# One's Complement: Binary -> Integer: 0000 0000 0000 0000 0000 0000 0000 1011 0111 1011 0011 1110 0111 1011 1010 0001 Signed Binary Number in One's Complement Representation, Converted and Written as a Decimal System Integer (in Base Ten)

## Signed binary in one's complement representation 0000 0000 0000 0000 0000 0000 0000 1011 0111 1011 0011 1110 0111 1011 1010 0001_{(2)} converted to an integer in decimal system (in base ten) = ?

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

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

#### 0000 0000 0000 0000 0000 0000 0000 1011 0111 1011 0011 1110 0111 1011 1010 0001 is the binary representation of a positive integer, on 64 bits (8 Bytes).

### 2. 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 *

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

2^{63}

0 2^{62}

0 2^{61}

0 2^{60}

0 2^{59}

0 2^{58}

0 2^{57}

0 2^{56}

0 2^{55}

0 2^{54}

0 2^{53}

0 2^{52}

0 2^{51}

0 2^{50}

0 2^{49}

0 2^{48}

0 2^{47}

0 2^{46}

0 2^{45}

0 2^{44}

0 2^{43}

0 2^{42}

0 2^{41}

0 2^{40}

0 2^{39}

0 2^{38}

0 2^{37}

0 2^{36}

0 2^{35}

1 2^{34}

0 2^{33}

1 2^{32}

1 2^{31}

0 2^{30}

1 2^{29}

1 2^{28}

1 2^{27}

1 2^{26}

0 2^{25}

1 2^{24}

1 2^{23}

0 2^{22}

0 2^{21}

1 2^{20}

1 2^{19}

1 2^{18}

1 2^{17}

1 2^{16}

0 2^{15}

0 2^{14}

1 2^{13}

1 2^{12}

1 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}

0 2^{0}

1

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

#### 0000 0000 0000 0000 0000 0000 0000 1011 0111 1011 0011 1110 0111 1011 1010 0001_{(2)} =

#### (0 × 2^{63} + 0 × 2^{62} + 0 × 2^{61} + 0 × 2^{60} + 0 × 2^{59} + 0 × 2^{58} + 0 × 2^{57} + 0 × 2^{56} + 0 × 2^{55} + 0 × 2^{54} + 0 × 2^{53} + 0 × 2^{52} + 0 × 2^{51} + 0 × 2^{50} + 0 × 2^{49} + 0 × 2^{48} + 0 × 2^{47} + 0 × 2^{46} + 0 × 2^{45} + 0 × 2^{44} + 0 × 2^{43} + 0 × 2^{42} + 0 × 2^{41} + 0 × 2^{40} + 0 × 2^{39} + 0 × 2^{38} + 0 × 2^{37} + 0 × 2^{36} + 1 × 2^{35} + 0 × 2^{34} + 1 × 2^{33} + 1 × 2^{32} + 0 × 2^{31} + 1 × 2^{30} + 1 × 2^{29} + 1 × 2^{28} + 1 × 2^{27} + 0 × 2^{26} + 1 × 2^{25} + 1 × 2^{24} + 0 × 2^{23} + 0 × 2^{22} + 1 × 2^{21} + 1 × 2^{20} + 1 × 2^{19} + 1 × 2^{18} + 1 × 2^{17} + 0 × 2^{16} + 0 × 2^{15} + 1 × 2^{14} + 1 × 2^{13} + 1 × 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} + 0 × 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 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 34 359 738 368 + 0 + 8 589 934 592 + 4 294 967 296 + 0 + 1 073 741 824 + 536 870 912 + 268 435 456 + 134 217 728 + 0 + 33 554 432 + 16 777 216 + 0 + 0 + 2 097 152 + 1 048 576 + 524 288 + 262 144 + 131 072 + 0 + 0 + 16 384 + 8 192 + 4 096 + 2 048 + 0 + 512 + 256 + 128 + 0 + 32 + 0 + 0 + 0 + 0 + 1)_{(10)} =

#### (34 359 738 368 + 8 589 934 592 + 4 294 967 296 + 1 073 741 824 + 536 870 912 + 268 435 456 + 134 217 728 + 33 554 432 + 16 777 216 + 2 097 152 + 1 048 576 + 524 288 + 262 144 + 131 072 + 16 384 + 8 192 + 4 096 + 2 048 + 512 + 256 + 128 + 32 + 1)_{(10)} =

#### 49 312 332 705_{(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 0000 0000 1011 0111 1011 0011 1110 0111 1011 1010 0001_{(2)} = 49 312 332 705_{(10)}

## The signed binary number in one's complement representation 0000 0000 0000 0000 0000 0000 0000 1011 0111 1011 0011 1110 0111 1011 1010 0001_{(2)} converted and written as an integer in decimal system (base ten):

0000 0000 0000 0000 0000 0000 0000 1011 0111 1011 0011 1110 0111 1011 1010 0001_{(2)} = 49 312 332 705_{(10)}

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

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

#### 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 in one's complement representation to an integer in base ten:

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

#### 2) Construct the unsigned binary number: flip all the bits in the signed binary one's complement representation (reversing the digits) - replace the bits set on 1 with 0s and the bits on 0 with 1s.

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

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

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