What determines what base the right side of this base coversion will be?

Question

Referring to this example of positional notation on Wikipedia:

There are several examples

$$465\;\;\text{(base 10)} = 465\;\;\text{(base 10)}$$

But then

$$465\;\;\text{(base 7)} = 243\;\;\text{(base 10)}$$

Why is the right hand side considered as base $10$? Wouldn't it be a base $7$ after the conversion using positional notation? Is the result always base $10$?

And Why doesn't the base-16 example result in a base 10 number

Answer 1

What determines it is the author's choice, nothing more. I can convert from base $285$ to base $57$ if I want, for example $$3\;\;\text{(base 285)}=3\;\;\text{(base 57)}$$ It's just that in current society, we think of numbers "by default" in base $10$, so writing something in base $10$ feels like the standard thing to convert to. There's nothing mathematically special about it.

Answer 2

Perhaps this will help you better understand the specific example you bring up: $$ (465)_\color{red}{7}=4(\color{red}{7^2})+6(\color{red}{7})+5=(243)_{\color{blue}{10}}=2(\color{blue}{10^2})+4(\color{blue}{10})+3. $$ Does it make sense now? Converting something from base-$16$ to base-$10$ is not an issue--simply do as above; however, converting a base-$10$ number (or a lower number base) to base-$16$ is slightly problematic only in the sense that you have to actually invent or come up with the symbols that you will use for bases $10,11,\ldots,15$.