Convert 1 100 110 111 012 527 to a Signed Binary in One's (1's) Complement Representation

How to convert decimal number 1 100 110 111 012 527(10) to a signed binary in one's (1's) complement representation

What are the steps to convert decimal number
1 100 110 111 012 527 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;
  • 1 100 110 111 012 527 ÷ 2 = 550 055 055 506 263 + 1;
  • 550 055 055 506 263 ÷ 2 = 275 027 527 753 131 + 1;
  • 275 027 527 753 131 ÷ 2 = 137 513 763 876 565 + 1;
  • 137 513 763 876 565 ÷ 2 = 68 756 881 938 282 + 1;
  • 68 756 881 938 282 ÷ 2 = 34 378 440 969 141 + 0;
  • 34 378 440 969 141 ÷ 2 = 17 189 220 484 570 + 1;
  • 17 189 220 484 570 ÷ 2 = 8 594 610 242 285 + 0;
  • 8 594 610 242 285 ÷ 2 = 4 297 305 121 142 + 1;
  • 4 297 305 121 142 ÷ 2 = 2 148 652 560 571 + 0;
  • 2 148 652 560 571 ÷ 2 = 1 074 326 280 285 + 1;
  • 1 074 326 280 285 ÷ 2 = 537 163 140 142 + 1;
  • 537 163 140 142 ÷ 2 = 268 581 570 071 + 0;
  • 268 581 570 071 ÷ 2 = 134 290 785 035 + 1;
  • 134 290 785 035 ÷ 2 = 67 145 392 517 + 1;
  • 67 145 392 517 ÷ 2 = 33 572 696 258 + 1;
  • 33 572 696 258 ÷ 2 = 16 786 348 129 + 0;
  • 16 786 348 129 ÷ 2 = 8 393 174 064 + 1;
  • 8 393 174 064 ÷ 2 = 4 196 587 032 + 0;
  • 4 196 587 032 ÷ 2 = 2 098 293 516 + 0;
  • 2 098 293 516 ÷ 2 = 1 049 146 758 + 0;
  • 1 049 146 758 ÷ 2 = 524 573 379 + 0;
  • 524 573 379 ÷ 2 = 262 286 689 + 1;
  • 262 286 689 ÷ 2 = 131 143 344 + 1;
  • 131 143 344 ÷ 2 = 65 571 672 + 0;
  • 65 571 672 ÷ 2 = 32 785 836 + 0;
  • 32 785 836 ÷ 2 = 16 392 918 + 0;
  • 16 392 918 ÷ 2 = 8 196 459 + 0;
  • 8 196 459 ÷ 2 = 4 098 229 + 1;
  • 4 098 229 ÷ 2 = 2 049 114 + 1;
  • 2 049 114 ÷ 2 = 1 024 557 + 0;
  • 1 024 557 ÷ 2 = 512 278 + 1;
  • 512 278 ÷ 2 = 256 139 + 0;
  • 256 139 ÷ 2 = 128 069 + 1;
  • 128 069 ÷ 2 = 64 034 + 1;
  • 64 034 ÷ 2 = 32 017 + 0;
  • 32 017 ÷ 2 = 16 008 + 1;
  • 16 008 ÷ 2 = 8 004 + 0;
  • 8 004 ÷ 2 = 4 002 + 0;
  • 4 002 ÷ 2 = 2 001 + 0;
  • 2 001 ÷ 2 = 1 000 + 1;
  • 1 000 ÷ 2 = 500 + 0;
  • 500 ÷ 2 = 250 + 0;
  • 250 ÷ 2 = 125 + 0;
  • 125 ÷ 2 = 62 + 1;
  • 62 ÷ 2 = 31 + 0;
  • 31 ÷ 2 = 15 + 1;
  • 15 ÷ 2 = 7 + 1;
  • 7 ÷ 2 = 3 + 1;
  • 3 ÷ 2 = 1 + 1;
  • 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.

1 100 110 111 012 527(10) = 11 1110 1000 1000 1011 0101 1000 0110 0001 0111 0110 1010 1111(2)

3. Determine the signed binary number bit length:

  • The base 2 number's actual length, in bits: 50.

  • 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, 50,

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 1 100 110 111 012 527(10) converted to signed binary in one's complement representation:

1 100 110 111 012 527(10) = 0000 0000 0000 0011 1110 1000 1000 1011 0101 1000 0110 0001 0111 0110 1010 1111

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