Convert -36 028 797 010 575 361 to a Signed Binary in Two's (2's) Complement Representation

How to convert decimal number -36 028 797 010 575 361(10) to a signed binary in two's (2's) complement representation

What are the steps to convert decimal number
-36 028 797 010 575 361 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:

|-36 028 797 010 575 361| = 36 028 797 010 575 361

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;
  • 36 028 797 010 575 361 ÷ 2 = 18 014 398 505 287 680 + 1;
  • 18 014 398 505 287 680 ÷ 2 = 9 007 199 252 643 840 + 0;
  • 9 007 199 252 643 840 ÷ 2 = 4 503 599 626 321 920 + 0;
  • 4 503 599 626 321 920 ÷ 2 = 2 251 799 813 160 960 + 0;
  • 2 251 799 813 160 960 ÷ 2 = 1 125 899 906 580 480 + 0;
  • 1 125 899 906 580 480 ÷ 2 = 562 949 953 290 240 + 0;
  • 562 949 953 290 240 ÷ 2 = 281 474 976 645 120 + 0;
  • 281 474 976 645 120 ÷ 2 = 140 737 488 322 560 + 0;
  • 140 737 488 322 560 ÷ 2 = 70 368 744 161 280 + 0;
  • 70 368 744 161 280 ÷ 2 = 35 184 372 080 640 + 0;
  • 35 184 372 080 640 ÷ 2 = 17 592 186 040 320 + 0;
  • 17 592 186 040 320 ÷ 2 = 8 796 093 020 160 + 0;
  • 8 796 093 020 160 ÷ 2 = 4 398 046 510 080 + 0;
  • 4 398 046 510 080 ÷ 2 = 2 199 023 255 040 + 0;
  • 2 199 023 255 040 ÷ 2 = 1 099 511 627 520 + 0;
  • 1 099 511 627 520 ÷ 2 = 549 755 813 760 + 0;
  • 549 755 813 760 ÷ 2 = 274 877 906 880 + 0;
  • 274 877 906 880 ÷ 2 = 137 438 953 440 + 0;
  • 137 438 953 440 ÷ 2 = 68 719 476 720 + 0;
  • 68 719 476 720 ÷ 2 = 34 359 738 360 + 0;
  • 34 359 738 360 ÷ 2 = 17 179 869 180 + 0;
  • 17 179 869 180 ÷ 2 = 8 589 934 590 + 0;
  • 8 589 934 590 ÷ 2 = 4 294 967 295 + 0;
  • 4 294 967 295 ÷ 2 = 2 147 483 647 + 1;
  • 2 147 483 647 ÷ 2 = 1 073 741 823 + 1;
  • 1 073 741 823 ÷ 2 = 536 870 911 + 1;
  • 536 870 911 ÷ 2 = 268 435 455 + 1;
  • 268 435 455 ÷ 2 = 134 217 727 + 1;
  • 134 217 727 ÷ 2 = 67 108 863 + 1;
  • 67 108 863 ÷ 2 = 33 554 431 + 1;
  • 33 554 431 ÷ 2 = 16 777 215 + 1;
  • 16 777 215 ÷ 2 = 8 388 607 + 1;
  • 8 388 607 ÷ 2 = 4 194 303 + 1;
  • 4 194 303 ÷ 2 = 2 097 151 + 1;
  • 2 097 151 ÷ 2 = 1 048 575 + 1;
  • 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.

36 028 797 010 575 361(10) = 111 1111 1111 1111 1111 1111 1111 1111 1000 0000 0000 0000 0000 0001(2)

4. Determine the signed binary number bit length:

  • The base 2 number's actual length, in bits: 55.

  • 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, 55,

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.


36 028 797 010 575 361(10) = 0000 0000 0111 1111 1111 1111 1111 1111 1111 1111 1000 0000 0000 0000 0000 0001

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.

!(0000 0000 0111 1111 1111 1111 1111 1111 1111 1111 1000 0000 0000 0000 0000 0001)

= 1111 1111 1000 0000 0000 0000 0000 0000 0000 0000 0111 1111 1111 1111 1111 1110


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 1111 1111 1000 0000 0000 0000 0000 0000 0000 0000 0111 1111 1111 1111 1111 1110 (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):

-36 028 797 010 575 361 =

1111 1111 1000 0000 0000 0000 0000 0000 0000 0000 0111 1111 1111 1111 1111 1110 + 1


Decimal Number -36 028 797 010 575 361(10) converted to signed binary in two's complement representation:

-36 028 797 010 575 361(10) = 1111 1111 1000 0000 0000 0000 0000 0000 0000 0000 0111 1111 1111 1111 1111 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