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 - 0 1 _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. 4_t_9)76_t_4_e_08(16687 705)22610744(32071 4_t_9 1460 ----- 5074 2814 1394 2546 689 ----- 28_te_ 2546 ------ 3650 3320 ----- 3308 2_t_33 ------ 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: