Convert -109 209 807 313 to a Signed Binary (Base 2)

How to convert -109 209 807 313(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 -109 209 807 313 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. Start with the positive version of the number:

|-109 209 807 313| = 109 209 807 313

2. 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;
  • 109 209 807 313 ÷ 2 = 54 604 903 656 + 1;
  • 54 604 903 656 ÷ 2 = 27 302 451 828 + 0;
  • 27 302 451 828 ÷ 2 = 13 651 225 914 + 0;
  • 13 651 225 914 ÷ 2 = 6 825 612 957 + 0;
  • 6 825 612 957 ÷ 2 = 3 412 806 478 + 1;
  • 3 412 806 478 ÷ 2 = 1 706 403 239 + 0;
  • 1 706 403 239 ÷ 2 = 853 201 619 + 1;
  • 853 201 619 ÷ 2 = 426 600 809 + 1;
  • 426 600 809 ÷ 2 = 213 300 404 + 1;
  • 213 300 404 ÷ 2 = 106 650 202 + 0;
  • 106 650 202 ÷ 2 = 53 325 101 + 0;
  • 53 325 101 ÷ 2 = 26 662 550 + 1;
  • 26 662 550 ÷ 2 = 13 331 275 + 0;
  • 13 331 275 ÷ 2 = 6 665 637 + 1;
  • 6 665 637 ÷ 2 = 3 332 818 + 1;
  • 3 332 818 ÷ 2 = 1 666 409 + 0;
  • 1 666 409 ÷ 2 = 833 204 + 1;
  • 833 204 ÷ 2 = 416 602 + 0;
  • 416 602 ÷ 2 = 208 301 + 0;
  • 208 301 ÷ 2 = 104 150 + 1;
  • 104 150 ÷ 2 = 52 075 + 0;
  • 52 075 ÷ 2 = 26 037 + 1;
  • 26 037 ÷ 2 = 13 018 + 1;
  • 13 018 ÷ 2 = 6 509 + 0;
  • 6 509 ÷ 2 = 3 254 + 1;
  • 3 254 ÷ 2 = 1 627 + 0;
  • 1 627 ÷ 2 = 813 + 1;
  • 813 ÷ 2 = 406 + 1;
  • 406 ÷ 2 = 203 + 0;
  • 203 ÷ 2 = 101 + 1;
  • 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;

3. Construct the base 2 representation of the positive number:

Take all the remainders starting from the bottom of the list constructed above.

109 209 807 313(10) = 1 1001 0110 1101 0110 1001 0110 1001 1101 0001(2)


4. Determine the signed binary number bit length:

  • The base 2 number's actual length, in bits: 37.

  • 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, 37,

3) so that the first bit (leftmost) could be zero
(we deal with a positive number at this moment)

=== is: 64.


5. 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:


109 209 807 313(10) = 0000 0000 0000 0000 0000 0000 0001 1001 0110 1101 0110 1001 0110 1001 1101 0001

6. Get the negative integer number representation:

To get the negative integer number representation on 64 bits (8 Bytes),

... change the first bit (the leftmost), from 0 to 1...


-109 209 807 313(10) Base 10 integer number converted and written as a signed binary code (in base 2):

-109 209 807 313(10) = 1000 0000 0000 0000 0000 0000 0001 1001 0110 1101 0110 1001 0110 1001 1101 0001

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