# Signed: Binary -> Integer: 0000 0000 0000 0000 0000 1011 1010 0011 Signed Binary Number Converted and Written as a Decimal System Integer (in Base Ten)

## The signed binary (in base two) 0000 0000 0000 0000 0000 1011 1010 0011_{(2)} to an integer (with sign) in decimal system (in base ten) = ?

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

#### In a signed binary, the first bit (the leftmost) is reserved for the sign,

#### 1 = negative, 0 = positive. This bit does not count when calculating the absolute value.

#### 0000 0000 0000 0000 0000 1011 1010 0011 is the binary representation of a positive integer, on 32 bits (4 Bytes).

### 2. Construct the unsigned binary number.

#### Exclude the first bit (the leftmost), that is reserved for the sign:

#### 0000 0000 0000 0000 0000 1011 1010 0011 = 000 0000 0000 0000 0000 1011 1010 0011

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

2^{30}

0 2^{29}

0 2^{28}

0 2^{27}

0 2^{26}

0 2^{25}

0 2^{24}

0 2^{23}

0 2^{22}

0 2^{21}

0 2^{20}

0 2^{19}

0 2^{18}

0 2^{17}

0 2^{16}

0 2^{15}

0 2^{14}

0 2^{13}

0 2^{12}

0 2^{11}

1 2^{10}

0 2^{9}

1 2^{8}

1 2^{7}

1 2^{6}

0 2^{5}

1 2^{4}

0 2^{3}

0 2^{2}

0 2^{1}

1 2^{0}

1

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

#### 000 0000 0000 0000 0000 1011 1010 0011_{(2)} =

#### (0 × 2^{30} + 0 × 2^{29} + 0 × 2^{28} + 0 × 2^{27} + 0 × 2^{26} + 0 × 2^{25} + 0 × 2^{24} + 0 × 2^{23} + 0 × 2^{22} + 0 × 2^{21} + 0 × 2^{20} + 0 × 2^{19} + 0 × 2^{18} + 0 × 2^{17} + 0 × 2^{16} + 0 × 2^{15} + 0 × 2^{14} + 0 × 2^{13} + 0 × 2^{12} + 1 × 2^{11} + 0 × 2^{10} + 1 × 2^{9} + 1 × 2^{8} + 1 × 2^{7} + 0 × 2^{6} + 1 × 2^{5} + 0 × 2^{4} + 0 × 2^{3} + 0 × 2^{2} + 1 × 2^{1} + 1 × 2^{0})_{(10)} =

#### (0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 2 048 + 0 + 512 + 256 + 128 + 0 + 32 + 0 + 0 + 0 + 2 + 1)_{(10)} =

#### (2 048 + 512 + 256 + 128 + 32 + 2 + 1)_{(10)} =

#### 2 979_{(10)}

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

#### 0000 0000 0000 0000 0000 1011 1010 0011_{(2)} = 2 979_{(10)}

## The number 0000 0000 0000 0000 0000 1011 1010 0011_{(2)} converted from a signed binary (base two) and written as an integer in decimal system (base ten):

0000 0000 0000 0000 0000 1011 1010 0011_{(2)} = 2 979_{(10)}

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

## Convert signed binary numbers to integers in decimal system (in base ten)

#### The first bit (the leftmost) is reserved for the sign (1 = negative, 0 = positive) and it doesn't count when calculating the absolute value.

#### Binary number's length must be: 2, 4, 8, 16, 32, 64 - or else extra bits on 0 are added in front (to the left).

### How to convert a signed binary number to an integer in base ten:

#### 1) Construct the unsigned binary number: exclude the first bit (the leftmost); this bit is reserved for the sign, 1 = negative, 0 = positive and does not count when calculating the absolute value (without sign).

#### 2) Multiply each bit of the binary number by its corresponding power of 2 that its place value represents.

#### 3) Add all the terms up to get the positive integer number in base ten.

#### 4) Adjust the sign of the integer number by the first bit of the initial binary number.