Convert 123 456 789 987 653 471 to a Signed Binary (Base 2)

How to convert 123 456 789 987 653 471(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 123 456 789 987 653 471 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;
  • 123 456 789 987 653 471 ÷ 2 = 61 728 394 993 826 735 + 1;
  • 61 728 394 993 826 735 ÷ 2 = 30 864 197 496 913 367 + 1;
  • 30 864 197 496 913 367 ÷ 2 = 15 432 098 748 456 683 + 1;
  • 15 432 098 748 456 683 ÷ 2 = 7 716 049 374 228 341 + 1;
  • 7 716 049 374 228 341 ÷ 2 = 3 858 024 687 114 170 + 1;
  • 3 858 024 687 114 170 ÷ 2 = 1 929 012 343 557 085 + 0;
  • 1 929 012 343 557 085 ÷ 2 = 964 506 171 778 542 + 1;
  • 964 506 171 778 542 ÷ 2 = 482 253 085 889 271 + 0;
  • 482 253 085 889 271 ÷ 2 = 241 126 542 944 635 + 1;
  • 241 126 542 944 635 ÷ 2 = 120 563 271 472 317 + 1;
  • 120 563 271 472 317 ÷ 2 = 60 281 635 736 158 + 1;
  • 60 281 635 736 158 ÷ 2 = 30 140 817 868 079 + 0;
  • 30 140 817 868 079 ÷ 2 = 15 070 408 934 039 + 1;
  • 15 070 408 934 039 ÷ 2 = 7 535 204 467 019 + 1;
  • 7 535 204 467 019 ÷ 2 = 3 767 602 233 509 + 1;
  • 3 767 602 233 509 ÷ 2 = 1 883 801 116 754 + 1;
  • 1 883 801 116 754 ÷ 2 = 941 900 558 377 + 0;
  • 941 900 558 377 ÷ 2 = 470 950 279 188 + 1;
  • 470 950 279 188 ÷ 2 = 235 475 139 594 + 0;
  • 235 475 139 594 ÷ 2 = 117 737 569 797 + 0;
  • 117 737 569 797 ÷ 2 = 58 868 784 898 + 1;
  • 58 868 784 898 ÷ 2 = 29 434 392 449 + 0;
  • 29 434 392 449 ÷ 2 = 14 717 196 224 + 1;
  • 14 717 196 224 ÷ 2 = 7 358 598 112 + 0;
  • 7 358 598 112 ÷ 2 = 3 679 299 056 + 0;
  • 3 679 299 056 ÷ 2 = 1 839 649 528 + 0;
  • 1 839 649 528 ÷ 2 = 919 824 764 + 0;
  • 919 824 764 ÷ 2 = 459 912 382 + 0;
  • 459 912 382 ÷ 2 = 229 956 191 + 0;
  • 229 956 191 ÷ 2 = 114 978 095 + 1;
  • 114 978 095 ÷ 2 = 57 489 047 + 1;
  • 57 489 047 ÷ 2 = 28 744 523 + 1;
  • 28 744 523 ÷ 2 = 14 372 261 + 1;
  • 14 372 261 ÷ 2 = 7 186 130 + 1;
  • 7 186 130 ÷ 2 = 3 593 065 + 0;
  • 3 593 065 ÷ 2 = 1 796 532 + 1;
  • 1 796 532 ÷ 2 = 898 266 + 0;
  • 898 266 ÷ 2 = 449 133 + 0;
  • 449 133 ÷ 2 = 224 566 + 1;
  • 224 566 ÷ 2 = 112 283 + 0;
  • 112 283 ÷ 2 = 56 141 + 1;
  • 56 141 ÷ 2 = 28 070 + 1;
  • 28 070 ÷ 2 = 14 035 + 0;
  • 14 035 ÷ 2 = 7 017 + 1;
  • 7 017 ÷ 2 = 3 508 + 1;
  • 3 508 ÷ 2 = 1 754 + 0;
  • 1 754 ÷ 2 = 877 + 0;
  • 877 ÷ 2 = 438 + 1;
  • 438 ÷ 2 = 219 + 0;
  • 219 ÷ 2 = 109 + 1;
  • 109 ÷ 2 = 54 + 1;
  • 54 ÷ 2 = 27 + 0;
  • 27 ÷ 2 = 13 + 1;
  • 13 ÷ 2 = 6 + 1;
  • 6 ÷ 2 = 3 + 0;
  • 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.

123 456 789 987 653 471(10) = 1 1011 0110 1001 1011 0100 1011 1110 0000 0101 0010 1111 0111 0101 1111(2)


3. Determine the signed binary number bit length:

  • The base 2 number's actual length, in bits: 57.

  • 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, 57,

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:


123 456 789 987 653 471(10) Base 10 integer number converted and written as a signed binary code (in base 2):

123 456 789 987 653 471(10) = 0000 0001 1011 0110 1001 1011 0100 1011 1110 0000 0101 0010 1111 0111 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