Convert 10 101 001 555 558 to a Signed Binary in Two's (2's) Complement Representation

How to convert decimal number 10 101 001 555 558(10) to a signed binary in two's (2's) complement representation

What are the steps to convert decimal number
10 101 001 555 558 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;
  • 10 101 001 555 558 ÷ 2 = 5 050 500 777 779 + 0;
  • 5 050 500 777 779 ÷ 2 = 2 525 250 388 889 + 1;
  • 2 525 250 388 889 ÷ 2 = 1 262 625 194 444 + 1;
  • 1 262 625 194 444 ÷ 2 = 631 312 597 222 + 0;
  • 631 312 597 222 ÷ 2 = 315 656 298 611 + 0;
  • 315 656 298 611 ÷ 2 = 157 828 149 305 + 1;
  • 157 828 149 305 ÷ 2 = 78 914 074 652 + 1;
  • 78 914 074 652 ÷ 2 = 39 457 037 326 + 0;
  • 39 457 037 326 ÷ 2 = 19 728 518 663 + 0;
  • 19 728 518 663 ÷ 2 = 9 864 259 331 + 1;
  • 9 864 259 331 ÷ 2 = 4 932 129 665 + 1;
  • 4 932 129 665 ÷ 2 = 2 466 064 832 + 1;
  • 2 466 064 832 ÷ 2 = 1 233 032 416 + 0;
  • 1 233 032 416 ÷ 2 = 616 516 208 + 0;
  • 616 516 208 ÷ 2 = 308 258 104 + 0;
  • 308 258 104 ÷ 2 = 154 129 052 + 0;
  • 154 129 052 ÷ 2 = 77 064 526 + 0;
  • 77 064 526 ÷ 2 = 38 532 263 + 0;
  • 38 532 263 ÷ 2 = 19 266 131 + 1;
  • 19 266 131 ÷ 2 = 9 633 065 + 1;
  • 9 633 065 ÷ 2 = 4 816 532 + 1;
  • 4 816 532 ÷ 2 = 2 408 266 + 0;
  • 2 408 266 ÷ 2 = 1 204 133 + 0;
  • 1 204 133 ÷ 2 = 602 066 + 1;
  • 602 066 ÷ 2 = 301 033 + 0;
  • 301 033 ÷ 2 = 150 516 + 1;
  • 150 516 ÷ 2 = 75 258 + 0;
  • 75 258 ÷ 2 = 37 629 + 0;
  • 37 629 ÷ 2 = 18 814 + 1;
  • 18 814 ÷ 2 = 9 407 + 0;
  • 9 407 ÷ 2 = 4 703 + 1;
  • 4 703 ÷ 2 = 2 351 + 1;
  • 2 351 ÷ 2 = 1 175 + 1;
  • 1 175 ÷ 2 = 587 + 1;
  • 587 ÷ 2 = 293 + 1;
  • 293 ÷ 2 = 146 + 1;
  • 146 ÷ 2 = 73 + 0;
  • 73 ÷ 2 = 36 + 1;
  • 36 ÷ 2 = 18 + 0;
  • 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.

10 101 001 555 558(10) = 1001 0010 1111 1101 0010 1001 1100 0000 1110 0110 0110(2)

3. Determine the signed binary number bit length:

  • The base 2 number's actual length, in bits: 44.

  • 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, 44,

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 10 101 001 555 558(10) converted to signed binary in two's complement representation:

10 101 001 555 558(10) = 0000 0000 0000 0000 0000 1001 0010 1111 1101 0010 1001 1100 0000 1110 0110 0110

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