Convert 1 111 001 000 100 121 to a Signed Binary (Base 2)

How to convert 1 111 001 000 100 121(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 1 111 001 000 100 121 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;
  • 1 111 001 000 100 121 ÷ 2 = 555 500 500 050 060 + 1;
  • 555 500 500 050 060 ÷ 2 = 277 750 250 025 030 + 0;
  • 277 750 250 025 030 ÷ 2 = 138 875 125 012 515 + 0;
  • 138 875 125 012 515 ÷ 2 = 69 437 562 506 257 + 1;
  • 69 437 562 506 257 ÷ 2 = 34 718 781 253 128 + 1;
  • 34 718 781 253 128 ÷ 2 = 17 359 390 626 564 + 0;
  • 17 359 390 626 564 ÷ 2 = 8 679 695 313 282 + 0;
  • 8 679 695 313 282 ÷ 2 = 4 339 847 656 641 + 0;
  • 4 339 847 656 641 ÷ 2 = 2 169 923 828 320 + 1;
  • 2 169 923 828 320 ÷ 2 = 1 084 961 914 160 + 0;
  • 1 084 961 914 160 ÷ 2 = 542 480 957 080 + 0;
  • 542 480 957 080 ÷ 2 = 271 240 478 540 + 0;
  • 271 240 478 540 ÷ 2 = 135 620 239 270 + 0;
  • 135 620 239 270 ÷ 2 = 67 810 119 635 + 0;
  • 67 810 119 635 ÷ 2 = 33 905 059 817 + 1;
  • 33 905 059 817 ÷ 2 = 16 952 529 908 + 1;
  • 16 952 529 908 ÷ 2 = 8 476 264 954 + 0;
  • 8 476 264 954 ÷ 2 = 4 238 132 477 + 0;
  • 4 238 132 477 ÷ 2 = 2 119 066 238 + 1;
  • 2 119 066 238 ÷ 2 = 1 059 533 119 + 0;
  • 1 059 533 119 ÷ 2 = 529 766 559 + 1;
  • 529 766 559 ÷ 2 = 264 883 279 + 1;
  • 264 883 279 ÷ 2 = 132 441 639 + 1;
  • 132 441 639 ÷ 2 = 66 220 819 + 1;
  • 66 220 819 ÷ 2 = 33 110 409 + 1;
  • 33 110 409 ÷ 2 = 16 555 204 + 1;
  • 16 555 204 ÷ 2 = 8 277 602 + 0;
  • 8 277 602 ÷ 2 = 4 138 801 + 0;
  • 4 138 801 ÷ 2 = 2 069 400 + 1;
  • 2 069 400 ÷ 2 = 1 034 700 + 0;
  • 1 034 700 ÷ 2 = 517 350 + 0;
  • 517 350 ÷ 2 = 258 675 + 0;
  • 258 675 ÷ 2 = 129 337 + 1;
  • 129 337 ÷ 2 = 64 668 + 1;
  • 64 668 ÷ 2 = 32 334 + 0;
  • 32 334 ÷ 2 = 16 167 + 0;
  • 16 167 ÷ 2 = 8 083 + 1;
  • 8 083 ÷ 2 = 4 041 + 1;
  • 4 041 ÷ 2 = 2 020 + 1;
  • 2 020 ÷ 2 = 1 010 + 0;
  • 1 010 ÷ 2 = 505 + 0;
  • 505 ÷ 2 = 252 + 1;
  • 252 ÷ 2 = 126 + 0;
  • 126 ÷ 2 = 63 + 0;
  • 63 ÷ 2 = 31 + 1;
  • 31 ÷ 2 = 15 + 1;
  • 15 ÷ 2 = 7 + 1;
  • 7 ÷ 2 = 3 + 1;
  • 3 ÷ 2 = 1 + 1;
  • 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.

1 111 001 000 100 121(10) = 11 1111 0010 0111 0011 0001 0011 1111 0100 1100 0001 0001 1001(2)


3. Determine the signed binary number bit length:

  • The base 2 number's actual length, in bits: 50.

  • 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, 50,

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:


1 111 001 000 100 121(10) Base 10 integer number converted and written as a signed binary code (in base 2):

1 111 001 000 100 121(10) = 0000 0000 0000 0011 1111 0010 0111 0011 0001 0011 1111 0100 1100 0001 0001 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