Convert 1 001 000 010 101 025 to a Signed Binary in Two's (2's) Complement Representation

How to convert decimal number 1 001 000 010 101 025(10) to a signed binary in two's (2's) complement representation

What are the steps to convert decimal number
1 001 000 010 101 025 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;
  • 1 001 000 010 101 025 ÷ 2 = 500 500 005 050 512 + 1;
  • 500 500 005 050 512 ÷ 2 = 250 250 002 525 256 + 0;
  • 250 250 002 525 256 ÷ 2 = 125 125 001 262 628 + 0;
  • 125 125 001 262 628 ÷ 2 = 62 562 500 631 314 + 0;
  • 62 562 500 631 314 ÷ 2 = 31 281 250 315 657 + 0;
  • 31 281 250 315 657 ÷ 2 = 15 640 625 157 828 + 1;
  • 15 640 625 157 828 ÷ 2 = 7 820 312 578 914 + 0;
  • 7 820 312 578 914 ÷ 2 = 3 910 156 289 457 + 0;
  • 3 910 156 289 457 ÷ 2 = 1 955 078 144 728 + 1;
  • 1 955 078 144 728 ÷ 2 = 977 539 072 364 + 0;
  • 977 539 072 364 ÷ 2 = 488 769 536 182 + 0;
  • 488 769 536 182 ÷ 2 = 244 384 768 091 + 0;
  • 244 384 768 091 ÷ 2 = 122 192 384 045 + 1;
  • 122 192 384 045 ÷ 2 = 61 096 192 022 + 1;
  • 61 096 192 022 ÷ 2 = 30 548 096 011 + 0;
  • 30 548 096 011 ÷ 2 = 15 274 048 005 + 1;
  • 15 274 048 005 ÷ 2 = 7 637 024 002 + 1;
  • 7 637 024 002 ÷ 2 = 3 818 512 001 + 0;
  • 3 818 512 001 ÷ 2 = 1 909 256 000 + 1;
  • 1 909 256 000 ÷ 2 = 954 628 000 + 0;
  • 954 628 000 ÷ 2 = 477 314 000 + 0;
  • 477 314 000 ÷ 2 = 238 657 000 + 0;
  • 238 657 000 ÷ 2 = 119 328 500 + 0;
  • 119 328 500 ÷ 2 = 59 664 250 + 0;
  • 59 664 250 ÷ 2 = 29 832 125 + 0;
  • 29 832 125 ÷ 2 = 14 916 062 + 1;
  • 14 916 062 ÷ 2 = 7 458 031 + 0;
  • 7 458 031 ÷ 2 = 3 729 015 + 1;
  • 3 729 015 ÷ 2 = 1 864 507 + 1;
  • 1 864 507 ÷ 2 = 932 253 + 1;
  • 932 253 ÷ 2 = 466 126 + 1;
  • 466 126 ÷ 2 = 233 063 + 0;
  • 233 063 ÷ 2 = 116 531 + 1;
  • 116 531 ÷ 2 = 58 265 + 1;
  • 58 265 ÷ 2 = 29 132 + 1;
  • 29 132 ÷ 2 = 14 566 + 0;
  • 14 566 ÷ 2 = 7 283 + 0;
  • 7 283 ÷ 2 = 3 641 + 1;
  • 3 641 ÷ 2 = 1 820 + 1;
  • 1 820 ÷ 2 = 910 + 0;
  • 910 ÷ 2 = 455 + 0;
  • 455 ÷ 2 = 227 + 1;
  • 227 ÷ 2 = 113 + 1;
  • 113 ÷ 2 = 56 + 1;
  • 56 ÷ 2 = 28 + 0;
  • 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.

1 001 000 010 101 025(10) = 11 1000 1110 0110 0111 0111 1010 0000 0101 1011 0001 0010 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 1 001 000 010 101 025(10) converted to signed binary in two's complement representation:

1 001 000 010 101 025(10) = 0000 0000 0000 0011 1000 1110 0110 0111 0111 1010 0000 0101 1011 0001 0010 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