[Math] Decimal to negabinary conversion

arithmeticnumber-systems

I found in Wikipedia that you can convert base 10 to a negative base simple by dividing to a base and keeping remainder like this:
enter image description here

Source: http://en.wikipedia.org/wiki/Negative_base

Now I'm stuck here.

I want to convert two digits -6 and 6 no base(-2)

Now example:

I do this:

6 / -2 = -3, remainder 1
-3/ -2 = 1, remainder 1
1/ -2 = 0, remainder 0

so, the final result is 0011 which is not quite right, because 1x(-2) + 1 = -1, not six.

HELP!

Update:

And one more question concerning negabinary:

Suppose we use 2n bits to represent integers using base −2 encoding,
for n > 0. What is the largest integer we can represent? What is the
smallest?

What I'm thinking is that the smallest number is 0, because we cannot represent -1 in negative numbers of bits.
2x(-1) = -2.
ANd there's no limit to a max number since 2x(10000) will fit in 200000 bytes.

Best Answer

6/-2=-3 with remainder 0, not 1

-3/-2=2 with remainder 1 (remember that your remainders have to be positive, like the -5/-3 in the example)

2/-2=-1 with remainder 0

-1/-2=1 with remainder 1

1/-2=0 with remainder 1

giving $11010_{-2}=6_{10}$

For your question about largest and smallest numbers representable in $2n$ bits, the largest positive number has all the positive bits set, so you have $1+4+16+\ldots$ This is $\sum_{i=0}^{n-1}4^i=\frac{4^n-1}{3}$. The smallest number has all the negative bits set and is twice this large in absolute value, so it is $-\frac{2(4^n-1)}{3}$

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