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

#### 0011 0001 0000 1011 is the binary representation of a positive integer, on 16 bits (2 Bytes).

#### In a signed binary in two's complement representation, the first bit (the leftmost) indicates the sign, 1 = negative, 0 = positive.

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

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

#### Note on binary subtraction rules:

#### 11 - 1 = 10; 10 - 1 = 1; 1 - 0 = 1; 1 - 1 = 0.

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

#### * Not the case - the number is positive *

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

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

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

#### * Not the case - the number is positive *

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

2^{15}

0 2^{14}

0 2^{13}

1 2^{12}

1 2^{11}

0 2^{10}

0 2^{9}

0 2^{8}

1 2^{7}

0 2^{6}

0 2^{5}

0 2^{4}

0 2^{3}

1 2^{2}

0 2^{1}

1 2^{0}

1

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

#### 0011 0001 0000 1011_{(2)} =

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

#### (0 + 0 + 8 192 + 4 096 + 0 + 0 + 0 + 256 + 0 + 0 + 0 + 0 + 8 + 0 + 2 + 1)_{(10)} =

#### (8 192 + 4 096 + 256 + 8 + 2 + 1)_{(10)} =

#### 12 555_{(10)}

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

#### 0011 0001 0000 1011_{(2)} = 12 555_{(10)}

## The signed binary number in two's complement representation 0011 0001 0000 1011_{(2)} converted and written as an integer in decimal system (base ten):

0011 0001 0000 1011_{(2)} = 12 555_{(10)}

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