Numeral systems and bit shifting quick overview

Numeral Systems

Numeral system	Radix/root	Digits	Example	In Decimal system
Binary		0,1	0,1
Byte	2	Groupings of 8 binary digits (representation of a byte or integer (16/32/64 bits))	01010001	(02 × 27) + (12 × 26) + (02 × 25) + (12 × 24) + (03 × 23) + (02 × 22) + (02 × 21) + (12 × 20)
Decimal	10	0-9	124	(110 × 102) + (210 × 101) + (410 × 100)
Octal	8	0-7	02732	(28 × 83) + (78 × 82) + (38 × 81) + (28 × 80)
Hexadecimal	16	0-9,A-F (corresponding to 10-15)	0x2AF3	(216 × 163) + (A16 × 162) + (F16 × 161) + (316 × 160)

Bit Shifting

 

128	64	32	16	8	4	2	1
27	26	25	24	23	22	21	20
0	1	0	1	0	0	0	1

0     1   0    1    0   0  0   1 => 1*64 + 1*16 + 1*1 = 81

 

 

Arithmetic bit shifting to the right with >>

 

Makes bits fall of the right and adds zero padding to the left. This is equivalent to arithmetic division.

01010001 = 1*64 + 1*16 + 1*1 = 81

$y =  0b01010001 >> 1

>> 1 … shifting by one position to the right:

01010001 → 00101000 = 1*32 + 1*8 = 40   (ie int(81/2))

Arithmetic bit shifting to the left with <<

Makes bits fall of the left and adds zero padding to the right. This is equivalent to arithmetic multiplication by 2 to the number on the right of the operator.

01010001 = 1*64 + 1*16 + 1*1 = 81

$y =  0b01010001 << 1

<< 1 … shifting by one position to the left, we are multiplying by 2 to 1:

01010001 → 10100010 = 1*128 + 1*32 + 1*2 = 162 (ie 81 * 2)

NOTE

 

The number of shift positions needs to result in a value within the allowed range of the original value type:

Wrong:

<< 2:
01010001 (81) → 01000100 = 1*64 + 1*4 =68