Convert 100 011 101 018 to a Signed Binary in One's (1's) Complement Representation

How to convert decimal number 100 011 101 018(10) to a signed binary in one's (1's) complement representation

What are the steps to convert decimal number
100 011 101 018 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;
  • 100 011 101 018 ÷ 2 = 50 005 550 509 + 0;
  • 50 005 550 509 ÷ 2 = 25 002 775 254 + 1;
  • 25 002 775 254 ÷ 2 = 12 501 387 627 + 0;
  • 12 501 387 627 ÷ 2 = 6 250 693 813 + 1;
  • 6 250 693 813 ÷ 2 = 3 125 346 906 + 1;
  • 3 125 346 906 ÷ 2 = 1 562 673 453 + 0;
  • 1 562 673 453 ÷ 2 = 781 336 726 + 1;
  • 781 336 726 ÷ 2 = 390 668 363 + 0;
  • 390 668 363 ÷ 2 = 195 334 181 + 1;
  • 195 334 181 ÷ 2 = 97 667 090 + 1;
  • 97 667 090 ÷ 2 = 48 833 545 + 0;
  • 48 833 545 ÷ 2 = 24 416 772 + 1;
  • 24 416 772 ÷ 2 = 12 208 386 + 0;
  • 12 208 386 ÷ 2 = 6 104 193 + 0;
  • 6 104 193 ÷ 2 = 3 052 096 + 1;
  • 3 052 096 ÷ 2 = 1 526 048 + 0;
  • 1 526 048 ÷ 2 = 763 024 + 0;
  • 763 024 ÷ 2 = 381 512 + 0;
  • 381 512 ÷ 2 = 190 756 + 0;
  • 190 756 ÷ 2 = 95 378 + 0;
  • 95 378 ÷ 2 = 47 689 + 0;
  • 47 689 ÷ 2 = 23 844 + 1;
  • 23 844 ÷ 2 = 11 922 + 0;
  • 11 922 ÷ 2 = 5 961 + 0;
  • 5 961 ÷ 2 = 2 980 + 1;
  • 2 980 ÷ 2 = 1 490 + 0;
  • 1 490 ÷ 2 = 745 + 0;
  • 745 ÷ 2 = 372 + 1;
  • 372 ÷ 2 = 186 + 0;
  • 186 ÷ 2 = 93 + 0;
  • 93 ÷ 2 = 46 + 1;
  • 46 ÷ 2 = 23 + 0;
  • 23 ÷ 2 = 11 + 1;
  • 11 ÷ 2 = 5 + 1;
  • 5 ÷ 2 = 2 + 1;
  • 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.

100 011 101 018(10) = 1 0111 0100 1001 0010 0000 0100 1011 0101 1010(2)

3. Determine the signed binary number bit length:

  • The base 2 number's actual length, in bits: 37.

  • 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, 37,

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 100 011 101 018(10) converted to signed binary in one's complement representation:

100 011 101 018(10) = 0000 0000 0000 0000 0000 0000 0001 0111 0100 1001 0010 0000 0100 1011 0101 1010

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