## How to convert a signed integer in decimal system (in base 10):

-87 713_{(10)}

to a signed binary one's complement representation

### 1. Start with the positive version of the number:

#### |-87 713| = 87 713

### 2. Divide the number repeatedly by 2:

#### Keep track of each remainder.

#### We stop when we get a quotient that is equal to zero.

- division = quotient +
**remainder**; - 87 713 ÷ 2 = 43 856 +
**1**; - 43 856 ÷ 2 = 21 928 +
**0**; - 21 928 ÷ 2 = 10 964 +
**0**; - 10 964 ÷ 2 = 5 482 +
**0**; - 5 482 ÷ 2 = 2 741 +
**0**; - 2 741 ÷ 2 = 1 370 +
**1**; - 1 370 ÷ 2 = 685 +
**0**; - 685 ÷ 2 = 342 +
**1**; - 342 ÷ 2 = 171 +
**0**; - 171 ÷ 2 = 85 +
**1**; - 85 ÷ 2 = 42 +
**1**; - 42 ÷ 2 = 21 +
**0**; - 21 ÷ 2 = 10 +
**1**; - 10 ÷ 2 = 5 +
**0**; - 5 ÷ 2 = 2 +
**1**; - 2 ÷ 2 = 1 +
**0**; - 1 ÷ 2 = 0 +
**1**;

### 3. Construct the base 2 representation of the positive number:

#### Take all the remainders starting from the bottom of the list constructed above.

#### 87 713_{(10)} = 1 0101 0110 1010 0001_{(2)}

### 4. Determine the signed binary number bit length:

#### The base 2 number's actual length, in bits: 17.

#### A signed binary's bit length must be equal to a power of 2, as of:

2^{1} = 2; 2^{2} = 4; 2^{3} = 8; 2^{4} = 16; 2^{5} = 32; 2^{6} = 64; ...

#### First bit (the leftmost) indicates the sign,

1 = negative, 0 = positive.

#### The least number that is:

#### a power of 2

#### and is larger than the actual length, 17,

#### so that the first bit (leftmost) could be zero

#### (we deal with a positive number at this moment)

#### is: 32.

### 5. Positive binary computer representation on 32 bits (4 Bytes):

#### If needed, add extra 0s in front (to the left) of the base 2 number, up to the required length, 32:

#### 87 713_{(10)} = 0000 0000 0000 0001 0101 0110 1010 0001

### 6. Get the negative integer number representation:

#### To get the negative integer number representation on 32 bits (4 Bytes),

#### signed binary one's complement,

#### replace all the bits on 0 with 1s

#### and all the bits set on 1 with 0s

#### (reverse the digits, flip the digits)

#### !(0000 0000 0000 0001 0101 0110 1010 0001) =

#### 1111 1111 1111 1110 1010 1001 0101 1110

## Conclusion:

## Number -87 713, a signed integer, converted from decimal system (base 10) to a signed binary one's complement representation:

## -87 713_{(10)} = 1111 1111 1111 1110 1010 1001 0101 1110

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

### More operations of this kind:

## Convert signed integer numbers from the decimal system (base ten) to signed binary one's complement representation