Convert 100 000 000 039 137 to a Signed Binary in Two's (2's) Complement Representation

How to convert decimal number 100 000 000 039 137(10) to a signed binary in two's (2's) complement representation

What are the steps to convert decimal number
100 000 000 039 137 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;
  • 100 000 000 039 137 ÷ 2 = 50 000 000 019 568 + 1;
  • 50 000 000 019 568 ÷ 2 = 25 000 000 009 784 + 0;
  • 25 000 000 009 784 ÷ 2 = 12 500 000 004 892 + 0;
  • 12 500 000 004 892 ÷ 2 = 6 250 000 002 446 + 0;
  • 6 250 000 002 446 ÷ 2 = 3 125 000 001 223 + 0;
  • 3 125 000 001 223 ÷ 2 = 1 562 500 000 611 + 1;
  • 1 562 500 000 611 ÷ 2 = 781 250 000 305 + 1;
  • 781 250 000 305 ÷ 2 = 390 625 000 152 + 1;
  • 390 625 000 152 ÷ 2 = 195 312 500 076 + 0;
  • 195 312 500 076 ÷ 2 = 97 656 250 038 + 0;
  • 97 656 250 038 ÷ 2 = 48 828 125 019 + 0;
  • 48 828 125 019 ÷ 2 = 24 414 062 509 + 1;
  • 24 414 062 509 ÷ 2 = 12 207 031 254 + 1;
  • 12 207 031 254 ÷ 2 = 6 103 515 627 + 0;
  • 6 103 515 627 ÷ 2 = 3 051 757 813 + 1;
  • 3 051 757 813 ÷ 2 = 1 525 878 906 + 1;
  • 1 525 878 906 ÷ 2 = 762 939 453 + 0;
  • 762 939 453 ÷ 2 = 381 469 726 + 1;
  • 381 469 726 ÷ 2 = 190 734 863 + 0;
  • 190 734 863 ÷ 2 = 95 367 431 + 1;
  • 95 367 431 ÷ 2 = 47 683 715 + 1;
  • 47 683 715 ÷ 2 = 23 841 857 + 1;
  • 23 841 857 ÷ 2 = 11 920 928 + 1;
  • 11 920 928 ÷ 2 = 5 960 464 + 0;
  • 5 960 464 ÷ 2 = 2 980 232 + 0;
  • 2 980 232 ÷ 2 = 1 490 116 + 0;
  • 1 490 116 ÷ 2 = 745 058 + 0;
  • 745 058 ÷ 2 = 372 529 + 0;
  • 372 529 ÷ 2 = 186 264 + 1;
  • 186 264 ÷ 2 = 93 132 + 0;
  • 93 132 ÷ 2 = 46 566 + 0;
  • 46 566 ÷ 2 = 23 283 + 0;
  • 23 283 ÷ 2 = 11 641 + 1;
  • 11 641 ÷ 2 = 5 820 + 1;
  • 5 820 ÷ 2 = 2 910 + 0;
  • 2 910 ÷ 2 = 1 455 + 0;
  • 1 455 ÷ 2 = 727 + 1;
  • 727 ÷ 2 = 363 + 1;
  • 363 ÷ 2 = 181 + 1;
  • 181 ÷ 2 = 90 + 1;
  • 90 ÷ 2 = 45 + 0;
  • 45 ÷ 2 = 22 + 1;
  • 22 ÷ 2 = 11 + 0;
  • 11 ÷ 2 = 5 + 1;
  • 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.

100 000 000 039 137(10) = 101 1010 1111 0011 0001 0000 0111 1010 1101 1000 1110 0001(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 100 000 000 039 137(10) converted to signed binary in two's complement representation:

100 000 000 039 137(10) = 0000 0000 0000 0000 0101 1010 1111 0011 0001 0000 0111 1010 1101 1000 1110 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