Convert 10 000 000 000 000 265 to a Signed Binary (Base 2)

How to convert 10 000 000 000 000 265(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 10 000 000 000 000 265 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;
  • 10 000 000 000 000 265 ÷ 2 = 5 000 000 000 000 132 + 1;
  • 5 000 000 000 000 132 ÷ 2 = 2 500 000 000 000 066 + 0;
  • 2 500 000 000 000 066 ÷ 2 = 1 250 000 000 000 033 + 0;
  • 1 250 000 000 000 033 ÷ 2 = 625 000 000 000 016 + 1;
  • 625 000 000 000 016 ÷ 2 = 312 500 000 000 008 + 0;
  • 312 500 000 000 008 ÷ 2 = 156 250 000 000 004 + 0;
  • 156 250 000 000 004 ÷ 2 = 78 125 000 000 002 + 0;
  • 78 125 000 000 002 ÷ 2 = 39 062 500 000 001 + 0;
  • 39 062 500 000 001 ÷ 2 = 19 531 250 000 000 + 1;
  • 19 531 250 000 000 ÷ 2 = 9 765 625 000 000 + 0;
  • 9 765 625 000 000 ÷ 2 = 4 882 812 500 000 + 0;
  • 4 882 812 500 000 ÷ 2 = 2 441 406 250 000 + 0;
  • 2 441 406 250 000 ÷ 2 = 1 220 703 125 000 + 0;
  • 1 220 703 125 000 ÷ 2 = 610 351 562 500 + 0;
  • 610 351 562 500 ÷ 2 = 305 175 781 250 + 0;
  • 305 175 781 250 ÷ 2 = 152 587 890 625 + 0;
  • 152 587 890 625 ÷ 2 = 76 293 945 312 + 1;
  • 76 293 945 312 ÷ 2 = 38 146 972 656 + 0;
  • 38 146 972 656 ÷ 2 = 19 073 486 328 + 0;
  • 19 073 486 328 ÷ 2 = 9 536 743 164 + 0;
  • 9 536 743 164 ÷ 2 = 4 768 371 582 + 0;
  • 4 768 371 582 ÷ 2 = 2 384 185 791 + 0;
  • 2 384 185 791 ÷ 2 = 1 192 092 895 + 1;
  • 1 192 092 895 ÷ 2 = 596 046 447 + 1;
  • 596 046 447 ÷ 2 = 298 023 223 + 1;
  • 298 023 223 ÷ 2 = 149 011 611 + 1;
  • 149 011 611 ÷ 2 = 74 505 805 + 1;
  • 74 505 805 ÷ 2 = 37 252 902 + 1;
  • 37 252 902 ÷ 2 = 18 626 451 + 0;
  • 18 626 451 ÷ 2 = 9 313 225 + 1;
  • 9 313 225 ÷ 2 = 4 656 612 + 1;
  • 4 656 612 ÷ 2 = 2 328 306 + 0;
  • 2 328 306 ÷ 2 = 1 164 153 + 0;
  • 1 164 153 ÷ 2 = 582 076 + 1;
  • 582 076 ÷ 2 = 291 038 + 0;
  • 291 038 ÷ 2 = 145 519 + 0;
  • 145 519 ÷ 2 = 72 759 + 1;
  • 72 759 ÷ 2 = 36 379 + 1;
  • 36 379 ÷ 2 = 18 189 + 1;
  • 18 189 ÷ 2 = 9 094 + 1;
  • 9 094 ÷ 2 = 4 547 + 0;
  • 4 547 ÷ 2 = 2 273 + 1;
  • 2 273 ÷ 2 = 1 136 + 1;
  • 1 136 ÷ 2 = 568 + 0;
  • 568 ÷ 2 = 284 + 0;
  • 284 ÷ 2 = 142 + 0;
  • 142 ÷ 2 = 71 + 0;
  • 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.

10 000 000 000 000 265(10) = 10 0011 1000 0110 1111 0010 0110 1111 1100 0001 0000 0001 0000 1001(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:


10 000 000 000 000 265(10) Base 10 integer number converted and written as a signed binary code (in base 2):

10 000 000 000 000 265(10) = 0000 0000 0010 0011 1000 0110 1111 0010 0110 1111 1100 0001 0000 0001 0000 1001

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