Convert -41 231 713 988 641 to a Signed Binary in Two's (2's) Complement Representation

How to convert decimal number -41 231 713 988 641(10) to a signed binary in two's (2's) complement representation

What are the steps to convert decimal number
-41 231 713 988 641 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. Start with the positive version of the number:

|-41 231 713 988 641| = 41 231 713 988 641

2. 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;
  • 41 231 713 988 641 ÷ 2 = 20 615 856 994 320 + 1;
  • 20 615 856 994 320 ÷ 2 = 10 307 928 497 160 + 0;
  • 10 307 928 497 160 ÷ 2 = 5 153 964 248 580 + 0;
  • 5 153 964 248 580 ÷ 2 = 2 576 982 124 290 + 0;
  • 2 576 982 124 290 ÷ 2 = 1 288 491 062 145 + 0;
  • 1 288 491 062 145 ÷ 2 = 644 245 531 072 + 1;
  • 644 245 531 072 ÷ 2 = 322 122 765 536 + 0;
  • 322 122 765 536 ÷ 2 = 161 061 382 768 + 0;
  • 161 061 382 768 ÷ 2 = 80 530 691 384 + 0;
  • 80 530 691 384 ÷ 2 = 40 265 345 692 + 0;
  • 40 265 345 692 ÷ 2 = 20 132 672 846 + 0;
  • 20 132 672 846 ÷ 2 = 10 066 336 423 + 0;
  • 10 066 336 423 ÷ 2 = 5 033 168 211 + 1;
  • 5 033 168 211 ÷ 2 = 2 516 584 105 + 1;
  • 2 516 584 105 ÷ 2 = 1 258 292 052 + 1;
  • 1 258 292 052 ÷ 2 = 629 146 026 + 0;
  • 629 146 026 ÷ 2 = 314 573 013 + 0;
  • 314 573 013 ÷ 2 = 157 286 506 + 1;
  • 157 286 506 ÷ 2 = 78 643 253 + 0;
  • 78 643 253 ÷ 2 = 39 321 626 + 1;
  • 39 321 626 ÷ 2 = 19 660 813 + 0;
  • 19 660 813 ÷ 2 = 9 830 406 + 1;
  • 9 830 406 ÷ 2 = 4 915 203 + 0;
  • 4 915 203 ÷ 2 = 2 457 601 + 1;
  • 2 457 601 ÷ 2 = 1 228 800 + 1;
  • 1 228 800 ÷ 2 = 614 400 + 0;
  • 614 400 ÷ 2 = 307 200 + 0;
  • 307 200 ÷ 2 = 153 600 + 0;
  • 153 600 ÷ 2 = 76 800 + 0;
  • 76 800 ÷ 2 = 38 400 + 0;
  • 38 400 ÷ 2 = 19 200 + 0;
  • 19 200 ÷ 2 = 9 600 + 0;
  • 9 600 ÷ 2 = 4 800 + 0;
  • 4 800 ÷ 2 = 2 400 + 0;
  • 2 400 ÷ 2 = 1 200 + 0;
  • 1 200 ÷ 2 = 600 + 0;
  • 600 ÷ 2 = 300 + 0;
  • 300 ÷ 2 = 150 + 0;
  • 150 ÷ 2 = 75 + 0;
  • 75 ÷ 2 = 37 + 1;
  • 37 ÷ 2 = 18 + 1;
  • 18 ÷ 2 = 9 + 0;
  • 9 ÷ 2 = 4 + 1;
  • 4 ÷ 2 = 2 + 0;
  • 2 ÷ 2 = 1 + 0;
  • 1 ÷ 2 = 0 + 1;

3. Construct the base 2 representation of the positive number:

Take all the remainders starting from the bottom of the list constructed above.

41 231 713 988 641(10) = 10 0101 1000 0000 0000 0001 1010 1010 0111 0000 0010 0001(2)

4. Determine the signed binary number bit length:

  • The base 2 number's actual length, in bits: 46.

  • 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, 46,

3) so that the first bit (leftmost) could be zero
(we deal with a positive number at this moment)

=== is: 64.


5. 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.


41 231 713 988 641(10) = 0000 0000 0000 0000 0010 0101 1000 0000 0000 0001 1010 1010 0111 0000 0010 0001

6. Get the negative integer number representation. Part 1:

  • To write the negative integer number on 64 bits (8 Bytes), as a signed binary in one's complement representation, replace all the bits on 0 with 1s and all the bits set on 1 with 0s.

Reverse the digits, flip the digits:

Replace the bits set on 0 with 1s and the bits set on 1 with 0s.

!(0000 0000 0000 0000 0010 0101 1000 0000 0000 0001 1010 1010 0111 0000 0010 0001)

= 1111 1111 1111 1111 1101 1010 0111 1111 1111 1110 0101 0101 1000 1111 1101 1110


7. Get the negative integer number representation. Part 2:

  • To write the negative integer number on 64 bits (8 Bytes), as a signed binary in two's complement representation, add 1 to the number calculated above 1111 1111 1111 1111 1101 1010 0111 1111 1111 1110 0101 0101 1000 1111 1101 1110 (to the signed binary in one's complement representation).

Binary addition carries on a value of 2:

  • 0 + 0 = 0
  • 0 + 1 = 1
  • 1 + 1 = 10
  • 1 + 10 = 11
  • 1 + 11 = 100

Add 1 to the number calculated above
(to the signed binary number in one's complement representation):

-41 231 713 988 641 =

1111 1111 1111 1111 1101 1010 0111 1111 1111 1110 0101 0101 1000 1111 1101 1110 + 1


Decimal Number -41 231 713 988 641(10) converted to signed binary in two's complement representation:

-41 231 713 988 641(10) = 1111 1111 1111 1111 1101 1010 0111 1111 1111 1110 0101 0101 1000 1111 1101 1111

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