Convert -9 151 384 811 561 092 136 to a Signed Binary in Two's (2's) Complement Representation

How to convert decimal number -9 151 384 811 561 092 136(10) to a signed binary in two's (2's) complement representation

What are the steps to convert decimal number
-9 151 384 811 561 092 136 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:

|-9 151 384 811 561 092 136| = 9 151 384 811 561 092 136

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;
  • 9 151 384 811 561 092 136 ÷ 2 = 4 575 692 405 780 546 068 + 0;
  • 4 575 692 405 780 546 068 ÷ 2 = 2 287 846 202 890 273 034 + 0;
  • 2 287 846 202 890 273 034 ÷ 2 = 1 143 923 101 445 136 517 + 0;
  • 1 143 923 101 445 136 517 ÷ 2 = 571 961 550 722 568 258 + 1;
  • 571 961 550 722 568 258 ÷ 2 = 285 980 775 361 284 129 + 0;
  • 285 980 775 361 284 129 ÷ 2 = 142 990 387 680 642 064 + 1;
  • 142 990 387 680 642 064 ÷ 2 = 71 495 193 840 321 032 + 0;
  • 71 495 193 840 321 032 ÷ 2 = 35 747 596 920 160 516 + 0;
  • 35 747 596 920 160 516 ÷ 2 = 17 873 798 460 080 258 + 0;
  • 17 873 798 460 080 258 ÷ 2 = 8 936 899 230 040 129 + 0;
  • 8 936 899 230 040 129 ÷ 2 = 4 468 449 615 020 064 + 1;
  • 4 468 449 615 020 064 ÷ 2 = 2 234 224 807 510 032 + 0;
  • 2 234 224 807 510 032 ÷ 2 = 1 117 112 403 755 016 + 0;
  • 1 117 112 403 755 016 ÷ 2 = 558 556 201 877 508 + 0;
  • 558 556 201 877 508 ÷ 2 = 279 278 100 938 754 + 0;
  • 279 278 100 938 754 ÷ 2 = 139 639 050 469 377 + 0;
  • 139 639 050 469 377 ÷ 2 = 69 819 525 234 688 + 1;
  • 69 819 525 234 688 ÷ 2 = 34 909 762 617 344 + 0;
  • 34 909 762 617 344 ÷ 2 = 17 454 881 308 672 + 0;
  • 17 454 881 308 672 ÷ 2 = 8 727 440 654 336 + 0;
  • 8 727 440 654 336 ÷ 2 = 4 363 720 327 168 + 0;
  • 4 363 720 327 168 ÷ 2 = 2 181 860 163 584 + 0;
  • 2 181 860 163 584 ÷ 2 = 1 090 930 081 792 + 0;
  • 1 090 930 081 792 ÷ 2 = 545 465 040 896 + 0;
  • 545 465 040 896 ÷ 2 = 272 732 520 448 + 0;
  • 272 732 520 448 ÷ 2 = 136 366 260 224 + 0;
  • 136 366 260 224 ÷ 2 = 68 183 130 112 + 0;
  • 68 183 130 112 ÷ 2 = 34 091 565 056 + 0;
  • 34 091 565 056 ÷ 2 = 17 045 782 528 + 0;
  • 17 045 782 528 ÷ 2 = 8 522 891 264 + 0;
  • 8 522 891 264 ÷ 2 = 4 261 445 632 + 0;
  • 4 261 445 632 ÷ 2 = 2 130 722 816 + 0;
  • 2 130 722 816 ÷ 2 = 1 065 361 408 + 0;
  • 1 065 361 408 ÷ 2 = 532 680 704 + 0;
  • 532 680 704 ÷ 2 = 266 340 352 + 0;
  • 266 340 352 ÷ 2 = 133 170 176 + 0;
  • 133 170 176 ÷ 2 = 66 585 088 + 0;
  • 66 585 088 ÷ 2 = 33 292 544 + 0;
  • 33 292 544 ÷ 2 = 16 646 272 + 0;
  • 16 646 272 ÷ 2 = 8 323 136 + 0;
  • 8 323 136 ÷ 2 = 4 161 568 + 0;
  • 4 161 568 ÷ 2 = 2 080 784 + 0;
  • 2 080 784 ÷ 2 = 1 040 392 + 0;
  • 1 040 392 ÷ 2 = 520 196 + 0;
  • 520 196 ÷ 2 = 260 098 + 0;
  • 260 098 ÷ 2 = 130 049 + 0;
  • 130 049 ÷ 2 = 65 024 + 1;
  • 65 024 ÷ 2 = 32 512 + 0;
  • 32 512 ÷ 2 = 16 256 + 0;
  • 16 256 ÷ 2 = 8 128 + 0;
  • 8 128 ÷ 2 = 4 064 + 0;
  • 4 064 ÷ 2 = 2 032 + 0;
  • 2 032 ÷ 2 = 1 016 + 0;
  • 1 016 ÷ 2 = 508 + 0;
  • 508 ÷ 2 = 254 + 0;
  • 254 ÷ 2 = 127 + 0;
  • 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.

9 151 384 811 561 092 136(10) = 111 1111 0000 0000 0100 0000 0000 0000 0000 0000 0000 0001 0000 0100 0010 1000(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.


9 151 384 811 561 092 136(10) = 0111 1111 0000 0000 0100 0000 0000 0000 0000 0000 0000 0001 0000 0100 0010 1000

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 0000 0000 0100 0000 0000 0000 0000 0000 0000 0001 0000 0100 0010 1000)

= 1000 0000 1111 1111 1011 1111 1111 1111 1111 1111 1111 1110 1111 1011 1101 0111


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 1111 1111 1011 1111 1111 1111 1111 1111 1111 1110 1111 1011 1101 0111 (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):

-9 151 384 811 561 092 136 =

1000 0000 1111 1111 1011 1111 1111 1111 1111 1111 1111 1110 1111 1011 1101 0111 + 1


Decimal Number -9 151 384 811 561 092 136(10) converted to signed binary in two's complement representation:

-9 151 384 811 561 092 136(10) = 1000 0000 1111 1111 1011 1111 1111 1111 1111 1111 1111 1110 1111 1011 1101 1000

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