Convert -725 621 712 967 255 to a Signed Binary in One's (1's) Complement Representation

How to convert decimal number -725 621 712 967 255(10) to a signed binary in one's (1's) complement representation

What are the steps to convert decimal number
-725 621 712 967 255 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:

|-725 621 712 967 255| = 725 621 712 967 255

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;
  • 725 621 712 967 255 ÷ 2 = 362 810 856 483 627 + 1;
  • 362 810 856 483 627 ÷ 2 = 181 405 428 241 813 + 1;
  • 181 405 428 241 813 ÷ 2 = 90 702 714 120 906 + 1;
  • 90 702 714 120 906 ÷ 2 = 45 351 357 060 453 + 0;
  • 45 351 357 060 453 ÷ 2 = 22 675 678 530 226 + 1;
  • 22 675 678 530 226 ÷ 2 = 11 337 839 265 113 + 0;
  • 11 337 839 265 113 ÷ 2 = 5 668 919 632 556 + 1;
  • 5 668 919 632 556 ÷ 2 = 2 834 459 816 278 + 0;
  • 2 834 459 816 278 ÷ 2 = 1 417 229 908 139 + 0;
  • 1 417 229 908 139 ÷ 2 = 708 614 954 069 + 1;
  • 708 614 954 069 ÷ 2 = 354 307 477 034 + 1;
  • 354 307 477 034 ÷ 2 = 177 153 738 517 + 0;
  • 177 153 738 517 ÷ 2 = 88 576 869 258 + 1;
  • 88 576 869 258 ÷ 2 = 44 288 434 629 + 0;
  • 44 288 434 629 ÷ 2 = 22 144 217 314 + 1;
  • 22 144 217 314 ÷ 2 = 11 072 108 657 + 0;
  • 11 072 108 657 ÷ 2 = 5 536 054 328 + 1;
  • 5 536 054 328 ÷ 2 = 2 768 027 164 + 0;
  • 2 768 027 164 ÷ 2 = 1 384 013 582 + 0;
  • 1 384 013 582 ÷ 2 = 692 006 791 + 0;
  • 692 006 791 ÷ 2 = 346 003 395 + 1;
  • 346 003 395 ÷ 2 = 173 001 697 + 1;
  • 173 001 697 ÷ 2 = 86 500 848 + 1;
  • 86 500 848 ÷ 2 = 43 250 424 + 0;
  • 43 250 424 ÷ 2 = 21 625 212 + 0;
  • 21 625 212 ÷ 2 = 10 812 606 + 0;
  • 10 812 606 ÷ 2 = 5 406 303 + 0;
  • 5 406 303 ÷ 2 = 2 703 151 + 1;
  • 2 703 151 ÷ 2 = 1 351 575 + 1;
  • 1 351 575 ÷ 2 = 675 787 + 1;
  • 675 787 ÷ 2 = 337 893 + 1;
  • 337 893 ÷ 2 = 168 946 + 1;
  • 168 946 ÷ 2 = 84 473 + 0;
  • 84 473 ÷ 2 = 42 236 + 1;
  • 42 236 ÷ 2 = 21 118 + 0;
  • 21 118 ÷ 2 = 10 559 + 0;
  • 10 559 ÷ 2 = 5 279 + 1;
  • 5 279 ÷ 2 = 2 639 + 1;
  • 2 639 ÷ 2 = 1 319 + 1;
  • 1 319 ÷ 2 = 659 + 1;
  • 659 ÷ 2 = 329 + 1;
  • 329 ÷ 2 = 164 + 1;
  • 164 ÷ 2 = 82 + 0;
  • 82 ÷ 2 = 41 + 0;
  • 41 ÷ 2 = 20 + 1;
  • 20 ÷ 2 = 10 + 0;
  • 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.

725 621 712 967 255(10) = 10 1001 0011 1111 0010 1111 1000 0111 0001 0101 0110 0101 0111(2)

4. Determine the signed binary number bit length:

  • The base 2 number's actual length, in bits: 50.

  • 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, 50,

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.


725 621 712 967 255(10) = 0000 0000 0000 0010 1001 0011 1111 0010 1111 1000 0111 0001 0101 0110 0101 0111

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).


-725 621 712 967 255(10) = !(0000 0000 0000 0010 1001 0011 1111 0010 1111 1000 0111 0001 0101 0110 0101 0111)


Decimal Number -725 621 712 967 255(10) converted to signed binary in one's complement representation:

-725 621 712 967 255(10) = 1111 1111 1111 1101 0110 1100 0000 1101 0000 0111 1000 1110 1010 1001 1010 1000

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