Convert 10 101 001 025 to a Signed Binary in One's (1's) Complement Representation

How to convert decimal number 10 101 001 025(10) to a signed binary in one's (1's) complement representation

What are the steps to convert decimal number
10 101 001 025 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;
  • 10 101 001 025 ÷ 2 = 5 050 500 512 + 1;
  • 5 050 500 512 ÷ 2 = 2 525 250 256 + 0;
  • 2 525 250 256 ÷ 2 = 1 262 625 128 + 0;
  • 1 262 625 128 ÷ 2 = 631 312 564 + 0;
  • 631 312 564 ÷ 2 = 315 656 282 + 0;
  • 315 656 282 ÷ 2 = 157 828 141 + 0;
  • 157 828 141 ÷ 2 = 78 914 070 + 1;
  • 78 914 070 ÷ 2 = 39 457 035 + 0;
  • 39 457 035 ÷ 2 = 19 728 517 + 1;
  • 19 728 517 ÷ 2 = 9 864 258 + 1;
  • 9 864 258 ÷ 2 = 4 932 129 + 0;
  • 4 932 129 ÷ 2 = 2 466 064 + 1;
  • 2 466 064 ÷ 2 = 1 233 032 + 0;
  • 1 233 032 ÷ 2 = 616 516 + 0;
  • 616 516 ÷ 2 = 308 258 + 0;
  • 308 258 ÷ 2 = 154 129 + 0;
  • 154 129 ÷ 2 = 77 064 + 1;
  • 77 064 ÷ 2 = 38 532 + 0;
  • 38 532 ÷ 2 = 19 266 + 0;
  • 19 266 ÷ 2 = 9 633 + 0;
  • 9 633 ÷ 2 = 4 816 + 1;
  • 4 816 ÷ 2 = 2 408 + 0;
  • 2 408 ÷ 2 = 1 204 + 0;
  • 1 204 ÷ 2 = 602 + 0;
  • 602 ÷ 2 = 301 + 0;
  • 301 ÷ 2 = 150 + 1;
  • 150 ÷ 2 = 75 + 0;
  • 75 ÷ 2 = 37 + 1;
  • 37 ÷ 2 = 18 + 1;
  • 18 ÷ 2 = 9 + 0;
  • 9 ÷ 2 = 4 + 1;
  • 4 ÷ 2 = 2 + 0;
  • 2 ÷ 2 = 1 + 0;
  • 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.

10 101 001 025(10) = 10 0101 1010 0001 0001 0000 1011 0100 0001(2)

3. Determine the signed binary number bit length:

  • The base 2 number's actual length, in bits: 34.

  • 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, 34,

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 10 101 001 025(10) converted to signed binary in one's complement representation:

10 101 001 025(10) = 0000 0000 0000 0000 0000 0000 0000 0010 0101 1010 0001 0001 0000 1011 0100 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