Convert 1 111 011 000 719 to a Signed Binary in Two's (2's) Complement Representation

How to convert decimal number 1 111 011 000 719(10) to a signed binary in two's (2's) complement representation

What are the steps to convert decimal number
1 111 011 000 719 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. 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;
  • 1 111 011 000 719 ÷ 2 = 555 505 500 359 + 1;
  • 555 505 500 359 ÷ 2 = 277 752 750 179 + 1;
  • 277 752 750 179 ÷ 2 = 138 876 375 089 + 1;
  • 138 876 375 089 ÷ 2 = 69 438 187 544 + 1;
  • 69 438 187 544 ÷ 2 = 34 719 093 772 + 0;
  • 34 719 093 772 ÷ 2 = 17 359 546 886 + 0;
  • 17 359 546 886 ÷ 2 = 8 679 773 443 + 0;
  • 8 679 773 443 ÷ 2 = 4 339 886 721 + 1;
  • 4 339 886 721 ÷ 2 = 2 169 943 360 + 1;
  • 2 169 943 360 ÷ 2 = 1 084 971 680 + 0;
  • 1 084 971 680 ÷ 2 = 542 485 840 + 0;
  • 542 485 840 ÷ 2 = 271 242 920 + 0;
  • 271 242 920 ÷ 2 = 135 621 460 + 0;
  • 135 621 460 ÷ 2 = 67 810 730 + 0;
  • 67 810 730 ÷ 2 = 33 905 365 + 0;
  • 33 905 365 ÷ 2 = 16 952 682 + 1;
  • 16 952 682 ÷ 2 = 8 476 341 + 0;
  • 8 476 341 ÷ 2 = 4 238 170 + 1;
  • 4 238 170 ÷ 2 = 2 119 085 + 0;
  • 2 119 085 ÷ 2 = 1 059 542 + 1;
  • 1 059 542 ÷ 2 = 529 771 + 0;
  • 529 771 ÷ 2 = 264 885 + 1;
  • 264 885 ÷ 2 = 132 442 + 1;
  • 132 442 ÷ 2 = 66 221 + 0;
  • 66 221 ÷ 2 = 33 110 + 1;
  • 33 110 ÷ 2 = 16 555 + 0;
  • 16 555 ÷ 2 = 8 277 + 1;
  • 8 277 ÷ 2 = 4 138 + 1;
  • 4 138 ÷ 2 = 2 069 + 0;
  • 2 069 ÷ 2 = 1 034 + 1;
  • 1 034 ÷ 2 = 517 + 0;
  • 517 ÷ 2 = 258 + 1;
  • 258 ÷ 2 = 129 + 0;
  • 129 ÷ 2 = 64 + 1;
  • 64 ÷ 2 = 32 + 0;
  • 32 ÷ 2 = 16 + 0;
  • 16 ÷ 2 = 8 + 0;
  • 8 ÷ 2 = 4 + 0;
  • 4 ÷ 2 = 2 + 0;
  • 2 ÷ 2 = 1 + 0;
  • 1 ÷ 2 = 0 + 1;

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

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

1 111 011 000 719(10) = 1 0000 0010 1010 1101 0110 1010 1000 0001 1000 1111(2)

3. Determine the signed binary number bit length:

  • The base 2 number's actual length, in bits: 41.

  • 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, 41,

3) so that the first bit (leftmost) could be zero
(we deal with a positive number at this moment)

=== is: 64.


4. 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.


Decimal Number 1 111 011 000 719(10) converted to signed binary in two's complement representation:

1 111 011 000 719(10) = 0000 0000 0000 0000 0000 0001 0000 0010 1010 1101 0110 1010 1000 0001 1000 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