Shortcut Conversions

Within these castle walls be forged Mavens of Computer Science ...

— Merlin, The Coder

Because both the octal and hexadecimal bases contain $2^{3}$ and $2^{4}$ possible digits, respectively, and the binary base contains $2^{1}$ possible digits, we're able to utilize a shortcut when converting between these systems. Note that each octal digit is comprised of three binary digits and each hexadecimal digit is comprised of four binary digits. We can therefore establish (and memorize) the following table:

Decimal	Binary	Octal	Hexadecimal
00	0000	00	0
01	0001	01	1
02	0010	02	2
03	0011	03	3
04	0100	04	4
05	0101	05	5
06	0110	06	6
07	0111	07	7
08	1000	10	8
09	1001	11	9
10	1010	12	A
11	1011	13	B
12	1100	14	C
13	1101	15	D
14	1110	16	E
15	1111	17	F

The shortcut is very straight-forward yet a significant time saver and based on the above information. When converting between binary, octal, and hexadecimal, there's no need to use an intermediate, decimal step.

Binary to Hexadecimal

Group the binary number in sets of four digits, from right to left. Each binary set represents a single hexadecimal digit. (It's sometimes helpful to pad the final set with zeroes.) For example, given the number $11000111\;\;00100101_{2}$, regroup the binary digits in sets of four:
$11000111\;\;00100101_{2}$
$\color {Orange}1100\color {Blue}0111\;\;\color {Orange}0010\color {Blue}0101\color {Black}_{2}$
$\color {Orange}1100\;\;\color {Blue}0111\;\;\color {Orange}0010\;\;\color {Blue}0101\color {Black}_{2}$.

Then, use the memorized table to translate each set of four binary digits to a single hexadecimal digit:

1100	0111	0010	0101
C	7	2	5

Hexadecimal to Binary

Simply follow the above procedure in reverse. Use the memorized table to convert each hexadecimal digit to a set of four binary digits.

Binary to Octal

Group the binary number in sets of three digits, from right to left. Each binary set represents a single octal digit. (It's sometimes helpful to pad the final set with zeroes.) For example, given the number $11000111\;\;00100101_{2}$, regroup the binary digits in sets of three:
$11000111\;\;00100101_{2}$
$\color {Orange}1\color {Blue}100\color {Orange}011\color {Blue}1\;\;00\color {Orange}100\color {Blue}101\color {Black}_{2}$
$00\color {Orange}1\;\;\color {Blue}100\;\;\color {Orange}011\;\;\color {Blue}100\;\;\color {Orange}100\;\;\color {Blue}101\color {Black}_{2}$.

Then, use the memorized table to translate each set of three binary digits to a single octal digit:

001	100	011	100	100	101
1	4	3	4	4	5

Octal to Binary

Simply follow the above procedure in reverse. Use the memorized table to convert each octal digit to a set of three binary digits.

Octal to Hexadecimal (and vice versa)

This requires slightly more work. Convert the number in the source base to binary, and then convert from binary to the target base.

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