Convert 10 101 011 010 559 to a Signed Binary in Two's (2's) Complement Representation

How to convert decimal number 10 101 011 010 559(10) to a signed binary in two's (2's) complement representation

What are the steps to convert decimal number
10 101 011 010 559 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;
  • 10 101 011 010 559 ÷ 2 = 5 050 505 505 279 + 1;
  • 5 050 505 505 279 ÷ 2 = 2 525 252 752 639 + 1;
  • 2 525 252 752 639 ÷ 2 = 1 262 626 376 319 + 1;
  • 1 262 626 376 319 ÷ 2 = 631 313 188 159 + 1;
  • 631 313 188 159 ÷ 2 = 315 656 594 079 + 1;
  • 315 656 594 079 ÷ 2 = 157 828 297 039 + 1;
  • 157 828 297 039 ÷ 2 = 78 914 148 519 + 1;
  • 78 914 148 519 ÷ 2 = 39 457 074 259 + 1;
  • 39 457 074 259 ÷ 2 = 19 728 537 129 + 1;
  • 19 728 537 129 ÷ 2 = 9 864 268 564 + 1;
  • 9 864 268 564 ÷ 2 = 4 932 134 282 + 0;
  • 4 932 134 282 ÷ 2 = 2 466 067 141 + 0;
  • 2 466 067 141 ÷ 2 = 1 233 033 570 + 1;
  • 1 233 033 570 ÷ 2 = 616 516 785 + 0;
  • 616 516 785 ÷ 2 = 308 258 392 + 1;
  • 308 258 392 ÷ 2 = 154 129 196 + 0;
  • 154 129 196 ÷ 2 = 77 064 598 + 0;
  • 77 064 598 ÷ 2 = 38 532 299 + 0;
  • 38 532 299 ÷ 2 = 19 266 149 + 1;
  • 19 266 149 ÷ 2 = 9 633 074 + 1;
  • 9 633 074 ÷ 2 = 4 816 537 + 0;
  • 4 816 537 ÷ 2 = 2 408 268 + 1;
  • 2 408 268 ÷ 2 = 1 204 134 + 0;
  • 1 204 134 ÷ 2 = 602 067 + 0;
  • 602 067 ÷ 2 = 301 033 + 1;
  • 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.

10 101 011 010 559(10) = 1001 0010 1111 1101 0011 0010 1100 0101 0011 1111 1111(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 10 101 011 010 559(10) converted to signed binary in two's complement representation:

10 101 011 010 559(10) = 0000 0000 0000 0000 0000 1001 0010 1111 1101 0011 0010 1100 0101 0011 1111 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