Convert 111 111 100 000 153 to a Signed Binary in Two's (2's) Complement Representation

How to convert decimal number 111 111 100 000 153(10) to a signed binary in two's (2's) complement representation

What are the steps to convert decimal number
111 111 100 000 153 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;
  • 111 111 100 000 153 ÷ 2 = 55 555 550 000 076 + 1;
  • 55 555 550 000 076 ÷ 2 = 27 777 775 000 038 + 0;
  • 27 777 775 000 038 ÷ 2 = 13 888 887 500 019 + 0;
  • 13 888 887 500 019 ÷ 2 = 6 944 443 750 009 + 1;
  • 6 944 443 750 009 ÷ 2 = 3 472 221 875 004 + 1;
  • 3 472 221 875 004 ÷ 2 = 1 736 110 937 502 + 0;
  • 1 736 110 937 502 ÷ 2 = 868 055 468 751 + 0;
  • 868 055 468 751 ÷ 2 = 434 027 734 375 + 1;
  • 434 027 734 375 ÷ 2 = 217 013 867 187 + 1;
  • 217 013 867 187 ÷ 2 = 108 506 933 593 + 1;
  • 108 506 933 593 ÷ 2 = 54 253 466 796 + 1;
  • 54 253 466 796 ÷ 2 = 27 126 733 398 + 0;
  • 27 126 733 398 ÷ 2 = 13 563 366 699 + 0;
  • 13 563 366 699 ÷ 2 = 6 781 683 349 + 1;
  • 6 781 683 349 ÷ 2 = 3 390 841 674 + 1;
  • 3 390 841 674 ÷ 2 = 1 695 420 837 + 0;
  • 1 695 420 837 ÷ 2 = 847 710 418 + 1;
  • 847 710 418 ÷ 2 = 423 855 209 + 0;
  • 423 855 209 ÷ 2 = 211 927 604 + 1;
  • 211 927 604 ÷ 2 = 105 963 802 + 0;
  • 105 963 802 ÷ 2 = 52 981 901 + 0;
  • 52 981 901 ÷ 2 = 26 490 950 + 1;
  • 26 490 950 ÷ 2 = 13 245 475 + 0;
  • 13 245 475 ÷ 2 = 6 622 737 + 1;
  • 6 622 737 ÷ 2 = 3 311 368 + 1;
  • 3 311 368 ÷ 2 = 1 655 684 + 0;
  • 1 655 684 ÷ 2 = 827 842 + 0;
  • 827 842 ÷ 2 = 413 921 + 0;
  • 413 921 ÷ 2 = 206 960 + 1;
  • 206 960 ÷ 2 = 103 480 + 0;
  • 103 480 ÷ 2 = 51 740 + 0;
  • 51 740 ÷ 2 = 25 870 + 0;
  • 25 870 ÷ 2 = 12 935 + 0;
  • 12 935 ÷ 2 = 6 467 + 1;
  • 6 467 ÷ 2 = 3 233 + 1;
  • 3 233 ÷ 2 = 1 616 + 1;
  • 1 616 ÷ 2 = 808 + 0;
  • 808 ÷ 2 = 404 + 0;
  • 404 ÷ 2 = 202 + 0;
  • 202 ÷ 2 = 101 + 0;
  • 101 ÷ 2 = 50 + 1;
  • 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.

111 111 100 000 153(10) = 110 0101 0000 1110 0001 0001 1010 0101 0110 0111 1001 1001(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 111 111 100 000 153(10) converted to signed binary in two's complement representation:

111 111 100 000 153(10) = 0000 0000 0000 0000 0110 0101 0000 1110 0001 0001 1010 0101 0110 0111 1001 1001

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