Convert 87 160 001 121 to a Signed Binary in Two's (2's) Complement Representation

How to convert decimal number 87 160 001 121(10) to a signed binary in two's (2's) complement representation

What are the steps to convert decimal number
87 160 001 121 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;
  • 87 160 001 121 ÷ 2 = 43 580 000 560 + 1;
  • 43 580 000 560 ÷ 2 = 21 790 000 280 + 0;
  • 21 790 000 280 ÷ 2 = 10 895 000 140 + 0;
  • 10 895 000 140 ÷ 2 = 5 447 500 070 + 0;
  • 5 447 500 070 ÷ 2 = 2 723 750 035 + 0;
  • 2 723 750 035 ÷ 2 = 1 361 875 017 + 1;
  • 1 361 875 017 ÷ 2 = 680 937 508 + 1;
  • 680 937 508 ÷ 2 = 340 468 754 + 0;
  • 340 468 754 ÷ 2 = 170 234 377 + 0;
  • 170 234 377 ÷ 2 = 85 117 188 + 1;
  • 85 117 188 ÷ 2 = 42 558 594 + 0;
  • 42 558 594 ÷ 2 = 21 279 297 + 0;
  • 21 279 297 ÷ 2 = 10 639 648 + 1;
  • 10 639 648 ÷ 2 = 5 319 824 + 0;
  • 5 319 824 ÷ 2 = 2 659 912 + 0;
  • 2 659 912 ÷ 2 = 1 329 956 + 0;
  • 1 329 956 ÷ 2 = 664 978 + 0;
  • 664 978 ÷ 2 = 332 489 + 0;
  • 332 489 ÷ 2 = 166 244 + 1;
  • 166 244 ÷ 2 = 83 122 + 0;
  • 83 122 ÷ 2 = 41 561 + 0;
  • 41 561 ÷ 2 = 20 780 + 1;
  • 20 780 ÷ 2 = 10 390 + 0;
  • 10 390 ÷ 2 = 5 195 + 0;
  • 5 195 ÷ 2 = 2 597 + 1;
  • 2 597 ÷ 2 = 1 298 + 1;
  • 1 298 ÷ 2 = 649 + 0;
  • 649 ÷ 2 = 324 + 1;
  • 324 ÷ 2 = 162 + 0;
  • 162 ÷ 2 = 81 + 0;
  • 81 ÷ 2 = 40 + 1;
  • 40 ÷ 2 = 20 + 0;
  • 20 ÷ 2 = 10 + 0;
  • 10 ÷ 2 = 5 + 0;
  • 5 ÷ 2 = 2 + 1;
  • 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.

87 160 001 121(10) = 1 0100 0100 1011 0010 0100 0001 0010 0110 0001(2)

3. Determine the signed binary number bit length:

  • The base 2 number's actual length, in bits: 37.

  • 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, 37,

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 87 160 001 121(10) converted to signed binary in two's complement representation:

87 160 001 121(10) = 0000 0000 0000 0000 0000 0000 0001 0100 0100 1011 0010 0100 0001 0010 0110 0001

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