Convert 11 010 101 010 457 to a Signed Binary in Two's (2's) Complement Representation

How to convert decimal number 11 010 101 010 457(10) to a signed binary in two's (2's) complement representation

What are the steps to convert decimal number
11 010 101 010 457 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;
  • 11 010 101 010 457 ÷ 2 = 5 505 050 505 228 + 1;
  • 5 505 050 505 228 ÷ 2 = 2 752 525 252 614 + 0;
  • 2 752 525 252 614 ÷ 2 = 1 376 262 626 307 + 0;
  • 1 376 262 626 307 ÷ 2 = 688 131 313 153 + 1;
  • 688 131 313 153 ÷ 2 = 344 065 656 576 + 1;
  • 344 065 656 576 ÷ 2 = 172 032 828 288 + 0;
  • 172 032 828 288 ÷ 2 = 86 016 414 144 + 0;
  • 86 016 414 144 ÷ 2 = 43 008 207 072 + 0;
  • 43 008 207 072 ÷ 2 = 21 504 103 536 + 0;
  • 21 504 103 536 ÷ 2 = 10 752 051 768 + 0;
  • 10 752 051 768 ÷ 2 = 5 376 025 884 + 0;
  • 5 376 025 884 ÷ 2 = 2 688 012 942 + 0;
  • 2 688 012 942 ÷ 2 = 1 344 006 471 + 0;
  • 1 344 006 471 ÷ 2 = 672 003 235 + 1;
  • 672 003 235 ÷ 2 = 336 001 617 + 1;
  • 336 001 617 ÷ 2 = 168 000 808 + 1;
  • 168 000 808 ÷ 2 = 84 000 404 + 0;
  • 84 000 404 ÷ 2 = 42 000 202 + 0;
  • 42 000 202 ÷ 2 = 21 000 101 + 0;
  • 21 000 101 ÷ 2 = 10 500 050 + 1;
  • 10 500 050 ÷ 2 = 5 250 025 + 0;
  • 5 250 025 ÷ 2 = 2 625 012 + 1;
  • 2 625 012 ÷ 2 = 1 312 506 + 0;
  • 1 312 506 ÷ 2 = 656 253 + 0;
  • 656 253 ÷ 2 = 328 126 + 1;
  • 328 126 ÷ 2 = 164 063 + 0;
  • 164 063 ÷ 2 = 82 031 + 1;
  • 82 031 ÷ 2 = 41 015 + 1;
  • 41 015 ÷ 2 = 20 507 + 1;
  • 20 507 ÷ 2 = 10 253 + 1;
  • 10 253 ÷ 2 = 5 126 + 1;
  • 5 126 ÷ 2 = 2 563 + 0;
  • 2 563 ÷ 2 = 1 281 + 1;
  • 1 281 ÷ 2 = 640 + 1;
  • 640 ÷ 2 = 320 + 0;
  • 320 ÷ 2 = 160 + 0;
  • 160 ÷ 2 = 80 + 0;
  • 80 ÷ 2 = 40 + 0;
  • 40 ÷ 2 = 20 + 0;
  • 20 ÷ 2 = 10 + 0;
  • 10 ÷ 2 = 5 + 0;
  • 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.

11 010 101 010 457(10) = 1010 0000 0011 0111 1101 0010 1000 1110 0000 0001 1001(2)

3. Determine the signed binary number bit length:

  • The base 2 number's actual length, in bits: 44.

  • 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, 44,

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 11 010 101 010 457(10) converted to signed binary in two's complement representation:

11 010 101 010 457(10) = 0000 0000 0000 0000 0000 1010 0000 0011 0111 1101 0010 1000 1110 0000 0001 1001

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