Convert 164 311 266 871 182 to a Signed Binary in Two's (2's) Complement Representation

How to convert decimal number 164 311 266 871 182(10) to a signed binary in two's (2's) complement representation

What are the steps to convert decimal number
164 311 266 871 182 to a signed binary in two's (2'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.

We stop when we get a quotient that is equal to zero.


  • division = quotient + remainder;
  • 164 311 266 871 182 ÷ 2 = 82 155 633 435 591 + 0;
  • 82 155 633 435 591 ÷ 2 = 41 077 816 717 795 + 1;
  • 41 077 816 717 795 ÷ 2 = 20 538 908 358 897 + 1;
  • 20 538 908 358 897 ÷ 2 = 10 269 454 179 448 + 1;
  • 10 269 454 179 448 ÷ 2 = 5 134 727 089 724 + 0;
  • 5 134 727 089 724 ÷ 2 = 2 567 363 544 862 + 0;
  • 2 567 363 544 862 ÷ 2 = 1 283 681 772 431 + 0;
  • 1 283 681 772 431 ÷ 2 = 641 840 886 215 + 1;
  • 641 840 886 215 ÷ 2 = 320 920 443 107 + 1;
  • 320 920 443 107 ÷ 2 = 160 460 221 553 + 1;
  • 160 460 221 553 ÷ 2 = 80 230 110 776 + 1;
  • 80 230 110 776 ÷ 2 = 40 115 055 388 + 0;
  • 40 115 055 388 ÷ 2 = 20 057 527 694 + 0;
  • 20 057 527 694 ÷ 2 = 10 028 763 847 + 0;
  • 10 028 763 847 ÷ 2 = 5 014 381 923 + 1;
  • 5 014 381 923 ÷ 2 = 2 507 190 961 + 1;
  • 2 507 190 961 ÷ 2 = 1 253 595 480 + 1;
  • 1 253 595 480 ÷ 2 = 626 797 740 + 0;
  • 626 797 740 ÷ 2 = 313 398 870 + 0;
  • 313 398 870 ÷ 2 = 156 699 435 + 0;
  • 156 699 435 ÷ 2 = 78 349 717 + 1;
  • 78 349 717 ÷ 2 = 39 174 858 + 1;
  • 39 174 858 ÷ 2 = 19 587 429 + 0;
  • 19 587 429 ÷ 2 = 9 793 714 + 1;
  • 9 793 714 ÷ 2 = 4 896 857 + 0;
  • 4 896 857 ÷ 2 = 2 448 428 + 1;
  • 2 448 428 ÷ 2 = 1 224 214 + 0;
  • 1 224 214 ÷ 2 = 612 107 + 0;
  • 612 107 ÷ 2 = 306 053 + 1;
  • 306 053 ÷ 2 = 153 026 + 1;
  • 153 026 ÷ 2 = 76 513 + 0;
  • 76 513 ÷ 2 = 38 256 + 1;
  • 38 256 ÷ 2 = 19 128 + 0;
  • 19 128 ÷ 2 = 9 564 + 0;
  • 9 564 ÷ 2 = 4 782 + 0;
  • 4 782 ÷ 2 = 2 391 + 0;
  • 2 391 ÷ 2 = 1 195 + 1;
  • 1 195 ÷ 2 = 597 + 1;
  • 597 ÷ 2 = 298 + 1;
  • 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.

164 311 266 871 182(10) = 1001 0101 0111 0000 1011 0010 1011 0001 1100 0111 1000 1110(2)

3. Determine the signed binary number bit length:

  • The base 2 number's actual length, in bits: 48.

  • 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, 48,

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 164 311 266 871 182(10) converted to signed binary in two's complement representation:

164 311 266 871 182(10) = 0000 0000 0000 0000 1001 0101 0111 0000 1011 0010 1011 0001 1100 0111 1000 1110

Spaces were used to group digits: for binary, by 4, for decimal, by 3.

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How to convert signed integers from decimal system to signed binary in two's complement representation

Follow the steps below to convert a signed base 10 integer number to signed binary in two'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, keeping track of each remainder, until we get a quotient that is 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, add extra bits on 0 in front (to the left) of the base 2 number above, up to the required length, so that the first bit (the leftmost) will 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 1s and all 1 bits with 0s (reversing the digits).
  • 6. To get the negative integer number, in signed binary two's complement representation, add 1 to the number above.

Example: convert the negative number -60 from the decimal system (base ten) to signed binary in two's complement:

  • 1. Start with the positive version of the number: |-60| = 60
  • 2. Divide repeatedly 60 by 2, keeping track of each remainder:
    • division = quotient + remainder
    • 60 ÷ 2 = 30 + 0
    • 30 ÷ 2 = 15 + 0
    • 15 ÷ 2 = 7 + 1
    • 7 ÷ 2 = 3 + 1
    • 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:
    60(10) = 11 1100(2)
  • 4. Bit length of base 2 representation number is 6, so the positive binary computer representation of a signed binary will take in this particular case 8 bits (the least power of 2 larger than 6) - add extra 0 digits in front of the base 2 number, up to the required length:
    60(10) = 0011 1100(2)
  • 5. To get the negative integer number representation in signed binary one's complement, replace all the 0 bits with 1s and all 1 bits with 0s (reversing the digits):
    !(0011 1100) = 1100 0011
  • 6. To get the negative integer number, signed binary in two's complement representation, add 1 to the number above:
    -60(10) = 1100 0011 + 1 = 1100 0100
  • Number -60(10), signed integer, converted from decimal system (base 10) to signed binary two's complement representation = 1100 0100