Convert 3 405 698 977 to a Signed Binary in One's (1's) Complement Representation

How to convert decimal number 3 405 698 977(10) to a signed binary in one's (1's) complement representation

What are the steps to convert decimal number
3 405 698 977 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;
  • 3 405 698 977 ÷ 2 = 1 702 849 488 + 1;
  • 1 702 849 488 ÷ 2 = 851 424 744 + 0;
  • 851 424 744 ÷ 2 = 425 712 372 + 0;
  • 425 712 372 ÷ 2 = 212 856 186 + 0;
  • 212 856 186 ÷ 2 = 106 428 093 + 0;
  • 106 428 093 ÷ 2 = 53 214 046 + 1;
  • 53 214 046 ÷ 2 = 26 607 023 + 0;
  • 26 607 023 ÷ 2 = 13 303 511 + 1;
  • 13 303 511 ÷ 2 = 6 651 755 + 1;
  • 6 651 755 ÷ 2 = 3 325 877 + 1;
  • 3 325 877 ÷ 2 = 1 662 938 + 1;
  • 1 662 938 ÷ 2 = 831 469 + 0;
  • 831 469 ÷ 2 = 415 734 + 1;
  • 415 734 ÷ 2 = 207 867 + 0;
  • 207 867 ÷ 2 = 103 933 + 1;
  • 103 933 ÷ 2 = 51 966 + 1;
  • 51 966 ÷ 2 = 25 983 + 0;
  • 25 983 ÷ 2 = 12 991 + 1;
  • 12 991 ÷ 2 = 6 495 + 1;
  • 6 495 ÷ 2 = 3 247 + 1;
  • 3 247 ÷ 2 = 1 623 + 1;
  • 1 623 ÷ 2 = 811 + 1;
  • 811 ÷ 2 = 405 + 1;
  • 405 ÷ 2 = 202 + 1;
  • 202 ÷ 2 = 101 + 0;
  • 101 ÷ 2 = 50 + 1;
  • 50 ÷ 2 = 25 + 0;
  • 25 ÷ 2 = 12 + 1;
  • 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.

3 405 698 977(10) = 1100 1010 1111 1110 1101 0111 1010 0001(2)

3. Determine the signed binary number bit length:

  • The base 2 number's actual length, in bits: 32.

  • 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, 32,

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 3 405 698 977(10) converted to signed binary in one's complement representation:

3 405 698 977(10) = 0000 0000 0000 0000 0000 0000 0000 0000 1100 1010 1111 1110 1101 0111 1010 0001

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