Convert -397 801 709 542 425 to a Signed Binary (Base 2)

How to convert -397 801 709 542 425(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 -397 801 709 542 425 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:

|-397 801 709 542 425| = 397 801 709 542 425

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;
  • 397 801 709 542 425 ÷ 2 = 198 900 854 771 212 + 1;
  • 198 900 854 771 212 ÷ 2 = 99 450 427 385 606 + 0;
  • 99 450 427 385 606 ÷ 2 = 49 725 213 692 803 + 0;
  • 49 725 213 692 803 ÷ 2 = 24 862 606 846 401 + 1;
  • 24 862 606 846 401 ÷ 2 = 12 431 303 423 200 + 1;
  • 12 431 303 423 200 ÷ 2 = 6 215 651 711 600 + 0;
  • 6 215 651 711 600 ÷ 2 = 3 107 825 855 800 + 0;
  • 3 107 825 855 800 ÷ 2 = 1 553 912 927 900 + 0;
  • 1 553 912 927 900 ÷ 2 = 776 956 463 950 + 0;
  • 776 956 463 950 ÷ 2 = 388 478 231 975 + 0;
  • 388 478 231 975 ÷ 2 = 194 239 115 987 + 1;
  • 194 239 115 987 ÷ 2 = 97 119 557 993 + 1;
  • 97 119 557 993 ÷ 2 = 48 559 778 996 + 1;
  • 48 559 778 996 ÷ 2 = 24 279 889 498 + 0;
  • 24 279 889 498 ÷ 2 = 12 139 944 749 + 0;
  • 12 139 944 749 ÷ 2 = 6 069 972 374 + 1;
  • 6 069 972 374 ÷ 2 = 3 034 986 187 + 0;
  • 3 034 986 187 ÷ 2 = 1 517 493 093 + 1;
  • 1 517 493 093 ÷ 2 = 758 746 546 + 1;
  • 758 746 546 ÷ 2 = 379 373 273 + 0;
  • 379 373 273 ÷ 2 = 189 686 636 + 1;
  • 189 686 636 ÷ 2 = 94 843 318 + 0;
  • 94 843 318 ÷ 2 = 47 421 659 + 0;
  • 47 421 659 ÷ 2 = 23 710 829 + 1;
  • 23 710 829 ÷ 2 = 11 855 414 + 1;
  • 11 855 414 ÷ 2 = 5 927 707 + 0;
  • 5 927 707 ÷ 2 = 2 963 853 + 1;
  • 2 963 853 ÷ 2 = 1 481 926 + 1;
  • 1 481 926 ÷ 2 = 740 963 + 0;
  • 740 963 ÷ 2 = 370 481 + 1;
  • 370 481 ÷ 2 = 185 240 + 1;
  • 185 240 ÷ 2 = 92 620 + 0;
  • 92 620 ÷ 2 = 46 310 + 0;
  • 46 310 ÷ 2 = 23 155 + 0;
  • 23 155 ÷ 2 = 11 577 + 1;
  • 11 577 ÷ 2 = 5 788 + 1;
  • 5 788 ÷ 2 = 2 894 + 0;
  • 2 894 ÷ 2 = 1 447 + 0;
  • 1 447 ÷ 2 = 723 + 1;
  • 723 ÷ 2 = 361 + 1;
  • 361 ÷ 2 = 180 + 1;
  • 180 ÷ 2 = 90 + 0;
  • 90 ÷ 2 = 45 + 0;
  • 45 ÷ 2 = 22 + 1;
  • 22 ÷ 2 = 11 + 0;
  • 11 ÷ 2 = 5 + 1;
  • 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.

397 801 709 542 425(10) = 1 0110 1001 1100 1100 0110 1101 1001 0110 1001 1100 0001 1001(2)


4. Determine the signed binary number bit length:

  • The base 2 number's actual length, in bits: 49.

  • 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, 49,

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:


397 801 709 542 425(10) = 0000 0000 0000 0001 0110 1001 1100 1100 0110 1101 1001 0110 1001 1100 0001 1001

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...


-397 801 709 542 425(10) Base 10 integer number converted and written as a signed binary code (in base 2):

-397 801 709 542 425(10) = 1000 0000 0000 0001 0110 1001 1100 1100 0110 1101 1001 0110 1001 1100 0001 1001

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