Convert 1 110 111 011 101 115 to a Signed Binary in Two's (2's) Complement Representation

How to convert decimal number 1 110 111 011 101 115(10) to a signed binary in two's (2's) complement representation

What are the steps to convert decimal number
1 110 111 011 101 115 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;
  • 1 110 111 011 101 115 ÷ 2 = 555 055 505 550 557 + 1;
  • 555 055 505 550 557 ÷ 2 = 277 527 752 775 278 + 1;
  • 277 527 752 775 278 ÷ 2 = 138 763 876 387 639 + 0;
  • 138 763 876 387 639 ÷ 2 = 69 381 938 193 819 + 1;
  • 69 381 938 193 819 ÷ 2 = 34 690 969 096 909 + 1;
  • 34 690 969 096 909 ÷ 2 = 17 345 484 548 454 + 1;
  • 17 345 484 548 454 ÷ 2 = 8 672 742 274 227 + 0;
  • 8 672 742 274 227 ÷ 2 = 4 336 371 137 113 + 1;
  • 4 336 371 137 113 ÷ 2 = 2 168 185 568 556 + 1;
  • 2 168 185 568 556 ÷ 2 = 1 084 092 784 278 + 0;
  • 1 084 092 784 278 ÷ 2 = 542 046 392 139 + 0;
  • 542 046 392 139 ÷ 2 = 271 023 196 069 + 1;
  • 271 023 196 069 ÷ 2 = 135 511 598 034 + 1;
  • 135 511 598 034 ÷ 2 = 67 755 799 017 + 0;
  • 67 755 799 017 ÷ 2 = 33 877 899 508 + 1;
  • 33 877 899 508 ÷ 2 = 16 938 949 754 + 0;
  • 16 938 949 754 ÷ 2 = 8 469 474 877 + 0;
  • 8 469 474 877 ÷ 2 = 4 234 737 438 + 1;
  • 4 234 737 438 ÷ 2 = 2 117 368 719 + 0;
  • 2 117 368 719 ÷ 2 = 1 058 684 359 + 1;
  • 1 058 684 359 ÷ 2 = 529 342 179 + 1;
  • 529 342 179 ÷ 2 = 264 671 089 + 1;
  • 264 671 089 ÷ 2 = 132 335 544 + 1;
  • 132 335 544 ÷ 2 = 66 167 772 + 0;
  • 66 167 772 ÷ 2 = 33 083 886 + 0;
  • 33 083 886 ÷ 2 = 16 541 943 + 0;
  • 16 541 943 ÷ 2 = 8 270 971 + 1;
  • 8 270 971 ÷ 2 = 4 135 485 + 1;
  • 4 135 485 ÷ 2 = 2 067 742 + 1;
  • 2 067 742 ÷ 2 = 1 033 871 + 0;
  • 1 033 871 ÷ 2 = 516 935 + 1;
  • 516 935 ÷ 2 = 258 467 + 1;
  • 258 467 ÷ 2 = 129 233 + 1;
  • 129 233 ÷ 2 = 64 616 + 1;
  • 64 616 ÷ 2 = 32 308 + 0;
  • 32 308 ÷ 2 = 16 154 + 0;
  • 16 154 ÷ 2 = 8 077 + 0;
  • 8 077 ÷ 2 = 4 038 + 1;
  • 4 038 ÷ 2 = 2 019 + 0;
  • 2 019 ÷ 2 = 1 009 + 1;
  • 1 009 ÷ 2 = 504 + 1;
  • 504 ÷ 2 = 252 + 0;
  • 252 ÷ 2 = 126 + 0;
  • 126 ÷ 2 = 63 + 0;
  • 63 ÷ 2 = 31 + 1;
  • 31 ÷ 2 = 15 + 1;
  • 15 ÷ 2 = 7 + 1;
  • 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.

1 110 111 011 101 115(10) = 11 1111 0001 1010 0011 1101 1100 0111 1010 0101 1001 1011 1011(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 1 110 111 011 101 115(10) converted to signed binary in two's complement representation:

1 110 111 011 101 115(10) = 0000 0000 0000 0011 1111 0001 1010 0011 1101 1100 0111 1010 0101 1001 1011 1011

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