Convert 100 111 000 101 641 to a Signed Binary in One's (1's) Complement Representation

How to convert decimal number 100 111 000 101 641(10) to a signed binary in one's (1's) complement representation

What are the steps to convert decimal number
100 111 000 101 641 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;
  • 100 111 000 101 641 ÷ 2 = 50 055 500 050 820 + 1;
  • 50 055 500 050 820 ÷ 2 = 25 027 750 025 410 + 0;
  • 25 027 750 025 410 ÷ 2 = 12 513 875 012 705 + 0;
  • 12 513 875 012 705 ÷ 2 = 6 256 937 506 352 + 1;
  • 6 256 937 506 352 ÷ 2 = 3 128 468 753 176 + 0;
  • 3 128 468 753 176 ÷ 2 = 1 564 234 376 588 + 0;
  • 1 564 234 376 588 ÷ 2 = 782 117 188 294 + 0;
  • 782 117 188 294 ÷ 2 = 391 058 594 147 + 0;
  • 391 058 594 147 ÷ 2 = 195 529 297 073 + 1;
  • 195 529 297 073 ÷ 2 = 97 764 648 536 + 1;
  • 97 764 648 536 ÷ 2 = 48 882 324 268 + 0;
  • 48 882 324 268 ÷ 2 = 24 441 162 134 + 0;
  • 24 441 162 134 ÷ 2 = 12 220 581 067 + 0;
  • 12 220 581 067 ÷ 2 = 6 110 290 533 + 1;
  • 6 110 290 533 ÷ 2 = 3 055 145 266 + 1;
  • 3 055 145 266 ÷ 2 = 1 527 572 633 + 0;
  • 1 527 572 633 ÷ 2 = 763 786 316 + 1;
  • 763 786 316 ÷ 2 = 381 893 158 + 0;
  • 381 893 158 ÷ 2 = 190 946 579 + 0;
  • 190 946 579 ÷ 2 = 95 473 289 + 1;
  • 95 473 289 ÷ 2 = 47 736 644 + 1;
  • 47 736 644 ÷ 2 = 23 868 322 + 0;
  • 23 868 322 ÷ 2 = 11 934 161 + 0;
  • 11 934 161 ÷ 2 = 5 967 080 + 1;
  • 5 967 080 ÷ 2 = 2 983 540 + 0;
  • 2 983 540 ÷ 2 = 1 491 770 + 0;
  • 1 491 770 ÷ 2 = 745 885 + 0;
  • 745 885 ÷ 2 = 372 942 + 1;
  • 372 942 ÷ 2 = 186 471 + 0;
  • 186 471 ÷ 2 = 93 235 + 1;
  • 93 235 ÷ 2 = 46 617 + 1;
  • 46 617 ÷ 2 = 23 308 + 1;
  • 23 308 ÷ 2 = 11 654 + 0;
  • 11 654 ÷ 2 = 5 827 + 0;
  • 5 827 ÷ 2 = 2 913 + 1;
  • 2 913 ÷ 2 = 1 456 + 1;
  • 1 456 ÷ 2 = 728 + 0;
  • 728 ÷ 2 = 364 + 0;
  • 364 ÷ 2 = 182 + 0;
  • 182 ÷ 2 = 91 + 0;
  • 91 ÷ 2 = 45 + 1;
  • 45 ÷ 2 = 22 + 1;
  • 22 ÷ 2 = 11 + 0;
  • 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.

100 111 000 101 641(10) = 101 1011 0000 1100 1110 1000 1001 1001 0110 0011 0000 1001(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 100 111 000 101 641(10) converted to signed binary in one's complement representation:

100 111 000 101 641(10) = 0000 0000 0000 0000 0101 1011 0000 1100 1110 1000 1001 1001 0110 0011 0000 1001

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