Convert 10 100 001 109 377 to a Signed Binary in One's (1's) Complement Representation

How to convert decimal number 10 100 001 109 377(10) to a signed binary in one's (1's) complement representation

What are the steps to convert decimal number
10 100 001 109 377 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;
  • 10 100 001 109 377 ÷ 2 = 5 050 000 554 688 + 1;
  • 5 050 000 554 688 ÷ 2 = 2 525 000 277 344 + 0;
  • 2 525 000 277 344 ÷ 2 = 1 262 500 138 672 + 0;
  • 1 262 500 138 672 ÷ 2 = 631 250 069 336 + 0;
  • 631 250 069 336 ÷ 2 = 315 625 034 668 + 0;
  • 315 625 034 668 ÷ 2 = 157 812 517 334 + 0;
  • 157 812 517 334 ÷ 2 = 78 906 258 667 + 0;
  • 78 906 258 667 ÷ 2 = 39 453 129 333 + 1;
  • 39 453 129 333 ÷ 2 = 19 726 564 666 + 1;
  • 19 726 564 666 ÷ 2 = 9 863 282 333 + 0;
  • 9 863 282 333 ÷ 2 = 4 931 641 166 + 1;
  • 4 931 641 166 ÷ 2 = 2 465 820 583 + 0;
  • 2 465 820 583 ÷ 2 = 1 232 910 291 + 1;
  • 1 232 910 291 ÷ 2 = 616 455 145 + 1;
  • 616 455 145 ÷ 2 = 308 227 572 + 1;
  • 308 227 572 ÷ 2 = 154 113 786 + 0;
  • 154 113 786 ÷ 2 = 77 056 893 + 0;
  • 77 056 893 ÷ 2 = 38 528 446 + 1;
  • 38 528 446 ÷ 2 = 19 264 223 + 0;
  • 19 264 223 ÷ 2 = 9 632 111 + 1;
  • 9 632 111 ÷ 2 = 4 816 055 + 1;
  • 4 816 055 ÷ 2 = 2 408 027 + 1;
  • 2 408 027 ÷ 2 = 1 204 013 + 1;
  • 1 204 013 ÷ 2 = 602 006 + 1;
  • 602 006 ÷ 2 = 301 003 + 0;
  • 301 003 ÷ 2 = 150 501 + 1;
  • 150 501 ÷ 2 = 75 250 + 1;
  • 75 250 ÷ 2 = 37 625 + 0;
  • 37 625 ÷ 2 = 18 812 + 1;
  • 18 812 ÷ 2 = 9 406 + 0;
  • 9 406 ÷ 2 = 4 703 + 0;
  • 4 703 ÷ 2 = 2 351 + 1;
  • 2 351 ÷ 2 = 1 175 + 1;
  • 1 175 ÷ 2 = 587 + 1;
  • 587 ÷ 2 = 293 + 1;
  • 293 ÷ 2 = 146 + 1;
  • 146 ÷ 2 = 73 + 0;
  • 73 ÷ 2 = 36 + 1;
  • 36 ÷ 2 = 18 + 0;
  • 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.

10 100 001 109 377(10) = 1001 0010 1111 1001 0110 1111 1010 0111 0101 1000 0001(2)

3. Determine the signed binary number bit length:

  • The base 2 number's actual length, in bits: 44.

  • 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, 44,

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 10 100 001 109 377(10) converted to signed binary in one's complement representation:

10 100 001 109 377(10) = 0000 0000 0000 0000 0000 1001 0010 1111 1001 0110 1111 1010 0111 0101 1000 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