Convert 36 028 795 421 899 831 to a Signed Binary (Base 2)

How to convert 36 028 795 421 899 831(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 36 028 795 421 899 831 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;
  • 36 028 795 421 899 831 ÷ 2 = 18 014 397 710 949 915 + 1;
  • 18 014 397 710 949 915 ÷ 2 = 9 007 198 855 474 957 + 1;
  • 9 007 198 855 474 957 ÷ 2 = 4 503 599 427 737 478 + 1;
  • 4 503 599 427 737 478 ÷ 2 = 2 251 799 713 868 739 + 0;
  • 2 251 799 713 868 739 ÷ 2 = 1 125 899 856 934 369 + 1;
  • 1 125 899 856 934 369 ÷ 2 = 562 949 928 467 184 + 1;
  • 562 949 928 467 184 ÷ 2 = 281 474 964 233 592 + 0;
  • 281 474 964 233 592 ÷ 2 = 140 737 482 116 796 + 0;
  • 140 737 482 116 796 ÷ 2 = 70 368 741 058 398 + 0;
  • 70 368 741 058 398 ÷ 2 = 35 184 370 529 199 + 0;
  • 35 184 370 529 199 ÷ 2 = 17 592 185 264 599 + 1;
  • 17 592 185 264 599 ÷ 2 = 8 796 092 632 299 + 1;
  • 8 796 092 632 299 ÷ 2 = 4 398 046 316 149 + 1;
  • 4 398 046 316 149 ÷ 2 = 2 199 023 158 074 + 1;
  • 2 199 023 158 074 ÷ 2 = 1 099 511 579 037 + 0;
  • 1 099 511 579 037 ÷ 2 = 549 755 789 518 + 1;
  • 549 755 789 518 ÷ 2 = 274 877 894 759 + 0;
  • 274 877 894 759 ÷ 2 = 137 438 947 379 + 1;
  • 137 438 947 379 ÷ 2 = 68 719 473 689 + 1;
  • 68 719 473 689 ÷ 2 = 34 359 736 844 + 1;
  • 34 359 736 844 ÷ 2 = 17 179 868 422 + 0;
  • 17 179 868 422 ÷ 2 = 8 589 934 211 + 0;
  • 8 589 934 211 ÷ 2 = 4 294 967 105 + 1;
  • 4 294 967 105 ÷ 2 = 2 147 483 552 + 1;
  • 2 147 483 552 ÷ 2 = 1 073 741 776 + 0;
  • 1 073 741 776 ÷ 2 = 536 870 888 + 0;
  • 536 870 888 ÷ 2 = 268 435 444 + 0;
  • 268 435 444 ÷ 2 = 134 217 722 + 0;
  • 134 217 722 ÷ 2 = 67 108 861 + 0;
  • 67 108 861 ÷ 2 = 33 554 430 + 1;
  • 33 554 430 ÷ 2 = 16 777 215 + 0;
  • 16 777 215 ÷ 2 = 8 388 607 + 1;
  • 8 388 607 ÷ 2 = 4 194 303 + 1;
  • 4 194 303 ÷ 2 = 2 097 151 + 1;
  • 2 097 151 ÷ 2 = 1 048 575 + 1;
  • 1 048 575 ÷ 2 = 524 287 + 1;
  • 524 287 ÷ 2 = 262 143 + 1;
  • 262 143 ÷ 2 = 131 071 + 1;
  • 131 071 ÷ 2 = 65 535 + 1;
  • 65 535 ÷ 2 = 32 767 + 1;
  • 32 767 ÷ 2 = 16 383 + 1;
  • 16 383 ÷ 2 = 8 191 + 1;
  • 8 191 ÷ 2 = 4 095 + 1;
  • 4 095 ÷ 2 = 2 047 + 1;
  • 2 047 ÷ 2 = 1 023 + 1;
  • 1 023 ÷ 2 = 511 + 1;
  • 511 ÷ 2 = 255 + 1;
  • 255 ÷ 2 = 127 + 1;
  • 127 ÷ 2 = 63 + 1;
  • 63 ÷ 2 = 31 + 1;
  • 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.

36 028 795 421 899 831(10) = 111 1111 1111 1111 1111 1111 1010 0000 1100 1110 1011 1100 0011 0111(2)


3. Determine the signed binary number bit length:

  • The base 2 number's actual length, in bits: 55.

  • 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, 55,

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:


36 028 795 421 899 831(10) Base 10 integer number converted and written as a signed binary code (in base 2):

36 028 795 421 899 831(10) = 0000 0000 0111 1111 1111 1111 1111 1111 1010 0000 1100 1110 1011 1100 0011 0111

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