Convert 1 111 111 000 101 271 to a Signed Binary in Two's (2's) Complement Representation

How to convert decimal number 1 111 111 000 101 271(10) to a signed binary in two's (2's) complement representation

What are the steps to convert decimal number
1 111 111 000 101 271 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;
  • 1 111 111 000 101 271 ÷ 2 = 555 555 500 050 635 + 1;
  • 555 555 500 050 635 ÷ 2 = 277 777 750 025 317 + 1;
  • 277 777 750 025 317 ÷ 2 = 138 888 875 012 658 + 1;
  • 138 888 875 012 658 ÷ 2 = 69 444 437 506 329 + 0;
  • 69 444 437 506 329 ÷ 2 = 34 722 218 753 164 + 1;
  • 34 722 218 753 164 ÷ 2 = 17 361 109 376 582 + 0;
  • 17 361 109 376 582 ÷ 2 = 8 680 554 688 291 + 0;
  • 8 680 554 688 291 ÷ 2 = 4 340 277 344 145 + 1;
  • 4 340 277 344 145 ÷ 2 = 2 170 138 672 072 + 1;
  • 2 170 138 672 072 ÷ 2 = 1 085 069 336 036 + 0;
  • 1 085 069 336 036 ÷ 2 = 542 534 668 018 + 0;
  • 542 534 668 018 ÷ 2 = 271 267 334 009 + 0;
  • 271 267 334 009 ÷ 2 = 135 633 667 004 + 1;
  • 135 633 667 004 ÷ 2 = 67 816 833 502 + 0;
  • 67 816 833 502 ÷ 2 = 33 908 416 751 + 0;
  • 33 908 416 751 ÷ 2 = 16 954 208 375 + 1;
  • 16 954 208 375 ÷ 2 = 8 477 104 187 + 1;
  • 8 477 104 187 ÷ 2 = 4 238 552 093 + 1;
  • 4 238 552 093 ÷ 2 = 2 119 276 046 + 1;
  • 2 119 276 046 ÷ 2 = 1 059 638 023 + 0;
  • 1 059 638 023 ÷ 2 = 529 819 011 + 1;
  • 529 819 011 ÷ 2 = 264 909 505 + 1;
  • 264 909 505 ÷ 2 = 132 454 752 + 1;
  • 132 454 752 ÷ 2 = 66 227 376 + 0;
  • 66 227 376 ÷ 2 = 33 113 688 + 0;
  • 33 113 688 ÷ 2 = 16 556 844 + 0;
  • 16 556 844 ÷ 2 = 8 278 422 + 0;
  • 8 278 422 ÷ 2 = 4 139 211 + 0;
  • 4 139 211 ÷ 2 = 2 069 605 + 1;
  • 2 069 605 ÷ 2 = 1 034 802 + 1;
  • 1 034 802 ÷ 2 = 517 401 + 0;
  • 517 401 ÷ 2 = 258 700 + 1;
  • 258 700 ÷ 2 = 129 350 + 0;
  • 129 350 ÷ 2 = 64 675 + 0;
  • 64 675 ÷ 2 = 32 337 + 1;
  • 32 337 ÷ 2 = 16 168 + 1;
  • 16 168 ÷ 2 = 8 084 + 0;
  • 8 084 ÷ 2 = 4 042 + 0;
  • 4 042 ÷ 2 = 2 021 + 0;
  • 2 021 ÷ 2 = 1 010 + 1;
  • 1 010 ÷ 2 = 505 + 0;
  • 505 ÷ 2 = 252 + 1;
  • 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.

1 111 111 000 101 271(10) = 11 1111 0010 1000 1100 1011 0000 0111 0111 1001 0001 1001 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 1 111 111 000 101 271(10) converted to signed binary in two's complement representation:

1 111 111 000 101 271(10) = 0000 0000 0000 0011 1111 0010 1000 1100 1011 0000 0111 0111 1001 0001 1001 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