Convert 1 644 167 142 to a Signed Binary in One's (1's) Complement Representation

How to convert decimal number 1 644 167 142(10) to a signed binary in one's (1's) complement representation

What are the steps to convert decimal number
1 644 167 142 to a signed binary in one's (1's) complement representation?

  • A signed integer, written in base ten, or a decimal system number, is a number written using the digits 0 through 9 and the sign, which can be positive (+) or negative (-). If positive, the sign is usually not written. A number written in base two, or binary, is a number written using only the digits 0 and 1.

1. Divide the number repeatedly by 2:

Keep track of each remainder.

Stop when you get a quotient that is equal to zero.


  • division = quotient + remainder;
  • 1 644 167 142 ÷ 2 = 822 083 571 + 0;
  • 822 083 571 ÷ 2 = 411 041 785 + 1;
  • 411 041 785 ÷ 2 = 205 520 892 + 1;
  • 205 520 892 ÷ 2 = 102 760 446 + 0;
  • 102 760 446 ÷ 2 = 51 380 223 + 0;
  • 51 380 223 ÷ 2 = 25 690 111 + 1;
  • 25 690 111 ÷ 2 = 12 845 055 + 1;
  • 12 845 055 ÷ 2 = 6 422 527 + 1;
  • 6 422 527 ÷ 2 = 3 211 263 + 1;
  • 3 211 263 ÷ 2 = 1 605 631 + 1;
  • 1 605 631 ÷ 2 = 802 815 + 1;
  • 802 815 ÷ 2 = 401 407 + 1;
  • 401 407 ÷ 2 = 200 703 + 1;
  • 200 703 ÷ 2 = 100 351 + 1;
  • 100 351 ÷ 2 = 50 175 + 1;
  • 50 175 ÷ 2 = 25 087 + 1;
  • 25 087 ÷ 2 = 12 543 + 1;
  • 12 543 ÷ 2 = 6 271 + 1;
  • 6 271 ÷ 2 = 3 135 + 1;
  • 3 135 ÷ 2 = 1 567 + 1;
  • 1 567 ÷ 2 = 783 + 1;
  • 783 ÷ 2 = 391 + 1;
  • 391 ÷ 2 = 195 + 1;
  • 195 ÷ 2 = 97 + 1;
  • 97 ÷ 2 = 48 + 1;
  • 48 ÷ 2 = 24 + 0;
  • 24 ÷ 2 = 12 + 0;
  • 12 ÷ 2 = 6 + 0;
  • 6 ÷ 2 = 3 + 0;
  • 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.

1 644 167 142(10) = 110 0001 1111 1111 1111 1111 1110 0110(2)

3. Determine the signed binary number bit length:

  • The base 2 number's actual length, in bits: 31.

  • A signed binary's bit length must be equal to a power of 2, as of:
  • 21 = 2; 22 = 4; 23 = 8; 24 = 16; 25 = 32; 26 = 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, 31,

3) so that the first bit (leftmost) could be zero
(we deal with a positive number at this moment)

=== is: 32.


4. Get the 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.


Decimal Number 1 644 167 142(10) converted to signed binary in one's complement representation:

1 644 167 142(10) = 0110 0001 1111 1111 1111 1111 1110 0110

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

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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