Convert 10 100 100 100 857 to a Signed Binary in One's (1's) Complement Representation

How to convert decimal number 10 100 100 100 857(10) to a signed binary in one's (1's) complement representation

What are the steps to convert decimal number
10 100 100 100 857 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;
  • 10 100 100 100 857 ÷ 2 = 5 050 050 050 428 + 1;
  • 5 050 050 050 428 ÷ 2 = 2 525 025 025 214 + 0;
  • 2 525 025 025 214 ÷ 2 = 1 262 512 512 607 + 0;
  • 1 262 512 512 607 ÷ 2 = 631 256 256 303 + 1;
  • 631 256 256 303 ÷ 2 = 315 628 128 151 + 1;
  • 315 628 128 151 ÷ 2 = 157 814 064 075 + 1;
  • 157 814 064 075 ÷ 2 = 78 907 032 037 + 1;
  • 78 907 032 037 ÷ 2 = 39 453 516 018 + 1;
  • 39 453 516 018 ÷ 2 = 19 726 758 009 + 0;
  • 19 726 758 009 ÷ 2 = 9 863 379 004 + 1;
  • 9 863 379 004 ÷ 2 = 4 931 689 502 + 0;
  • 4 931 689 502 ÷ 2 = 2 465 844 751 + 0;
  • 2 465 844 751 ÷ 2 = 1 232 922 375 + 1;
  • 1 232 922 375 ÷ 2 = 616 461 187 + 1;
  • 616 461 187 ÷ 2 = 308 230 593 + 1;
  • 308 230 593 ÷ 2 = 154 115 296 + 1;
  • 154 115 296 ÷ 2 = 77 057 648 + 0;
  • 77 057 648 ÷ 2 = 38 528 824 + 0;
  • 38 528 824 ÷ 2 = 19 264 412 + 0;
  • 19 264 412 ÷ 2 = 9 632 206 + 0;
  • 9 632 206 ÷ 2 = 4 816 103 + 0;
  • 4 816 103 ÷ 2 = 2 408 051 + 1;
  • 2 408 051 ÷ 2 = 1 204 025 + 1;
  • 1 204 025 ÷ 2 = 602 012 + 1;
  • 602 012 ÷ 2 = 301 006 + 0;
  • 301 006 ÷ 2 = 150 503 + 0;
  • 150 503 ÷ 2 = 75 251 + 1;
  • 75 251 ÷ 2 = 37 625 + 1;
  • 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.

10 100 100 100 857(10) = 1001 0010 1111 1001 1100 1110 0000 1111 0010 1111 1001(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 10 100 100 100 857(10) converted to signed binary in one's complement representation:

10 100 100 100 857(10) = 0000 0000 0000 0000 0000 1001 0010 1111 1001 1100 1110 0000 1111 0010 1111 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