Convert -2 724 918 814 159 767 302 to a Signed Binary in Two's (2's) Complement Representation

How to convert decimal number -2 724 918 814 159 767 302(10) to a signed binary in two's (2's) complement representation

What are the steps to convert decimal number
-2 724 918 814 159 767 302 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. Start with the positive version of the number:

|-2 724 918 814 159 767 302| = 2 724 918 814 159 767 302

2. 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;
  • 2 724 918 814 159 767 302 ÷ 2 = 1 362 459 407 079 883 651 + 0;
  • 1 362 459 407 079 883 651 ÷ 2 = 681 229 703 539 941 825 + 1;
  • 681 229 703 539 941 825 ÷ 2 = 340 614 851 769 970 912 + 1;
  • 340 614 851 769 970 912 ÷ 2 = 170 307 425 884 985 456 + 0;
  • 170 307 425 884 985 456 ÷ 2 = 85 153 712 942 492 728 + 0;
  • 85 153 712 942 492 728 ÷ 2 = 42 576 856 471 246 364 + 0;
  • 42 576 856 471 246 364 ÷ 2 = 21 288 428 235 623 182 + 0;
  • 21 288 428 235 623 182 ÷ 2 = 10 644 214 117 811 591 + 0;
  • 10 644 214 117 811 591 ÷ 2 = 5 322 107 058 905 795 + 1;
  • 5 322 107 058 905 795 ÷ 2 = 2 661 053 529 452 897 + 1;
  • 2 661 053 529 452 897 ÷ 2 = 1 330 526 764 726 448 + 1;
  • 1 330 526 764 726 448 ÷ 2 = 665 263 382 363 224 + 0;
  • 665 263 382 363 224 ÷ 2 = 332 631 691 181 612 + 0;
  • 332 631 691 181 612 ÷ 2 = 166 315 845 590 806 + 0;
  • 166 315 845 590 806 ÷ 2 = 83 157 922 795 403 + 0;
  • 83 157 922 795 403 ÷ 2 = 41 578 961 397 701 + 1;
  • 41 578 961 397 701 ÷ 2 = 20 789 480 698 850 + 1;
  • 20 789 480 698 850 ÷ 2 = 10 394 740 349 425 + 0;
  • 10 394 740 349 425 ÷ 2 = 5 197 370 174 712 + 1;
  • 5 197 370 174 712 ÷ 2 = 2 598 685 087 356 + 0;
  • 2 598 685 087 356 ÷ 2 = 1 299 342 543 678 + 0;
  • 1 299 342 543 678 ÷ 2 = 649 671 271 839 + 0;
  • 649 671 271 839 ÷ 2 = 324 835 635 919 + 1;
  • 324 835 635 919 ÷ 2 = 162 417 817 959 + 1;
  • 162 417 817 959 ÷ 2 = 81 208 908 979 + 1;
  • 81 208 908 979 ÷ 2 = 40 604 454 489 + 1;
  • 40 604 454 489 ÷ 2 = 20 302 227 244 + 1;
  • 20 302 227 244 ÷ 2 = 10 151 113 622 + 0;
  • 10 151 113 622 ÷ 2 = 5 075 556 811 + 0;
  • 5 075 556 811 ÷ 2 = 2 537 778 405 + 1;
  • 2 537 778 405 ÷ 2 = 1 268 889 202 + 1;
  • 1 268 889 202 ÷ 2 = 634 444 601 + 0;
  • 634 444 601 ÷ 2 = 317 222 300 + 1;
  • 317 222 300 ÷ 2 = 158 611 150 + 0;
  • 158 611 150 ÷ 2 = 79 305 575 + 0;
  • 79 305 575 ÷ 2 = 39 652 787 + 1;
  • 39 652 787 ÷ 2 = 19 826 393 + 1;
  • 19 826 393 ÷ 2 = 9 913 196 + 1;
  • 9 913 196 ÷ 2 = 4 956 598 + 0;
  • 4 956 598 ÷ 2 = 2 478 299 + 0;
  • 2 478 299 ÷ 2 = 1 239 149 + 1;
  • 1 239 149 ÷ 2 = 619 574 + 1;
  • 619 574 ÷ 2 = 309 787 + 0;
  • 309 787 ÷ 2 = 154 893 + 1;
  • 154 893 ÷ 2 = 77 446 + 1;
  • 77 446 ÷ 2 = 38 723 + 0;
  • 38 723 ÷ 2 = 19 361 + 1;
  • 19 361 ÷ 2 = 9 680 + 1;
  • 9 680 ÷ 2 = 4 840 + 0;
  • 4 840 ÷ 2 = 2 420 + 0;
  • 2 420 ÷ 2 = 1 210 + 0;
  • 1 210 ÷ 2 = 605 + 0;
  • 605 ÷ 2 = 302 + 1;
  • 302 ÷ 2 = 151 + 0;
  • 151 ÷ 2 = 75 + 1;
  • 75 ÷ 2 = 37 + 1;
  • 37 ÷ 2 = 18 + 1;
  • 18 ÷ 2 = 9 + 0;
  • 9 ÷ 2 = 4 + 1;
  • 4 ÷ 2 = 2 + 0;
  • 2 ÷ 2 = 1 + 0;
  • 1 ÷ 2 = 0 + 1;

3. Construct the base 2 representation of the positive number:

Take all the remainders starting from the bottom of the list constructed above.

2 724 918 814 159 767 302(10) = 10 0101 1101 0000 1101 1011 0011 1001 0110 0111 1100 0101 1000 0111 0000 0110(2)

4. Determine the signed binary number bit length:

  • The base 2 number's actual length, in bits: 62.

  • 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, 62,

3) so that the first bit (leftmost) could be zero
(we deal with a positive number at this moment)

=== is: 64.


5. 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.


2 724 918 814 159 767 302(10) = 0010 0101 1101 0000 1101 1011 0011 1001 0110 0111 1100 0101 1000 0111 0000 0110

6. Get the negative integer number representation. Part 1:

  • To write the negative integer number on 64 bits (8 Bytes), as a signed binary in one's complement representation, replace all the bits on 0 with 1s and all the bits set on 1 with 0s.

Reverse the digits, flip the digits:

Replace the bits set on 0 with 1s and the bits set on 1 with 0s.

!(0010 0101 1101 0000 1101 1011 0011 1001 0110 0111 1100 0101 1000 0111 0000 0110)

= 1101 1010 0010 1111 0010 0100 1100 0110 1001 1000 0011 1010 0111 1000 1111 1001


7. Get the negative integer number representation. Part 2:

  • To write the negative integer number on 64 bits (8 Bytes), as a signed binary in two's complement representation, add 1 to the number calculated above 1101 1010 0010 1111 0010 0100 1100 0110 1001 1000 0011 1010 0111 1000 1111 1001 (to the signed binary in one's complement representation).

Binary addition carries on a value of 2:

  • 0 + 0 = 0
  • 0 + 1 = 1
  • 1 + 1 = 10
  • 1 + 10 = 11
  • 1 + 11 = 100

Add 1 to the number calculated above
(to the signed binary number in one's complement representation):

-2 724 918 814 159 767 302 =

1101 1010 0010 1111 0010 0100 1100 0110 1001 1000 0011 1010 0111 1000 1111 1001 + 1


Decimal Number -2 724 918 814 159 767 302(10) converted to signed binary in two's complement representation:

-2 724 918 814 159 767 302(10) = 1101 1010 0010 1111 0010 0100 1100 0110 1001 1000 0011 1010 0111 1000 1111 1010

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