Convert 1 010 111 009 837 to a Signed Binary (Base 2)

How to convert 1 010 111 009 837(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 1 010 111 009 837 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;
  • 1 010 111 009 837 ÷ 2 = 505 055 504 918 + 1;
  • 505 055 504 918 ÷ 2 = 252 527 752 459 + 0;
  • 252 527 752 459 ÷ 2 = 126 263 876 229 + 1;
  • 126 263 876 229 ÷ 2 = 63 131 938 114 + 1;
  • 63 131 938 114 ÷ 2 = 31 565 969 057 + 0;
  • 31 565 969 057 ÷ 2 = 15 782 984 528 + 1;
  • 15 782 984 528 ÷ 2 = 7 891 492 264 + 0;
  • 7 891 492 264 ÷ 2 = 3 945 746 132 + 0;
  • 3 945 746 132 ÷ 2 = 1 972 873 066 + 0;
  • 1 972 873 066 ÷ 2 = 986 436 533 + 0;
  • 986 436 533 ÷ 2 = 493 218 266 + 1;
  • 493 218 266 ÷ 2 = 246 609 133 + 0;
  • 246 609 133 ÷ 2 = 123 304 566 + 1;
  • 123 304 566 ÷ 2 = 61 652 283 + 0;
  • 61 652 283 ÷ 2 = 30 826 141 + 1;
  • 30 826 141 ÷ 2 = 15 413 070 + 1;
  • 15 413 070 ÷ 2 = 7 706 535 + 0;
  • 7 706 535 ÷ 2 = 3 853 267 + 1;
  • 3 853 267 ÷ 2 = 1 926 633 + 1;
  • 1 926 633 ÷ 2 = 963 316 + 1;
  • 963 316 ÷ 2 = 481 658 + 0;
  • 481 658 ÷ 2 = 240 829 + 0;
  • 240 829 ÷ 2 = 120 414 + 1;
  • 120 414 ÷ 2 = 60 207 + 0;
  • 60 207 ÷ 2 = 30 103 + 1;
  • 30 103 ÷ 2 = 15 051 + 1;
  • 15 051 ÷ 2 = 7 525 + 1;
  • 7 525 ÷ 2 = 3 762 + 1;
  • 3 762 ÷ 2 = 1 881 + 0;
  • 1 881 ÷ 2 = 940 + 1;
  • 940 ÷ 2 = 470 + 0;
  • 470 ÷ 2 = 235 + 0;
  • 235 ÷ 2 = 117 + 1;
  • 117 ÷ 2 = 58 + 1;
  • 58 ÷ 2 = 29 + 0;
  • 29 ÷ 2 = 14 + 1;
  • 14 ÷ 2 = 7 + 0;
  • 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.

1 010 111 009 837(10) = 1110 1011 0010 1111 0100 1110 1101 0100 0010 1101(2)


3. 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.


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:


1 010 111 009 837(10) Base 10 integer number converted and written as a signed binary code (in base 2):

1 010 111 009 837(10) = 0000 0000 0000 0000 0000 0000 1110 1011 0010 1111 0100 1110 1101 0100 0010 1101

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