# One's Complement: Integer -> Binary: 15 538 Convert the Integer Number to a Signed Binary in One's Complement Representation. Write the Base Ten Decimal System Number as a Binary Code (Written in Base Two)

## Signed integer number 15 538_{(10)} converted and written as a signed binary in one's complement representation (base 2) = ?

### 1. 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**; - 15 538 ÷ 2 = 7 769 +
**0**; - 7 769 ÷ 2 = 3 884 +
**1**; - 3 884 ÷ 2 = 1 942 +
**0**; - 1 942 ÷ 2 = 971 +
**0**; - 971 ÷ 2 = 485 +
**1**; - 485 ÷ 2 = 242 +
**1**; - 242 ÷ 2 = 121 +
**0**; - 121 ÷ 2 = 60 +
**1**; - 60 ÷ 2 = 30 +
**0**; - 30 ÷ 2 = 15 +
**0**; - 15 ÷ 2 = 7 +
**1**; - 7 ÷ 2 = 3 +
**1**; - 3 ÷ 2 = 1 +
**1**; - 1 ÷ 2 = 0 +
**1**;

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

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

#### 15 538_{(10)} = 11 1100 1011 0010_{(2)}

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

#### The base 2 number's actual length, in bits: 14.

#### 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; ...

#### The first bit (the leftmost) indicates the sign:

#### 0 = positive integer number, 1 = negative integer number

#### The least number that is:

#### 1) a power of 2

#### 2) and is larger than the actual length, 14,

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

(we deal with a positive number at this moment)

#### === is: 16.

### 4. Get the positive binary computer representation on 16 bits (2 Bytes):

#### If needed, add extra 0s in front (to the left) of the base 2 number, up to the required length, 16.

## Number 15 538_{(10)}, a signed integer number (with sign), converted from decimal system (from base 10) and written as a signed binary in one's complement representation:

## 15 538_{(10)} = 0011 1100 1011 0010

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

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

### How to convert a base 10 signed integer number to signed binary in one's complement representation:

#### 1) Divide the positive version of the number repeatedly by 2, keeping track of each remainder, till getting a quotient that is 0.

#### 2) Construct the base 2 representation by taking the previously calculated remainders starting from the last remainder up to the first one, in that order.

#### 3) Construct the positive binary computer representation so that the first bit is 0.

#### 4) Only if the initial number is negative, switch all the bits from 0 to 1 and from 1 to 0 (reversing the digits).