Convert -9 218 868 437 227 405 454 to a Signed Binary in Two's (2's) Complement Representation

How to convert decimal number -9 218 868 437 227 405 454(10) to a signed binary in two's (2's) complement representation

What are the steps to convert decimal number
-9 218 868 437 227 405 454 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. Start with the positive version of the number:

|-9 218 868 437 227 405 454| = 9 218 868 437 227 405 454

2. 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;
  • 9 218 868 437 227 405 454 ÷ 2 = 4 609 434 218 613 702 727 + 0;
  • 4 609 434 218 613 702 727 ÷ 2 = 2 304 717 109 306 851 363 + 1;
  • 2 304 717 109 306 851 363 ÷ 2 = 1 152 358 554 653 425 681 + 1;
  • 1 152 358 554 653 425 681 ÷ 2 = 576 179 277 326 712 840 + 1;
  • 576 179 277 326 712 840 ÷ 2 = 288 089 638 663 356 420 + 0;
  • 288 089 638 663 356 420 ÷ 2 = 144 044 819 331 678 210 + 0;
  • 144 044 819 331 678 210 ÷ 2 = 72 022 409 665 839 105 + 0;
  • 72 022 409 665 839 105 ÷ 2 = 36 011 204 832 919 552 + 1;
  • 36 011 204 832 919 552 ÷ 2 = 18 005 602 416 459 776 + 0;
  • 18 005 602 416 459 776 ÷ 2 = 9 002 801 208 229 888 + 0;
  • 9 002 801 208 229 888 ÷ 2 = 4 501 400 604 114 944 + 0;
  • 4 501 400 604 114 944 ÷ 2 = 2 250 700 302 057 472 + 0;
  • 2 250 700 302 057 472 ÷ 2 = 1 125 350 151 028 736 + 0;
  • 1 125 350 151 028 736 ÷ 2 = 562 675 075 514 368 + 0;
  • 562 675 075 514 368 ÷ 2 = 281 337 537 757 184 + 0;
  • 281 337 537 757 184 ÷ 2 = 140 668 768 878 592 + 0;
  • 140 668 768 878 592 ÷ 2 = 70 334 384 439 296 + 0;
  • 70 334 384 439 296 ÷ 2 = 35 167 192 219 648 + 0;
  • 35 167 192 219 648 ÷ 2 = 17 583 596 109 824 + 0;
  • 17 583 596 109 824 ÷ 2 = 8 791 798 054 912 + 0;
  • 8 791 798 054 912 ÷ 2 = 4 395 899 027 456 + 0;
  • 4 395 899 027 456 ÷ 2 = 2 197 949 513 728 + 0;
  • 2 197 949 513 728 ÷ 2 = 1 098 974 756 864 + 0;
  • 1 098 974 756 864 ÷ 2 = 549 487 378 432 + 0;
  • 549 487 378 432 ÷ 2 = 274 743 689 216 + 0;
  • 274 743 689 216 ÷ 2 = 137 371 844 608 + 0;
  • 137 371 844 608 ÷ 2 = 68 685 922 304 + 0;
  • 68 685 922 304 ÷ 2 = 34 342 961 152 + 0;
  • 34 342 961 152 ÷ 2 = 17 171 480 576 + 0;
  • 17 171 480 576 ÷ 2 = 8 585 740 288 + 0;
  • 8 585 740 288 ÷ 2 = 4 292 870 144 + 0;
  • 4 292 870 144 ÷ 2 = 2 146 435 072 + 0;
  • 2 146 435 072 ÷ 2 = 1 073 217 536 + 0;
  • 1 073 217 536 ÷ 2 = 536 608 768 + 0;
  • 536 608 768 ÷ 2 = 268 304 384 + 0;
  • 268 304 384 ÷ 2 = 134 152 192 + 0;
  • 134 152 192 ÷ 2 = 67 076 096 + 0;
  • 67 076 096 ÷ 2 = 33 538 048 + 0;
  • 33 538 048 ÷ 2 = 16 769 024 + 0;
  • 16 769 024 ÷ 2 = 8 384 512 + 0;
  • 8 384 512 ÷ 2 = 4 192 256 + 0;
  • 4 192 256 ÷ 2 = 2 096 128 + 0;
  • 2 096 128 ÷ 2 = 1 048 064 + 0;
  • 1 048 064 ÷ 2 = 524 032 + 0;
  • 524 032 ÷ 2 = 262 016 + 0;
  • 262 016 ÷ 2 = 131 008 + 0;
  • 131 008 ÷ 2 = 65 504 + 0;
  • 65 504 ÷ 2 = 32 752 + 0;
  • 32 752 ÷ 2 = 16 376 + 0;
  • 16 376 ÷ 2 = 8 188 + 0;
  • 8 188 ÷ 2 = 4 094 + 0;
  • 4 094 ÷ 2 = 2 047 + 0;
  • 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;

3. Construct the base 2 representation of the positive number:

Take all the remainders starting from the bottom of the list constructed above.

9 218 868 437 227 405 454(10) = 111 1111 1111 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 1000 1110(2)

4. 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.


5. 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.


9 218 868 437 227 405 454(10) = 0111 1111 1111 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 1000 1110

6. Get the negative integer number representation. Part 1:

  • To write the negative integer number on 64 bits (8 Bytes), as a signed binary in one's complement representation, replace all the bits on 0 with 1s and all the bits set on 1 with 0s.

Reverse the digits, flip the digits:

Replace the bits set on 0 with 1s and the bits set on 1 with 0s.

!(0111 1111 1111 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 1000 1110)

= 1000 0000 0000 1111 1111 1111 1111 1111 1111 1111 1111 1111 1111 1111 0111 0001


7. Get the negative integer number representation. Part 2:

  • To write the negative integer number on 64 bits (8 Bytes), as a signed binary in two's complement representation, add 1 to the number calculated above 1000 0000 0000 1111 1111 1111 1111 1111 1111 1111 1111 1111 1111 1111 0111 0001 (to the signed binary in one's complement representation).

Binary addition carries on a value of 2:

  • 0 + 0 = 0
  • 0 + 1 = 1
  • 1 + 1 = 10
  • 1 + 10 = 11
  • 1 + 11 = 100

Add 1 to the number calculated above
(to the signed binary number in one's complement representation):

-9 218 868 437 227 405 454 =

1000 0000 0000 1111 1111 1111 1111 1111 1111 1111 1111 1111 1111 1111 0111 0001 + 1


Decimal Number -9 218 868 437 227 405 454(10) converted to signed binary in two's complement representation:

-9 218 868 437 227 405 454(10) = 1000 0000 0000 1111 1111 1111 1111 1111 1111 1111 1111 1111 1111 1111 0111 0010

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