Convert -8 157 989 233 041 780 536 to a Signed Binary (Base 2)

How to convert -8 157 989 233 041 780 536(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 -8 157 989 233 041 780 536 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:

|-8 157 989 233 041 780 536| = 8 157 989 233 041 780 536

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;
  • 8 157 989 233 041 780 536 ÷ 2 = 4 078 994 616 520 890 268 + 0;
  • 4 078 994 616 520 890 268 ÷ 2 = 2 039 497 308 260 445 134 + 0;
  • 2 039 497 308 260 445 134 ÷ 2 = 1 019 748 654 130 222 567 + 0;
  • 1 019 748 654 130 222 567 ÷ 2 = 509 874 327 065 111 283 + 1;
  • 509 874 327 065 111 283 ÷ 2 = 254 937 163 532 555 641 + 1;
  • 254 937 163 532 555 641 ÷ 2 = 127 468 581 766 277 820 + 1;
  • 127 468 581 766 277 820 ÷ 2 = 63 734 290 883 138 910 + 0;
  • 63 734 290 883 138 910 ÷ 2 = 31 867 145 441 569 455 + 0;
  • 31 867 145 441 569 455 ÷ 2 = 15 933 572 720 784 727 + 1;
  • 15 933 572 720 784 727 ÷ 2 = 7 966 786 360 392 363 + 1;
  • 7 966 786 360 392 363 ÷ 2 = 3 983 393 180 196 181 + 1;
  • 3 983 393 180 196 181 ÷ 2 = 1 991 696 590 098 090 + 1;
  • 1 991 696 590 098 090 ÷ 2 = 995 848 295 049 045 + 0;
  • 995 848 295 049 045 ÷ 2 = 497 924 147 524 522 + 1;
  • 497 924 147 524 522 ÷ 2 = 248 962 073 762 261 + 0;
  • 248 962 073 762 261 ÷ 2 = 124 481 036 881 130 + 1;
  • 124 481 036 881 130 ÷ 2 = 62 240 518 440 565 + 0;
  • 62 240 518 440 565 ÷ 2 = 31 120 259 220 282 + 1;
  • 31 120 259 220 282 ÷ 2 = 15 560 129 610 141 + 0;
  • 15 560 129 610 141 ÷ 2 = 7 780 064 805 070 + 1;
  • 7 780 064 805 070 ÷ 2 = 3 890 032 402 535 + 0;
  • 3 890 032 402 535 ÷ 2 = 1 945 016 201 267 + 1;
  • 1 945 016 201 267 ÷ 2 = 972 508 100 633 + 1;
  • 972 508 100 633 ÷ 2 = 486 254 050 316 + 1;
  • 486 254 050 316 ÷ 2 = 243 127 025 158 + 0;
  • 243 127 025 158 ÷ 2 = 121 563 512 579 + 0;
  • 121 563 512 579 ÷ 2 = 60 781 756 289 + 1;
  • 60 781 756 289 ÷ 2 = 30 390 878 144 + 1;
  • 30 390 878 144 ÷ 2 = 15 195 439 072 + 0;
  • 15 195 439 072 ÷ 2 = 7 597 719 536 + 0;
  • 7 597 719 536 ÷ 2 = 3 798 859 768 + 0;
  • 3 798 859 768 ÷ 2 = 1 899 429 884 + 0;
  • 1 899 429 884 ÷ 2 = 949 714 942 + 0;
  • 949 714 942 ÷ 2 = 474 857 471 + 0;
  • 474 857 471 ÷ 2 = 237 428 735 + 1;
  • 237 428 735 ÷ 2 = 118 714 367 + 1;
  • 118 714 367 ÷ 2 = 59 357 183 + 1;
  • 59 357 183 ÷ 2 = 29 678 591 + 1;
  • 29 678 591 ÷ 2 = 14 839 295 + 1;
  • 14 839 295 ÷ 2 = 7 419 647 + 1;
  • 7 419 647 ÷ 2 = 3 709 823 + 1;
  • 3 709 823 ÷ 2 = 1 854 911 + 1;
  • 1 854 911 ÷ 2 = 927 455 + 1;
  • 927 455 ÷ 2 = 463 727 + 1;
  • 463 727 ÷ 2 = 231 863 + 1;
  • 231 863 ÷ 2 = 115 931 + 1;
  • 115 931 ÷ 2 = 57 965 + 1;
  • 57 965 ÷ 2 = 28 982 + 1;
  • 28 982 ÷ 2 = 14 491 + 0;
  • 14 491 ÷ 2 = 7 245 + 1;
  • 7 245 ÷ 2 = 3 622 + 1;
  • 3 622 ÷ 2 = 1 811 + 0;
  • 1 811 ÷ 2 = 905 + 1;
  • 905 ÷ 2 = 452 + 1;
  • 452 ÷ 2 = 226 + 0;
  • 226 ÷ 2 = 113 + 0;
  • 113 ÷ 2 = 56 + 1;
  • 56 ÷ 2 = 28 + 0;
  • 28 ÷ 2 = 14 + 0;
  • 14 ÷ 2 = 7 + 0;
  • 7 ÷ 2 = 3 + 1;
  • 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.

8 157 989 233 041 780 536(10) = 111 0001 0011 0110 1111 1111 1111 1100 0000 1100 1110 1010 1010 1111 0011 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:


8 157 989 233 041 780 536(10) = 0111 0001 0011 0110 1111 1111 1111 1100 0000 1100 1110 1010 1010 1111 0011 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...


-8 157 989 233 041 780 536(10) Base 10 integer number converted and written as a signed binary code (in base 2):

-8 157 989 233 041 780 536(10) = 1111 0001 0011 0110 1111 1111 1111 1100 0000 1100 1110 1010 1010 1111 0011 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