Convert 110 000 001 111 160 to a Signed Binary in Two's (2's) Complement Representation

How to convert decimal number 110 000 001 111 160(10) to a signed binary in two's (2's) complement representation

What are the steps to convert decimal number
110 000 001 111 160 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;
  • 110 000 001 111 160 ÷ 2 = 55 000 000 555 580 + 0;
  • 55 000 000 555 580 ÷ 2 = 27 500 000 277 790 + 0;
  • 27 500 000 277 790 ÷ 2 = 13 750 000 138 895 + 0;
  • 13 750 000 138 895 ÷ 2 = 6 875 000 069 447 + 1;
  • 6 875 000 069 447 ÷ 2 = 3 437 500 034 723 + 1;
  • 3 437 500 034 723 ÷ 2 = 1 718 750 017 361 + 1;
  • 1 718 750 017 361 ÷ 2 = 859 375 008 680 + 1;
  • 859 375 008 680 ÷ 2 = 429 687 504 340 + 0;
  • 429 687 504 340 ÷ 2 = 214 843 752 170 + 0;
  • 214 843 752 170 ÷ 2 = 107 421 876 085 + 0;
  • 107 421 876 085 ÷ 2 = 53 710 938 042 + 1;
  • 53 710 938 042 ÷ 2 = 26 855 469 021 + 0;
  • 26 855 469 021 ÷ 2 = 13 427 734 510 + 1;
  • 13 427 734 510 ÷ 2 = 6 713 867 255 + 0;
  • 6 713 867 255 ÷ 2 = 3 356 933 627 + 1;
  • 3 356 933 627 ÷ 2 = 1 678 466 813 + 1;
  • 1 678 466 813 ÷ 2 = 839 233 406 + 1;
  • 839 233 406 ÷ 2 = 419 616 703 + 0;
  • 419 616 703 ÷ 2 = 209 808 351 + 1;
  • 209 808 351 ÷ 2 = 104 904 175 + 1;
  • 104 904 175 ÷ 2 = 52 452 087 + 1;
  • 52 452 087 ÷ 2 = 26 226 043 + 1;
  • 26 226 043 ÷ 2 = 13 113 021 + 1;
  • 13 113 021 ÷ 2 = 6 556 510 + 1;
  • 6 556 510 ÷ 2 = 3 278 255 + 0;
  • 3 278 255 ÷ 2 = 1 639 127 + 1;
  • 1 639 127 ÷ 2 = 819 563 + 1;
  • 819 563 ÷ 2 = 409 781 + 1;
  • 409 781 ÷ 2 = 204 890 + 1;
  • 204 890 ÷ 2 = 102 445 + 0;
  • 102 445 ÷ 2 = 51 222 + 1;
  • 51 222 ÷ 2 = 25 611 + 0;
  • 25 611 ÷ 2 = 12 805 + 1;
  • 12 805 ÷ 2 = 6 402 + 1;
  • 6 402 ÷ 2 = 3 201 + 0;
  • 3 201 ÷ 2 = 1 600 + 1;
  • 1 600 ÷ 2 = 800 + 0;
  • 800 ÷ 2 = 400 + 0;
  • 400 ÷ 2 = 200 + 0;
  • 200 ÷ 2 = 100 + 0;
  • 100 ÷ 2 = 50 + 0;
  • 50 ÷ 2 = 25 + 0;
  • 25 ÷ 2 = 12 + 1;
  • 12 ÷ 2 = 6 + 0;
  • 6 ÷ 2 = 3 + 0;
  • 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.

110 000 001 111 160(10) = 110 0100 0000 1011 0101 1110 1111 1101 1101 0100 0111 1000(2)

3. Determine the signed binary number bit length:

  • The base 2 number's actual length, in bits: 47.

  • 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, 47,

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 110 000 001 111 160(10) converted to signed binary in two's complement representation:

110 000 001 111 160(10) = 0000 0000 0000 0000 0110 0100 0000 1011 0101 1110 1111 1101 1101 0100 0111 1000

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