Convert 10 977 675 159 to a Signed Binary in Two's (2's) Complement Representation

How to convert decimal number 10 977 675 159(10) to a signed binary in two's (2's) complement representation

What are the steps to convert decimal number
10 977 675 159 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;
  • 10 977 675 159 ÷ 2 = 5 488 837 579 + 1;
  • 5 488 837 579 ÷ 2 = 2 744 418 789 + 1;
  • 2 744 418 789 ÷ 2 = 1 372 209 394 + 1;
  • 1 372 209 394 ÷ 2 = 686 104 697 + 0;
  • 686 104 697 ÷ 2 = 343 052 348 + 1;
  • 343 052 348 ÷ 2 = 171 526 174 + 0;
  • 171 526 174 ÷ 2 = 85 763 087 + 0;
  • 85 763 087 ÷ 2 = 42 881 543 + 1;
  • 42 881 543 ÷ 2 = 21 440 771 + 1;
  • 21 440 771 ÷ 2 = 10 720 385 + 1;
  • 10 720 385 ÷ 2 = 5 360 192 + 1;
  • 5 360 192 ÷ 2 = 2 680 096 + 0;
  • 2 680 096 ÷ 2 = 1 340 048 + 0;
  • 1 340 048 ÷ 2 = 670 024 + 0;
  • 670 024 ÷ 2 = 335 012 + 0;
  • 335 012 ÷ 2 = 167 506 + 0;
  • 167 506 ÷ 2 = 83 753 + 0;
  • 83 753 ÷ 2 = 41 876 + 1;
  • 41 876 ÷ 2 = 20 938 + 0;
  • 20 938 ÷ 2 = 10 469 + 0;
  • 10 469 ÷ 2 = 5 234 + 1;
  • 5 234 ÷ 2 = 2 617 + 0;
  • 2 617 ÷ 2 = 1 308 + 1;
  • 1 308 ÷ 2 = 654 + 0;
  • 654 ÷ 2 = 327 + 0;
  • 327 ÷ 2 = 163 + 1;
  • 163 ÷ 2 = 81 + 1;
  • 81 ÷ 2 = 40 + 1;
  • 40 ÷ 2 = 20 + 0;
  • 20 ÷ 2 = 10 + 0;
  • 10 ÷ 2 = 5 + 0;
  • 5 ÷ 2 = 2 + 1;
  • 2 ÷ 2 = 1 + 0;
  • 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.

10 977 675 159(10) = 10 1000 1110 0101 0010 0000 0111 1001 0111(2)

3. Determine the signed binary number bit length:

  • The base 2 number's actual length, in bits: 34.

  • 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, 34,

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 10 977 675 159(10) converted to signed binary in two's complement representation:

10 977 675 159(10) = 0000 0000 0000 0000 0000 0000 0000 0010 1000 1110 0101 0010 0000 0111 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