Convert 1 001 001 000 991 to a Signed Binary in Two's (2's) Complement Representation

How to convert decimal number 1 001 001 000 991(10) to a signed binary in two's (2's) complement representation

What are the steps to convert decimal number
1 001 001 000 991 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;
  • 1 001 001 000 991 ÷ 2 = 500 500 500 495 + 1;
  • 500 500 500 495 ÷ 2 = 250 250 250 247 + 1;
  • 250 250 250 247 ÷ 2 = 125 125 125 123 + 1;
  • 125 125 125 123 ÷ 2 = 62 562 562 561 + 1;
  • 62 562 562 561 ÷ 2 = 31 281 281 280 + 1;
  • 31 281 281 280 ÷ 2 = 15 640 640 640 + 0;
  • 15 640 640 640 ÷ 2 = 7 820 320 320 + 0;
  • 7 820 320 320 ÷ 2 = 3 910 160 160 + 0;
  • 3 910 160 160 ÷ 2 = 1 955 080 080 + 0;
  • 1 955 080 080 ÷ 2 = 977 540 040 + 0;
  • 977 540 040 ÷ 2 = 488 770 020 + 0;
  • 488 770 020 ÷ 2 = 244 385 010 + 0;
  • 244 385 010 ÷ 2 = 122 192 505 + 0;
  • 122 192 505 ÷ 2 = 61 096 252 + 1;
  • 61 096 252 ÷ 2 = 30 548 126 + 0;
  • 30 548 126 ÷ 2 = 15 274 063 + 0;
  • 15 274 063 ÷ 2 = 7 637 031 + 1;
  • 7 637 031 ÷ 2 = 3 818 515 + 1;
  • 3 818 515 ÷ 2 = 1 909 257 + 1;
  • 1 909 257 ÷ 2 = 954 628 + 1;
  • 954 628 ÷ 2 = 477 314 + 0;
  • 477 314 ÷ 2 = 238 657 + 0;
  • 238 657 ÷ 2 = 119 328 + 1;
  • 119 328 ÷ 2 = 59 664 + 0;
  • 59 664 ÷ 2 = 29 832 + 0;
  • 29 832 ÷ 2 = 14 916 + 0;
  • 14 916 ÷ 2 = 7 458 + 0;
  • 7 458 ÷ 2 = 3 729 + 0;
  • 3 729 ÷ 2 = 1 864 + 1;
  • 1 864 ÷ 2 = 932 + 0;
  • 932 ÷ 2 = 466 + 0;
  • 466 ÷ 2 = 233 + 0;
  • 233 ÷ 2 = 116 + 1;
  • 116 ÷ 2 = 58 + 0;
  • 58 ÷ 2 = 29 + 0;
  • 29 ÷ 2 = 14 + 1;
  • 14 ÷ 2 = 7 + 0;
  • 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.

1 001 001 000 991(10) = 1110 1001 0001 0000 0100 1111 0010 0000 0001 1111(2)

3. Determine the signed binary number bit length:

  • The base 2 number's actual length, in bits: 40.

  • 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, 40,

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 1 001 001 000 991(10) converted to signed binary in two's complement representation:

1 001 001 000 991(10) = 0000 0000 0000 0000 0000 0000 1110 1001 0001 0000 0100 1111 0010 0000 0001 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