Convert 9 218 868 437 227 405 231 to a Signed Binary in Two's (2's) Complement Representation

How to convert decimal number 9 218 868 437 227 405 231(10) to a signed binary in two's (2's) complement representation

What are the steps to convert decimal number
9 218 868 437 227 405 231 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;
  • 9 218 868 437 227 405 231 ÷ 2 = 4 609 434 218 613 702 615 + 1;
  • 4 609 434 218 613 702 615 ÷ 2 = 2 304 717 109 306 851 307 + 1;
  • 2 304 717 109 306 851 307 ÷ 2 = 1 152 358 554 653 425 653 + 1;
  • 1 152 358 554 653 425 653 ÷ 2 = 576 179 277 326 712 826 + 1;
  • 576 179 277 326 712 826 ÷ 2 = 288 089 638 663 356 413 + 0;
  • 288 089 638 663 356 413 ÷ 2 = 144 044 819 331 678 206 + 1;
  • 144 044 819 331 678 206 ÷ 2 = 72 022 409 665 839 103 + 0;
  • 72 022 409 665 839 103 ÷ 2 = 36 011 204 832 919 551 + 1;
  • 36 011 204 832 919 551 ÷ 2 = 18 005 602 416 459 775 + 1;
  • 18 005 602 416 459 775 ÷ 2 = 9 002 801 208 229 887 + 1;
  • 9 002 801 208 229 887 ÷ 2 = 4 501 400 604 114 943 + 1;
  • 4 501 400 604 114 943 ÷ 2 = 2 250 700 302 057 471 + 1;
  • 2 250 700 302 057 471 ÷ 2 = 1 125 350 151 028 735 + 1;
  • 1 125 350 151 028 735 ÷ 2 = 562 675 075 514 367 + 1;
  • 562 675 075 514 367 ÷ 2 = 281 337 537 757 183 + 1;
  • 281 337 537 757 183 ÷ 2 = 140 668 768 878 591 + 1;
  • 140 668 768 878 591 ÷ 2 = 70 334 384 439 295 + 1;
  • 70 334 384 439 295 ÷ 2 = 35 167 192 219 647 + 1;
  • 35 167 192 219 647 ÷ 2 = 17 583 596 109 823 + 1;
  • 17 583 596 109 823 ÷ 2 = 8 791 798 054 911 + 1;
  • 8 791 798 054 911 ÷ 2 = 4 395 899 027 455 + 1;
  • 4 395 899 027 455 ÷ 2 = 2 197 949 513 727 + 1;
  • 2 197 949 513 727 ÷ 2 = 1 098 974 756 863 + 1;
  • 1 098 974 756 863 ÷ 2 = 549 487 378 431 + 1;
  • 549 487 378 431 ÷ 2 = 274 743 689 215 + 1;
  • 274 743 689 215 ÷ 2 = 137 371 844 607 + 1;
  • 137 371 844 607 ÷ 2 = 68 685 922 303 + 1;
  • 68 685 922 303 ÷ 2 = 34 342 961 151 + 1;
  • 34 342 961 151 ÷ 2 = 17 171 480 575 + 1;
  • 17 171 480 575 ÷ 2 = 8 585 740 287 + 1;
  • 8 585 740 287 ÷ 2 = 4 292 870 143 + 1;
  • 4 292 870 143 ÷ 2 = 2 146 435 071 + 1;
  • 2 146 435 071 ÷ 2 = 1 073 217 535 + 1;
  • 1 073 217 535 ÷ 2 = 536 608 767 + 1;
  • 536 608 767 ÷ 2 = 268 304 383 + 1;
  • 268 304 383 ÷ 2 = 134 152 191 + 1;
  • 134 152 191 ÷ 2 = 67 076 095 + 1;
  • 67 076 095 ÷ 2 = 33 538 047 + 1;
  • 33 538 047 ÷ 2 = 16 769 023 + 1;
  • 16 769 023 ÷ 2 = 8 384 511 + 1;
  • 8 384 511 ÷ 2 = 4 192 255 + 1;
  • 4 192 255 ÷ 2 = 2 096 127 + 1;
  • 2 096 127 ÷ 2 = 1 048 063 + 1;
  • 1 048 063 ÷ 2 = 524 031 + 1;
  • 524 031 ÷ 2 = 262 015 + 1;
  • 262 015 ÷ 2 = 131 007 + 1;
  • 131 007 ÷ 2 = 65 503 + 1;
  • 65 503 ÷ 2 = 32 751 + 1;
  • 32 751 ÷ 2 = 16 375 + 1;
  • 16 375 ÷ 2 = 8 187 + 1;
  • 8 187 ÷ 2 = 4 093 + 1;
  • 4 093 ÷ 2 = 2 046 + 1;
  • 2 046 ÷ 2 = 1 023 + 0;
  • 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.

9 218 868 437 227 405 231(10) = 111 1111 1110 1111 1111 1111 1111 1111 1111 1111 1111 1111 1111 1111 1010 1111(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) 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, 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.


Decimal Number 9 218 868 437 227 405 231(10) converted to signed binary in two's complement representation:

9 218 868 437 227 405 231(10) = 0111 1111 1110 1111 1111 1111 1111 1111 1111 1111 1111 1111 1111 1111 1010 1111

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