Base Conversion

See also: Integer Representation

Use the following algorithm to construct base b expansions of an integer n.

1. First divide n by b to get a quotient and remainder $n=b{q}_0+a_0,0\le a_0<b$ The remainder $a_0$ is the first digit in the base b expansion of n
2. Next divide $q_0$ by b to obtains $q_0=b{q}_1+a_1,0\le a_1<b.$ The remainder $a_1$ is the 2nd digit in the base b expansion of n
3. Repeat the steps above until you get 0 as a quotient.

Example

Find the find binary expansion of $(241{)}_{10}$ $241/2=120r1$ $120/2=60r0$ $60/2=30r0$ $30/2=15r0$ $15/2=7r1$ $7/2=3r1$ $3/2=1r1$ $1/2=0r0$

Conversion between binary, octal, hexadecimal

Example

Find the octal and hex expansions of $(11111010111100{)}_2$ 011|111|0 10|11 1|100 3 | 7 | 2 | 7 | 4 $(37274{)}_8$

0011|1110|1011|1100 3 | 14 | 11 | 12 $(3EBC{)}_{16}$

Example

Find the binary expansions of $(765{)}_8$ and $(A8D{)}_{1}6$ (7 6 5${)}_8$ $(111110101{)}_2$

(A 8 D${)}_{16}$ $(101010001101{)}_2$