Encoding - BIN, OCT, DEC, HEX

Numbering systems

Types: Decimal, Oct, Hex, Binary
Syntax: Subscript is used to show the base.

$10_{2}$ = 10 in BIN = $2_{10}$
$10_{8}$ = 10 in OCT = $8_{10}$
$10_{10}$ = 10 in DEC = $10_{10}$
$10_{16}$ = 10 in HEX = $16_{10}$

Adding in other systems

Example: In Oct

Converting between systems

BIN to DEC

Multiply wherever it has 1 by $2^{bit pos}$ then add everything. (bit pos start from 0)

Example

Binary	1	0	1	0	1	1	0	1	Result
Multiply by	128	64	32	16	8	4	2	1	-
Decimal	128	0	32	0	8	4	0	1	$128 + 32 + 8 + 4 + 1 = 173$

DEC to BIN

Divide decimal number by 2
Save remainder (It represents the binary number on the right (Least significant bit))
If quotient is > 0, repeat step 1 but divide by quotient
Binary result = the remainders of Step 2 from the last

Example

$\begin{aligned} Division & = Quotient (Remainder) \\ \frac{10}{2} & = 5 (0) \\ \frac{5}{2} & = 2 (1) \\ \frac{2}{2} & = 1 (0) \\ \frac{1}{2} & = 0 (1) \end{aligned}$

As shown, need to divide until Quotient == 0
$10_{10} = 1010_{2}$

Another example

From 72 (DEC) $\to$ BIN
72/2 = 36, 0
36/2 = 18, 0
18/2 = 9, 0
9/2 = 4, 1
4/2 = 2, 0
2/2 = 1, 0
1/2 = 0, 1

$72_{10} = 1001000_{2}$

Lookup table

Useful for converting between BIN $\Leftrightarrow$ HEX or BIN $\Leftrightarrow$ OCT

4 digits of BIN $=$ 1 digit of HEX

BIN	OCT	HEX	DEC
0000	0	0	0
0001	1	1	1
0010	2	2	2
0011	3	3	3
0100	4	4	4
0101	5	5	5
0110	6	6	6
0111	7	7	7
1000	10	8	8
1001	11	9	9
1010	12	A	10
1011	13	B	11
1100	14	C	12
1101	15	D	13
1110	16	E	14
1111	17	F	15
Fun way of converting $A_{16}$ to $1010_{2}$ . $A_{16} = 10_{10}$ . And in binary, there's 2 10s