Convert -9 223 371 976 725 203 141 to a Signed Binary (Base 2)

How to convert -9 223 371 976 725 203 141(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 -9 223 371 976 725 203 141 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:

|-9 223 371 976 725 203 141| = 9 223 371 976 725 203 141

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;
  • 9 223 371 976 725 203 141 ÷ 2 = 4 611 685 988 362 601 570 + 1;
  • 4 611 685 988 362 601 570 ÷ 2 = 2 305 842 994 181 300 785 + 0;
  • 2 305 842 994 181 300 785 ÷ 2 = 1 152 921 497 090 650 392 + 1;
  • 1 152 921 497 090 650 392 ÷ 2 = 576 460 748 545 325 196 + 0;
  • 576 460 748 545 325 196 ÷ 2 = 288 230 374 272 662 598 + 0;
  • 288 230 374 272 662 598 ÷ 2 = 144 115 187 136 331 299 + 0;
  • 144 115 187 136 331 299 ÷ 2 = 72 057 593 568 165 649 + 1;
  • 72 057 593 568 165 649 ÷ 2 = 36 028 796 784 082 824 + 1;
  • 36 028 796 784 082 824 ÷ 2 = 18 014 398 392 041 412 + 0;
  • 18 014 398 392 041 412 ÷ 2 = 9 007 199 196 020 706 + 0;
  • 9 007 199 196 020 706 ÷ 2 = 4 503 599 598 010 353 + 0;
  • 4 503 599 598 010 353 ÷ 2 = 2 251 799 799 005 176 + 1;
  • 2 251 799 799 005 176 ÷ 2 = 1 125 899 899 502 588 + 0;
  • 1 125 899 899 502 588 ÷ 2 = 562 949 949 751 294 + 0;
  • 562 949 949 751 294 ÷ 2 = 281 474 974 875 647 + 0;
  • 281 474 974 875 647 ÷ 2 = 140 737 487 437 823 + 1;
  • 140 737 487 437 823 ÷ 2 = 70 368 743 718 911 + 1;
  • 70 368 743 718 911 ÷ 2 = 35 184 371 859 455 + 1;
  • 35 184 371 859 455 ÷ 2 = 17 592 185 929 727 + 1;
  • 17 592 185 929 727 ÷ 2 = 8 796 092 964 863 + 1;
  • 8 796 092 964 863 ÷ 2 = 4 398 046 482 431 + 1;
  • 4 398 046 482 431 ÷ 2 = 2 199 023 241 215 + 1;
  • 2 199 023 241 215 ÷ 2 = 1 099 511 620 607 + 1;
  • 1 099 511 620 607 ÷ 2 = 549 755 810 303 + 1;
  • 549 755 810 303 ÷ 2 = 274 877 905 151 + 1;
  • 274 877 905 151 ÷ 2 = 137 438 952 575 + 1;
  • 137 438 952 575 ÷ 2 = 68 719 476 287 + 1;
  • 68 719 476 287 ÷ 2 = 34 359 738 143 + 1;
  • 34 359 738 143 ÷ 2 = 17 179 869 071 + 1;
  • 17 179 869 071 ÷ 2 = 8 589 934 535 + 1;
  • 8 589 934 535 ÷ 2 = 4 294 967 267 + 1;
  • 4 294 967 267 ÷ 2 = 2 147 483 633 + 1;
  • 2 147 483 633 ÷ 2 = 1 073 741 816 + 1;
  • 1 073 741 816 ÷ 2 = 536 870 908 + 0;
  • 536 870 908 ÷ 2 = 268 435 454 + 0;
  • 268 435 454 ÷ 2 = 134 217 727 + 0;
  • 134 217 727 ÷ 2 = 67 108 863 + 1;
  • 67 108 863 ÷ 2 = 33 554 431 + 1;
  • 33 554 431 ÷ 2 = 16 777 215 + 1;
  • 16 777 215 ÷ 2 = 8 388 607 + 1;
  • 8 388 607 ÷ 2 = 4 194 303 + 1;
  • 4 194 303 ÷ 2 = 2 097 151 + 1;
  • 2 097 151 ÷ 2 = 1 048 575 + 1;
  • 1 048 575 ÷ 2 = 524 287 + 1;
  • 524 287 ÷ 2 = 262 143 + 1;
  • 262 143 ÷ 2 = 131 071 + 1;
  • 131 071 ÷ 2 = 65 535 + 1;
  • 65 535 ÷ 2 = 32 767 + 1;
  • 32 767 ÷ 2 = 16 383 + 1;
  • 16 383 ÷ 2 = 8 191 + 1;
  • 8 191 ÷ 2 = 4 095 + 1;
  • 4 095 ÷ 2 = 2 047 + 1;
  • 2 047 ÷ 2 = 1 023 + 1;
  • 1 023 ÷ 2 = 511 + 1;
  • 511 ÷ 2 = 255 + 1;
  • 255 ÷ 2 = 127 + 1;
  • 127 ÷ 2 = 63 + 1;
  • 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:

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

9 223 371 976 725 203 141(10) = 111 1111 1111 1111 1111 1111 1111 0001 1111 1111 1111 1111 1000 1000 1100 0101(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:


9 223 371 976 725 203 141(10) = 0111 1111 1111 1111 1111 1111 1111 0001 1111 1111 1111 1111 1000 1000 1100 0101

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...


-9 223 371 976 725 203 141(10) Base 10 integer number converted and written as a signed binary code (in base 2):

-9 223 371 976 725 203 141(10) = 1111 1111 1111 1111 1111 1111 1111 0001 1111 1111 1111 1111 1000 1000 1100 0101

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