Binary Number Representations

Unsigned binary represents non-negative integers (zero and above). With $n$ bits, the representable range is $0$ to $2^n-1$.

Bits ($n$)	Range	Max value
4	0 to 15	$2^4-1=15$
8	0 to 255	$2^8-1=255$
16	0 to 65 535	$2^{16}-1=65535$
32	0 to $\approx 4.3\times 10^9$	$2^{32}-1$

Converting unsigned binary to decimal:

$10110101_2$: working right to left, bit $k$ has value $2^k$.

Bit	7	6	5	4	3	2	1	0
Value	128	64	32	16	8	4	2	1
Digit	1	0	1	1	0	1	0	1
Contribution	128	—	32	16	—	4	—	1

$128+32+16+4+1=181_{10}$

Converting decimal to binary: see the number systems lesson (repeated division by 2).

Binary addition follows the same rules as decimal addition but in base 2:

A	B	Carry in	Sum bit	Carry out
0	0	0	0	0
0	1	0	1	0
1	0	0	1	0
1	1	0	0	1
1	1	1	1	1

Worked example — add $01111010_2$ (122) + $00110101_2$ (53):

  0111 1010   (122)
+ 0011 0101   ( 53)
-----------
  1010 1111   (175)

Carry trace (right to left): 0+1=1; 1+0=1; 0+1=1; 1+0=1; 1+1=10 (0 carry 1); 1+1+1=11 (1 carry 1); 1+0+1=10 (0 carry 1); 0+0+1=1.

Result: $10101111_2=128+32+8+4+2+1=175_{10}$ ✓

Multiplication by 2: shift all bits one place to the left and insert a 0 in the rightmost position. This is because each bit's place value doubles when shifted left.

$0011_2(3)\stackrel{\times 2}{\to }0110_2(6)$

Two's complement is the standard way to represent positive and negative integers in binary. With $n$ bits:

• Range: $-2^{n-1}$ to $2^{n-1}-1$
• The most significant bit (MSB) acts as the sign bit: 0 = positive, 1 = negative

For 8 bits: range is $-128$ to $+127$.

Interpreting a two's complement number:

The MSB has a negative place value: $-2^{n-1}$.

$\underset{-128}{\underbrace{1}}0011010_2=-128+16+8+2=-102$

$\underset{+0}{\underbrace{0}}1101001_2=0+64+32+8+1=105$

To negate a number (flip its sign): invert all bits, then add 1.

Example — represent $-37$ in 8-bit two's complement:

Step 1: write $+37$ in binary: $00100101_2$

Step 2: invert all bits: $11011010_2$

Step 3: add 1: $11011010+00000001=11011011_2$

So $-37=11011011_2$.

Verify using the sign-bit method: $-128+64+16+8+2+1=-128+91=-37$ ✓

Converting negative two's complement to decimal:

Given $11110011_2$: apply the same process — invert and add 1: Invert: $00001100_2$ Add 1: $00001101_2=13_{10}$ Therefore $11110011_2=-13_{10}$.

Binary subtraction is performed by adding the two's complement of the subtrahend:

$A-B=A+(-B)$

Worked example — compute $10-6$ in 8-bit two's complement:

$10=00001010_2$

$-6$: start with $+6=00000110$; invert: $11111001$; add 1: $11111010_2$

  0000 1010   (+10)
+ 1111 1010   ( -6)
-----------
1 0000 0100   (carry-out discarded)

Result: $00000100_2=4_{10}$ ✓ (carry out of the MSB is discarded for correct results)

Overflow occurs when the result exceeds the range of $n$ bits. In two's complement: overflow is detected when the carry into the MSB differs from the carry out of the MSB.

	Unsigned binary	Two's complement
8-bit range	0 to 255	−128 to 127
Negative values	Not representable	Representable
MSB role	Place value $2^{n-1}$	Negative place value $-2^{n-1}$
Addition same?	Yes	Yes
Overflow detection	Carry out of MSB	Carry in ≠ carry out of MSB

The same 8-bit pattern has different meanings under unsigned vs two's complement:

• $10000000_2$ unsigned = $128_{10}$
• $10000000_2$ two's complement = $-128_{10}$

1. Using the wrong range formula

Unsigned $n$-bit range: $0$ to $2^n-1$. Two's complement $n$-bit range: $-2^{n-1}$ to $2^{n-1}-1$. Note the asymmetry — there is one more negative value than positive.

2. Forgetting to add 1 when negating

Inverting the bits alone gives the one's complement, not the two's complement. The +1 step is essential. Omitting it gives the wrong result.

3. Not discarding the carry-out in subtraction

When subtracting with two's complement, a carry-out of the MSB is expected for a correct positive result. It must be discarded, not included in the answer.

4. Misreading the sign of a two's complement number

A leading 1 does not mean the number is a large positive. $11111111_2$ two's complement = $-1$, not $255$.