The number 10 is a base for
the positive integers because every positive integer can be written
uniquely as
d_n 10^n + d_{n-1}10^{n-1} + \cdots + d_1\cdot
10 + d_0,
where each $d_i$ is one of the digits $0,1,2,\dots 9$. The number $-2$
is a base for all integers using the digits 0 or 1. For example $1101$
represents $-3$ since
1(-2)^3 + 1(-2)^2 + 0(-2) + 1 =-3.
Find the representation in base $-2$ for the decimal number $-2374$.
Solution. In
base 10, we would normally find an expansion by dividing the number
successively by 10 and recording the remainders. A similar algorithm
works in base $-2$, bearing in mind that the remainders must be $0$ or
$+1$. So
-2374_{(10)} = 101111001110_{(-2)}.