Algebra

Inequalities

342 If 𝑁 be a whole number of 𝑛+1 digits, and 𝑟 the radix of the scale, 𝑁=𝑝_𝑛𝑟^𝑛+𝑝_𝑛−1𝑟^𝑛−1+𝑝_𝑛−2𝑟^𝑛−2+⋯+𝑝₁𝑟+𝑝₀0 where 𝑝_𝑛, 𝑝_𝑛−1, ⋯, 𝑝₀ are the digits. 343 Similarly a radix-fraction will be expressed by 𝑝₁𝑟+𝑝₂𝑟²+𝑝₃𝑟³+⋯ where 𝑝₁, 𝑝₂, ⋯ are the digits.

Examples

3426 in the scale of 7=3⋅7³+4⋅7²+2⋅7+6;
0.1045 is the same scale=17+07²+47³+57⁴ 344

Example

To transform 34268 from the scale of 5 to the scale of 11 Rule:Divide successively by the new radix.

11|34268
 11|1343−𝑡
   11|40−3
       1−9

Result 193𝑡, in which 𝑡 stands for 10. 345 Example: To transform ⋅𝑡0𝑒1 from the scale of 12 to that of 7, 𝑒 standing for 11, and 𝑡 for 10. Rule. Mltiply successively by the new radix.

⋅𝑡0𝑒1
     7
------
5⋅𝑡657
     7        
------
6⋅1931
     7        
------
1⋅0497
     7        
------
0⋅2971

Result ⋅5610 346 Ex. In what scale does 2𝑡7 represent the number 475 in the scale of ten? Solve the equation 2𝑟+10𝑟+7=475178 Result 𝑟=13. 347 The sum of the digits of any number divided by 𝑟−1 leaves the same remainder as the number itself divided by 𝑟−1; 𝑟 being the radix of the scale 401 348 The difference between the sums of the digits in the even and odd places divided by 𝑟+1 leaves the same remainder as the number itself when divided by 𝑟+1.

Sources and References

https://archive.org/details/synopsis-of-elementary-results-in-pure-and-applied-mathematics-pdfdrive