Convert 13 640 301 709 537 784 to a Signed Binary in One's (1's) Complement Representation

How to convert decimal number 13 640 301 709 537 784(10) to a signed binary in one's (1's) complement representation

What are the steps to convert decimal number
13 640 301 709 537 784 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;
  • 13 640 301 709 537 784 ÷ 2 = 6 820 150 854 768 892 + 0;
  • 6 820 150 854 768 892 ÷ 2 = 3 410 075 427 384 446 + 0;
  • 3 410 075 427 384 446 ÷ 2 = 1 705 037 713 692 223 + 0;
  • 1 705 037 713 692 223 ÷ 2 = 852 518 856 846 111 + 1;
  • 852 518 856 846 111 ÷ 2 = 426 259 428 423 055 + 1;
  • 426 259 428 423 055 ÷ 2 = 213 129 714 211 527 + 1;
  • 213 129 714 211 527 ÷ 2 = 106 564 857 105 763 + 1;
  • 106 564 857 105 763 ÷ 2 = 53 282 428 552 881 + 1;
  • 53 282 428 552 881 ÷ 2 = 26 641 214 276 440 + 1;
  • 26 641 214 276 440 ÷ 2 = 13 320 607 138 220 + 0;
  • 13 320 607 138 220 ÷ 2 = 6 660 303 569 110 + 0;
  • 6 660 303 569 110 ÷ 2 = 3 330 151 784 555 + 0;
  • 3 330 151 784 555 ÷ 2 = 1 665 075 892 277 + 1;
  • 1 665 075 892 277 ÷ 2 = 832 537 946 138 + 1;
  • 832 537 946 138 ÷ 2 = 416 268 973 069 + 0;
  • 416 268 973 069 ÷ 2 = 208 134 486 534 + 1;
  • 208 134 486 534 ÷ 2 = 104 067 243 267 + 0;
  • 104 067 243 267 ÷ 2 = 52 033 621 633 + 1;
  • 52 033 621 633 ÷ 2 = 26 016 810 816 + 1;
  • 26 016 810 816 ÷ 2 = 13 008 405 408 + 0;
  • 13 008 405 408 ÷ 2 = 6 504 202 704 + 0;
  • 6 504 202 704 ÷ 2 = 3 252 101 352 + 0;
  • 3 252 101 352 ÷ 2 = 1 626 050 676 + 0;
  • 1 626 050 676 ÷ 2 = 813 025 338 + 0;
  • 813 025 338 ÷ 2 = 406 512 669 + 0;
  • 406 512 669 ÷ 2 = 203 256 334 + 1;
  • 203 256 334 ÷ 2 = 101 628 167 + 0;
  • 101 628 167 ÷ 2 = 50 814 083 + 1;
  • 50 814 083 ÷ 2 = 25 407 041 + 1;
  • 25 407 041 ÷ 2 = 12 703 520 + 1;
  • 12 703 520 ÷ 2 = 6 351 760 + 0;
  • 6 351 760 ÷ 2 = 3 175 880 + 0;
  • 3 175 880 ÷ 2 = 1 587 940 + 0;
  • 1 587 940 ÷ 2 = 793 970 + 0;
  • 793 970 ÷ 2 = 396 985 + 0;
  • 396 985 ÷ 2 = 198 492 + 1;
  • 198 492 ÷ 2 = 99 246 + 0;
  • 99 246 ÷ 2 = 49 623 + 0;
  • 49 623 ÷ 2 = 24 811 + 1;
  • 24 811 ÷ 2 = 12 405 + 1;
  • 12 405 ÷ 2 = 6 202 + 1;
  • 6 202 ÷ 2 = 3 101 + 0;
  • 3 101 ÷ 2 = 1 550 + 1;
  • 1 550 ÷ 2 = 775 + 0;
  • 775 ÷ 2 = 387 + 1;
  • 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;

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

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

13 640 301 709 537 784(10) = 11 0000 0111 0101 1100 1000 0011 1010 0000 0110 1011 0001 1111 1000(2)

3. Determine the signed binary number bit length:

  • The base 2 number's actual length, in bits: 54.

  • 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, 54,

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

=== is: 64.


4. 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.


Decimal Number 13 640 301 709 537 784(10) converted to signed binary in one's complement representation:

13 640 301 709 537 784(10) = 0000 0000 0011 0000 0111 0101 1100 1000 0011 1010 0000 0110 1011 0001 1111 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