Convert -1 311 768 467 463 789 952 to a Signed Binary in Two's (2's) Complement Representation

How to convert decimal number -1 311 768 467 463 789 952(10) to a signed binary in two's (2's) complement representation

What are the steps to convert decimal number
-1 311 768 467 463 789 952 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:

|-1 311 768 467 463 789 952| = 1 311 768 467 463 789 952

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;
  • 1 311 768 467 463 789 952 ÷ 2 = 655 884 233 731 894 976 + 0;
  • 655 884 233 731 894 976 ÷ 2 = 327 942 116 865 947 488 + 0;
  • 327 942 116 865 947 488 ÷ 2 = 163 971 058 432 973 744 + 0;
  • 163 971 058 432 973 744 ÷ 2 = 81 985 529 216 486 872 + 0;
  • 81 985 529 216 486 872 ÷ 2 = 40 992 764 608 243 436 + 0;
  • 40 992 764 608 243 436 ÷ 2 = 20 496 382 304 121 718 + 0;
  • 20 496 382 304 121 718 ÷ 2 = 10 248 191 152 060 859 + 0;
  • 10 248 191 152 060 859 ÷ 2 = 5 124 095 576 030 429 + 1;
  • 5 124 095 576 030 429 ÷ 2 = 2 562 047 788 015 214 + 1;
  • 2 562 047 788 015 214 ÷ 2 = 1 281 023 894 007 607 + 0;
  • 1 281 023 894 007 607 ÷ 2 = 640 511 947 003 803 + 1;
  • 640 511 947 003 803 ÷ 2 = 320 255 973 501 901 + 1;
  • 320 255 973 501 901 ÷ 2 = 160 127 986 750 950 + 1;
  • 160 127 986 750 950 ÷ 2 = 80 063 993 375 475 + 0;
  • 80 063 993 375 475 ÷ 2 = 40 031 996 687 737 + 1;
  • 40 031 996 687 737 ÷ 2 = 20 015 998 343 868 + 1;
  • 20 015 998 343 868 ÷ 2 = 10 007 999 171 934 + 0;
  • 10 007 999 171 934 ÷ 2 = 5 003 999 585 967 + 0;
  • 5 003 999 585 967 ÷ 2 = 2 501 999 792 983 + 1;
  • 2 501 999 792 983 ÷ 2 = 1 250 999 896 491 + 1;
  • 1 250 999 896 491 ÷ 2 = 625 499 948 245 + 1;
  • 625 499 948 245 ÷ 2 = 312 749 974 122 + 1;
  • 312 749 974 122 ÷ 2 = 156 374 987 061 + 0;
  • 156 374 987 061 ÷ 2 = 78 187 493 530 + 1;
  • 78 187 493 530 ÷ 2 = 39 093 746 765 + 0;
  • 39 093 746 765 ÷ 2 = 19 546 873 382 + 1;
  • 19 546 873 382 ÷ 2 = 9 773 436 691 + 0;
  • 9 773 436 691 ÷ 2 = 4 886 718 345 + 1;
  • 4 886 718 345 ÷ 2 = 2 443 359 172 + 1;
  • 2 443 359 172 ÷ 2 = 1 221 679 586 + 0;
  • 1 221 679 586 ÷ 2 = 610 839 793 + 0;
  • 610 839 793 ÷ 2 = 305 419 896 + 1;
  • 305 419 896 ÷ 2 = 152 709 948 + 0;
  • 152 709 948 ÷ 2 = 76 354 974 + 0;
  • 76 354 974 ÷ 2 = 38 177 487 + 0;
  • 38 177 487 ÷ 2 = 19 088 743 + 1;
  • 19 088 743 ÷ 2 = 9 544 371 + 1;
  • 9 544 371 ÷ 2 = 4 772 185 + 1;
  • 4 772 185 ÷ 2 = 2 386 092 + 1;
  • 2 386 092 ÷ 2 = 1 193 046 + 0;
  • 1 193 046 ÷ 2 = 596 523 + 0;
  • 596 523 ÷ 2 = 298 261 + 1;
  • 298 261 ÷ 2 = 149 130 + 1;
  • 149 130 ÷ 2 = 74 565 + 0;
  • 74 565 ÷ 2 = 37 282 + 1;
  • 37 282 ÷ 2 = 18 641 + 0;
  • 18 641 ÷ 2 = 9 320 + 1;
  • 9 320 ÷ 2 = 4 660 + 0;
  • 4 660 ÷ 2 = 2 330 + 0;
  • 2 330 ÷ 2 = 1 165 + 0;
  • 1 165 ÷ 2 = 582 + 1;
  • 582 ÷ 2 = 291 + 0;
  • 291 ÷ 2 = 145 + 1;
  • 145 ÷ 2 = 72 + 1;
  • 72 ÷ 2 = 36 + 0;
  • 36 ÷ 2 = 18 + 0;
  • 18 ÷ 2 = 9 + 0;
  • 9 ÷ 2 = 4 + 1;
  • 4 ÷ 2 = 2 + 0;
  • 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.

1 311 768 467 463 789 952(10) = 1 0010 0011 0100 0101 0110 0111 1000 1001 1010 1011 1100 1101 1101 1000 0000(2)

4. Determine the signed binary number bit length:

  • The base 2 number's actual length, in bits: 61.

  • 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, 61,

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.


1 311 768 467 463 789 952(10) = 0001 0010 0011 0100 0101 0110 0111 1000 1001 1010 1011 1100 1101 1101 1000 0000

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.

!(0001 0010 0011 0100 0101 0110 0111 1000 1001 1010 1011 1100 1101 1101 1000 0000)

= 1110 1101 1100 1011 1010 1001 1000 0111 0110 0101 0100 0011 0010 0010 0111 1111


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 1110 1101 1100 1011 1010 1001 1000 0111 0110 0101 0100 0011 0010 0010 0111 1111 (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):

-1 311 768 467 463 789 952 =

1110 1101 1100 1011 1010 1001 1000 0111 0110 0101 0100 0011 0010 0010 0111 1111 + 1


Decimal Number -1 311 768 467 463 789 952(10) converted to signed binary in two's complement representation:

-1 311 768 467 463 789 952(10) = 1110 1101 1100 1011 1010 1001 1000 0111 0110 0101 0100 0011 0010 0010 1000 0000

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