Convert 3 414 648 673 to a Signed Binary in One's (1's) Complement Representation

How to convert decimal number 3 414 648 673(10) to a signed binary in one's (1's) complement representation

What are the steps to convert decimal number
3 414 648 673 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;
  • 3 414 648 673 ÷ 2 = 1 707 324 336 + 1;
  • 1 707 324 336 ÷ 2 = 853 662 168 + 0;
  • 853 662 168 ÷ 2 = 426 831 084 + 0;
  • 426 831 084 ÷ 2 = 213 415 542 + 0;
  • 213 415 542 ÷ 2 = 106 707 771 + 0;
  • 106 707 771 ÷ 2 = 53 353 885 + 1;
  • 53 353 885 ÷ 2 = 26 676 942 + 1;
  • 26 676 942 ÷ 2 = 13 338 471 + 0;
  • 13 338 471 ÷ 2 = 6 669 235 + 1;
  • 6 669 235 ÷ 2 = 3 334 617 + 1;
  • 3 334 617 ÷ 2 = 1 667 308 + 1;
  • 1 667 308 ÷ 2 = 833 654 + 0;
  • 833 654 ÷ 2 = 416 827 + 0;
  • 416 827 ÷ 2 = 208 413 + 1;
  • 208 413 ÷ 2 = 104 206 + 1;
  • 104 206 ÷ 2 = 52 103 + 0;
  • 52 103 ÷ 2 = 26 051 + 1;
  • 26 051 ÷ 2 = 13 025 + 1;
  • 13 025 ÷ 2 = 6 512 + 1;
  • 6 512 ÷ 2 = 3 256 + 0;
  • 3 256 ÷ 2 = 1 628 + 0;
  • 1 628 ÷ 2 = 814 + 0;
  • 814 ÷ 2 = 407 + 0;
  • 407 ÷ 2 = 203 + 1;
  • 203 ÷ 2 = 101 + 1;
  • 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.

3 414 648 673(10) = 1100 1011 1000 0111 0110 0111 0110 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 3 414 648 673(10) converted to signed binary in one's complement representation:

3 414 648 673(10) = 0000 0000 0000 0000 0000 0000 0000 0000 1100 1011 1000 0111 0110 0111 0110 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