Convert -999 999 999 939 to a Signed Binary (Base 2)

How to convert -999 999 999 939(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 -999 999 999 939 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:

|-999 999 999 939| = 999 999 999 939

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;
  • 999 999 999 939 ÷ 2 = 499 999 999 969 + 1;
  • 499 999 999 969 ÷ 2 = 249 999 999 984 + 1;
  • 249 999 999 984 ÷ 2 = 124 999 999 992 + 0;
  • 124 999 999 992 ÷ 2 = 62 499 999 996 + 0;
  • 62 499 999 996 ÷ 2 = 31 249 999 998 + 0;
  • 31 249 999 998 ÷ 2 = 15 624 999 999 + 0;
  • 15 624 999 999 ÷ 2 = 7 812 499 999 + 1;
  • 7 812 499 999 ÷ 2 = 3 906 249 999 + 1;
  • 3 906 249 999 ÷ 2 = 1 953 124 999 + 1;
  • 1 953 124 999 ÷ 2 = 976 562 499 + 1;
  • 976 562 499 ÷ 2 = 488 281 249 + 1;
  • 488 281 249 ÷ 2 = 244 140 624 + 1;
  • 244 140 624 ÷ 2 = 122 070 312 + 0;
  • 122 070 312 ÷ 2 = 61 035 156 + 0;
  • 61 035 156 ÷ 2 = 30 517 578 + 0;
  • 30 517 578 ÷ 2 = 15 258 789 + 0;
  • 15 258 789 ÷ 2 = 7 629 394 + 1;
  • 7 629 394 ÷ 2 = 3 814 697 + 0;
  • 3 814 697 ÷ 2 = 1 907 348 + 1;
  • 1 907 348 ÷ 2 = 953 674 + 0;
  • 953 674 ÷ 2 = 476 837 + 0;
  • 476 837 ÷ 2 = 238 418 + 1;
  • 238 418 ÷ 2 = 119 209 + 0;
  • 119 209 ÷ 2 = 59 604 + 1;
  • 59 604 ÷ 2 = 29 802 + 0;
  • 29 802 ÷ 2 = 14 901 + 0;
  • 14 901 ÷ 2 = 7 450 + 1;
  • 7 450 ÷ 2 = 3 725 + 0;
  • 3 725 ÷ 2 = 1 862 + 1;
  • 1 862 ÷ 2 = 931 + 0;
  • 931 ÷ 2 = 465 + 1;
  • 465 ÷ 2 = 232 + 1;
  • 232 ÷ 2 = 116 + 0;
  • 116 ÷ 2 = 58 + 0;
  • 58 ÷ 2 = 29 + 0;
  • 29 ÷ 2 = 14 + 1;
  • 14 ÷ 2 = 7 + 0;
  • 7 ÷ 2 = 3 + 1;
  • 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.

999 999 999 939(10) = 1110 1000 1101 0100 1010 0101 0000 1111 1100 0011(2)


4. Determine the signed binary number bit length:

  • The base 2 number's actual length, in bits: 40.

  • 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, 40,

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:


999 999 999 939(10) = 0000 0000 0000 0000 0000 0000 1110 1000 1101 0100 1010 0101 0000 1111 1100 0011

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...


-999 999 999 939(10) Base 10 integer number converted and written as a signed binary code (in base 2):

-999 999 999 939(10) = 1000 0000 0000 0000 0000 0000 1110 1000 1101 0100 1010 0101 0000 1111 1100 0011

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