Convert -3 533 601 920 529 063 516 to a Signed Binary in Two's (2's) Complement Representation

How to convert decimal number -3 533 601 920 529 063 516(10) to a signed binary in two's (2's) complement representation

What are the steps to convert decimal number
-3 533 601 920 529 063 516 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:

|-3 533 601 920 529 063 516| = 3 533 601 920 529 063 516

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;
  • 3 533 601 920 529 063 516 ÷ 2 = 1 766 800 960 264 531 758 + 0;
  • 1 766 800 960 264 531 758 ÷ 2 = 883 400 480 132 265 879 + 0;
  • 883 400 480 132 265 879 ÷ 2 = 441 700 240 066 132 939 + 1;
  • 441 700 240 066 132 939 ÷ 2 = 220 850 120 033 066 469 + 1;
  • 220 850 120 033 066 469 ÷ 2 = 110 425 060 016 533 234 + 1;
  • 110 425 060 016 533 234 ÷ 2 = 55 212 530 008 266 617 + 0;
  • 55 212 530 008 266 617 ÷ 2 = 27 606 265 004 133 308 + 1;
  • 27 606 265 004 133 308 ÷ 2 = 13 803 132 502 066 654 + 0;
  • 13 803 132 502 066 654 ÷ 2 = 6 901 566 251 033 327 + 0;
  • 6 901 566 251 033 327 ÷ 2 = 3 450 783 125 516 663 + 1;
  • 3 450 783 125 516 663 ÷ 2 = 1 725 391 562 758 331 + 1;
  • 1 725 391 562 758 331 ÷ 2 = 862 695 781 379 165 + 1;
  • 862 695 781 379 165 ÷ 2 = 431 347 890 689 582 + 1;
  • 431 347 890 689 582 ÷ 2 = 215 673 945 344 791 + 0;
  • 215 673 945 344 791 ÷ 2 = 107 836 972 672 395 + 1;
  • 107 836 972 672 395 ÷ 2 = 53 918 486 336 197 + 1;
  • 53 918 486 336 197 ÷ 2 = 26 959 243 168 098 + 1;
  • 26 959 243 168 098 ÷ 2 = 13 479 621 584 049 + 0;
  • 13 479 621 584 049 ÷ 2 = 6 739 810 792 024 + 1;
  • 6 739 810 792 024 ÷ 2 = 3 369 905 396 012 + 0;
  • 3 369 905 396 012 ÷ 2 = 1 684 952 698 006 + 0;
  • 1 684 952 698 006 ÷ 2 = 842 476 349 003 + 0;
  • 842 476 349 003 ÷ 2 = 421 238 174 501 + 1;
  • 421 238 174 501 ÷ 2 = 210 619 087 250 + 1;
  • 210 619 087 250 ÷ 2 = 105 309 543 625 + 0;
  • 105 309 543 625 ÷ 2 = 52 654 771 812 + 1;
  • 52 654 771 812 ÷ 2 = 26 327 385 906 + 0;
  • 26 327 385 906 ÷ 2 = 13 163 692 953 + 0;
  • 13 163 692 953 ÷ 2 = 6 581 846 476 + 1;
  • 6 581 846 476 ÷ 2 = 3 290 923 238 + 0;
  • 3 290 923 238 ÷ 2 = 1 645 461 619 + 0;
  • 1 645 461 619 ÷ 2 = 822 730 809 + 1;
  • 822 730 809 ÷ 2 = 411 365 404 + 1;
  • 411 365 404 ÷ 2 = 205 682 702 + 0;
  • 205 682 702 ÷ 2 = 102 841 351 + 0;
  • 102 841 351 ÷ 2 = 51 420 675 + 1;
  • 51 420 675 ÷ 2 = 25 710 337 + 1;
  • 25 710 337 ÷ 2 = 12 855 168 + 1;
  • 12 855 168 ÷ 2 = 6 427 584 + 0;
  • 6 427 584 ÷ 2 = 3 213 792 + 0;
  • 3 213 792 ÷ 2 = 1 606 896 + 0;
  • 1 606 896 ÷ 2 = 803 448 + 0;
  • 803 448 ÷ 2 = 401 724 + 0;
  • 401 724 ÷ 2 = 200 862 + 0;
  • 200 862 ÷ 2 = 100 431 + 0;
  • 100 431 ÷ 2 = 50 215 + 1;
  • 50 215 ÷ 2 = 25 107 + 1;
  • 25 107 ÷ 2 = 12 553 + 1;
  • 12 553 ÷ 2 = 6 276 + 1;
  • 6 276 ÷ 2 = 3 138 + 0;
  • 3 138 ÷ 2 = 1 569 + 0;
  • 1 569 ÷ 2 = 784 + 1;
  • 784 ÷ 2 = 392 + 0;
  • 392 ÷ 2 = 196 + 0;
  • 196 ÷ 2 = 98 + 0;
  • 98 ÷ 2 = 49 + 0;
  • 49 ÷ 2 = 24 + 1;
  • 24 ÷ 2 = 12 + 0;
  • 12 ÷ 2 = 6 + 0;
  • 6 ÷ 2 = 3 + 0;
  • 3 ÷ 2 = 1 + 1;
  • 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.

3 533 601 920 529 063 516(10) = 11 0001 0000 1001 1110 0000 0011 1001 1001 0010 1100 0101 1101 1110 0101 1100(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.


3 533 601 920 529 063 516(10) = 0011 0001 0000 1001 1110 0000 0011 1001 1001 0010 1100 0101 1101 1110 0101 1100

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.

!(0011 0001 0000 1001 1110 0000 0011 1001 1001 0010 1100 0101 1101 1110 0101 1100)

= 1100 1110 1111 0110 0001 1111 1100 0110 0110 1101 0011 1010 0010 0001 1010 0011


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 1100 1110 1111 0110 0001 1111 1100 0110 0110 1101 0011 1010 0010 0001 1010 0011 (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):

-3 533 601 920 529 063 516 =

1100 1110 1111 0110 0001 1111 1100 0110 0110 1101 0011 1010 0010 0001 1010 0011 + 1


Decimal Number -3 533 601 920 529 063 516(10) converted to signed binary in two's complement representation:

-3 533 601 920 529 063 516(10) = 1100 1110 1111 0110 0001 1111 1100 0110 0110 1101 0011 1010 0010 0001 1010 0100

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