Convert 1 101 101 011 101 076 to a Signed Binary in Two's (2's) Complement Representation

How to convert decimal number 1 101 101 011 101 076(10) to a signed binary in two's (2's) complement representation

What are the steps to convert decimal number
1 101 101 011 101 076 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;
  • 1 101 101 011 101 076 ÷ 2 = 550 550 505 550 538 + 0;
  • 550 550 505 550 538 ÷ 2 = 275 275 252 775 269 + 0;
  • 275 275 252 775 269 ÷ 2 = 137 637 626 387 634 + 1;
  • 137 637 626 387 634 ÷ 2 = 68 818 813 193 817 + 0;
  • 68 818 813 193 817 ÷ 2 = 34 409 406 596 908 + 1;
  • 34 409 406 596 908 ÷ 2 = 17 204 703 298 454 + 0;
  • 17 204 703 298 454 ÷ 2 = 8 602 351 649 227 + 0;
  • 8 602 351 649 227 ÷ 2 = 4 301 175 824 613 + 1;
  • 4 301 175 824 613 ÷ 2 = 2 150 587 912 306 + 1;
  • 2 150 587 912 306 ÷ 2 = 1 075 293 956 153 + 0;
  • 1 075 293 956 153 ÷ 2 = 537 646 978 076 + 1;
  • 537 646 978 076 ÷ 2 = 268 823 489 038 + 0;
  • 268 823 489 038 ÷ 2 = 134 411 744 519 + 0;
  • 134 411 744 519 ÷ 2 = 67 205 872 259 + 1;
  • 67 205 872 259 ÷ 2 = 33 602 936 129 + 1;
  • 33 602 936 129 ÷ 2 = 16 801 468 064 + 1;
  • 16 801 468 064 ÷ 2 = 8 400 734 032 + 0;
  • 8 400 734 032 ÷ 2 = 4 200 367 016 + 0;
  • 4 200 367 016 ÷ 2 = 2 100 183 508 + 0;
  • 2 100 183 508 ÷ 2 = 1 050 091 754 + 0;
  • 1 050 091 754 ÷ 2 = 525 045 877 + 0;
  • 525 045 877 ÷ 2 = 262 522 938 + 1;
  • 262 522 938 ÷ 2 = 131 261 469 + 0;
  • 131 261 469 ÷ 2 = 65 630 734 + 1;
  • 65 630 734 ÷ 2 = 32 815 367 + 0;
  • 32 815 367 ÷ 2 = 16 407 683 + 1;
  • 16 407 683 ÷ 2 = 8 203 841 + 1;
  • 8 203 841 ÷ 2 = 4 101 920 + 1;
  • 4 101 920 ÷ 2 = 2 050 960 + 0;
  • 2 050 960 ÷ 2 = 1 025 480 + 0;
  • 1 025 480 ÷ 2 = 512 740 + 0;
  • 512 740 ÷ 2 = 256 370 + 0;
  • 256 370 ÷ 2 = 128 185 + 0;
  • 128 185 ÷ 2 = 64 092 + 1;
  • 64 092 ÷ 2 = 32 046 + 0;
  • 32 046 ÷ 2 = 16 023 + 0;
  • 16 023 ÷ 2 = 8 011 + 1;
  • 8 011 ÷ 2 = 4 005 + 1;
  • 4 005 ÷ 2 = 2 002 + 1;
  • 2 002 ÷ 2 = 1 001 + 0;
  • 1 001 ÷ 2 = 500 + 1;
  • 500 ÷ 2 = 250 + 0;
  • 250 ÷ 2 = 125 + 0;
  • 125 ÷ 2 = 62 + 1;
  • 62 ÷ 2 = 31 + 0;
  • 31 ÷ 2 = 15 + 1;
  • 15 ÷ 2 = 7 + 1;
  • 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.

1 101 101 011 101 076(10) = 11 1110 1001 0111 0010 0000 1110 1010 0000 1110 0101 1001 0100(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 1 101 101 011 101 076(10) converted to signed binary in two's complement representation:

1 101 101 011 101 076(10) = 0000 0000 0000 0011 1110 1001 0111 0010 0000 1110 1010 0000 1110 0101 1001 0100

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