Convert -9 223 363 293 970 415 720 to a Signed Binary in Two's (2's) Complement Representation

How to convert decimal number -9 223 363 293 970 415 720(10) to a signed binary in two's (2's) complement representation

What are the steps to convert decimal number
-9 223 363 293 970 415 720 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:

|-9 223 363 293 970 415 720| = 9 223 363 293 970 415 720

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;
  • 9 223 363 293 970 415 720 ÷ 2 = 4 611 681 646 985 207 860 + 0;
  • 4 611 681 646 985 207 860 ÷ 2 = 2 305 840 823 492 603 930 + 0;
  • 2 305 840 823 492 603 930 ÷ 2 = 1 152 920 411 746 301 965 + 0;
  • 1 152 920 411 746 301 965 ÷ 2 = 576 460 205 873 150 982 + 1;
  • 576 460 205 873 150 982 ÷ 2 = 288 230 102 936 575 491 + 0;
  • 288 230 102 936 575 491 ÷ 2 = 144 115 051 468 287 745 + 1;
  • 144 115 051 468 287 745 ÷ 2 = 72 057 525 734 143 872 + 1;
  • 72 057 525 734 143 872 ÷ 2 = 36 028 762 867 071 936 + 0;
  • 36 028 762 867 071 936 ÷ 2 = 18 014 381 433 535 968 + 0;
  • 18 014 381 433 535 968 ÷ 2 = 9 007 190 716 767 984 + 0;
  • 9 007 190 716 767 984 ÷ 2 = 4 503 595 358 383 992 + 0;
  • 4 503 595 358 383 992 ÷ 2 = 2 251 797 679 191 996 + 0;
  • 2 251 797 679 191 996 ÷ 2 = 1 125 898 839 595 998 + 0;
  • 1 125 898 839 595 998 ÷ 2 = 562 949 419 797 999 + 0;
  • 562 949 419 797 999 ÷ 2 = 281 474 709 898 999 + 1;
  • 281 474 709 898 999 ÷ 2 = 140 737 354 949 499 + 1;
  • 140 737 354 949 499 ÷ 2 = 70 368 677 474 749 + 1;
  • 70 368 677 474 749 ÷ 2 = 35 184 338 737 374 + 1;
  • 35 184 338 737 374 ÷ 2 = 17 592 169 368 687 + 0;
  • 17 592 169 368 687 ÷ 2 = 8 796 084 684 343 + 1;
  • 8 796 084 684 343 ÷ 2 = 4 398 042 342 171 + 1;
  • 4 398 042 342 171 ÷ 2 = 2 199 021 171 085 + 1;
  • 2 199 021 171 085 ÷ 2 = 1 099 510 585 542 + 1;
  • 1 099 510 585 542 ÷ 2 = 549 755 292 771 + 0;
  • 549 755 292 771 ÷ 2 = 274 877 646 385 + 1;
  • 274 877 646 385 ÷ 2 = 137 438 823 192 + 1;
  • 137 438 823 192 ÷ 2 = 68 719 411 596 + 0;
  • 68 719 411 596 ÷ 2 = 34 359 705 798 + 0;
  • 34 359 705 798 ÷ 2 = 17 179 852 899 + 0;
  • 17 179 852 899 ÷ 2 = 8 589 926 449 + 1;
  • 8 589 926 449 ÷ 2 = 4 294 963 224 + 1;
  • 4 294 963 224 ÷ 2 = 2 147 481 612 + 0;
  • 2 147 481 612 ÷ 2 = 1 073 740 806 + 0;
  • 1 073 740 806 ÷ 2 = 536 870 403 + 0;
  • 536 870 403 ÷ 2 = 268 435 201 + 1;
  • 268 435 201 ÷ 2 = 134 217 600 + 1;
  • 134 217 600 ÷ 2 = 67 108 800 + 0;
  • 67 108 800 ÷ 2 = 33 554 400 + 0;
  • 33 554 400 ÷ 2 = 16 777 200 + 0;
  • 16 777 200 ÷ 2 = 8 388 600 + 0;
  • 8 388 600 ÷ 2 = 4 194 300 + 0;
  • 4 194 300 ÷ 2 = 2 097 150 + 0;
  • 2 097 150 ÷ 2 = 1 048 575 + 0;
  • 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;

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

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

9 223 363 293 970 415 720(10) = 111 1111 1111 1111 1111 1000 0000 1100 0110 0011 0111 1011 1100 0000 0110 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.


9 223 363 293 970 415 720(10) = 0111 1111 1111 1111 1111 1000 0000 1100 0110 0011 0111 1011 1100 0000 0110 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 1111 1111 1111 1000 0000 1100 0110 0011 0111 1011 1100 0000 0110 1000)

= 1000 0000 0000 0000 0000 0111 1111 0011 1001 1100 1000 0100 0011 1111 1001 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 0000 0000 0000 0111 1111 0011 1001 1100 1000 0100 0011 1111 1001 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):

-9 223 363 293 970 415 720 =

1000 0000 0000 0000 0000 0111 1111 0011 1001 1100 1000 0100 0011 1111 1001 0111 + 1


Decimal Number -9 223 363 293 970 415 720(10) converted to signed binary in two's complement representation:

-9 223 363 293 970 415 720(10) = 1000 0000 0000 0000 0000 0111 1111 0011 1001 1100 1000 0100 0011 1111 1001 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