Convert -51 999 999 634 to a Signed Binary in One's (1's) Complement Representation

How to convert decimal number -51 999 999 634(10) to a signed binary in one's (1's) complement representation

What are the steps to convert decimal number
-51 999 999 634 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. Start with the positive version of the number:

|-51 999 999 634| = 51 999 999 634

2. 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;
  • 51 999 999 634 ÷ 2 = 25 999 999 817 + 0;
  • 25 999 999 817 ÷ 2 = 12 999 999 908 + 1;
  • 12 999 999 908 ÷ 2 = 6 499 999 954 + 0;
  • 6 499 999 954 ÷ 2 = 3 249 999 977 + 0;
  • 3 249 999 977 ÷ 2 = 1 624 999 988 + 1;
  • 1 624 999 988 ÷ 2 = 812 499 994 + 0;
  • 812 499 994 ÷ 2 = 406 249 997 + 0;
  • 406 249 997 ÷ 2 = 203 124 998 + 1;
  • 203 124 998 ÷ 2 = 101 562 499 + 0;
  • 101 562 499 ÷ 2 = 50 781 249 + 1;
  • 50 781 249 ÷ 2 = 25 390 624 + 1;
  • 25 390 624 ÷ 2 = 12 695 312 + 0;
  • 12 695 312 ÷ 2 = 6 347 656 + 0;
  • 6 347 656 ÷ 2 = 3 173 828 + 0;
  • 3 173 828 ÷ 2 = 1 586 914 + 0;
  • 1 586 914 ÷ 2 = 793 457 + 0;
  • 793 457 ÷ 2 = 396 728 + 1;
  • 396 728 ÷ 2 = 198 364 + 0;
  • 198 364 ÷ 2 = 99 182 + 0;
  • 99 182 ÷ 2 = 49 591 + 0;
  • 49 591 ÷ 2 = 24 795 + 1;
  • 24 795 ÷ 2 = 12 397 + 1;
  • 12 397 ÷ 2 = 6 198 + 1;
  • 6 198 ÷ 2 = 3 099 + 0;
  • 3 099 ÷ 2 = 1 549 + 1;
  • 1 549 ÷ 2 = 774 + 1;
  • 774 ÷ 2 = 387 + 0;
  • 387 ÷ 2 = 193 + 1;
  • 193 ÷ 2 = 96 + 1;
  • 96 ÷ 2 = 48 + 0;
  • 48 ÷ 2 = 24 + 0;
  • 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:

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

51 999 999 634(10) = 1100 0001 1011 0111 0001 0000 0110 1001 0010(2)

4. Determine the signed binary number bit length:

  • The base 2 number's actual length, in bits: 36.

  • 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, 36,

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

=== is: 64.


5. Get the positive binary computer representation on 64 bits (8 Bytes):

If needed, add extra 0s in front (to the left) of the base 2 number, up to the required length, 64.


51 999 999 634(10) = 0000 0000 0000 0000 0000 0000 0000 1100 0001 1011 0111 0001 0000 0110 1001 0010

6. Get the negative integer number representation:

  • To write the negative integer number on 64 bits (8 Bytes), as a signed binary in one's complement representation,

  • ... Reverse all the bits from 0 to 1 and from 1 to 0 (flip the digits).


-51 999 999 634(10) = !(0000 0000 0000 0000 0000 0000 0000 1100 0001 1011 0111 0001 0000 0110 1001 0010)


Decimal Number -51 999 999 634(10) converted to signed binary in one's complement representation:

-51 999 999 634(10) = 1111 1111 1111 1111 1111 1111 1111 0011 1110 0100 1000 1110 1111 1001 0110 1101

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