268 



THE POPULAR EDUCATOE. 



powers of any other number than 10. 



For instance, if 4 were taken as the radix of the scale of 

 notation, 231 would denote 2x4 2 + 3x4 + 1. 



Similarly, if 8 were the radix of the scale, 5732 would mean 

 5x8" +7x8* + 3X8 + 2. 



The numbers here represented by 231 and 5732 are said to 

 be expressed in the scale of 4 and the scale of 8 respectively. 



2. It is evident that each of the figures or digits which occur 

 in any scale must be less than the radix of the scale. Thus, in 

 the scale of 4, the largest digit or figure which can occur is 3 ; 

 in that of 8, 7 ; and similarly for other scales. For suppose 8 

 to occur in the scale of 8, its effect would be to increase by 1 

 that power of 8 which it multiplied; and this would be the 

 same thing as increasing the digit immediately on its left 

 byl. 



Suppose we take 12 for the radix of the scale, we must then 

 have symbols to represent the numbers 10 and 11. If we use t 

 and e respectively for them, then such a number as the follow- 

 ing, 574e9, would mean 



5 x 12 s + 7 x 12* + 10 x 12 s + 4 x 12 2 + 11 x 12 + 9. 



The scale of which the radix is 12 is called the duodecimal or 

 duodenary scale. 



We do not mean here to go at length into the various pro- 

 positions connected with scales of notation, for they belong 

 properly to algebra, and cannot be satisfactorily explained with- 

 out its aid. We subjoin, however, one or two propositions, which 

 will bs useful in giving the student a clearer insight into the 

 principles of arithmetical notation. 



3. A number being given in the Decimal Scale, to express it in 

 the Duodecimal Scale. 



5 ... 6 



The answer therefore is 56789. 

 The division is performed as follows : 

 Ten in 28 (duodecimal scale) is really 10 in 2 x 12 + 8, or 32, which 



gives 3 and 2 over. 

 Ten in 24 is really 10 in 2 x 12 + 10, or 34, which gives 3 and 4 over. 

 Ten in 44 is really 10 in 4 x 12 + 4, or 52, which gives 5 and 2 over. 

 Ten in 25 is really 10 in 2 x 12 + 5, or 29, which gives 2 and 9 over. 

 And similarly for the other lines. 



This operation has proved to us that 



2 x 12* + 8 x 12* + 10 x 12 + 4 x 12 + 5 = 5 x 10* + 6 x 10 s + 7 x 

 10 9 + 8 x 10 + 9. 



5. Similarly, a number expressed in any scale whatever can 

 be transformed to any other scale, by successively dividing by 

 the radix of the scale into which it is to be transformed. 



We content ourselves with giving an example, with ita 

 explanation. 



EXAMPLE. Given 56435, expressed in the scale of 8, to 

 reduce it to the scale of 6. 



6 ) 56435 



6 ) 7604 ... 5 



6 ) 1226 ... 



6) 

 6) 



156 ... 2 

 22 ... 2 



3 ... 



The division is performed as follows : 



802205.- 'Answer. 



