## Signed binary two's complement 1000 1101 1111 1111_{(2)} to an integer in decimal system (in base 10) = ?

### 1. Is this a positive or a negative number?

#### In a signed binary two's complement, first bit (the leftmost) indicates the sign,

1 = negative, 0 = positive.

#### 1000 1101 1111 1111 is the binary representation of a negative integer, on 16 bits (2 Bytes).

### 2. Get the binary representation in one's complement:

#### * Run this step only if the number is negative *

#### Subtract 1 from the binary initial number:

#### 1000 1101 1111 1111 - 1 = 1000 1101 1111 1110

### 3. Get the binary representation of the positive (unsigned) number:

#### * Run this step only if the number is negative *

#### Flip all the bits in the signed binary one's complement representation (reverse the digits) - replace the bits set on 1 with 0s and the bits on 0 with 1s:

#### !(1000 1101 1111 1110) = 0111 0010 0000 0001

### 4. Map the unsigned binary number's digits versus the corresponding powers of 2 that their place value represent:

2^{15}

0 2^{14}

1 2^{13}

1 2^{12}

1 2^{11}

0 2^{10}

0 2^{9}

1 2^{8}

0 2^{7}

0 2^{6}

0 2^{5}

0 2^{4}

0 2^{3}

0 2^{2}

0 2^{1}

0 2^{0}

1

### 5. Multiply each bit by its corresponding power of 2 and add all the terms up:

#### 0111 0010 0000 0001_{(2)} =

#### (0 × 2^{15} + 1 × 2^{14} + 1 × 2^{13} + 1 × 2^{12} + 0 × 2^{11} + 0 × 2^{10} + 1 × 2^{9} + 0 × 2^{8} + 0 × 2^{7} + 0 × 2^{6} + 0 × 2^{5} + 0 × 2^{4} + 0 × 2^{3} + 0 × 2^{2} + 0 × 2^{1} + 1 × 2^{0})_{(10)} =

#### (0 + 16 384 + 8 192 + 4 096 + 0 + 0 + 512 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 1)_{(10)} =

#### (16 384 + 8 192 + 4 096 + 512 + 1)_{(10)} =

#### 29 185_{(10)}

### 6. If needed, adjust the sign of the integer number by the first digit (leftmost) of the signed binary:

#### 1000 1101 1111 1111_{(2)} = -29 185_{(10)}

## Number 1000 1101 1111 1111_{(2)} converted from signed binary two's complement representation to an integer in decimal system (in base 10):

1000 1101 1111 1111_{(2)} = -29 185_{(10)}

#### Spaces used to group digits: for binary, by 4; for decimal, by 3.

### More operations of this kind:

## Convert signed binary two's complement numbers to decimal system (base ten) integers

### Entered binary number length must be: 2, 4, 8, 16, 32, or 64 - otherwise extra bits on 0 will be added in front (to the left).