Convert 1 001 110 001 000 119 to a Signed Binary in Two's (2's) Complement Representation

How to convert decimal number 1 001 110 001 000 119(10) to a signed binary in two's (2's) complement representation

What are the steps to convert decimal number
1 001 110 001 000 119 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;
  • 1 001 110 001 000 119 ÷ 2 = 500 555 000 500 059 + 1;
  • 500 555 000 500 059 ÷ 2 = 250 277 500 250 029 + 1;
  • 250 277 500 250 029 ÷ 2 = 125 138 750 125 014 + 1;
  • 125 138 750 125 014 ÷ 2 = 62 569 375 062 507 + 0;
  • 62 569 375 062 507 ÷ 2 = 31 284 687 531 253 + 1;
  • 31 284 687 531 253 ÷ 2 = 15 642 343 765 626 + 1;
  • 15 642 343 765 626 ÷ 2 = 7 821 171 882 813 + 0;
  • 7 821 171 882 813 ÷ 2 = 3 910 585 941 406 + 1;
  • 3 910 585 941 406 ÷ 2 = 1 955 292 970 703 + 0;
  • 1 955 292 970 703 ÷ 2 = 977 646 485 351 + 1;
  • 977 646 485 351 ÷ 2 = 488 823 242 675 + 1;
  • 488 823 242 675 ÷ 2 = 244 411 621 337 + 1;
  • 244 411 621 337 ÷ 2 = 122 205 810 668 + 1;
  • 122 205 810 668 ÷ 2 = 61 102 905 334 + 0;
  • 61 102 905 334 ÷ 2 = 30 551 452 667 + 0;
  • 30 551 452 667 ÷ 2 = 15 275 726 333 + 1;
  • 15 275 726 333 ÷ 2 = 7 637 863 166 + 1;
  • 7 637 863 166 ÷ 2 = 3 818 931 583 + 0;
  • 3 818 931 583 ÷ 2 = 1 909 465 791 + 1;
  • 1 909 465 791 ÷ 2 = 954 732 895 + 1;
  • 954 732 895 ÷ 2 = 477 366 447 + 1;
  • 477 366 447 ÷ 2 = 238 683 223 + 1;
  • 238 683 223 ÷ 2 = 119 341 611 + 1;
  • 119 341 611 ÷ 2 = 59 670 805 + 1;
  • 59 670 805 ÷ 2 = 29 835 402 + 1;
  • 29 835 402 ÷ 2 = 14 917 701 + 0;
  • 14 917 701 ÷ 2 = 7 458 850 + 1;
  • 7 458 850 ÷ 2 = 3 729 425 + 0;
  • 3 729 425 ÷ 2 = 1 864 712 + 1;
  • 1 864 712 ÷ 2 = 932 356 + 0;
  • 932 356 ÷ 2 = 466 178 + 0;
  • 466 178 ÷ 2 = 233 089 + 0;
  • 233 089 ÷ 2 = 116 544 + 1;
  • 116 544 ÷ 2 = 58 272 + 0;
  • 58 272 ÷ 2 = 29 136 + 0;
  • 29 136 ÷ 2 = 14 568 + 0;
  • 14 568 ÷ 2 = 7 284 + 0;
  • 7 284 ÷ 2 = 3 642 + 0;
  • 3 642 ÷ 2 = 1 821 + 0;
  • 1 821 ÷ 2 = 910 + 1;
  • 910 ÷ 2 = 455 + 0;
  • 455 ÷ 2 = 227 + 1;
  • 227 ÷ 2 = 113 + 1;
  • 113 ÷ 2 = 56 + 1;
  • 56 ÷ 2 = 28 + 0;
  • 28 ÷ 2 = 14 + 0;
  • 14 ÷ 2 = 7 + 0;
  • 7 ÷ 2 = 3 + 1;
  • 3 ÷ 2 = 1 + 1;
  • 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.

1 001 110 001 000 119(10) = 11 1000 1110 1000 0001 0001 0101 1111 1101 1001 1110 1011 0111(2)

3. Determine the signed binary number bit length:

  • The base 2 number's actual length, in bits: 50.

  • 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, 50,

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 1 001 110 001 000 119(10) converted to signed binary in two's complement representation:

1 001 110 001 000 119(10) = 0000 0000 0000 0011 1000 1110 1000 0001 0001 0101 1111 1101 1001 1110 1011 0111

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