Convert 111 111 111 110 126 to a Signed Binary (Base 2)

How to convert 111 111 111 110 126(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 111 111 111 110 126 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;
  • 111 111 111 110 126 ÷ 2 = 55 555 555 555 063 + 0;
  • 55 555 555 555 063 ÷ 2 = 27 777 777 777 531 + 1;
  • 27 777 777 777 531 ÷ 2 = 13 888 888 888 765 + 1;
  • 13 888 888 888 765 ÷ 2 = 6 944 444 444 382 + 1;
  • 6 944 444 444 382 ÷ 2 = 3 472 222 222 191 + 0;
  • 3 472 222 222 191 ÷ 2 = 1 736 111 111 095 + 1;
  • 1 736 111 111 095 ÷ 2 = 868 055 555 547 + 1;
  • 868 055 555 547 ÷ 2 = 434 027 777 773 + 1;
  • 434 027 777 773 ÷ 2 = 217 013 888 886 + 1;
  • 217 013 888 886 ÷ 2 = 108 506 944 443 + 0;
  • 108 506 944 443 ÷ 2 = 54 253 472 221 + 1;
  • 54 253 472 221 ÷ 2 = 27 126 736 110 + 1;
  • 27 126 736 110 ÷ 2 = 13 563 368 055 + 0;
  • 13 563 368 055 ÷ 2 = 6 781 684 027 + 1;
  • 6 781 684 027 ÷ 2 = 3 390 842 013 + 1;
  • 3 390 842 013 ÷ 2 = 1 695 421 006 + 1;
  • 1 695 421 006 ÷ 2 = 847 710 503 + 0;
  • 847 710 503 ÷ 2 = 423 855 251 + 1;
  • 423 855 251 ÷ 2 = 211 927 625 + 1;
  • 211 927 625 ÷ 2 = 105 963 812 + 1;
  • 105 963 812 ÷ 2 = 52 981 906 + 0;
  • 52 981 906 ÷ 2 = 26 490 953 + 0;
  • 26 490 953 ÷ 2 = 13 245 476 + 1;
  • 13 245 476 ÷ 2 = 6 622 738 + 0;
  • 6 622 738 ÷ 2 = 3 311 369 + 0;
  • 3 311 369 ÷ 2 = 1 655 684 + 1;
  • 1 655 684 ÷ 2 = 827 842 + 0;
  • 827 842 ÷ 2 = 413 921 + 0;
  • 413 921 ÷ 2 = 206 960 + 1;
  • 206 960 ÷ 2 = 103 480 + 0;
  • 103 480 ÷ 2 = 51 740 + 0;
  • 51 740 ÷ 2 = 25 870 + 0;
  • 25 870 ÷ 2 = 12 935 + 0;
  • 12 935 ÷ 2 = 6 467 + 1;
  • 6 467 ÷ 2 = 3 233 + 1;
  • 3 233 ÷ 2 = 1 616 + 1;
  • 1 616 ÷ 2 = 808 + 0;
  • 808 ÷ 2 = 404 + 0;
  • 404 ÷ 2 = 202 + 0;
  • 202 ÷ 2 = 101 + 0;
  • 101 ÷ 2 = 50 + 1;
  • 50 ÷ 2 = 25 + 0;
  • 25 ÷ 2 = 12 + 1;
  • 12 ÷ 2 = 6 + 0;
  • 6 ÷ 2 = 3 + 0;
  • 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.

111 111 111 110 126(10) = 110 0101 0000 1110 0001 0010 0100 1110 1110 1101 1110 1110(2)


3. Determine the signed binary number bit length:

  • The base 2 number's actual length, in bits: 47.

  • 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, 47,

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:


111 111 111 110 126(10) Base 10 integer number converted and written as a signed binary code (in base 2):

111 111 111 110 126(10) = 0000 0000 0000 0000 0110 0101 0000 1110 0001 0010 0100 1110 1110 1101 1110 1110

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