Binary Number Representations

Bits, bytes, and words

• Bit = binary digit (0 or 1).
• Byte = 8 bits, e.g. 01010111.
• Modern computers deal with words, usually a power-of-2 number of bytes: 1, 2, 4, or 8 bytes = 8, 16, 32, 64 bits.

Representing whole (unsigned) numbers

Each bit position has a value — a power of 2, increasing from right (least significant) to left (most significant):

Bit position	9	8	7	6	5	4	3	2	1	0
Value	512	256	128	64	32	16	8	4	2	1

Binary → decimal: add the values of each position where the bit is 1. E.g. 10010001 = \(128 + 16 + 1 = 145\).

• Least significant bit (LSB) — the bit position worth the least (\(2^0 = 1\)).
• Most significant bit (MSB) — the bit position worth the most. For an \(n\)-bit unsigned word, the MSB is worth \(2^{n-1}\).

Converting decimal to binary

Two equivalent methods (example: convert 53 to binary):

• Method 1: rewrite \(n\) as a sum of powers of 2, by repeatedly subtracting the largest power of 2 not greater than \(n\). Assemble the binary number from 1’s in the bit positions corresponding to those powers of 2, 0’s elsewhere.
• Method 2 (build up from the right/LSB): divide \(n\) by 2; the remainder (0 or 1) is the next bit; repeat with \(n\) = the quotient, until \(n = 0\).

Number range (unsigned)

• Smallest representable value: all 0’s → 0.
• Largest representable value: all 1’s → for an \(n\)-bit word, \(2^n - 1\) (e.g. 255 for 8 bits).

Other radices

Radix = number system base. A radix-\(k\) number system has \(k\) distinct symbols for digits \(0\) to \(k-1\), and the value of each digit (from the right) is \(k^0, k^1, k^2, \dots\)

• Octal (radix-8): symbols 0–7. One octal digit corresponds to exactly 3 bits.
• Hexadecimal (radix-16): symbols 0–9, A–F. One hex digit corresponds to exactly 4 bits — very convenient for grouping binary.

Dec	0	1	2	3	4	5	6	7	8	9	10	11	12	13	14	15	16	17
Oct	0	1	2	3	4	5	6	7	10	11	12	13	14	15	16	17	20	21
Hex	0	1	2	3	4	5	6	7	8	9	A	B	C	D	E	F	10	11

Radix notation conventions

Since a bare number like 101 or 747 is ambiguous, a radix indicator is needed. A subscript works generically (e.g. \(101_2\), \(101_{16}\)); in code/assembly, conventions vary:

Radix	Convention	Example	Where used
Hex	leading 0x	0x101	C, Atmel AVR
Hex	trailing h	101h	some assembly languages
Hex	leading $	$747	Atmel AVR assembly
Octal	leading 0	0101	C, Atmel AVR
Octal	trailing q	101q	some assembly languages
Octal	leading @	@747	some assembly languages
Binary	leading 0b	0b101	Atmel AVR assembly, some C
Binary	trailing b	101b	some assembly languages
Binary	leading %	%101	some assembly languages