Convert -123 456 788 888 927 to a Signed Binary (Base 2)

How to convert -123 456 788 888 927(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 -123 456 788 888 927 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:

|-123 456 788 888 927| = 123 456 788 888 927

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;
  • 123 456 788 888 927 ÷ 2 = 61 728 394 444 463 + 1;
  • 61 728 394 444 463 ÷ 2 = 30 864 197 222 231 + 1;
  • 30 864 197 222 231 ÷ 2 = 15 432 098 611 115 + 1;
  • 15 432 098 611 115 ÷ 2 = 7 716 049 305 557 + 1;
  • 7 716 049 305 557 ÷ 2 = 3 858 024 652 778 + 1;
  • 3 858 024 652 778 ÷ 2 = 1 929 012 326 389 + 0;
  • 1 929 012 326 389 ÷ 2 = 964 506 163 194 + 1;
  • 964 506 163 194 ÷ 2 = 482 253 081 597 + 0;
  • 482 253 081 597 ÷ 2 = 241 126 540 798 + 1;
  • 241 126 540 798 ÷ 2 = 120 563 270 399 + 0;
  • 120 563 270 399 ÷ 2 = 60 281 635 199 + 1;
  • 60 281 635 199 ÷ 2 = 30 140 817 599 + 1;
  • 30 140 817 599 ÷ 2 = 15 070 408 799 + 1;
  • 15 070 408 799 ÷ 2 = 7 535 204 399 + 1;
  • 7 535 204 399 ÷ 2 = 3 767 602 199 + 1;
  • 3 767 602 199 ÷ 2 = 1 883 801 099 + 1;
  • 1 883 801 099 ÷ 2 = 941 900 549 + 1;
  • 941 900 549 ÷ 2 = 470 950 274 + 1;
  • 470 950 274 ÷ 2 = 235 475 137 + 0;
  • 235 475 137 ÷ 2 = 117 737 568 + 1;
  • 117 737 568 ÷ 2 = 58 868 784 + 0;
  • 58 868 784 ÷ 2 = 29 434 392 + 0;
  • 29 434 392 ÷ 2 = 14 717 196 + 0;
  • 14 717 196 ÷ 2 = 7 358 598 + 0;
  • 7 358 598 ÷ 2 = 3 679 299 + 0;
  • 3 679 299 ÷ 2 = 1 839 649 + 1;
  • 1 839 649 ÷ 2 = 919 824 + 1;
  • 919 824 ÷ 2 = 459 912 + 0;
  • 459 912 ÷ 2 = 229 956 + 0;
  • 229 956 ÷ 2 = 114 978 + 0;
  • 114 978 ÷ 2 = 57 489 + 0;
  • 57 489 ÷ 2 = 28 744 + 1;
  • 28 744 ÷ 2 = 14 372 + 0;
  • 14 372 ÷ 2 = 7 186 + 0;
  • 7 186 ÷ 2 = 3 593 + 0;
  • 3 593 ÷ 2 = 1 796 + 1;
  • 1 796 ÷ 2 = 898 + 0;
  • 898 ÷ 2 = 449 + 0;
  • 449 ÷ 2 = 224 + 1;
  • 224 ÷ 2 = 112 + 0;
  • 112 ÷ 2 = 56 + 0;
  • 56 ÷ 2 = 28 + 0;
  • 28 ÷ 2 = 14 + 0;
  • 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.

123 456 788 888 927(10) = 111 0000 0100 1000 1000 0110 0000 1011 1111 1101 0101 1111(2)


4. 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.


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:


123 456 788 888 927(10) = 0000 0000 0000 0000 0111 0000 0100 1000 1000 0110 0000 1011 1111 1101 0101 1111

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...


-123 456 788 888 927(10) Base 10 integer number converted and written as a signed binary code (in base 2):

-123 456 788 888 927(10) = 1000 0000 0000 0000 0111 0000 0100 1000 1000 0110 0000 1011 1111 1101 0101 1111

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