Convert 110 100 001 010 101 172 to a Signed Binary in One's (1's) Complement Representation

How to convert decimal number 110 100 001 010 101 172(10) to a signed binary in one's (1's) complement representation

What are the steps to convert decimal number
110 100 001 010 101 172 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;
  • 110 100 001 010 101 172 ÷ 2 = 55 050 000 505 050 586 + 0;
  • 55 050 000 505 050 586 ÷ 2 = 27 525 000 252 525 293 + 0;
  • 27 525 000 252 525 293 ÷ 2 = 13 762 500 126 262 646 + 1;
  • 13 762 500 126 262 646 ÷ 2 = 6 881 250 063 131 323 + 0;
  • 6 881 250 063 131 323 ÷ 2 = 3 440 625 031 565 661 + 1;
  • 3 440 625 031 565 661 ÷ 2 = 1 720 312 515 782 830 + 1;
  • 1 720 312 515 782 830 ÷ 2 = 860 156 257 891 415 + 0;
  • 860 156 257 891 415 ÷ 2 = 430 078 128 945 707 + 1;
  • 430 078 128 945 707 ÷ 2 = 215 039 064 472 853 + 1;
  • 215 039 064 472 853 ÷ 2 = 107 519 532 236 426 + 1;
  • 107 519 532 236 426 ÷ 2 = 53 759 766 118 213 + 0;
  • 53 759 766 118 213 ÷ 2 = 26 879 883 059 106 + 1;
  • 26 879 883 059 106 ÷ 2 = 13 439 941 529 553 + 0;
  • 13 439 941 529 553 ÷ 2 = 6 719 970 764 776 + 1;
  • 6 719 970 764 776 ÷ 2 = 3 359 985 382 388 + 0;
  • 3 359 985 382 388 ÷ 2 = 1 679 992 691 194 + 0;
  • 1 679 992 691 194 ÷ 2 = 839 996 345 597 + 0;
  • 839 996 345 597 ÷ 2 = 419 998 172 798 + 1;
  • 419 998 172 798 ÷ 2 = 209 999 086 399 + 0;
  • 209 999 086 399 ÷ 2 = 104 999 543 199 + 1;
  • 104 999 543 199 ÷ 2 = 52 499 771 599 + 1;
  • 52 499 771 599 ÷ 2 = 26 249 885 799 + 1;
  • 26 249 885 799 ÷ 2 = 13 124 942 899 + 1;
  • 13 124 942 899 ÷ 2 = 6 562 471 449 + 1;
  • 6 562 471 449 ÷ 2 = 3 281 235 724 + 1;
  • 3 281 235 724 ÷ 2 = 1 640 617 862 + 0;
  • 1 640 617 862 ÷ 2 = 820 308 931 + 0;
  • 820 308 931 ÷ 2 = 410 154 465 + 1;
  • 410 154 465 ÷ 2 = 205 077 232 + 1;
  • 205 077 232 ÷ 2 = 102 538 616 + 0;
  • 102 538 616 ÷ 2 = 51 269 308 + 0;
  • 51 269 308 ÷ 2 = 25 634 654 + 0;
  • 25 634 654 ÷ 2 = 12 817 327 + 0;
  • 12 817 327 ÷ 2 = 6 408 663 + 1;
  • 6 408 663 ÷ 2 = 3 204 331 + 1;
  • 3 204 331 ÷ 2 = 1 602 165 + 1;
  • 1 602 165 ÷ 2 = 801 082 + 1;
  • 801 082 ÷ 2 = 400 541 + 0;
  • 400 541 ÷ 2 = 200 270 + 1;
  • 200 270 ÷ 2 = 100 135 + 0;
  • 100 135 ÷ 2 = 50 067 + 1;
  • 50 067 ÷ 2 = 25 033 + 1;
  • 25 033 ÷ 2 = 12 516 + 1;
  • 12 516 ÷ 2 = 6 258 + 0;
  • 6 258 ÷ 2 = 3 129 + 0;
  • 3 129 ÷ 2 = 1 564 + 1;
  • 1 564 ÷ 2 = 782 + 0;
  • 782 ÷ 2 = 391 + 0;
  • 391 ÷ 2 = 195 + 1;
  • 195 ÷ 2 = 97 + 1;
  • 97 ÷ 2 = 48 + 1;
  • 48 ÷ 2 = 24 + 0;
  • 24 ÷ 2 = 12 + 0;
  • 12 ÷ 2 = 6 + 0;
  • 6 ÷ 2 = 3 + 0;
  • 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.

110 100 001 010 101 172(10) = 1 1000 0111 0010 0111 0101 1110 0001 1001 1111 1010 0010 1011 1011 0100(2)

3. Determine the signed binary number bit length:

  • The base 2 number's actual length, in bits: 57.

  • 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, 57,

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 110 100 001 010 101 172(10) converted to signed binary in one's complement representation:

110 100 001 010 101 172(10) = 0000 0001 1000 0111 0010 0111 0101 1110 0001 1001 1111 1010 0010 1011 1011 0100

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