Convert 1 111 111 010 100 041 to a Signed Binary in Two's (2's) Complement Representation

How to convert decimal number 1 111 111 010 100 041(10) to a signed binary in two's (2's) complement representation

What are the steps to convert decimal number
1 111 111 010 100 041 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;
  • 1 111 111 010 100 041 ÷ 2 = 555 555 505 050 020 + 1;
  • 555 555 505 050 020 ÷ 2 = 277 777 752 525 010 + 0;
  • 277 777 752 525 010 ÷ 2 = 138 888 876 262 505 + 0;
  • 138 888 876 262 505 ÷ 2 = 69 444 438 131 252 + 1;
  • 69 444 438 131 252 ÷ 2 = 34 722 219 065 626 + 0;
  • 34 722 219 065 626 ÷ 2 = 17 361 109 532 813 + 0;
  • 17 361 109 532 813 ÷ 2 = 8 680 554 766 406 + 1;
  • 8 680 554 766 406 ÷ 2 = 4 340 277 383 203 + 0;
  • 4 340 277 383 203 ÷ 2 = 2 170 138 691 601 + 1;
  • 2 170 138 691 601 ÷ 2 = 1 085 069 345 800 + 1;
  • 1 085 069 345 800 ÷ 2 = 542 534 672 900 + 0;
  • 542 534 672 900 ÷ 2 = 271 267 336 450 + 0;
  • 271 267 336 450 ÷ 2 = 135 633 668 225 + 0;
  • 135 633 668 225 ÷ 2 = 67 816 834 112 + 1;
  • 67 816 834 112 ÷ 2 = 33 908 417 056 + 0;
  • 33 908 417 056 ÷ 2 = 16 954 208 528 + 0;
  • 16 954 208 528 ÷ 2 = 8 477 104 264 + 0;
  • 8 477 104 264 ÷ 2 = 4 238 552 132 + 0;
  • 4 238 552 132 ÷ 2 = 2 119 276 066 + 0;
  • 2 119 276 066 ÷ 2 = 1 059 638 033 + 0;
  • 1 059 638 033 ÷ 2 = 529 819 016 + 1;
  • 529 819 016 ÷ 2 = 264 909 508 + 0;
  • 264 909 508 ÷ 2 = 132 454 754 + 0;
  • 132 454 754 ÷ 2 = 66 227 377 + 0;
  • 66 227 377 ÷ 2 = 33 113 688 + 1;
  • 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.

1 111 111 010 100 041(10) = 11 1111 0010 1000 1100 1011 0001 0001 0000 0010 0011 0100 1001(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 1 111 111 010 100 041(10) converted to signed binary in two's complement representation:

1 111 111 010 100 041(10) = 0000 0000 0000 0011 1111 0010 1000 1100 1011 0001 0001 0000 0010 0011 0100 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