Convert 123 456 789 013 829 to a Signed Binary in One's (1's) Complement Representation

How to convert decimal number 123 456 789 013 829(10) to a signed binary in one's (1's) complement representation

What are the steps to convert decimal number
123 456 789 013 829 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;
  • 123 456 789 013 829 ÷ 2 = 61 728 394 506 914 + 1;
  • 61 728 394 506 914 ÷ 2 = 30 864 197 253 457 + 0;
  • 30 864 197 253 457 ÷ 2 = 15 432 098 626 728 + 1;
  • 15 432 098 626 728 ÷ 2 = 7 716 049 313 364 + 0;
  • 7 716 049 313 364 ÷ 2 = 3 858 024 656 682 + 0;
  • 3 858 024 656 682 ÷ 2 = 1 929 012 328 341 + 0;
  • 1 929 012 328 341 ÷ 2 = 964 506 164 170 + 1;
  • 964 506 164 170 ÷ 2 = 482 253 082 085 + 0;
  • 482 253 082 085 ÷ 2 = 241 126 541 042 + 1;
  • 241 126 541 042 ÷ 2 = 120 563 270 521 + 0;
  • 120 563 270 521 ÷ 2 = 60 281 635 260 + 1;
  • 60 281 635 260 ÷ 2 = 30 140 817 630 + 0;
  • 30 140 817 630 ÷ 2 = 15 070 408 815 + 0;
  • 15 070 408 815 ÷ 2 = 7 535 204 407 + 1;
  • 7 535 204 407 ÷ 2 = 3 767 602 203 + 1;
  • 3 767 602 203 ÷ 2 = 1 883 801 101 + 1;
  • 1 883 801 101 ÷ 2 = 941 900 550 + 1;
  • 941 900 550 ÷ 2 = 470 950 275 + 0;
  • 470 950 275 ÷ 2 = 235 475 137 + 1;
  • 235 475 137 ÷ 2 = 117 737 568 + 1;
  • 117 737 568 ÷ 2 = 58 868 784 + 0;
  • 58 868 784 ÷ 2 = 29 434 392 + 0;
  • 29 434 392 ÷ 2 = 14 717 196 + 0;
  • 14 717 196 ÷ 2 = 7 358 598 + 0;
  • 7 358 598 ÷ 2 = 3 679 299 + 0;
  • 3 679 299 ÷ 2 = 1 839 649 + 1;
  • 1 839 649 ÷ 2 = 919 824 + 1;
  • 919 824 ÷ 2 = 459 912 + 0;
  • 459 912 ÷ 2 = 229 956 + 0;
  • 229 956 ÷ 2 = 114 978 + 0;
  • 114 978 ÷ 2 = 57 489 + 0;
  • 57 489 ÷ 2 = 28 744 + 1;
  • 28 744 ÷ 2 = 14 372 + 0;
  • 14 372 ÷ 2 = 7 186 + 0;
  • 7 186 ÷ 2 = 3 593 + 0;
  • 3 593 ÷ 2 = 1 796 + 1;
  • 1 796 ÷ 2 = 898 + 0;
  • 898 ÷ 2 = 449 + 0;
  • 449 ÷ 2 = 224 + 1;
  • 224 ÷ 2 = 112 + 0;
  • 112 ÷ 2 = 56 + 0;
  • 56 ÷ 2 = 28 + 0;
  • 28 ÷ 2 = 14 + 0;
  • 14 ÷ 2 = 7 + 0;
  • 7 ÷ 2 = 3 + 1;
  • 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.

123 456 789 013 829(10) = 111 0000 0100 1000 1000 0110 0000 1101 1110 0101 0100 0101(2)

3. Determine the signed binary number bit length:

  • The base 2 number's actual length, in bits: 47.

  • 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, 47,

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 123 456 789 013 829(10) converted to signed binary in one's complement representation:

123 456 789 013 829(10) = 0000 0000 0000 0000 0111 0000 0100 1000 1000 0110 0000 1101 1110 0101 0100 0101

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