Convert 25 602 300 050 966 to a Signed Binary in Two's (2's) Complement Representation

How to convert decimal number 25 602 300 050 966(10) to a signed binary in two's (2's) complement representation

What are the steps to convert decimal number
25 602 300 050 966 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;
  • 25 602 300 050 966 ÷ 2 = 12 801 150 025 483 + 0;
  • 12 801 150 025 483 ÷ 2 = 6 400 575 012 741 + 1;
  • 6 400 575 012 741 ÷ 2 = 3 200 287 506 370 + 1;
  • 3 200 287 506 370 ÷ 2 = 1 600 143 753 185 + 0;
  • 1 600 143 753 185 ÷ 2 = 800 071 876 592 + 1;
  • 800 071 876 592 ÷ 2 = 400 035 938 296 + 0;
  • 400 035 938 296 ÷ 2 = 200 017 969 148 + 0;
  • 200 017 969 148 ÷ 2 = 100 008 984 574 + 0;
  • 100 008 984 574 ÷ 2 = 50 004 492 287 + 0;
  • 50 004 492 287 ÷ 2 = 25 002 246 143 + 1;
  • 25 002 246 143 ÷ 2 = 12 501 123 071 + 1;
  • 12 501 123 071 ÷ 2 = 6 250 561 535 + 1;
  • 6 250 561 535 ÷ 2 = 3 125 280 767 + 1;
  • 3 125 280 767 ÷ 2 = 1 562 640 383 + 1;
  • 1 562 640 383 ÷ 2 = 781 320 191 + 1;
  • 781 320 191 ÷ 2 = 390 660 095 + 1;
  • 390 660 095 ÷ 2 = 195 330 047 + 1;
  • 195 330 047 ÷ 2 = 97 665 023 + 1;
  • 97 665 023 ÷ 2 = 48 832 511 + 1;
  • 48 832 511 ÷ 2 = 24 416 255 + 1;
  • 24 416 255 ÷ 2 = 12 208 127 + 1;
  • 12 208 127 ÷ 2 = 6 104 063 + 1;
  • 6 104 063 ÷ 2 = 3 052 031 + 1;
  • 3 052 031 ÷ 2 = 1 526 015 + 1;
  • 1 526 015 ÷ 2 = 763 007 + 1;
  • 763 007 ÷ 2 = 381 503 + 1;
  • 381 503 ÷ 2 = 190 751 + 1;
  • 190 751 ÷ 2 = 95 375 + 1;
  • 95 375 ÷ 2 = 47 687 + 1;
  • 47 687 ÷ 2 = 23 843 + 1;
  • 23 843 ÷ 2 = 11 921 + 1;
  • 11 921 ÷ 2 = 5 960 + 1;
  • 5 960 ÷ 2 = 2 980 + 0;
  • 2 980 ÷ 2 = 1 490 + 0;
  • 1 490 ÷ 2 = 745 + 0;
  • 745 ÷ 2 = 372 + 1;
  • 372 ÷ 2 = 186 + 0;
  • 186 ÷ 2 = 93 + 0;
  • 93 ÷ 2 = 46 + 1;
  • 46 ÷ 2 = 23 + 0;
  • 23 ÷ 2 = 11 + 1;
  • 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.

25 602 300 050 966(10) = 1 0111 0100 1000 1111 1111 1111 1111 1111 1110 0001 0110(2)

3. Determine the signed binary number bit length:

  • The base 2 number's actual length, in bits: 45.

  • 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, 45,

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 25 602 300 050 966(10) converted to signed binary in two's complement representation:

25 602 300 050 966(10) = 0000 0000 0000 0000 0001 0111 0100 1000 1111 1111 1111 1111 1111 1110 0001 0110

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