Convert -7 777 716 553 456 665 864 to a Signed Binary (Base 2)

How to convert -7 777 716 553 456 665 864(10), a signed base 10 integer number? How to write it as a signed binary code in base 2

What are the required steps to convert base 10 integer
number -7 777 716 553 456 665 864 to signed binary code (in base 2)?

  • 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:

|-7 777 716 553 456 665 864| = 7 777 716 553 456 665 864

2. Divide the number repeatedly by 2:

Keep track of each remainder.

Stop when you get a quotient that is equal to zero.


  • division = quotient + remainder;
  • 7 777 716 553 456 665 864 ÷ 2 = 3 888 858 276 728 332 932 + 0;
  • 3 888 858 276 728 332 932 ÷ 2 = 1 944 429 138 364 166 466 + 0;
  • 1 944 429 138 364 166 466 ÷ 2 = 972 214 569 182 083 233 + 0;
  • 972 214 569 182 083 233 ÷ 2 = 486 107 284 591 041 616 + 1;
  • 486 107 284 591 041 616 ÷ 2 = 243 053 642 295 520 808 + 0;
  • 243 053 642 295 520 808 ÷ 2 = 121 526 821 147 760 404 + 0;
  • 121 526 821 147 760 404 ÷ 2 = 60 763 410 573 880 202 + 0;
  • 60 763 410 573 880 202 ÷ 2 = 30 381 705 286 940 101 + 0;
  • 30 381 705 286 940 101 ÷ 2 = 15 190 852 643 470 050 + 1;
  • 15 190 852 643 470 050 ÷ 2 = 7 595 426 321 735 025 + 0;
  • 7 595 426 321 735 025 ÷ 2 = 3 797 713 160 867 512 + 1;
  • 3 797 713 160 867 512 ÷ 2 = 1 898 856 580 433 756 + 0;
  • 1 898 856 580 433 756 ÷ 2 = 949 428 290 216 878 + 0;
  • 949 428 290 216 878 ÷ 2 = 474 714 145 108 439 + 0;
  • 474 714 145 108 439 ÷ 2 = 237 357 072 554 219 + 1;
  • 237 357 072 554 219 ÷ 2 = 118 678 536 277 109 + 1;
  • 118 678 536 277 109 ÷ 2 = 59 339 268 138 554 + 1;
  • 59 339 268 138 554 ÷ 2 = 29 669 634 069 277 + 0;
  • 29 669 634 069 277 ÷ 2 = 14 834 817 034 638 + 1;
  • 14 834 817 034 638 ÷ 2 = 7 417 408 517 319 + 0;
  • 7 417 408 517 319 ÷ 2 = 3 708 704 258 659 + 1;
  • 3 708 704 258 659 ÷ 2 = 1 854 352 129 329 + 1;
  • 1 854 352 129 329 ÷ 2 = 927 176 064 664 + 1;
  • 927 176 064 664 ÷ 2 = 463 588 032 332 + 0;
  • 463 588 032 332 ÷ 2 = 231 794 016 166 + 0;
  • 231 794 016 166 ÷ 2 = 115 897 008 083 + 0;
  • 115 897 008 083 ÷ 2 = 57 948 504 041 + 1;
  • 57 948 504 041 ÷ 2 = 28 974 252 020 + 1;
  • 28 974 252 020 ÷ 2 = 14 487 126 010 + 0;
  • 14 487 126 010 ÷ 2 = 7 243 563 005 + 0;
  • 7 243 563 005 ÷ 2 = 3 621 781 502 + 1;
  • 3 621 781 502 ÷ 2 = 1 810 890 751 + 0;
  • 1 810 890 751 ÷ 2 = 905 445 375 + 1;
  • 905 445 375 ÷ 2 = 452 722 687 + 1;
  • 452 722 687 ÷ 2 = 226 361 343 + 1;
  • 226 361 343 ÷ 2 = 113 180 671 + 1;
  • 113 180 671 ÷ 2 = 56 590 335 + 1;
  • 56 590 335 ÷ 2 = 28 295 167 + 1;
  • 28 295 167 ÷ 2 = 14 147 583 + 1;
  • 14 147 583 ÷ 2 = 7 073 791 + 1;
  • 7 073 791 ÷ 2 = 3 536 895 + 1;
  • 3 536 895 ÷ 2 = 1 768 447 + 1;
  • 1 768 447 ÷ 2 = 884 223 + 1;
  • 884 223 ÷ 2 = 442 111 + 1;
  • 442 111 ÷ 2 = 221 055 + 1;
  • 221 055 ÷ 2 = 110 527 + 1;
  • 110 527 ÷ 2 = 55 263 + 1;
  • 55 263 ÷ 2 = 27 631 + 1;
  • 27 631 ÷ 2 = 13 815 + 1;
  • 13 815 ÷ 2 = 6 907 + 1;
  • 6 907 ÷ 2 = 3 453 + 1;
  • 3 453 ÷ 2 = 1 726 + 1;
  • 1 726 ÷ 2 = 863 + 0;
  • 863 ÷ 2 = 431 + 1;
  • 431 ÷ 2 = 215 + 1;
  • 215 ÷ 2 = 107 + 1;
  • 107 ÷ 2 = 53 + 1;
  • 53 ÷ 2 = 26 + 1;
  • 26 ÷ 2 = 13 + 0;
  • 13 ÷ 2 = 6 + 1;
  • 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.

