Convert -7 678 710 519 452 155 683 to a Signed Binary (Base 2)

How to convert -7 678 710 519 452 155 683(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 -7 678 710 519 452 155 683 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:

|-7 678 710 519 452 155 683| = 7 678 710 519 452 155 683

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;
  • 7 678 710 519 452 155 683 ÷ 2 = 3 839 355 259 726 077 841 + 1;
  • 3 839 355 259 726 077 841 ÷ 2 = 1 919 677 629 863 038 920 + 1;
  • 1 919 677 629 863 038 920 ÷ 2 = 959 838 814 931 519 460 + 0;
  • 959 838 814 931 519 460 ÷ 2 = 479 919 407 465 759 730 + 0;
  • 479 919 407 465 759 730 ÷ 2 = 239 959 703 732 879 865 + 0;
  • 239 959 703 732 879 865 ÷ 2 = 119 979 851 866 439 932 + 1;
  • 119 979 851 866 439 932 ÷ 2 = 59 989 925 933 219 966 + 0;
  • 59 989 925 933 219 966 ÷ 2 = 29 994 962 966 609 983 + 0;
  • 29 994 962 966 609 983 ÷ 2 = 14 997 481 483 304 991 + 1;
  • 14 997 481 483 304 991 ÷ 2 = 7 498 740 741 652 495 + 1;
  • 7 498 740 741 652 495 ÷ 2 = 3 749 370 370 826 247 + 1;
  • 3 749 370 370 826 247 ÷ 2 = 1 874 685 185 413 123 + 1;
  • 1 874 685 185 413 123 ÷ 2 = 937 342 592 706 561 + 1;
  • 937 342 592 706 561 ÷ 2 = 468 671 296 353 280 + 1;
  • 468 671 296 353 280 ÷ 2 = 234 335 648 176 640 + 0;
  • 234 335 648 176 640 ÷ 2 = 117 167 824 088 320 + 0;
  • 117 167 824 088 320 ÷ 2 = 58 583 912 044 160 + 0;
  • 58 583 912 044 160 ÷ 2 = 29 291 956 022 080 + 0;
  • 29 291 956 022 080 ÷ 2 = 14 645 978 011 040 + 0;
  • 14 645 978 011 040 ÷ 2 = 7 322 989 005 520 + 0;
  • 7 322 989 005 520 ÷ 2 = 3 661 494 502 760 + 0;
  • 3 661 494 502 760 ÷ 2 = 1 830 747 251 380 + 0;
  • 1 830 747 251 380 ÷ 2 = 915 373 625 690 + 0;
  • 915 373 625 690 ÷ 2 = 457 686 812 845 + 0;
  • 457 686 812 845 ÷ 2 = 228 843 406 422 + 1;
  • 228 843 406 422 ÷ 2 = 114 421 703 211 + 0;
  • 114 421 703 211 ÷ 2 = 57 210 851 605 + 1;
  • 57 210 851 605 ÷ 2 = 28 605 425 802 + 1;
  • 28 605 425 802 ÷ 2 = 14 302 712 901 + 0;
  • 14 302 712 901 ÷ 2 = 7 151 356 450 + 1;
  • 7 151 356 450 ÷ 2 = 3 575 678 225 + 0;
  • 3 575 678 225 ÷ 2 = 1 787 839 112 + 1;
  • 1 787 839 112 ÷ 2 = 893 919 556 + 0;
  • 893 919 556 ÷ 2 = 446 959 778 + 0;
  • 446 959 778 ÷ 2 = 223 479 889 + 0;
  • 223 479 889 ÷ 2 = 111 739 944 + 1;
  • 111 739 944 ÷ 2 = 55 869 972 + 0;
  • 55 869 972 ÷ 2 = 27 934 986 + 0;
  • 27 934 986 ÷ 2 = 13 967 493 + 0;
  • 13 967 493 ÷ 2 = 6 983 746 + 1;
  • 6 983 746 ÷ 2 = 3 491 873 + 0;
  • 3 491 873 ÷ 2 = 1 745 936 + 1;
  • 1 745 936 ÷ 2 = 872 968 + 0;
  • 872 968 ÷ 2 = 436 484 + 0;
  • 436 484 ÷ 2 = 218 242 + 0;
  • 218 242 ÷ 2 = 109 121 + 0;
  • 109 121 ÷ 2 = 54 560 + 1;
  • 54 560 ÷ 2 = 27 280 + 0;
  • 27 280 ÷ 2 = 13 640 + 0;
  • 13 640 ÷ 2 = 6 820 + 0;
  • 6 820 ÷ 2 = 3 410 + 0;
  • 3 410 ÷ 2 = 1 705 + 0;
  • 1 705 ÷ 2 = 852 + 1;
  • 852 ÷ 2 = 426 + 0;
  • 426 ÷ 2 = 213 + 0;
  • 213 ÷ 2 = 106 + 1;
  • 106 ÷ 2 = 53 + 0;
  • 53 ÷ 2 = 26 + 1;
  • 26 ÷ 2 = 13 + 0;
  • 13 ÷ 2 = 6 + 1;
  • 6 ÷ 2 = 3 + 0;
  • 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.

7 678 710 519 452 155 683(10) = 110 1010 1001 0000 0100 0010 1000 1000 1010 1101 0000 0000 0011 1111 0010 0011(2)


4. Determine the signed binary number bit length:

  • The base 2 number's actual length, in bits: 63.

  • 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, 63,

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:


7 678 710 519 452 155 683(10) = 0110 1010 1001 0000 0100 0010 1000 1000 1010 1101 0000 0000 0011 1111 0010 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...


-7 678 710 519 452 155 683(10) Base 10 integer number converted and written as a signed binary code (in base 2):

-7 678 710 519 452 155 683(10) = 1110 1010 1001 0000 0100 0010 1000 1000 1010 1101 0000 0000 0011 1111 0010 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