Convert 5 237 811 932 528 241 to a Signed Binary in Two's (2's) Complement Representation

How to convert decimal number 5 237 811 932 528 241(10) to a signed binary in two's (2's) complement representation

What are the steps to convert decimal number
5 237 811 932 528 241 to a signed binary in two's (2's) complement representation?

  • 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.

We stop when we get a quotient that is equal to zero.


  • division = quotient + remainder;
  • 5 237 811 932 528 241 ÷ 2 = 2 618 905 966 264 120 + 1;
  • 2 618 905 966 264 120 ÷ 2 = 1 309 452 983 132 060 + 0;
  • 1 309 452 983 132 060 ÷ 2 = 654 726 491 566 030 + 0;
  • 654 726 491 566 030 ÷ 2 = 327 363 245 783 015 + 0;
  • 327 363 245 783 015 ÷ 2 = 163 681 622 891 507 + 1;
  • 163 681 622 891 507 ÷ 2 = 81 840 811 445 753 + 1;
  • 81 840 811 445 753 ÷ 2 = 40 920 405 722 876 + 1;
  • 40 920 405 722 876 ÷ 2 = 20 460 202 861 438 + 0;
  • 20 460 202 861 438 ÷ 2 = 10 230 101 430 719 + 0;
  • 10 230 101 430 719 ÷ 2 = 5 115 050 715 359 + 1;
  • 5 115 050 715 359 ÷ 2 = 2 557 525 357 679 + 1;
  • 2 557 525 357 679 ÷ 2 = 1 278 762 678 839 + 1;
  • 1 278 762 678 839 ÷ 2 = 639 381 339 419 + 1;
  • 639 381 339 419 ÷ 2 = 319 690 669 709 + 1;
  • 319 690 669 709 ÷ 2 = 159 845 334 854 + 1;
  • 159 845 334 854 ÷ 2 = 79 922 667 427 + 0;
  • 79 922 667 427 ÷ 2 = 39 961 333 713 + 1;
  • 39 961 333 713 ÷ 2 = 19 980 666 856 + 1;
  • 19 980 666 856 ÷ 2 = 9 990 333 428 + 0;
  • 9 990 333 428 ÷ 2 = 4 995 166 714 + 0;
  • 4 995 166 714 ÷ 2 = 2 497 583 357 + 0;
  • 2 497 583 357 ÷ 2 = 1 248 791 678 + 1;
  • 1 248 791 678 ÷ 2 = 624 395 839 + 0;
  • 624 395 839 ÷ 2 = 312 197 919 + 1;
  • 312 197 919 ÷ 2 = 156 098 959 + 1;
  • 156 098 959 ÷ 2 = 78 049 479 + 1;
  • 78 049 479 ÷ 2 = 39 024 739 + 1;
  • 39 024 739 ÷ 2 = 19 512 369 + 1;
  • 19 512 369 ÷ 2 = 9 756 184 + 1;
  • 9 756 184 ÷ 2 = 4 878 092 + 0;
  • 4 878 092 ÷ 2 = 2 439 046 + 0;
  • 2 439 046 ÷ 2 = 1 219 523 + 0;
  • 1 219 523 ÷ 2 = 609 761 + 1;
  • 609 761 ÷ 2 = 304 880 + 1;
  • 304 880 ÷ 2 = 152 440 + 0;
  • 152 440 ÷ 2 = 76 220 + 0;
  • 76 220 ÷ 2 = 38 110 + 0;
  • 38 110 ÷ 2 = 19 055 + 0;
  • 19 055 ÷ 2 = 9 527 + 1;
  • 9 527 ÷ 2 = 4 763 + 1;
  • 4 763 ÷ 2 = 2 381 + 1;
  • 2 381 ÷ 2 = 1 190 + 1;
  • 1 190 ÷ 2 = 595 + 0;
  • 595 ÷ 2 = 297 + 1;
  • 297 ÷ 2 = 148 + 1;
  • 148 ÷ 2 = 74 + 0;
  • 74 ÷ 2 = 37 + 0;
  • 37 ÷ 2 = 18 + 1;
  • 18 ÷ 2 = 9 + 0;
  • 9 ÷ 2 = 4 + 1;
  • 4 ÷ 2 = 2 + 0;
  • 2 ÷ 2 = 1 + 0;
  • 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.

5 237 811 932 528 241(10) = 1 0010 1001 1011 1100 0011 0001 1111 1010 0011 0111 1110 0111 0001(2)

3. Determine the signed binary number bit length:

  • The base 2 number's actual length, in bits: 53.

  • 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) indicates 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, 53,

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.


Decimal Number 5 237 811 932 528 241(10) converted to signed binary in two's complement representation:

5 237 811 932 528 241(10) = 0000 0000 0001 0010 1001 1011 1100 0011 0001 1111 1010 0011 0111 1110 0111 0001

Spaces were used to group digits: for binary, by 4, for decimal, by 3.

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How to convert signed integers from decimal system to signed binary in two's complement representation

Follow the steps below to convert a signed base 10 integer number to signed binary in two's complement representation:

  • 1. If the number to be converted is negative, start with the positive version of the number.
  • 2. Divide repeatedly by 2 the positive representation of the integer number, keeping track of each remainder, until 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 must have 4, 8, 16, 32, 64, ... bit length (a power of 2) - if needed, add extra bits on 0 in front (to the left) of the base 2 number above, up to the required length, so that the first bit (the leftmost) will be 0, correctly representing a positive number.
  • 5. To get the negative integer number representation in signed binary one's complement, replace all 0 bits with 1s and all 1 bits with 0s (reversing the digits).
  • 6. To get the negative integer number, in signed binary two's complement representation, add 1 to the number above.

Example: convert the negative number -60 from the decimal system (base ten) to signed binary in two's complement:

  • 1. Start with the positive version of the number: |-60| = 60
  • 2. Divide repeatedly 60 by 2, keeping track of each remainder:
    • division = quotient + remainder
    • 60 ÷ 2 = 30 + 0
    • 30 ÷ 2 = 15 + 0
    • 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:
    60(10) = 11 1100(2)
  • 4. Bit length of base 2 representation number is 6, so the positive binary computer representation of a signed binary will take in this particular case 8 bits (the least power of 2 larger than 6) - add extra 0 digits in front of the base 2 number, up to the required length:
    60(10) = 0011 1100(2)
  • 5. To get the negative integer number representation in signed binary one's complement, replace all the 0 bits with 1s and all 1 bits with 0s (reversing the digits):
    !(0011 1100) = 1100 0011
  • 6. To get the negative integer number, signed binary in two's complement representation, add 1 to the number above:
    -60(10) = 1100 0011 + 1 = 1100 0100
  • Number -60(10), signed integer, converted from decimal system (base 10) to signed binary two's complement representation = 1100 0100