Convert 8 999 999 999 999 999 937 to a Signed Binary (Base 2)

How to convert 8 999 999 999 999 999 937(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 8 999 999 999 999 999 937 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;
  • 8 999 999 999 999 999 937 ÷ 2 = 4 499 999 999 999 999 968 + 1;
  • 4 499 999 999 999 999 968 ÷ 2 = 2 249 999 999 999 999 984 + 0;
  • 2 249 999 999 999 999 984 ÷ 2 = 1 124 999 999 999 999 992 + 0;
  • 1 124 999 999 999 999 992 ÷ 2 = 562 499 999 999 999 996 + 0;
  • 562 499 999 999 999 996 ÷ 2 = 281 249 999 999 999 998 + 0;
  • 281 249 999 999 999 998 ÷ 2 = 140 624 999 999 999 999 + 0;
  • 140 624 999 999 999 999 ÷ 2 = 70 312 499 999 999 999 + 1;
  • 70 312 499 999 999 999 ÷ 2 = 35 156 249 999 999 999 + 1;
  • 35 156 249 999 999 999 ÷ 2 = 17 578 124 999 999 999 + 1;
  • 17 578 124 999 999 999 ÷ 2 = 8 789 062 499 999 999 + 1;
  • 8 789 062 499 999 999 ÷ 2 = 4 394 531 249 999 999 + 1;
  • 4 394 531 249 999 999 ÷ 2 = 2 197 265 624 999 999 + 1;
  • 2 197 265 624 999 999 ÷ 2 = 1 098 632 812 499 999 + 1;
  • 1 098 632 812 499 999 ÷ 2 = 549 316 406 249 999 + 1;
  • 549 316 406 249 999 ÷ 2 = 274 658 203 124 999 + 1;
  • 274 658 203 124 999 ÷ 2 = 137 329 101 562 499 + 1;
  • 137 329 101 562 499 ÷ 2 = 68 664 550 781 249 + 1;
  • 68 664 550 781 249 ÷ 2 = 34 332 275 390 624 + 1;
  • 34 332 275 390 624 ÷ 2 = 17 166 137 695 312 + 0;
  • 17 166 137 695 312 ÷ 2 = 8 583 068 847 656 + 0;
  • 8 583 068 847 656 ÷ 2 = 4 291 534 423 828 + 0;
  • 4 291 534 423 828 ÷ 2 = 2 145 767 211 914 + 0;
  • 2 145 767 211 914 ÷ 2 = 1 072 883 605 957 + 0;
  • 1 072 883 605 957 ÷ 2 = 536 441 802 978 + 1;
  • 536 441 802 978 ÷ 2 = 268 220 901 489 + 0;
  • 268 220 901 489 ÷ 2 = 134 110 450 744 + 1;
  • 134 110 450 744 ÷ 2 = 67 055 225 372 + 0;
  • 67 055 225 372 ÷ 2 = 33 527 612 686 + 0;
  • 33 527 612 686 ÷ 2 = 16 763 806 343 + 0;
  • 16 763 806 343 ÷ 2 = 8 381 903 171 + 1;
  • 8 381 903 171 ÷ 2 = 4 190 951 585 + 1;
  • 4 190 951 585 ÷ 2 = 2 095 475 792 + 1;
  • 2 095 475 792 ÷ 2 = 1 047 737 896 + 0;
  • 1 047 737 896 ÷ 2 = 523 868 948 + 0;
  • 523 868 948 ÷ 2 = 261 934 474 + 0;
  • 261 934 474 ÷ 2 = 130 967 237 + 0;
  • 130 967 237 ÷ 2 = 65 483 618 + 1;
  • 65 483 618 ÷ 2 = 32 741 809 + 0;
  • 32 741 809 ÷ 2 = 16 370 904 + 1;
  • 16 370 904 ÷ 2 = 8 185 452 + 0;
  • 8 185 452 ÷ 2 = 4 092 726 + 0;
  • 4 092 726 ÷ 2 = 2 046 363 + 0;
  • 2 046 363 ÷ 2 = 1 023 181 + 1;
  • 1 023 181 ÷ 2 = 511 590 + 1;
  • 511 590 ÷ 2 = 255 795 + 0;
  • 255 795 ÷ 2 = 127 897 + 1;
  • 127 897 ÷ 2 = 63 948 + 1;
  • 63 948 ÷ 2 = 31 974 + 0;
  • 31 974 ÷ 2 = 15 987 + 0;
  • 15 987 ÷ 2 = 7 993 + 1;
  • 7 993 ÷ 2 = 3 996 + 1;
  • 3 996 ÷ 2 = 1 998 + 0;
  • 1 998 ÷ 2 = 999 + 0;
  • 999 ÷ 2 = 499 + 1;
  • 499 ÷ 2 = 249 + 1;
  • 249 ÷ 2 = 124 + 1;
  • 124 ÷ 2 = 62 + 0;
  • 62 ÷ 2 = 31 + 0;
  • 31 ÷ 2 = 15 + 1;
  • 15 ÷ 2 = 7 + 1;
  • 7 ÷ 2 = 3 + 1;
  • 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.

8 999 999 999 999 999 937(10) = 111 1100 1110 0110 0110 1100 0101 0000 1110 0010 1000 0011 1111 1111 1100 0001(2)


3. 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.


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:


8 999 999 999 999 999 937(10) Base 10 integer number converted and written as a signed binary code (in base 2):

8 999 999 999 999 999 937(10) = 0111 1100 1110 0110 0110 1100 0101 0000 1110 0010 1000 0011 1111 1111 1100 0001

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