Convert 5 646 473 765 376 359 004 to a Signed Binary in Two's (2's) Complement Representation

How to convert decimal number 5 646 473 765 376 359 004(10) to a signed binary in two's (2's) complement representation

What are the steps to convert decimal number
5 646 473 765 376 359 004 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;
  • 5 646 473 765 376 359 004 ÷ 2 = 2 823 236 882 688 179 502 + 0;
  • 2 823 236 882 688 179 502 ÷ 2 = 1 411 618 441 344 089 751 + 0;
  • 1 411 618 441 344 089 751 ÷ 2 = 705 809 220 672 044 875 + 1;
  • 705 809 220 672 044 875 ÷ 2 = 352 904 610 336 022 437 + 1;
  • 352 904 610 336 022 437 ÷ 2 = 176 452 305 168 011 218 + 1;
  • 176 452 305 168 011 218 ÷ 2 = 88 226 152 584 005 609 + 0;
  • 88 226 152 584 005 609 ÷ 2 = 44 113 076 292 002 804 + 1;
  • 44 113 076 292 002 804 ÷ 2 = 22 056 538 146 001 402 + 0;
  • 22 056 538 146 001 402 ÷ 2 = 11 028 269 073 000 701 + 0;
  • 11 028 269 073 000 701 ÷ 2 = 5 514 134 536 500 350 + 1;
  • 5 514 134 536 500 350 ÷ 2 = 2 757 067 268 250 175 + 0;
  • 2 757 067 268 250 175 ÷ 2 = 1 378 533 634 125 087 + 1;
  • 1 378 533 634 125 087 ÷ 2 = 689 266 817 062 543 + 1;
  • 689 266 817 062 543 ÷ 2 = 344 633 408 531 271 + 1;
  • 344 633 408 531 271 ÷ 2 = 172 316 704 265 635 + 1;
  • 172 316 704 265 635 ÷ 2 = 86 158 352 132 817 + 1;
  • 86 158 352 132 817 ÷ 2 = 43 079 176 066 408 + 1;
  • 43 079 176 066 408 ÷ 2 = 21 539 588 033 204 + 0;
  • 21 539 588 033 204 ÷ 2 = 10 769 794 016 602 + 0;
  • 10 769 794 016 602 ÷ 2 = 5 384 897 008 301 + 0;
  • 5 384 897 008 301 ÷ 2 = 2 692 448 504 150 + 1;
  • 2 692 448 504 150 ÷ 2 = 1 346 224 252 075 + 0;
  • 1 346 224 252 075 ÷ 2 = 673 112 126 037 + 1;
  • 673 112 126 037 ÷ 2 = 336 556 063 018 + 1;
  • 336 556 063 018 ÷ 2 = 168 278 031 509 + 0;
  • 168 278 031 509 ÷ 2 = 84 139 015 754 + 1;
  • 84 139 015 754 ÷ 2 = 42 069 507 877 + 0;
  • 42 069 507 877 ÷ 2 = 21 034 753 938 + 1;
  • 21 034 753 938 ÷ 2 = 10 517 376 969 + 0;
  • 10 517 376 969 ÷ 2 = 5 258 688 484 + 1;
  • 5 258 688 484 ÷ 2 = 2 629 344 242 + 0;
  • 2 629 344 242 ÷ 2 = 1 314 672 121 + 0;
  • 1 314 672 121 ÷ 2 = 657 336 060 + 1;
  • 657 336 060 ÷ 2 = 328 668 030 + 0;
  • 328 668 030 ÷ 2 = 164 334 015 + 0;
  • 164 334 015 ÷ 2 = 82 167 007 + 1;
  • 82 167 007 ÷ 2 = 41 083 503 + 1;
  • 41 083 503 ÷ 2 = 20 541 751 + 1;
  • 20 541 751 ÷ 2 = 10 270 875 + 1;
  • 10 270 875 ÷ 2 = 5 135 437 + 1;
  • 5 135 437 ÷ 2 = 2 567 718 + 1;
  • 2 567 718 ÷ 2 = 1 283 859 + 0;
  • 1 283 859 ÷ 2 = 641 929 + 1;
  • 641 929 ÷ 2 = 320 964 + 1;
  • 320 964 ÷ 2 = 160 482 + 0;
  • 160 482 ÷ 2 = 80 241 + 0;
  • 80 241 ÷ 2 = 40 120 + 1;
  • 40 120 ÷ 2 = 20 060 + 0;
  • 20 060 ÷ 2 = 10 030 + 0;
  • 10 030 ÷ 2 = 5 015 + 0;
  • 5 015 ÷ 2 = 2 507 + 1;
  • 2 507 ÷ 2 = 1 253 + 1;
  • 1 253 ÷ 2 = 626 + 1;
  • 626 ÷ 2 = 313 + 0;
  • 313 ÷ 2 = 156 + 1;
  • 156 ÷ 2 = 78 + 0;
  • 78 ÷ 2 = 39 + 0;
  • 39 ÷ 2 = 19 + 1;
  • 19 ÷ 2 = 9 + 1;
  • 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.

5 646 473 765 376 359 004(10) = 100 1110 0101 1100 0100 1101 1111 1001 0010 1010 1101 0001 1111 1010 0101 1100(2)

3. Determine the signed binary number bit length:

  • The base 2 number's actual length, in bits: 63.

  • 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, 63,

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 5 646 473 765 376 359 004(10) converted to signed binary in two's complement representation:

5 646 473 765 376 359 004(10) = 0100 1110 0101 1100 0100 1101 1111 1001 0010 1010 1101 0001 1111 1010 0101 1100

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