The number which 10 stands for is called the radix of the scale of notation. To change a number from one scale into another, divide the number, written as in the first scale, by the number which is to be the radix of the new scale; repeat this division again and again, and the remainders are the digits required. For example, what, in the quinary scale, is that number which, in the decimal scale, is 17036?
- 5)17036
- 5)3407 Remʳ. 1
- 5)681 2
- 5)136 1
- 5)27 1
- 5)5 2
- 5)1 0
- 01
| Answer 1021121 | ||
| Quinary. | Decimal. | |
| Verification, 1000000 | means | 15625 |
| 20000 | 1250 | |
| 1000 | 125 | |
| 100 | 25 | |
| 20 | 10 | |
| 1 | 1 | |
| 1021121 | 17036 | |
The reason of this rule is easy. Our process of division is nothing but telling off 17036 into 3407 fives and 1 over; we then find 3407 fives to be 681 fives of fives and 2 fives over. Next we form 681 fives of fives into 136 fives of fives of fives and 1 five of fives over; and so on.
It is a useful exercise to multiply and divide numbers represented in other scales of notation than the common or decimal one. The rules are in all respects the same for all systems, the number carried being always the radix of the system. Thus, in the quinary system we carry fives instead of tens. I now give an example of multiplication and division:
| Quinary. | Decimal. | |
| 42143 | means | 2798 |
| 1234 | 194 | |
| 324232 | 11192 | |
| 232034 | 25182 | |
| 134341 | 2798 | |
| 42143 | ||
| 114332222 | 542812 |
| Duodecimal. | Decimal. |
| 4t9)76t4e08(16687 | 705)22610744(32071 |
| 4t9 | 1460 |
| 2814 | 5074 |
| 2546 | 1394 |
| 28te | 689 |
| 2546 | |
| 3650 | |
| 3320 | |
| 3308 | |
| 2t33 | |
| 495 |
Another way of turning a number from one scale into another is as follows: Multiply the first digit by the old radix in the new scale, and add the next digit; multiply the result again by the old radix in the new scale, and take in the next digit, and so on to the end, always using the radix of the scale you want to leave, and the notation of the scale you want to end in.
Thus, suppose it required to turn 16687 (duodecimal) into the decimal scale, and 16432 (septenary) into the quaternary scale:
| 16687 | 16432 |
| Duodecimals into Decimals. | Septenaries into Quaternaries. |
| 1 × 12 + 6 = 18 | 1 × 7 + 6 = 31 |
| × 12 + 6 | × 7 + 4 |
| 222 | 1133 |
| × 12 + 8 | × 7 + 3 |
| 2672 | 22130 |
| × 12 + 7 | × 7 + 2 |
| Answer 32071 | 1021012 |