Convert 11 000 011 100 001 031 to a Signed Binary (Base 2)

How to convert 11 000 011 100 001 031(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 11 000 011 100 001 031 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. 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;
  • 11 000 011 100 001 031 ÷ 2 = 5 500 005 550 000 515 + 1;
  • 5 500 005 550 000 515 ÷ 2 = 2 750 002 775 000 257 + 1;
  • 2 750 002 775 000 257 ÷ 2 = 1 375 001 387 500 128 + 1;
  • 1 375 001 387 500 128 ÷ 2 = 687 500 693 750 064 + 0;
  • 687 500 693 750 064 ÷ 2 = 343 750 346 875 032 + 0;
  • 343 750 346 875 032 ÷ 2 = 171 875 173 437 516 + 0;
  • 171 875 173 437 516 ÷ 2 = 85 937 586 718 758 + 0;
  • 85 937 586 718 758 ÷ 2 = 42 968 793 359 379 + 0;
  • 42 968 793 359 379 ÷ 2 = 21 484 396 679 689 + 1;
  • 21 484 396 679 689 ÷ 2 = 10 742 198 339 844 + 1;
  • 10 742 198 339 844 ÷ 2 = 5 371 099 169 922 + 0;
  • 5 371 099 169 922 ÷ 2 = 2 685 549 584 961 + 0;
  • 2 685 549 584 961 ÷ 2 = 1 342 774 792 480 + 1;
  • 1 342 774 792 480 ÷ 2 = 671 387 396 240 + 0;
  • 671 387 396 240 ÷ 2 = 335 693 698 120 + 0;
  • 335 693 698 120 ÷ 2 = 167 846 849 060 + 0;
  • 167 846 849 060 ÷ 2 = 83 923 424 530 + 0;
  • 83 923 424 530 ÷ 2 = 41 961 712 265 + 0;
  • 41 961 712 265 ÷ 2 = 20 980 856 132 + 1;
  • 20 980 856 132 ÷ 2 = 10 490 428 066 + 0;
  • 10 490 428 066 ÷ 2 = 5 245 214 033 + 0;
  • 5 245 214 033 ÷ 2 = 2 622 607 016 + 1;
  • 2 622 607 016 ÷ 2 = 1 311 303 508 + 0;
  • 1 311 303 508 ÷ 2 = 655 651 754 + 0;
  • 655 651 754 ÷ 2 = 327 825 877 + 0;
  • 327 825 877 ÷ 2 = 163 912 938 + 1;
  • 163 912 938 ÷ 2 = 81 956 469 + 0;
  • 81 956 469 ÷ 2 = 40 978 234 + 1;
  • 40 978 234 ÷ 2 = 20 489 117 + 0;
  • 20 489 117 ÷ 2 = 10 244 558 + 1;
  • 10 244 558 ÷ 2 = 5 122 279 + 0;
  • 5 122 279 ÷ 2 = 2 561 139 + 1;
  • 2 561 139 ÷ 2 = 1 280 569 + 1;
  • 1 280 569 ÷ 2 = 640 284 + 1;
  • 640 284 ÷ 2 = 320 142 + 0;
  • 320 142 ÷ 2 = 160 071 + 0;
  • 160 071 ÷ 2 = 80 035 + 1;
  • 80 035 ÷ 2 = 40 017 + 1;
  • 40 017 ÷ 2 = 20 008 + 1;
  • 20 008 ÷ 2 = 10 004 + 0;
  • 10 004 ÷ 2 = 5 002 + 0;
  • 5 002 ÷ 2 = 2 501 + 0;
  • 2 501 ÷ 2 = 1 250 + 1;
  • 1 250 ÷ 2 = 625 + 0;
  • 625 ÷ 2 = 312 + 1;
  • 312 ÷ 2 = 156 + 0;
  • 156 ÷ 2 = 78 + 0;
  • 78 ÷ 2 = 39 + 0;
  • 39 ÷ 2 = 19 + 1;
  • 19 ÷ 2 = 9 + 1;
  • 9 ÷ 2 = 4 + 1;
  • 4 ÷ 2 = 2 + 0;
  • 2 ÷ 2 = 1 + 0;
  • 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.

11 000 011 100 001 031(10) = 10 0111 0001 0100 0111 0011 1010 1010 0010 0100 0001 0011 0000 0111(2)


3. Determine the signed binary number bit length:

  • The base 2 number's actual length, in bits: 54.

  • 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, 54,

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:


11 000 011 100 001 031(10) Base 10 integer number converted and written as a signed binary code (in base 2):

11 000 011 100 001 031(10) = 0000 0000 0010 0111 0001 0100 0111 0011 1010 1010 0010 0100 0001 0011 0000 0111

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