Convert 10 101 010 100 100 804 to a Signed Binary (Base 2)

How to convert 10 101 010 100 100 804(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 10 101 010 100 100 804 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;
  • 10 101 010 100 100 804 ÷ 2 = 5 050 505 050 050 402 + 0;
  • 5 050 505 050 050 402 ÷ 2 = 2 525 252 525 025 201 + 0;
  • 2 525 252 525 025 201 ÷ 2 = 1 262 626 262 512 600 + 1;
  • 1 262 626 262 512 600 ÷ 2 = 631 313 131 256 300 + 0;
  • 631 313 131 256 300 ÷ 2 = 315 656 565 628 150 + 0;
  • 315 656 565 628 150 ÷ 2 = 157 828 282 814 075 + 0;
  • 157 828 282 814 075 ÷ 2 = 78 914 141 407 037 + 1;
  • 78 914 141 407 037 ÷ 2 = 39 457 070 703 518 + 1;
  • 39 457 070 703 518 ÷ 2 = 19 728 535 351 759 + 0;
  • 19 728 535 351 759 ÷ 2 = 9 864 267 675 879 + 1;
  • 9 864 267 675 879 ÷ 2 = 4 932 133 837 939 + 1;
  • 4 932 133 837 939 ÷ 2 = 2 466 066 918 969 + 1;
  • 2 466 066 918 969 ÷ 2 = 1 233 033 459 484 + 1;
  • 1 233 033 459 484 ÷ 2 = 616 516 729 742 + 0;
  • 616 516 729 742 ÷ 2 = 308 258 364 871 + 0;
  • 308 258 364 871 ÷ 2 = 154 129 182 435 + 1;
  • 154 129 182 435 ÷ 2 = 77 064 591 217 + 1;
  • 77 064 591 217 ÷ 2 = 38 532 295 608 + 1;
  • 38 532 295 608 ÷ 2 = 19 266 147 804 + 0;
  • 19 266 147 804 ÷ 2 = 9 633 073 902 + 0;
  • 9 633 073 902 ÷ 2 = 4 816 536 951 + 0;
  • 4 816 536 951 ÷ 2 = 2 408 268 475 + 1;
  • 2 408 268 475 ÷ 2 = 1 204 134 237 + 1;
  • 1 204 134 237 ÷ 2 = 602 067 118 + 1;
  • 602 067 118 ÷ 2 = 301 033 559 + 0;
  • 301 033 559 ÷ 2 = 150 516 779 + 1;
  • 150 516 779 ÷ 2 = 75 258 389 + 1;
  • 75 258 389 ÷ 2 = 37 629 194 + 1;
  • 37 629 194 ÷ 2 = 18 814 597 + 0;
  • 18 814 597 ÷ 2 = 9 407 298 + 1;
  • 9 407 298 ÷ 2 = 4 703 649 + 0;
  • 4 703 649 ÷ 2 = 2 351 824 + 1;
  • 2 351 824 ÷ 2 = 1 175 912 + 0;
  • 1 175 912 ÷ 2 = 587 956 + 0;
  • 587 956 ÷ 2 = 293 978 + 0;
  • 293 978 ÷ 2 = 146 989 + 0;
  • 146 989 ÷ 2 = 73 494 + 1;
  • 73 494 ÷ 2 = 36 747 + 0;
  • 36 747 ÷ 2 = 18 373 + 1;
  • 18 373 ÷ 2 = 9 186 + 1;
  • 9 186 ÷ 2 = 4 593 + 0;
  • 4 593 ÷ 2 = 2 296 + 1;
  • 2 296 ÷ 2 = 1 148 + 0;
  • 1 148 ÷ 2 = 574 + 0;
  • 574 ÷ 2 = 287 + 0;
  • 287 ÷ 2 = 143 + 1;
  • 143 ÷ 2 = 71 + 1;
  • 71 ÷ 2 = 35 + 1;
  • 35 ÷ 2 = 17 + 1;
  • 17 ÷ 2 = 8 + 1;
  • 8 ÷ 2 = 4 + 0;
  • 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.

10 101 010 100 100 804(10) = 10 0011 1110 0010 1101 0000 1010 1110 1110 0011 1001 1110 1100 0100(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:


10 101 010 100 100 804(10) Base 10 integer number converted and written as a signed binary code (in base 2):

10 101 010 100 100 804(10) = 0000 0000 0010 0011 1110 0010 1101 0000 1010 1110 1110 0011 1001 1110 1100 0100

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