Convert -21 474 836 645 to a Signed Binary (Base 2)

How to convert -21 474 836 645(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 -21 474 836 645 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:

|-21 474 836 645| = 21 474 836 645

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;
  • 21 474 836 645 ÷ 2 = 10 737 418 322 + 1;
  • 10 737 418 322 ÷ 2 = 5 368 709 161 + 0;
  • 5 368 709 161 ÷ 2 = 2 684 354 580 + 1;
  • 2 684 354 580 ÷ 2 = 1 342 177 290 + 0;
  • 1 342 177 290 ÷ 2 = 671 088 645 + 0;
  • 671 088 645 ÷ 2 = 335 544 322 + 1;
  • 335 544 322 ÷ 2 = 167 772 161 + 0;
  • 167 772 161 ÷ 2 = 83 886 080 + 1;
  • 83 886 080 ÷ 2 = 41 943 040 + 0;
  • 41 943 040 ÷ 2 = 20 971 520 + 0;
  • 20 971 520 ÷ 2 = 10 485 760 + 0;
  • 10 485 760 ÷ 2 = 5 242 880 + 0;
  • 5 242 880 ÷ 2 = 2 621 440 + 0;
  • 2 621 440 ÷ 2 = 1 310 720 + 0;
  • 1 310 720 ÷ 2 = 655 360 + 0;
  • 655 360 ÷ 2 = 327 680 + 0;
  • 327 680 ÷ 2 = 163 840 + 0;
  • 163 840 ÷ 2 = 81 920 + 0;
  • 81 920 ÷ 2 = 40 960 + 0;
  • 40 960 ÷ 2 = 20 480 + 0;
  • 20 480 ÷ 2 = 10 240 + 0;
  • 10 240 ÷ 2 = 5 120 + 0;
  • 5 120 ÷ 2 = 2 560 + 0;
  • 2 560 ÷ 2 = 1 280 + 0;
  • 1 280 ÷ 2 = 640 + 0;
  • 640 ÷ 2 = 320 + 0;
  • 320 ÷ 2 = 160 + 0;
  • 160 ÷ 2 = 80 + 0;
  • 80 ÷ 2 = 40 + 0;
  • 40 ÷ 2 = 20 + 0;
  • 20 ÷ 2 = 10 + 0;
  • 10 ÷ 2 = 5 + 0;
  • 5 ÷ 2 = 2 + 1;
  • 2 ÷ 2 = 1 + 0;
  • 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.

21 474 836 645(10) = 101 0000 0000 0000 0000 0000 0000 1010 0101(2)


4. Determine the signed binary number bit length:

  • The base 2 number's actual length, in bits: 35.

  • 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, 35,

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:


21 474 836 645(10) = 0000 0000 0000 0000 0000 0000 0000 0101 0000 0000 0000 0000 0000 0000 1010 0101

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...


-21 474 836 645(10) Base 10 integer number converted and written as a signed binary code (in base 2):

-21 474 836 645(10) = 1000 0000 0000 0000 0000 0000 0000 0101 0000 0000 0000 0000 0000 0000 1010 0101

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