3.2.6 Changing Bases: Miscellaneous Topics

There are a handful of topics involving changing bases that rely on understanding other tricks previously discussed in this book. Take this problem for example:

Problem: Convert the decimal $.333 \dots_7$ into a base-10 fraction. Solution: The above problem relies on using the formula for the sum of an infinite geometric series:

$$.333 \dots_7 = \frac{3}{7} + \frac{3}{49} + \frac{3}{343} + \dots = \frac{\frac{3}{7}}{1 - \frac{1}{7}} = \frac{3}{7} \times \frac{7}{6} = \frac{1}{2}$$

Another problem which relies on understanding of how the derivation of finding the remainder of a number when dividing by 9, only in a different base is:

Problem: The number $123456_7 \div 6$ has what remainder? Solution: The origins of this is rooted in modular arithmetic (see Section 3.4) and noticing that: $7^n \cong 1 (\text{mod } 6)$. So our integer can be represented as:

$$123456_7 = 1 \cdot 7^5 + 2 \cdot 7^4 + 3 \cdot 7^3 + 4 \cdot 7^2 + 5 \cdot 7^1 + 6 \cdot 7^0 \cong (1 + 2 + 3 + 4 + 5 + 6) = 6 \cdot \frac{7}{2} = 21 \cong 3 (\text{mod } 6)$$

So an important result is that when you have a base-n number and divide it by $n - 1$, all you need to do is sum the digits and see what the remainder that is when dividing by $n - 1$.

Problem Set 3.2.6