, units of the third order. 21 units +2 units = 23 units. 23-12=1, with a remainder 11. 11 units of the third order = 77 units of the second order. 77 units +5 units 82 units. 82-12=6, with a second order 70 units of the first order. 12-6, with a remainder 4. Hence, the mainder 4. Treat this quotient in like tient is obtained, that is less than 12. remainder 10. 10 units of the 70 units +6 units=76 units. 76 first quotient is 166, with a remanner, and so on, until a quo I. What is the sum of 45324502 and 25405534, in the senary system? 45324502 115134440 Explanation.-4+2-6. 6-6-1, with no remainder. Write the 0 and carry the 1. 3+1=4. Write the 4. 5+5=10. 10÷÷÷6-1, with a remainder Write the 4 and carry the 1. 5+4+1=10. 10÷6-1, with a remainder 4. Write the 4 and carry the 1. 0+2+1=3. Write the 3. 4+3-7. 76+1 with a remainder 1. Write 1 and carry 1 5+5+1=11. 11÷6 4. 1, with a remainder 5. Write the 5 and carry the 1. 2+4+1=7. 7÷6= 1, with a remainder 1. Write 1 and carry 1. The result is 115134440. I. What is the difference between 24502 and 5534 in the octonary system? 24502 16746 Explanation.-4 cannot be taken from 2. Hence, borrow one unit from a higher denomination. Then (2+8)-6-4. (8—1)—3—4. 5 from (4+8) =7. 5 from (3+8)=6. Hence, the result is 16746. I. Transform 3413 from the scale of 6 to the scale of 7. (1. 7)3413 II.2. 7)310+3 III. 3. 7)24+3 2+2 .. 3413 in the senary system=2233 in the septenary sys tem. I. Multiply 24305 by 34120 in the senary system. 24305 34120 530140 24305 150032 121323 1411103040 4. Explanation-Multiplying 5 by 2 gives 10. 10÷6-1, with a remainder Write 4 and carry 1 to the next order. 2 times 0 0. 0+1=1. Write the 1. 2 times 3-6. 6-6-1, with a remainder 0. Write the 0 and carry the 1 to the next higher order. 2 times 4-8. 8+1-9. 9÷6=1, with a remainder 3. Write 3 and carry the 1 to the next higher order. 2 times 24. 4+1-5. Write 5. Multiply in like manner by 1, 4, and 3. Add the partial products, remembering that 6 units of one order, in the senary system, uniformly make one of the next higher. I. Multiply 2483 by 589 in the undenary system, or system whose radix is 11. Explanation. In the undenary system, 11 units of one order uniformly make one of the next higher order. 9 times 3 27. 27+11=2, with a remainder 5. Write 5 and carry the 2 to the next higher order, or second order. 9 times 8-72. 72+2=74. 74÷11=6, with a remainder 8. Write 8 and carry the 6 to the next higher order, or third order. 9 times 436. 36+6=42. 42÷11-3, with a remainder 9. Write 9 and carry the 3 to the next higher order, or the fourth order. 9 times 2-18. 18+3=21. 21÷11 =1, with a remainder t. Write t and carry the 1 to the next higher order. Multiply in like manner by 8 and 5. Add the partial products, remembering that 11 units of one order equals one of the next higher. Wherever 10 occurs, it must be represented by a single character t. I. Divide 1184323 by 589 in the duodenary system. In the duodenary system, we must have 12 characters; viz., 1, 2, 3, 4, 5, 6, 7, 8, 9, t, e, and 0. t represents 10 and e, 11. 589)1184323(2486 2213 1te0 3e32 3910 1523 Explanation. In the duodenary system, 12 units of one order make one of the next higher. 1184 will contain 589, 2 times. Then multiply the divisor, 589, by 2 thus: 2 times 9-18. 18÷12-1, with a remainder 6. Write the 6 and carry the 1. 2 times 8-16. 16+1=17. ' 17÷12=1, with a remainder 5. Write the 5 and carry the 1. 2 times 5 t. t+1=e. Write the e. Then subtract. 6 from (12+4)=t, 5 from 7=2, and e from (12 +1)=2. Hence, the first partial dividend is 22 t. contain 589, 4 times. Multiply as before. obtain 2483 for a quotient. Bring down 3. Then 22t3 will I. Divide 95088918 by tt4, in the duodenary system. tt4)95088918(t4tee 9074 4548 3754 9e49 9074 t951 9e58 t e58 t e58 I. Extract the square root of 11122441 in the senary system. Explanation. The greatest square in 11 expressed in the senary system is 4. Subtracting and bringing down the next period, we have 312 for the next partial dividend. Doubling the root already found and finding how many times it is contained in 312 expressed in the senary system, we find it is 4. Continuing the process the same as in the decimal system, the result is 2405. I. Extract the square root of 11000000100001 in the binary system. I. Find in what scale, or system, 95 is denoted by 137. II. III. 3. r2+3r-95-7-88, and |4. r2+3r+2=88+2=341, by completing the square. .. 95 is denoted by 137 in the octonary system. (Todhunter's Alg., p. 255, prob. 26.) I. Find in what system 1331 is denoted by 1000. 1. Let r=the radix of the system. Then II. 3. 4-1331. Whence 4. r=√1331=11, the radix of the system. III. .. 1331 is denoted by 1000 in the undenary system. 4 (Todhunter's Alg., p. 255, prob. 28.) |