7 777 716 553 456 665 864(10) = 110 1011 1110 1111 1111 1111 1111 1111 0100 1100 0111 0101 1100 0101 0000 1000(2)


4. 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) is reserved for 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.


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:


7 777 716 553 456 665 864(10) = 0110 1011 1110 1111 1111 1111 1111 1111 0100 1100 0111 0101 1100 0101 0000 1000

6. Get the negative integer number representation:

To get the negative integer number representation on 64 bits (8 Bytes),

... change the first bit (the leftmost), from 0 to 1...


-7 777 716 553 456 665 864(10) Base 10 integer number converted and written as a signed binary code (in base 2):

-7 777 716 553 456 665 864(10) = 1110 1011 1110 1111 1111 1111 1111 1111 0100 1100 0111 0101 1100 0101 0000 1000

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

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How to convert signed base 10 integers in decimal to binary code system

Follow the steps below to convert a signed base ten integer number to signed binary:

  • 1. In a signed binary, first bit (the leftmost) is reserved for sign: 0 = positive integer number, 1 = positive integer number. If the number to be converted is negative, start with its positive version.
  • 2. Divide repeatedly by 2 the positive integer number keeping track of each remainder. STOP when 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 have a length of 4, 8, 16, 32, 64, ... bits (power of 2) - if needed, fill in extra '0' bits in front of the base 2 number (to the left), up to the right length; this way the first bit (the leftmost one) is always '0', as for a positive representation.
  • 5. To get the negative reprezentation of the number, simply switch the first bit (the leftmost one), from '0' to '1'.

Example: convert the negative number -63 from decimal system (base ten) to signed binary code system:

  • 1. Start with the positive version of the number: |-63| = 63;
  • 2. Divide repeatedly 63 by 2, keeping track of each remainder, until we get a quotient that is equal to zero:
    • division = quotient + remainder
    • 63 ÷ 2 = 31 + 1
    • 31 ÷ 2 = 15 + 1
    • 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:
    63(10) = 11 1111(2)
  • 4. The actual length of base 2 representation number is 6, so the positive binary computer representation length of the signed binary will take in this case 8 bits (the least power of 2 higher than 6) - add extra '0's in front (to the left), up to the required length; this way the first bit (the leftmost one) is to be '0', as for a positive number:
    63(10) = 0011 1111(2)
  • 5. To get the negative integer number representation simply change the first bit (the leftmost), from '0' to '1':
    -63(10) = 1011 1111
  • Number -63(10), signed integer, converted from decimal system (base 10) to signed binary = 1011 1111