# Convert 657 106 to a signed binary in one's complement representation, from a base 10 decimal system signed integer number

## 657 106_{(10)} to a signed binary one's complement representation = ?

### 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**; - 657 106 ÷ 2 = 328 553 +
**0**; - 328 553 ÷ 2 = 164 276 +
**1**; - 164 276 ÷ 2 = 82 138 +
**0**; - 82 138 ÷ 2 = 41 069 +
**0**; - 41 069 ÷ 2 = 20 534 +
**1**; - 20 534 ÷ 2 = 10 267 +
**0**; - 10 267 ÷ 2 = 5 133 +
**1**; - 5 133 ÷ 2 = 2 566 +
**1**; - 2 566 ÷ 2 = 1 283 +
**0**; - 1 283 ÷ 2 = 641 +
**1**; - 641 ÷ 2 = 320 +
**1**; - 320 ÷ 2 = 160 +
**0**; - 160 ÷ 2 = 80 +
**0**; - 80 ÷ 2 = 40 +
**0**; - 40 ÷ 2 = 20 +
**0**; - 20 ÷ 2 = 10 +
**0**; - 10 ÷ 2 = 5 +
**0**; - 5 ÷ 2 = 2 +
**1**; - 2 ÷ 2 = 1 +
**0**; - 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.

#### 657 106_{(10)} = 1010 0000 0110 1101 0010_{(2)}

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

#### The base 2 number's actual length, in bits: 20.

#### 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,

#### 1 = negative, 0 = positive.

#### The least number that is:

#### 1) a power of 2

#### 2) and is larger than the actual length, 20,

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

(we deal with a positive number at this moment)

#### === is: 32.

### 4. 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.

## Number 657 106, a signed integer, converted from decimal system (base 10) to a signed binary one's complement representation:

## 657 106_{(10)} = 0000 0000 0000 1010 0000 0110 1101 0010

#### 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

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

#### 1) Divide the positive version of 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).

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## How to convert signed integers from the decimal system to signed binary in one's complement representation

### Follow the steps below to convert a signed base 10 integer number to signed binary in one's complement representation:

- 1. If the number to be converted is negative, start with the positive version of the number.
- 2. Divide repeatedly by 2 the positive representation of the integer number that is to be converted to binary, keeping track of each remainder, until we get a quotient that is equal to ZERO.
- 3. Construct the base 2 representation of the positive number, by taking all the remainders starting from the bottom of the list constructed above. Thus, the last remainder of the divisions becomes the first symbol (the leftmost) of the base two number, while the first remainder becomes the last symbol (the rightmost).
- 4. Binary numbers represented in computer language must have 4, 8, 16, 32, 64, ... bit length (a power of 2) - if needed, fill in '0' bits in front (to the left) of the base 2 number calculated above, up to the right length; this way the first bit (leftmost) will always be '0', correctly representing a positive number.
- 5. To get the negative integer number representation in signed binary one's complement, replace all '0' bits with '1's and all '1' bits with '0's.

### Example: convert the negative number -49 from the decimal system (base ten) to signed binary one's complement:

- 1. Start with the positive version of the number: |-49| = 49
- 2. Divide repeatedly 49 by 2, keeping track of each remainder:
- division = quotient +
**remainder**
- 49 ÷ 2 = 24 +
**1**
- 24 ÷ 2 = 12 +
**0**
- 12 ÷ 2 = 6 +
**0**
- 6 ÷ 2 = 3 +
**0**
- 3 ÷ 2 = 1 +
**1**
- 1 ÷ 2 = 0 +
**1**

- 3. Construct the base 2 representation of the positive number, by taking all the remainders starting from the bottom of the list constructed above:

49_{(10)} = 11 0001_{(2)} - 4. The actual bit length of base 2 representation is 6, so the positive binary computer representation of a signed binary will take in this case 8 bits (the least power of 2 that is larger than 6) - add '0's in front of the base 2 number, up to the required length:

49_{(10)} = 0011 0001_{(2)} - 5. To get the negative integer number representation in signed binary one's complement, replace all '0' bits with '1's and all '1' bits with '0's:

-49_{(10)} = 1100 1110 #### Number -49_{(10)}, signed integer, converted from the decimal system (base 10) to signed binary in one's complement representation = 1100 1110