Convert -1 456 777 255 815 009 to a Signed Binary in Two's (2's) Complement Representation

How to convert decimal number -1 456 777 255 815 009(10) to a signed binary in two's (2's) complement representation

What are the steps to convert decimal number
-1 456 777 255 815 009 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:

|-1 456 777 255 815 009| = 1 456 777 255 815 009

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;
  • 1 456 777 255 815 009 ÷ 2 = 728 388 627 907 504 + 1;
  • 728 388 627 907 504 ÷ 2 = 364 194 313 953 752 + 0;
  • 364 194 313 953 752 ÷ 2 = 182 097 156 976 876 + 0;
  • 182 097 156 976 876 ÷ 2 = 91 048 578 488 438 + 0;
  • 91 048 578 488 438 ÷ 2 = 45 524 289 244 219 + 0;
  • 45 524 289 244 219 ÷ 2 = 22 762 144 622 109 + 1;
  • 22 762 144 622 109 ÷ 2 = 11 381 072 311 054 + 1;
  • 11 381 072 311 054 ÷ 2 = 5 690 536 155 527 + 0;
  • 5 690 536 155 527 ÷ 2 = 2 845 268 077 763 + 1;
  • 2 845 268 077 763 ÷ 2 = 1 422 634 038 881 + 1;
  • 1 422 634 038 881 ÷ 2 = 711 317 019 440 + 1;
  • 711 317 019 440 ÷ 2 = 355 658 509 720 + 0;
  • 355 658 509 720 ÷ 2 = 177 829 254 860 + 0;
  • 177 829 254 860 ÷ 2 = 88 914 627 430 + 0;
  • 88 914 627 430 ÷ 2 = 44 457 313 715 + 0;
  • 44 457 313 715 ÷ 2 = 22 228 656 857 + 1;
  • 22 228 656 857 ÷ 2 = 11 114 328 428 + 1;
  • 11 114 328 428 ÷ 2 = 5 557 164 214 + 0;
  • 5 557 164 214 ÷ 2 = 2 778 582 107 + 0;
  • 2 778 582 107 ÷ 2 = 1 389 291 053 + 1;
  • 1 389 291 053 ÷ 2 = 694 645 526 + 1;
  • 694 645 526 ÷ 2 = 347 322 763 + 0;
  • 347 322 763 ÷ 2 = 173 661 381 + 1;
  • 173 661 381 ÷ 2 = 86 830 690 + 1;
  • 86 830 690 ÷ 2 = 43 415 345 + 0;
  • 43 415 345 ÷ 2 = 21 707 672 + 1;
  • 21 707 672 ÷ 2 = 10 853 836 + 0;
  • 10 853 836 ÷ 2 = 5 426 918 + 0;
  • 5 426 918 ÷ 2 = 2 713 459 + 0;
  • 2 713 459 ÷ 2 = 1 356 729 + 1;
  • 1 356 729 ÷ 2 = 678 364 + 1;
  • 678 364 ÷ 2 = 339 182 + 0;
  • 339 182 ÷ 2 = 169 591 + 0;
  • 169 591 ÷ 2 = 84 795 + 1;
  • 84 795 ÷ 2 = 42 397 + 1;
  • 42 397 ÷ 2 = 21 198 + 1;
  • 21 198 ÷ 2 = 10 599 + 0;
  • 10 599 ÷ 2 = 5 299 + 1;
  • 5 299 ÷ 2 = 2 649 + 1;
  • 2 649 ÷ 2 = 1 324 + 1;
  • 1 324 ÷ 2 = 662 + 0;
  • 662 ÷ 2 = 331 + 0;
  • 331 ÷ 2 = 165 + 1;
  • 165 ÷ 2 = 82 + 1;
  • 82 ÷ 2 = 41 + 0;
  • 41 ÷ 2 = 20 + 1;
  • 20 ÷ 2 = 10 + 0;
  • 10 ÷ 2 = 5 + 0;
  • 5 ÷ 2 = 2 + 1;
  • 2 ÷ 2 = 1 + 0;
  • 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.

1 456 777 255 815 009(10) = 101 0010 1100 1110 1110 0110 0010 1101 1001 1000 0111 0110 0001(2)

4. Determine the signed binary number bit length:

  • The base 2 number's actual length, in bits: 51.

  • 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, 51,

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.


1 456 777 255 815 009(10) = 0000 0000 0000 0101 0010 1100 1110 1110 0110 0010 1101 1001 1000 0111 0110 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 0000 0101 0010 1100 1110 1110 0110 0010 1101 1001 1000 0111 0110 0001)

= 1111 1111 1111 1010 1101 0011 0001 0001 1001 1101 0010 0110 0111 1000 1001 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 1111 1010 1101 0011 0001 0001 1001 1101 0010 0110 0111 1000 1001 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):

-1 456 777 255 815 009 =

1111 1111 1111 1010 1101 0011 0001 0001 1001 1101 0010 0110 0111 1000 1001 1110 + 1


Decimal Number -1 456 777 255 815 009(10) converted to signed binary in two's complement representation:

-1 456 777 255 815 009(10) = 1111 1111 1111 1010 1101 0011 0001 0001 1001 1101 0010 0110 0111 1000 1001 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