Convert 5 646 473 765 375 591 to a Signed Binary in Two's (2's) Complement Representation

How to convert decimal number 5 646 473 765 375 591(10) to a signed binary in two's (2's) complement representation

What are the steps to convert decimal number
5 646 473 765 375 591 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;
  • 5 646 473 765 375 591 ÷ 2 = 2 823 236 882 687 795 + 1;
  • 2 823 236 882 687 795 ÷ 2 = 1 411 618 441 343 897 + 1;
  • 1 411 618 441 343 897 ÷ 2 = 705 809 220 671 948 + 1;
  • 705 809 220 671 948 ÷ 2 = 352 904 610 335 974 + 0;
  • 352 904 610 335 974 ÷ 2 = 176 452 305 167 987 + 0;
  • 176 452 305 167 987 ÷ 2 = 88 226 152 583 993 + 1;
  • 88 226 152 583 993 ÷ 2 = 44 113 076 291 996 + 1;
  • 44 113 076 291 996 ÷ 2 = 22 056 538 145 998 + 0;
  • 22 056 538 145 998 ÷ 2 = 11 028 269 072 999 + 0;
  • 11 028 269 072 999 ÷ 2 = 5 514 134 536 499 + 1;
  • 5 514 134 536 499 ÷ 2 = 2 757 067 268 249 + 1;
  • 2 757 067 268 249 ÷ 2 = 1 378 533 634 124 + 1;
  • 1 378 533 634 124 ÷ 2 = 689 266 817 062 + 0;
  • 689 266 817 062 ÷ 2 = 344 633 408 531 + 0;
  • 344 633 408 531 ÷ 2 = 172 316 704 265 + 1;
  • 172 316 704 265 ÷ 2 = 86 158 352 132 + 1;
  • 86 158 352 132 ÷ 2 = 43 079 176 066 + 0;
  • 43 079 176 066 ÷ 2 = 21 539 588 033 + 0;
  • 21 539 588 033 ÷ 2 = 10 769 794 016 + 1;
  • 10 769 794 016 ÷ 2 = 5 384 897 008 + 0;
  • 5 384 897 008 ÷ 2 = 2 692 448 504 + 0;
  • 2 692 448 504 ÷ 2 = 1 346 224 252 + 0;
  • 1 346 224 252 ÷ 2 = 673 112 126 + 0;
  • 673 112 126 ÷ 2 = 336 556 063 + 0;
  • 336 556 063 ÷ 2 = 168 278 031 + 1;
  • 168 278 031 ÷ 2 = 84 139 015 + 1;
  • 84 139 015 ÷ 2 = 42 069 507 + 1;
  • 42 069 507 ÷ 2 = 21 034 753 + 1;
  • 21 034 753 ÷ 2 = 10 517 376 + 1;
  • 10 517 376 ÷ 2 = 5 258 688 + 0;
  • 5 258 688 ÷ 2 = 2 629 344 + 0;
  • 2 629 344 ÷ 2 = 1 314 672 + 0;
  • 1 314 672 ÷ 2 = 657 336 + 0;
  • 657 336 ÷ 2 = 328 668 + 0;
  • 328 668 ÷ 2 = 164 334 + 0;
  • 164 334 ÷ 2 = 82 167 + 0;
  • 82 167 ÷ 2 = 41 083 + 1;
  • 41 083 ÷ 2 = 20 541 + 1;
  • 20 541 ÷ 2 = 10 270 + 1;
  • 10 270 ÷ 2 = 5 135 + 0;
  • 5 135 ÷ 2 = 2 567 + 1;
  • 2 567 ÷ 2 = 1 283 + 1;
  • 1 283 ÷ 2 = 641 + 1;
  • 641 ÷ 2 = 320 + 1;
  • 320 ÷ 2 = 160 + 0;
  • 160 ÷ 2 = 80 + 0;
  • 80 ÷ 2 = 40 + 0;
  • 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.

5 646 473 765 375 591(10) = 1 0100 0000 1111 0111 0000 0001 1111 0000 0100 1100 1110 0110 0111(2)

3. Determine the signed binary number bit length:

  • The base 2 number's actual length, in bits: 53.

  • 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, 53,

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 5 646 473 765 375 591(10) converted to signed binary in two's complement representation:

5 646 473 765 375 591(10) = 0000 0000 0001 0100 0000 1111 0111 0000 0001 1111 0000 0100 1100 1110 0110 0111

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