Convert 110 001 101 110 906 to a Signed Binary in Two's (2's) Complement Representation

How to convert decimal number 110 001 101 110 906(10) to a signed binary in two's (2's) complement representation

What are the steps to convert decimal number
110 001 101 110 906 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;
  • 110 001 101 110 906 ÷ 2 = 55 000 550 555 453 + 0;
  • 55 000 550 555 453 ÷ 2 = 27 500 275 277 726 + 1;
  • 27 500 275 277 726 ÷ 2 = 13 750 137 638 863 + 0;
  • 13 750 137 638 863 ÷ 2 = 6 875 068 819 431 + 1;
  • 6 875 068 819 431 ÷ 2 = 3 437 534 409 715 + 1;
  • 3 437 534 409 715 ÷ 2 = 1 718 767 204 857 + 1;
  • 1 718 767 204 857 ÷ 2 = 859 383 602 428 + 1;
  • 859 383 602 428 ÷ 2 = 429 691 801 214 + 0;
  • 429 691 801 214 ÷ 2 = 214 845 900 607 + 0;
  • 214 845 900 607 ÷ 2 = 107 422 950 303 + 1;
  • 107 422 950 303 ÷ 2 = 53 711 475 151 + 1;
  • 53 711 475 151 ÷ 2 = 26 855 737 575 + 1;
  • 26 855 737 575 ÷ 2 = 13 427 868 787 + 1;
  • 13 427 868 787 ÷ 2 = 6 713 934 393 + 1;
  • 6 713 934 393 ÷ 2 = 3 356 967 196 + 1;
  • 3 356 967 196 ÷ 2 = 1 678 483 598 + 0;
  • 1 678 483 598 ÷ 2 = 839 241 799 + 0;
  • 839 241 799 ÷ 2 = 419 620 899 + 1;
  • 419 620 899 ÷ 2 = 209 810 449 + 1;
  • 209 810 449 ÷ 2 = 104 905 224 + 1;
  • 104 905 224 ÷ 2 = 52 452 612 + 0;
  • 52 452 612 ÷ 2 = 26 226 306 + 0;
  • 26 226 306 ÷ 2 = 13 113 153 + 0;
  • 13 113 153 ÷ 2 = 6 556 576 + 1;
  • 6 556 576 ÷ 2 = 3 278 288 + 0;
  • 3 278 288 ÷ 2 = 1 639 144 + 0;
  • 1 639 144 ÷ 2 = 819 572 + 0;
  • 819 572 ÷ 2 = 409 786 + 0;
  • 409 786 ÷ 2 = 204 893 + 0;
  • 204 893 ÷ 2 = 102 446 + 1;
  • 102 446 ÷ 2 = 51 223 + 0;
  • 51 223 ÷ 2 = 25 611 + 1;
  • 25 611 ÷ 2 = 12 805 + 1;
  • 12 805 ÷ 2 = 6 402 + 1;
  • 6 402 ÷ 2 = 3 201 + 0;
  • 3 201 ÷ 2 = 1 600 + 1;
  • 1 600 ÷ 2 = 800 + 0;
  • 800 ÷ 2 = 400 + 0;
  • 400 ÷ 2 = 200 + 0;
  • 200 ÷ 2 = 100 + 0;
  • 100 ÷ 2 = 50 + 0;
  • 50 ÷ 2 = 25 + 0;
  • 25 ÷ 2 = 12 + 1;
  • 12 ÷ 2 = 6 + 0;
  • 6 ÷ 2 = 3 + 0;
  • 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.

110 001 101 110 906(10) = 110 0100 0000 1011 1010 0000 1000 1110 0111 1110 0111 1010(2)

3. Determine the signed binary number bit length:

  • The base 2 number's actual length, in bits: 47.

  • 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, 47,

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 110 001 101 110 906(10) converted to signed binary in two's complement representation:

110 001 101 110 906(10) = 0000 0000 0000 0000 0110 0100 0000 1011 1010 0000 1000 1110 0111 1110 0111 1010

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