Convert 110 111 101 101 087 to a Signed Binary in Two's (2's) Complement Representation

How to convert decimal number 110 111 101 101 087(10) to a signed binary in two's (2's) complement representation

What are the steps to convert decimal number
110 111 101 101 087 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;
  • 110 111 101 101 087 ÷ 2 = 55 055 550 550 543 + 1;
  • 55 055 550 550 543 ÷ 2 = 27 527 775 275 271 + 1;
  • 27 527 775 275 271 ÷ 2 = 13 763 887 637 635 + 1;
  • 13 763 887 637 635 ÷ 2 = 6 881 943 818 817 + 1;
  • 6 881 943 818 817 ÷ 2 = 3 440 971 909 408 + 1;
  • 3 440 971 909 408 ÷ 2 = 1 720 485 954 704 + 0;
  • 1 720 485 954 704 ÷ 2 = 860 242 977 352 + 0;
  • 860 242 977 352 ÷ 2 = 430 121 488 676 + 0;
  • 430 121 488 676 ÷ 2 = 215 060 744 338 + 0;
  • 215 060 744 338 ÷ 2 = 107 530 372 169 + 0;
  • 107 530 372 169 ÷ 2 = 53 765 186 084 + 1;
  • 53 765 186 084 ÷ 2 = 26 882 593 042 + 0;
  • 26 882 593 042 ÷ 2 = 13 441 296 521 + 0;
  • 13 441 296 521 ÷ 2 = 6 720 648 260 + 1;
  • 6 720 648 260 ÷ 2 = 3 360 324 130 + 0;
  • 3 360 324 130 ÷ 2 = 1 680 162 065 + 0;
  • 1 680 162 065 ÷ 2 = 840 081 032 + 1;
  • 840 081 032 ÷ 2 = 420 040 516 + 0;
  • 420 040 516 ÷ 2 = 210 020 258 + 0;
  • 210 020 258 ÷ 2 = 105 010 129 + 0;
  • 105 010 129 ÷ 2 = 52 505 064 + 1;
  • 52 505 064 ÷ 2 = 26 252 532 + 0;
  • 26 252 532 ÷ 2 = 13 126 266 + 0;
  • 13 126 266 ÷ 2 = 6 563 133 + 0;
  • 6 563 133 ÷ 2 = 3 281 566 + 1;
  • 3 281 566 ÷ 2 = 1 640 783 + 0;
  • 1 640 783 ÷ 2 = 820 391 + 1;
  • 820 391 ÷ 2 = 410 195 + 1;
  • 410 195 ÷ 2 = 205 097 + 1;
  • 205 097 ÷ 2 = 102 548 + 1;
  • 102 548 ÷ 2 = 51 274 + 0;
  • 51 274 ÷ 2 = 25 637 + 0;
  • 25 637 ÷ 2 = 12 818 + 1;
  • 12 818 ÷ 2 = 6 409 + 0;
  • 6 409 ÷ 2 = 3 204 + 1;
  • 3 204 ÷ 2 = 1 602 + 0;
  • 1 602 ÷ 2 = 801 + 0;
  • 801 ÷ 2 = 400 + 1;
  • 400 ÷ 2 = 200 + 0;
  • 200 ÷ 2 = 100 + 0;
  • 100 ÷ 2 = 50 + 0;
  • 50 ÷ 2 = 25 + 0;
  • 25 ÷ 2 = 12 + 1;
  • 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.

110 111 101 101 087(10) = 110 0100 0010 0101 0011 1101 0001 0001 0010 0100 0001 1111(2)

3. Determine the signed binary number bit length:

  • The base 2 number's actual length, in bits: 47.

  • 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, 47,

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 110 111 101 101 087(10) converted to signed binary in two's complement representation:

110 111 101 101 087(10) = 0000 0000 0000 0000 0110 0100 0010 0101 0011 1101 0001 0001 0010 0100 0001 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 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