Convert 1 010 100 100 000 017 to a Signed Binary in Two's (2's) Complement Representation

How to convert decimal number 1 010 100 100 000 017(10) to a signed binary in two's (2's) complement representation

What are the steps to convert decimal number
1 010 100 100 000 017 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;
  • 1 010 100 100 000 017 ÷ 2 = 505 050 050 000 008 + 1;
  • 505 050 050 000 008 ÷ 2 = 252 525 025 000 004 + 0;
  • 252 525 025 000 004 ÷ 2 = 126 262 512 500 002 + 0;
  • 126 262 512 500 002 ÷ 2 = 63 131 256 250 001 + 0;
  • 63 131 256 250 001 ÷ 2 = 31 565 628 125 000 + 1;
  • 31 565 628 125 000 ÷ 2 = 15 782 814 062 500 + 0;
  • 15 782 814 062 500 ÷ 2 = 7 891 407 031 250 + 0;
  • 7 891 407 031 250 ÷ 2 = 3 945 703 515 625 + 0;
  • 3 945 703 515 625 ÷ 2 = 1 972 851 757 812 + 1;
  • 1 972 851 757 812 ÷ 2 = 986 425 878 906 + 0;
  • 986 425 878 906 ÷ 2 = 493 212 939 453 + 0;
  • 493 212 939 453 ÷ 2 = 246 606 469 726 + 1;
  • 246 606 469 726 ÷ 2 = 123 303 234 863 + 0;
  • 123 303 234 863 ÷ 2 = 61 651 617 431 + 1;
  • 61 651 617 431 ÷ 2 = 30 825 808 715 + 1;
  • 30 825 808 715 ÷ 2 = 15 412 904 357 + 1;
  • 15 412 904 357 ÷ 2 = 7 706 452 178 + 1;
  • 7 706 452 178 ÷ 2 = 3 853 226 089 + 0;
  • 3 853 226 089 ÷ 2 = 1 926 613 044 + 1;
  • 1 926 613 044 ÷ 2 = 963 306 522 + 0;
  • 963 306 522 ÷ 2 = 481 653 261 + 0;
  • 481 653 261 ÷ 2 = 240 826 630 + 1;
  • 240 826 630 ÷ 2 = 120 413 315 + 0;
  • 120 413 315 ÷ 2 = 60 206 657 + 1;
  • 60 206 657 ÷ 2 = 30 103 328 + 1;
  • 30 103 328 ÷ 2 = 15 051 664 + 0;
  • 15 051 664 ÷ 2 = 7 525 832 + 0;
  • 7 525 832 ÷ 2 = 3 762 916 + 0;
  • 3 762 916 ÷ 2 = 1 881 458 + 0;
  • 1 881 458 ÷ 2 = 940 729 + 0;
  • 940 729 ÷ 2 = 470 364 + 1;
  • 470 364 ÷ 2 = 235 182 + 0;
  • 235 182 ÷ 2 = 117 591 + 0;
  • 117 591 ÷ 2 = 58 795 + 1;
  • 58 795 ÷ 2 = 29 397 + 1;
  • 29 397 ÷ 2 = 14 698 + 1;
  • 14 698 ÷ 2 = 7 349 + 0;
  • 7 349 ÷ 2 = 3 674 + 1;
  • 3 674 ÷ 2 = 1 837 + 0;
  • 1 837 ÷ 2 = 918 + 1;
  • 918 ÷ 2 = 459 + 0;
  • 459 ÷ 2 = 229 + 1;
  • 229 ÷ 2 = 114 + 1;
  • 114 ÷ 2 = 57 + 0;
  • 57 ÷ 2 = 28 + 1;
  • 28 ÷ 2 = 14 + 0;
  • 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.

1 010 100 100 000 017(10) = 11 1001 0110 1010 1110 0100 0001 1010 0101 1110 1001 0001 0001(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 1 010 100 100 000 017(10) converted to signed binary in two's complement representation:

1 010 100 100 000 017(10) = 0000 0000 0000 0011 1001 0110 1010 1110 0100 0001 1010 0101 1110 1001 0001 0001

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