Base Number Arithmetic
Arithmetic in different bases works exactly the same, but takes some getting used to. It’s easy to forget when to carry and make compounding mistakes.
Addition
Addition in other bases works the exact same as base 10. We could either add the numbers directly,
21_6 + 14_6 = 25_6 + 10_6 = 35_6
Or convert to base 10 and preform our addition normally
21_6 + 14_6 = 13_{10} + 10_{10} = 23_{10} = 35_6
Subtraction
Subtraction unsuprisingly is no different. In the example below we seperate the number into groups of 9.
\begin{align*} 517_9 - 214_9 &= (5 - 2) \cdot 9^2 + (1 - 1) \cdot 9^1 + (7 - 4) \cdot 9^0\\ &= 3 \cdot 9^2 + 0 \cdot 9^1 + 3 \cdot 9^0\\ &= 303_9 \end{align*}
Multiplication
When multiplying two numbers of any base, it is almost always easier to regroup the multiplicands into bundles.
\begin{align*} 435_6 &= 4 \cdot 6^2 + 3 \cdot 6^1 + 5 \cdot 6^0\\ 12_6 &= 0 \cdot 6^2 + 1 \cdot 6^1 + 2 \cdot 6^0 \end{align*}
We can then multiply the numbers using normal methods
(4 \cdot 36 + 3 \cdot 6 + 5) \cdot (6 + 2) = \\ = 167 \cdot 8\\ = 1336
Division
The process of long division is the same in every base, so I won’t waste your time with needless examples (I don’t wanna Tex them).