What insight is there on negative number bases? I don't really have much to say, other than that I was curious about how negative number bases work, and I was wondering if I could get some insight on how to understand them.
I know how to convert negative base numbers to base 10, but have no clue as to how to write a base 10 number to an arbitrary negative base, at least not easily.
I have also noticed that there seems to be 2 ways to write every number in a negative base, for example $37_{10}$ can be written as $177_{-10}$ or $-43_{-10}$ and think there could be some way to use that to your advantage, although I still haven't gotten anywhere myself.
 A: 
I know how to convert negative base numbers to base 10, but have no clue as to how to write a base 10 number to an arbitrary negative base, at least not easily.

Write it to the positive absolute value base first.  subtract the odd terms from $|b|$ and carry to even terms.
Example:  To write $1234$ in base $-6$.
$1234 = 205\times 6 + 4 =$
$34\times 6^2 + 1\times 6 + 4=$
$5\times 6^3 + 4\times 6^2 + 1\times 6 + 4 =5414_6$.
$=5\times 6^3 + 4\times 6^2 + 1\times 6 + 4$
$=(6-1)\times 6^3 + 4\times 6^2 + (6-5)\times 6 + 4=$
$=6^4 - 6^3 + (4+1)\times 6^2 -5\times 6 + 4 = $
$(-6)^4 + (-6)^3 + 5\times (-6)^2 + 5\times(-6) + 4 = 11554_{-6}$
With practice we can do it directly.
$1234= -205(-6) + 4=$
$(35\cdot (-6)+5)(-6) + 4=$
$35(-6)^2 + 5(-6) + 4=$
$(-5\times -6 + 5)(-6)^2 + 5(-6) + 4=$
$-5(-6)^3 + 5(-6)^2 + 5(-6) + 4=$
$(-6 + 1)(-6)^3 + 5(-6)^2 + 5(-6) + 4=$
$(-6)^4 + (-6)^3 + 5(-6)^2 + 5(-6) + 4=11554_{-6}$.

I have also noticed that there seems to be 2 ways to write every number in a negative base, for example 3710 can be written as 177−10 or −43−10 and think there could be some way to use that to your advantage, although I still haven't gotten anywhere myself.

Yes that's true.  But I find the negative sign in front of a number that is ultimately positive confusing.
Note a negative sign in front of an even number of digits or a lack of negative sign in front of an odd number of digits means the number is positive (and vice versa).
Actually, that could be your rule.  All numbers in negative base are expressed with positive digits.  If the number has an even number of digits the number is positive  (as $neg^{highest\ even} > 0$ will overpower the number).  But if the number has an odd number of digits the number is negative (as $neg^{highest\ odd} < 0$ will overpower the numbers).
