Convert 167 772 761 709 537 111 to a Signed Binary in One's (1's) Complement Representation

How to convert decimal number 167 772 761 709 537 111(10) to a signed binary in one's (1's) complement representation

What are the steps to convert decimal number
167 772 761 709 537 111 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;
  • 167 772 761 709 537 111 ÷ 2 = 83 886 380 854 768 555 + 1;
  • 83 886 380 854 768 555 ÷ 2 = 41 943 190 427 384 277 + 1;
  • 41 943 190 427 384 277 ÷ 2 = 20 971 595 213 692 138 + 1;
  • 20 971 595 213 692 138 ÷ 2 = 10 485 797 606 846 069 + 0;
  • 10 485 797 606 846 069 ÷ 2 = 5 242 898 803 423 034 + 1;
  • 5 242 898 803 423 034 ÷ 2 = 2 621 449 401 711 517 + 0;
  • 2 621 449 401 711 517 ÷ 2 = 1 310 724 700 855 758 + 1;
  • 1 310 724 700 855 758 ÷ 2 = 655 362 350 427 879 + 0;
  • 655 362 350 427 879 ÷ 2 = 327 681 175 213 939 + 1;
  • 327 681 175 213 939 ÷ 2 = 163 840 587 606 969 + 1;
  • 163 840 587 606 969 ÷ 2 = 81 920 293 803 484 + 1;
  • 81 920 293 803 484 ÷ 2 = 40 960 146 901 742 + 0;
  • 40 960 146 901 742 ÷ 2 = 20 480 073 450 871 + 0;
  • 20 480 073 450 871 ÷ 2 = 10 240 036 725 435 + 1;
  • 10 240 036 725 435 ÷ 2 = 5 120 018 362 717 + 1;
  • 5 120 018 362 717 ÷ 2 = 2 560 009 181 358 + 1;
  • 2 560 009 181 358 ÷ 2 = 1 280 004 590 679 + 0;
  • 1 280 004 590 679 ÷ 2 = 640 002 295 339 + 1;
  • 640 002 295 339 ÷ 2 = 320 001 147 669 + 1;
  • 320 001 147 669 ÷ 2 = 160 000 573 834 + 1;
  • 160 000 573 834 ÷ 2 = 80 000 286 917 + 0;
  • 80 000 286 917 ÷ 2 = 40 000 143 458 + 1;
  • 40 000 143 458 ÷ 2 = 20 000 071 729 + 0;
  • 20 000 071 729 ÷ 2 = 10 000 035 864 + 1;
  • 10 000 035 864 ÷ 2 = 5 000 017 932 + 0;
  • 5 000 017 932 ÷ 2 = 2 500 008 966 + 0;
  • 2 500 008 966 ÷ 2 = 1 250 004 483 + 0;
  • 1 250 004 483 ÷ 2 = 625 002 241 + 1;
  • 625 002 241 ÷ 2 = 312 501 120 + 1;
  • 312 501 120 ÷ 2 = 156 250 560 + 0;
  • 156 250 560 ÷ 2 = 78 125 280 + 0;
  • 78 125 280 ÷ 2 = 39 062 640 + 0;
  • 39 062 640 ÷ 2 = 19 531 320 + 0;
  • 19 531 320 ÷ 2 = 9 765 660 + 0;
  • 9 765 660 ÷ 2 = 4 882 830 + 0;
  • 4 882 830 ÷ 2 = 2 441 415 + 0;
  • 2 441 415 ÷ 2 = 1 220 707 + 1;
  • 1 220 707 ÷ 2 = 610 353 + 1;
  • 610 353 ÷ 2 = 305 176 + 1;
  • 305 176 ÷ 2 = 152 588 + 0;
  • 152 588 ÷ 2 = 76 294 + 0;
  • 76 294 ÷ 2 = 38 147 + 0;
  • 38 147 ÷ 2 = 19 073 + 1;
  • 19 073 ÷ 2 = 9 536 + 1;
  • 9 536 ÷ 2 = 4 768 + 0;
  • 4 768 ÷ 2 = 2 384 + 0;
  • 2 384 ÷ 2 = 1 192 + 0;
  • 1 192 ÷ 2 = 596 + 0;
  • 596 ÷ 2 = 298 + 0;
  • 298 ÷ 2 = 149 + 0;
  • 149 ÷ 2 = 74 + 1;
  • 74 ÷ 2 = 37 + 0;
  • 37 ÷ 2 = 18 + 1;
  • 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.

167 772 761 709 537 111(10) = 10 0101 0100 0000 1100 0111 0000 0001 1000 1010 1110 1110 0111 0101 0111(2)

3. Determine the signed binary number bit length:

  • The base 2 number's actual length, in bits: 58.

  • 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, 58,

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 167 772 761 709 537 111(10) converted to signed binary in one's complement representation:

167 772 761 709 537 111(10) = 0000 0010 0101 0100 0000 1100 0111 0000 0001 1000 1010 1110 1110 0111 0101 0111

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