Two's Complement: Integer ↗ Binary: 3 506 643 727 934 778 987 Convert the Integer Number to a Signed Binary in Two's Complement Representation. Write the Base Ten Decimal System Number as a Binary Code (Written in Base Two)

Signed integer number 3 506 643 727 934 778 987(10) converted and written as a signed binary in two's complement representation (base 2) = ?

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;
  • 3 506 643 727 934 778 987 ÷ 2 = 1 753 321 863 967 389 493 + 1;
  • 1 753 321 863 967 389 493 ÷ 2 = 876 660 931 983 694 746 + 1;
  • 876 660 931 983 694 746 ÷ 2 = 438 330 465 991 847 373 + 0;
  • 438 330 465 991 847 373 ÷ 2 = 219 165 232 995 923 686 + 1;
  • 219 165 232 995 923 686 ÷ 2 = 109 582 616 497 961 843 + 0;
  • 109 582 616 497 961 843 ÷ 2 = 54 791 308 248 980 921 + 1;
  • 54 791 308 248 980 921 ÷ 2 = 27 395 654 124 490 460 + 1;
  • 27 395 654 124 490 460 ÷ 2 = 13 697 827 062 245 230 + 0;
  • 13 697 827 062 245 230 ÷ 2 = 6 848 913 531 122 615 + 0;
  • 6 848 913 531 122 615 ÷ 2 = 3 424 456 765 561 307 + 1;
  • 3 424 456 765 561 307 ÷ 2 = 1 712 228 382 780 653 + 1;
  • 1 712 228 382 780 653 ÷ 2 = 856 114 191 390 326 + 1;
  • 856 114 191 390 326 ÷ 2 = 428 057 095 695 163 + 0;
  • 428 057 095 695 163 ÷ 2 = 214 028 547 847 581 + 1;
  • 214 028 547 847 581 ÷ 2 = 107 014 273 923 790 + 1;
  • 107 014 273 923 790 ÷ 2 = 53 507 136 961 895 + 0;
  • 53 507 136 961 895 ÷ 2 = 26 753 568 480 947 + 1;
  • 26 753 568 480 947 ÷ 2 = 13 376 784 240 473 + 1;
  • 13 376 784 240 473 ÷ 2 = 6 688 392 120 236 + 1;
  • 6 688 392 120 236 ÷ 2 = 3 344 196 060 118 + 0;
  • 3 344 196 060 118 ÷ 2 = 1 672 098 030 059 + 0;
  • 1 672 098 030 059 ÷ 2 = 836 049 015 029 + 1;
  • 836 049 015 029 ÷ 2 = 418 024 507 514 + 1;
  • 418 024 507 514 ÷ 2 = 209 012 253 757 + 0;
  • 209 012 253 757 ÷ 2 = 104 506 126 878 + 1;
  • 104 506 126 878 ÷ 2 = 52 253 063 439 + 0;
  • 52 253 063 439 ÷ 2 = 26 126 531 719 + 1;
  • 26 126 531 719 ÷ 2 = 13 063 265 859 + 1;
  • 13 063 265 859 ÷ 2 = 6 531 632 929 + 1;
  • 6 531 632 929 ÷ 2 = 3 265 816 464 + 1;
  • 3 265 816 464 ÷ 2 = 1 632 908 232 + 0;
  • 1 632 908 232 ÷ 2 = 816 454 116 + 0;
  • 816 454 116 ÷ 2 = 408 227 058 + 0;
  • 408 227 058 ÷ 2 = 204 113 529 + 0;
  • 204 113 529 ÷ 2 = 102 056 764 + 1;
  • 102 056 764 ÷ 2 = 51 028 382 + 0;
  • 51 028 382 ÷ 2 = 25 514 191 + 0;
  • 25 514 191 ÷ 2 = 12 757 095 + 1;
  • 12 757 095 ÷ 2 = 6 378 547 + 1;
  • 6 378 547 ÷ 2 = 3 189 273 + 1;
  • 3 189 273 ÷ 2 = 1 594 636 + 1;
  • 1 594 636 ÷ 2 = 797 318 + 0;
  • 797 318 ÷ 2 = 398 659 + 0;
  • 398 659 ÷ 2 = 199 329 + 1;
  • 199 329 ÷ 2 = 99 664 + 1;
  • 99 664 ÷ 2 = 49 832 + 0;
  • 49 832 ÷ 2 = 24 916 + 0;
  • 24 916 ÷ 2 = 12 458 + 0;
  • 12 458 ÷ 2 = 6 229 + 0;
  • 6 229 ÷ 2 = 3 114 + 1;
  • 3 114 ÷ 2 = 1 557 + 0;
  • 1 557 ÷ 2 = 778 + 1;
  • 778 ÷ 2 = 389 + 0;
  • 389 ÷ 2 = 194 + 1;
  • 194 ÷ 2 = 97 + 0;
  • 97 ÷ 2 = 48 + 1;
  • 48 ÷ 2 = 24 + 0;
  • 24 ÷ 2 = 12 + 0;
  • 12 ÷ 2 = 6 + 0;
  • 6 ÷ 2 = 3 + 0;
  • 3 ÷ 2 = 1 + 1;
  • 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.


3 506 643 727 934 778 987(10) = 11 0000 1010 1010 0001 1001 1110 0100 0011 1101 0110 0111 0110 1110 0110 1011(2)


3. 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.


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.


Number 3 506 643 727 934 778 987(10), a signed integer number (with sign), converted from decimal system (from base 10) and written as a signed binary in two's complement representation:

3 506 643 727 934 778 987(10) = 0011 0000 1010 1010 0001 1001 1110 0100 0011 1101 0110 0111 0110 1110 0110 1011

Spaces were used to group digits: for binary, by 4, for decimal, by 3.

More operations on integers written as signed binary numbers in two's complement representation:

» Signed integer number 3 506 643 727 934 778 986 converted from decimal system (from base 10) and written as a signed binary in two's complement representation (written in base 2) = ?

» Signed integer number 3 506 643 727 934 778 988 converted from decimal system (from base 10) and written as a signed binary in two's complement representation (written in base 2) = ?

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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